lbmpy/boundaries/boundaries_in_kernel.py → src/lbmpy/boundaries/boundaries_in_kernel.py

 import sympy as sp
 
-from lbmpy.boundaries.boundaryhandling import LbmWeightInfo
+from .._compat import IS_PYSTENCILS_2
+
 from lbmpy.advanced_streaming.indexing import BetweenTimestepsIndexing
 from lbmpy.advanced_streaming.utility import Timestep, get_accessor
 from pystencils.boundaries.boundaryhandling import BoundaryOffsetInfo
-from pystencils.assignment import Assignment
-from pystencils.astnodes import Block, Conditional, LoopOverCoordinate, SympyAssignment
-from pystencils.data_types import type_all_numbers
-from pystencils.simp.assignment_collection import AssignmentCollection
-from pystencils.simp.simplifications import sympy_cse_on_assignment_list
+from pystencils import Assignment
+from pystencils.simp import AssignmentCollection, sympy_cse_on_assignment_list
 from pystencils.stencil import inverse_direction
 from pystencils.sympyextensions import fast_subs
 
+if IS_PYSTENCILS_2:
+    from lbmpy.lookup_tables import LbmWeightInfo
+else:
+    from lbmpy.custom_code_nodes import LbmWeightInfo
+    from pystencils.astnodes import Block, Conditional, LoopOverCoordinate, SympyAssignment  # TODO replace
+
 
 def direction_indices_in_direction(direction, stencil):
     for i, offset in enumerate(stencil):
@@ -48,17 +52,20 @@ def border_conditions(direction, field, ghost_layers=1):
     border_condition = sp.Eq(loop_ctr, gl if val < 0 else field.shape[idx] - gl - 1)
 
     if ghost_layers == 0:
-        return type_all_numbers(border_condition, loop_ctr.dtype)
+        return border_condition
     else:
         other_min = [sp.Ge(c, gl)
                      for c in loop_ctrs if c != loop_ctr]
         other_max = [sp.Lt(c, field.shape[i] - gl)
                      for i, c in enumerate(loop_ctrs) if c != loop_ctr]
         result = sp.And(border_condition, *other_min, *other_max)
-        return type_all_numbers(result, loop_ctr.dtype)
+        return result
 
 
 def boundary_conditional(boundary, direction, streaming_pattern, prev_timestep, lb_method, output_field, cse=False):
+    if IS_PYSTENCILS_2:
+        raise NotImplementedError("In-Kernel Boundaries are not yet available on pystencils 2.0")
+
     stencil = lb_method.stencil
 
     dir_indices = direction_indices_in_direction(direction, stencil)
@@ -68,7 +75,7 @@ def boundary_conditional(boundary, direction, streaming_pattern, prev_timestep,
 
     assignments = []
     for direction_idx in dir_indices:
-        rule = boundary(f_out, f_in, direction_idx, inv_dir, lb_method, index_field=None)
+        rule = boundary(f_out, f_in, direction_idx, inv_dir, lb_method, index_field=None, force_vector=None)
 
         #   rhs: replace f_out by post collision symbols.
         rhs_substitutions = {f_out(i): sym for i, sym in enumerate(lb_method.post_collision_pdf_symbols)}

lbmpy/boundaries/boundaryconditions.py → src/lbmpy/boundaries/boundaryconditions.py

+import abc
+from enum import Enum, auto
+from warnings import warn
+
+from pystencils import Assignment, AssignmentCollection, Field, TypedSymbol
+from pystencils.stencil import offset_to_direction_string, direction_string_to_offset, inverse_direction
+from pystencils.sympyextensions import get_symmetric_part, simplify_by_equality, scalar_product
+
 from lbmpy.advanced_streaming.utility import AccessPdfValues, Timestep
-from pystencils.simp.assignment_collection import AssignmentCollection
-from pystencils import Assignment, Field
-from lbmpy.boundaries.boundaryhandling import LbmWeightInfo
-from pystencils.data_types import create_type
-from pystencils.sympyextensions import get_symmetric_part
+from lbmpy.maxwellian_equilibrium import discrete_equilibrium
 from lbmpy.simplificationfactory import create_simplification_strategy
-from lbmpy.advanced_streaming.indexing import NeighbourOffsetArrays
-from pystencils.stencil import offset_to_direction_string, direction_string_to_offset, inverse_direction
 
 import sympy as sp
-
-
-class LbBoundary:
+import numpy as np
+
+from .._compat import IS_PYSTENCILS_2
+
+if IS_PYSTENCILS_2:
+    from pystencils import create_type
+    from pystencils.sympyextensions.typed_sympy import CastFunc
+    from pystencils.types.quick import Arr
+    from lbmpy.lookup_tables import (
+        NeighbourOffsetArrays,
+        MirroredStencilDirections,
+        LbmWeightInfo,
+        TranslationArraysNode
+    )
+else:
+    from pystencils.typing import create_type, CastFunc
+    from lbmpy.custom_code_nodes import (
+        NeighbourOffsetArrays,
+        MirroredStencilDirections,
+        LbmWeightInfo,
+        TranslationArraysNode
+    )
+
+
+class LbBoundary(abc.ABC):
     """Base class that all boundaries should derive from.
 
     Args:
@@ -21,10 +45,11 @@ class LbBoundary:
     inner_or_boundary = True
     single_link = False
 
-    def __init__(self, name=None):
+    def __init__(self, name=None, calculate_force_on_boundary=False):
         self._name = name
+        self.calculate_force_on_boundary = calculate_force_on_boundary
 
-    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field):
+    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector):
         """
         This function defines the boundary behavior and must therefore be implemented by all boundaries.
         The boundary is defined through a list of sympy equations from which a boundary kernel is generated.
@@ -41,6 +66,8 @@ class LbBoundary:
         lb_method:      an instance of the LB method used. Use this to adapt the boundary to the method
                         (e.g. compressibility)
         index_field:    the boundary index field that can be used to retrieve and update boundary data
+        force_vector:   vector to store the force on the boundary. Has the same size as the index field and
+                        D-entries per cell
 
         Returns:
             list of pystencils assignments, or pystencils.AssignmentCollection
@@ -92,31 +119,674 @@ class LbBoundary:
 
 
 class NoSlip(LbBoundary):
-    """
+    r"""
     No-Slip, (half-way) simple bounce back boundary condition, enforcing zero velocity at obstacle.
-    Extended for use with any streaming pattern.
+    Populations leaving the boundary node :math:`\mathbf{x}_b` at time :math:`t` are reflected
+    back with :math:`\mathbf{c}_{\overline{i}} = -\mathbf{c}_{i}`
+
+    .. math ::
+        f_{\overline{i}}(\mathbf{x}_b, t + \Delta t) = f^{\star}_{i}(\mathbf{x}_b, t)
+
+    Args:
+        name: optional name of the boundary.
+        calculate_force_on_boundary: stores the force for each PDF at the boundary in a force vector
+    """
+
+    def __init__(self, name=None, calculate_force_on_boundary=False):
+        """Set an optional name here, to mark boundaries, for example for force evaluations"""
+        super(NoSlip, self).__init__(name, calculate_force_on_boundary)
+
+    def get_additional_code_nodes(self, lb_method):
+        if self.calculate_force_on_boundary:
+            return [NeighbourOffsetArrays(lb_method.stencil)]
+        else:
+            return []
+
+    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector):
+        if self.calculate_force_on_boundary:
+            force = sp.Symbol("f")
+            subexpressions = [Assignment(force, sp.Float(2.0) * f_out(dir_symbol))]
+            offset = NeighbourOffsetArrays.neighbour_offset(dir_symbol, lb_method.stencil)
+            if IS_PYSTENCILS_2:
+                offset = [CastFunc.as_numeric(o) for o in offset]
+            for i in range(lb_method.stencil.D):
+                subexpressions.append(Assignment(force_vector[0](f'F_{i}'), force * offset[i]))
+        else:
+            subexpressions = []
+
+        boundary_assignments = [Assignment(f_in(inv_dir[dir_symbol]), f_out(dir_symbol))]
+        return AssignmentCollection(boundary_assignments, subexpressions=subexpressions)
+
+
+class NoSlipLinearBouzidi(LbBoundary):
+    """
+    No-Slip, (half-way) simple bounce back boundary condition with interpolation
+    to increase accuracy: :cite:`BouzidiBC`. In order to make the boundary condition work properly a
+    Python callback function needs to be provided to calculate the distance from the wall for each cell near to the
+    boundary. If this is not done the boundary condition will fall back to a simple NoSlip boundary.
+    Furthermore, for this boundary condition a second fluid cell away from the wall is needed. If the second fluid
+    cell is not available (e.g. because it is marked as boundary as well), the boundary condition should fall back to
+    a NoSlip boundary as well.
+
+    Args:
+        name: optional name of the boundary.
+        init_wall_distance: Python callback function to calculate the wall distance for each cell near to the  boundary
+        data_type: data type of the wall distance q
+    """
+
+    def __init__(self, name=None, init_wall_distance=None, data_type='double', calculate_force_on_boundary=False):
+        self.data_type = data_type
+        self.init_wall_distance = init_wall_distance
+        super(NoSlipLinearBouzidi, self).__init__(name, calculate_force_on_boundary)
+
+    @property
+    def additional_data(self):
+        """Used internally only. For the NoSlipLinearBouzidi boundary the distance to the obstacle of every
+        direction is needed. This information is stored in the index vector."""
+        return [('q', create_type(self.data_type))]
+
+    def get_additional_code_nodes(self, lb_method):
+        if self.calculate_force_on_boundary:
+            return [NeighbourOffsetArrays(lb_method.stencil)]
+        else:
+            return []
+
+    @property
+    def additional_data_init_callback(self):
+        def default_callback(boundary_data, **_):
+            for cell in boundary_data.index_array:
+                cell['q'] = -1
+
+        if self.init_wall_distance:
+            return self.init_wall_distance
+        else:
+            warn("No callback function provided to initialise the wall distance for each cell "
+                 "(init_wall_distance=None). The boundary condition will fall back to a simple NoSlip BC")
+            return default_callback
+
+    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector):
+        f_xf = sp.Symbol("f_xf")
+        f_xf_inv = sp.Symbol("f_xf_inv")
+        d_x2f = sp.Symbol("d_x2f")
+        q = sp.Symbol("q")
+        one = sp.Float(1.0)
+        two = sp.Float(2.0)
+        half = sp.Rational(1, 2)
+
+        subexpressions = [Assignment(f_xf, f_out(dir_symbol)),
+                          Assignment(f_xf_inv, f_out(inv_dir[dir_symbol])),
+                          Assignment(d_x2f, f_in(dir_symbol)),
+                          Assignment(q, index_field[0]('q'))]
+
+        case_one = (half * (f_xf + f_xf_inv * (two * q - one))) / q
+        case_two = two * q * f_xf + (one - two * q) * d_x2f
+        case_three = f_xf
+
+        rhs = sp.Piecewise((case_one, sp.Ge(q, 0.5)),
+                           (case_two, sp.And(sp.Gt(q, 0), sp.Lt(q, 0.5))),
+                           (case_three, True))
+
+        if self.calculate_force_on_boundary:
+            force = sp.Symbol("f")
+            subexpressions.append(Assignment(force, f_xf + rhs))
+            offset = NeighbourOffsetArrays.neighbour_offset(dir_symbol, lb_method.stencil)
+            if IS_PYSTENCILS_2:
+                offset = [CastFunc.as_numeric(o) for o in offset]
+            for i in range(lb_method.stencil.D):
+                subexpressions.append(Assignment(force_vector[0](f'F_{i}'), force * offset[i]))
+
+        boundary_assignments = [Assignment(f_in(inv_dir[dir_symbol]), rhs)]
+
+        return AssignmentCollection(boundary_assignments, subexpressions=subexpressions)
+
+
+# end class NoSlipLinearBouzidi
+
+class QuadraticBounceBack(LbBoundary):
+    """
+    Second order accurate bounce back boundary condition. Implementation details are provided in a demo notebook here:
+    https://pycodegen.pages.i10git.cs.fau.de/lbmpy/notebooks/demo_interpolation_boundary_conditions.html
 
     Args:
+        relaxation_rate: relaxation rate to realise a BGK scheme for recovering the pre collision PDF value.
         name: optional name of the boundary.
+        init_wall_distance: Python callback function to calculate the wall distance for each cell near to the  boundary
+        data_type: data type of the wall distance q
+    """
+
+    def __init__(self, relaxation_rate, name=None, init_wall_distance=None, data_type='double',
+                 calculate_force_on_boundary=False):
+        self.relaxation_rate = relaxation_rate
+        self.data_type = data_type
+        self.init_wall_distance = init_wall_distance
+        self.equilibrium_values_name = "f_eq"
+
+        super(QuadraticBounceBack, self).__init__(name, calculate_force_on_boundary)
+
+    def inv_dir_symbol(self, stencil):
+        if IS_PYSTENCILS_2:
+            return TypedSymbol("inv_dir", Arr(create_type("int32"), stencil.Q))
+        else:
+            return TypedSymbol("inv_dir", create_type("int32"))
+
+    @property
+    def additional_data(self):
+        """Used internally only. For the NoSlipLinearBouzidi boundary the distance to the obstacle of every
+        direction is needed. This information is stored in the index vector."""
+        return [('q', create_type(self.data_type))]
+
+    @property
+    def additional_data_init_callback(self):
+        def default_callback(boundary_data, **_):
+            for cell in boundary_data.index_array:
+                cell['q'] = 0.5
+
+        if self.init_wall_distance:
+            return self.init_wall_distance
+        else:
+            warn("No callback function provided to initialise the wall distance for each cell "
+                 "(init_wall_distance=None). The boundary condition will fall back to a simple NoSlip BC")
+            return default_callback
+
+    def get_additional_code_nodes(self, lb_method):
+        """Return a list of code nodes that will be added in the generated code before the index field loop.
+
+        Args:
+            lb_method: Lattice Boltzmann method. See :func:`lbmpy.creationfunctions.create_lb_method`
+
+        Returns:
+            list containing LbmWeightInfo
+        """
+        stencil = lb_method.stencil
+        inv_directions = [str(stencil.index(inverse_direction(direction))) for direction in stencil]
+        
+        if IS_PYSTENCILS_2:
+            inverse_dir_node = TranslationArraysNode([(self.inv_dir_symbol(stencil), inv_directions), ])
+        else:
+            inv_dir_symbol = self.inv_dir_symbol(stencil)
+            dtype = inv_dir_symbol.dtype
+            name = inv_dir_symbol.name
+            inverse_dir_node = TranslationArraysNode([(dtype, name, inv_directions), ], {inv_dir_symbol})
+        
+        return [LbmWeightInfo(lb_method, self.data_type), inverse_dir_node, NeighbourOffsetArrays(lb_method.stencil)]
+
+    @staticmethod
+    def get_equilibrium(v, u, rho, drho, weight, compressible, zero_centered):
+        rho_background = sp.Integer(1)
+
+        result = discrete_equilibrium(v, u, rho, weight,
+                                      order=2, c_s_sq=sp.Rational(1, 3), compressible=compressible)
+        if zero_centered:
+            shift = discrete_equilibrium(v, [0] * len(u), rho_background, weight,
+                                         order=0, c_s_sq=sp.Rational(1, 3), compressible=False)
+            result = simplify_by_equality(result - shift, rho, drho, rho_background)
+
+        return result
+
+    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector):
+        omega = self.relaxation_rate
+        inv = sp.IndexedBase(self.inv_dir_symbol(lb_method.stencil), shape=(1,))[dir_symbol]
+        weight_info = LbmWeightInfo(lb_method, data_type=self.data_type)
+        weight_of_direction = weight_info.weight_of_direction
+        pdf_field_accesses = [f_out(i) for i in range(len(lb_method.stencil))]
+
+        pdf_symbols = [sp.Symbol(f"pdf_{i}") for i in range(len(lb_method.stencil))]
+        f_xf = sp.Symbol("f_xf")
+        f_xf_inv = sp.Symbol("f_xf_inv")
+        q = sp.Symbol("q")
+        feq = sp.Symbol("f_eq")
+        weight = sp.Symbol("w")
+        weight_inv = sp.Symbol("w_inv")
+        v = [TypedSymbol(f"c_{i}", self.data_type) for i in range(lb_method.stencil.D)]
+        v_inv = [TypedSymbol(f"c_inv_{i}", self.data_type) for i in range(lb_method.stencil.D)]
+        one = sp.Float(1.0)
+        half = sp.Rational(1, 2)
+
+        subexpressions = [Assignment(pdf_symbols[i], pdf) for i, pdf in enumerate(pdf_field_accesses)]
+        subexpressions.append(Assignment(f_xf, f_out(dir_symbol)))
+        subexpressions.append(Assignment(f_xf_inv, f_out(inv_dir[dir_symbol])))
+        subexpressions.append(Assignment(q, index_field[0]('q')))
+        subexpressions.append(Assignment(weight, weight_of_direction(dir_symbol, lb_method)))
+        subexpressions.append(Assignment(weight_inv, weight_of_direction(inv, lb_method)))
+
+        if IS_PYSTENCILS_2:
+            cast_offset = CastFunc.as_numeric
+        else:
+            def cast_offset(x):
+                return x
+
+        for i in range(lb_method.stencil.D):
+            offset = NeighbourOffsetArrays.neighbour_offset(dir_symbol, lb_method.stencil)
+            subexpressions.append(Assignment(v[i], cast_offset(offset[i])))
+
+        for i in range(lb_method.stencil.D):
+            offset = NeighbourOffsetArrays.neighbour_offset(inv, lb_method.stencil)
+            subexpressions.append(Assignment(v_inv[i], cast_offset(offset[i])))
+
+        cqc = lb_method.conserved_quantity_computation
+        rho = cqc.density_symbol
+        drho = cqc.density_deviation_symbol
+        u = sp.Matrix(cqc.velocity_symbols)
+        compressible = cqc.compressible
+        zero_centered = cqc.zero_centered_pdfs
+
+        cqe = cqc.equilibrium_input_equations_from_pdfs(pdf_symbols, False)
+        subexpressions.append(cqe.all_assignments)
+
+        eq_dir = self.get_equilibrium(v, u, rho, drho, weight, compressible, zero_centered)
+        eq_inv = self.get_equilibrium(v_inv, u, rho, drho, weight_inv, compressible, zero_centered)
+
+        subexpressions.append(Assignment(feq, eq_dir + eq_inv))
+
+        t1 = (f_xf - f_xf_inv + (f_xf + f_xf_inv - feq * omega) / (one - omega))
+        t2 = (q * (f_xf + f_xf_inv)) / (one + q)
+        result = (one - q) / (one + q) * t1 * half + t2
+
+        if self.calculate_force_on_boundary:
+            force = sp.Symbol("f")
+            subexpressions.append(Assignment(force, f_xf + result))
+            offset = NeighbourOffsetArrays.neighbour_offset(dir_symbol, lb_method.stencil)
+            if IS_PYSTENCILS_2:
+                offset = [CastFunc.as_numeric(o) for o in offset]
+            for i in range(lb_method.stencil.D):
+                subexpressions.append(Assignment(force_vector[0](f'F_{i}'), force * offset[i]))
+
+        boundary_assignments = [Assignment(f_in(inv_dir[dir_symbol]), result)]
+        return AssignmentCollection(boundary_assignments, subexpressions=subexpressions)
+
+
+# end class QuadraticBounceBack
+
+class FreeSlip(LbBoundary):
+    """
+    Free-Slip boundary condition, which enforces a zero normal fluid velocity :math:`u_n = 0` but places no restrictions
+    on the tangential fluid velocity :math:`u_t`.
+
+    Args:
+        stencil: LBM stencil which is used for the simulation
+        normal_direction: optional normal direction pointing from wall to fluid.
+                          If the Free slip boundary is applied to a certain side in the domain it is not necessary
+                          to calculate the normal direction since it can be stated for all boundary cells.
+                          This reduces the memory space for the index array significantly.
+        name: optional name of the boundary.
+    """
+
+    def __init__(self, stencil, normal_direction=None, name=None):
+        """Set an optional name here, to mark boundaries, for example for force evaluations"""
+        self.stencil = stencil
+
+        if normal_direction and len(normal_direction) - normal_direction.count(0) != 1:
+            raise ValueError("It is only possible to pre specify the normal direction for simple situations."
+                             "This means if the free slip boundary is applied to a straight wall or side in the "
+                             "simulation domain. A possible value for example would be (0, 1, 0) if the "
+                             "free slip boundary is applied to the northern wall. For more complex situations "
+                             "the normal direction has to be calculated for each cell. This is done when "
+                             "the normal direction is not defined for this class")
+
+        if normal_direction:
+            normal_direction = tuple([int(n) for n in normal_direction])
+            assert all([n in [-1, 0, 1] for n in normal_direction]), \
+                "Only -1, 0 and 1 allowed for defining the normal direction"
+            self.mirror_axis = normal_direction.index(*[d for d in normal_direction if d != 0])
+
+        self.normal_direction = normal_direction
+        self.dim = len(stencil[0])
+
+        if name is None and normal_direction:
+            name = f"Free Slip : {offset_to_direction_string([-x for x in normal_direction])}"
+
+        super(FreeSlip, self).__init__(name, calculate_force_on_boundary=False)
+
+    def init_callback(self, boundary_data, **_):
+        if len(boundary_data.index_array) > 1e6:
+            warn(f"The calculation of the normal direction for each cell might take a long time, because "
+                 f"{len(boundary_data.index_array)} cells are marked as Free Slip boundary cells. Consider specifying "
+                 f" the normal direction beforehand, which is possible if it is equal for all cells (e.g. at a wall)")
+
+        dim = boundary_data.dim
+        coords = [coord for coord, _ in zip(['x', 'y', 'z'], range(dim))]
+
+        boundary_cells = set()
+
+        # get a set containing all boundary cells
+        for cell in boundary_data.index_array:
+            fluid_cell = tuple([cell[coord] for coord in coords])
+            direction = self.stencil[cell['dir']]
+            boundary_cell = tuple([i + d for i, d in zip(fluid_cell, direction)])
+            boundary_cells.add(boundary_cell)
+
+        for cell in boundary_data.index_array:
+            fluid_cell = tuple([cell[coord] for coord in coords])
+            direction = self.stencil[cell['dir']]
+            ref_direction = direction
+            normal_direction = [0] * dim
+
+            for i in range(dim):
+                sub_direction = [0] * dim
+                sub_direction[i] = direction[i]
+                test_cell = tuple([x + y for x, y in zip(fluid_cell, sub_direction)])
+
+                if test_cell in boundary_cells:
+                    normal_direction[i] = direction[i]
+                    ref_direction = MirroredStencilDirections.mirror_stencil(ref_direction, i)
+
+            # convex corner special case:
+            if all(n == 0 for n in normal_direction):
+                normal_direction = direction
+            else:
+                ref_direction = inverse_direction(ref_direction)
+
+            for i, cell_name in zip(range(dim), self.additional_data):
+                cell[cell_name[0]] = -normal_direction[i]
+            cell['ref_dir'] = self.stencil.index(ref_direction)
+
+    @property
+    def additional_data(self):
+        """Used internally only. For the FreeSlip boundary the information of the normal direction for each pdf
+        direction is needed. This information is stored in the index vector."""
+        if self.normal_direction:
+            return []
+        else:
+            data_type = create_type('int32')
+            wnz = [] if self.dim == 2 else [('wnz', data_type)]
+            data = [('wnx', data_type), ('wny', data_type)] + wnz
+            return data + [('ref_dir', data_type)]
+
+    @property
+    def additional_data_init_callback(self):
+        if self.normal_direction:
+            return None
+        else:
+            return self.init_callback
+
+    def get_additional_code_nodes(self, lb_method):
+        if self.normal_direction:
+            return [MirroredStencilDirections(self.stencil, self.mirror_axis), NeighbourOffsetArrays(lb_method.stencil)]
+        else:
+            return [NeighbourOffsetArrays(lb_method.stencil)]
+
+    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector):
+        neighbor_offset = NeighbourOffsetArrays.neighbour_offset(dir_symbol, lb_method.stencil)
+        if self.normal_direction:
+            tangential_offset = tuple(offset + normal for offset, normal in zip(neighbor_offset, self.normal_direction))
+            mirrored_stencil_symbol = MirroredStencilDirections._mirrored_symbol(self.mirror_axis, self.stencil)
+            mirrored_direction = inv_dir[sp.IndexedBase(mirrored_stencil_symbol, shape=(1,))[dir_symbol]]
+        else:
+            normal_direction = list()
+            for i, cell_name in zip(range(self.dim), self.additional_data):
+                normal_direction.append(index_field[0](cell_name[0]))
+            normal_direction = tuple(normal_direction)
+            tangential_offset = tuple(offset + normal for offset, normal in zip(neighbor_offset, normal_direction))
+
+            mirrored_direction = index_field[0]('ref_dir')
+
+        return Assignment(f_in.center(inv_dir[dir_symbol]), f_out[tangential_offset](mirrored_direction))
+
+
+# end class FreeSlip
+
+
+class WallFunctionBounce(LbBoundary):
     """
+    Wall function based on the bounce back idea, cf. :cite:`Han2021`. Its implementation is extended to the D3Q27
+    stencil, whereas different weights of the drag distribution are proposed.
 
-    def __init__(self, name=None):
+    Args:
+        lb_method: LB method which is used for the simulation
+        pdfs: Symbolic representation of the particle distribution functions.
+        normal_direction: Normal direction of the wall. Currently, only straight and axis-aligned walls are supported.
+        wall_function_model: Wall function that is used to retrieve the wall stress :math:`tau_w` during the simulation.
+                             See :class:`lbmpy.boundaries.wall_treatment.WallFunctionModel` for more details
+        mean_velocity: Optional field or field access for the mean velocity. As wall functions are typically defined
+                       in terms of the mean velocity, it is recommended to provide this variable. Per default, the
+                       instantaneous velocity obtained from pdfs is used for the wall function.
+        sampling_shift: Optional sampling shift for the velocity sampling. Can be provided as symbolic variable or
+                        integer. In both cases, the user must assure that the sampling shift is at least 1, as sampling
+                        in boundary cells is not physical. Per default, a sampling shift of 1 is employed which
+                        corresponds to a sampling in the first fluid cell normal to the wall. For lower friction
+                        Reynolds numbers, choosing a sampling shift >1 has shown to improve the results for higher
+                        resolutions.
+                        Mutually exclusive with the Maronga sampling shift.
+        maronga_sampling_shift: Optionally, apply a correction factor to the wall shear stress proposed by Maronga et
+                                al. :cite:`Maronga2020`. Has only been tested and validated for the MOST wall function.
+                                No guarantee is given that it also works with other wall functions.
+                                Mutually exclusive with the standard sampling shift.
+        dt: time discretisation. Usually one in LB units
+        dy: space discretisation. Usually one in LB units
+        y: distance from the wall
+        target_friction_velocity: A target friction velocity can be given if an estimate is known a priori. This target
+                                  friction velocity will be used as initial guess for implicit wall functions to ensure
+                                  convergence of the Newton algorithm.
+        weight_method: The extension of the WFB to a D3Q27 stencil is non-unique. Different weights can be chosen to
+                       define the drag distribution onto the pdfs. Per default, weights corresponding to the weights
+                       in the D3Q27 stencil are chosen.
+        name: Optional name of the boundary.
+        data_type: Floating-point precision. Per default, double.
+    """
+
+    class WeightMethod(Enum):
+        LATTICE_WEIGHT = auto(),
+        GEOMETRIC_WEIGHT = auto()
+
+    def __init__(self, lb_method, pdfs, normal_direction, wall_function_model,
+                 mean_velocity=None, sampling_shift=1, maronga_sampling_shift=None,
+                 dt=1, dy=1, y=0.5,
+                 target_friction_velocity=None,
+                 weight_method=WeightMethod.LATTICE_WEIGHT,
+                 name=None, data_type='double'):
         """Set an optional name here, to mark boundaries, for example for force evaluations"""
-        super(NoSlip, self).__init__(name)
+        self.stencil = lb_method.stencil
+        if not (self.stencil.Q == 19 or self.stencil.Q == 27):
+            raise ValueError("WFB boundary is currently only defined for D3Q19 and D3Q27 stencils.")
+        self.pdfs = pdfs
+
+        self.wall_function_model = wall_function_model
 
-    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field):
-        return Assignment(f_in(inv_dir[dir_symbol]), f_out(dir_symbol))
+        if mean_velocity:
+            if isinstance(mean_velocity, Field):
+                self.mean_velocity = mean_velocity.center_vector
+            elif isinstance(mean_velocity, Field.Access):
+                self.mean_velocity = mean_velocity.field.neighbor_vector(mean_velocity.offsets)
+            else:
+                raise ValueError("Mean velocity field has to be a pystencils Field or Field.Access")
+        else:
+            self.mean_velocity = None
+
+        if not isinstance(sampling_shift, int):
+            self.sampling_shift = TypedSymbol(sampling_shift.name, np.uint32)
+        else:
+            assert sampling_shift >= 1, "The sampling shift must be greater than 1."
+            self.sampling_shift = sampling_shift
+
+        if maronga_sampling_shift:
+            assert self.mean_velocity, "Mean velocity field must be provided when using the Maronga correction"
+            if not isinstance(maronga_sampling_shift, int):
+                self.maronga_sampling_shift = TypedSymbol(maronga_sampling_shift.name, np.uint32)
+            else:
+                assert maronga_sampling_shift >= 1, "The Maronga sampling shift must be greater than 1."
+                self.maronga_sampling_shift = maronga_sampling_shift
+        else:
+            self.maronga_sampling_shift = None
+
+        if (self.sampling_shift != 1) and self.maronga_sampling_shift:
+            raise ValueError("Both sampling shift and Maronga offset are set. This is currently not supported.")
+
+        self.dt = dt
+        self.dy = dy
+        self.y = y
+        self.data_type = data_type
+
+        self.target_friction_velocity = target_friction_velocity
+
+        self.weight_method = weight_method
+
+        if len(normal_direction) - normal_direction.count(0) != 1:
+            raise ValueError("Only normal directions for straight walls are supported for example (0, 1, 0) for "
+                             "a WallFunctionBounce applied to the southern boundary of the domain")
 
-# end class NoSlip
+        self.mirror_axis = normal_direction.index(*[direction for direction in normal_direction if direction != 0])
+
+        self.normal_direction = normal_direction
+        assert all([n in [-1, 0, 1] for n in self.normal_direction]), \
+            "Only -1, 0 and 1 allowed for defining the normal direction"
+        tangential_component = [int(not n) for n in self.normal_direction]
+        self.normal_axis = tangential_component.index(0)
+        self.tangential_axis = [0, 1, 2]
+        self.tangential_axis.remove(self.normal_axis)
+
+        self.dim = self.stencil.D
+
+        if name is None:
+            name = f"WFB : {offset_to_direction_string([-x for x in normal_direction])}"
+
+        super(WallFunctionBounce, self).__init__(name, calculate_force_on_boundary=False)
+
+    def get_additional_code_nodes(self, lb_method):
+        return [MirroredStencilDirections(self.stencil, self.mirror_axis),
+                NeighbourOffsetArrays(lb_method.stencil)]
+
+    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector):
+        # needed symbols for offsets and indices
+        # neighbour offset symbols are basically the stencil directions defined in stencils.py:L130ff.
+        neighbor_offset = NeighbourOffsetArrays.neighbour_offset(dir_symbol, lb_method.stencil)
+        tangential_offset = tuple(offset + normal for offset, normal in zip(neighbor_offset, self.normal_direction))
+        mirrored_stencil_symbol = MirroredStencilDirections._mirrored_symbol(self.mirror_axis, self.stencil)
+        mirrored_direction = inv_dir[sp.IndexedBase(mirrored_stencil_symbol, shape=(1,))[dir_symbol]]
+
+        name_base = "f_in_inv_offsets_"
+        offset_array_symbols = [TypedSymbol(name_base + d, mirrored_stencil_symbol.dtype) for d in ['x', 'y', 'z']]
+        mirrored_offset = sp.IndexedBase(mirrored_stencil_symbol, shape=(1,))[dir_symbol]
+        offsets = tuple(sp.IndexedBase(s, shape=(1,))[mirrored_offset] for s in offset_array_symbols)
+
+        # needed symbols in the Assignments
+        u_m = sp.Symbol("u_m")
+        tau_w = sp.Symbol("tau_w")
+        wall_stress = sp.symbols("tau_w_x tau_w_y tau_w_z")
+
+        # if the mean velocity field is not given, or the Maronga correction is applied, density and velocity values
+        # will be calculated from pdfs
+        cqc = lb_method.conserved_quantity_computation
+
+        result = []
+        if (not self.mean_velocity) or self.maronga_sampling_shift:
+            pdf_center_vector = sp.Matrix([0] * self.stencil.Q)
+
+            for i in range(self.stencil.Q):
+                pdf_center_vector[i] = self.pdfs[offsets[0] + self.normal_direction[0],
+                                                 offsets[1] + self.normal_direction[1],
+                                                 offsets[2] + self.normal_direction[2]](i)
+
+            eq_equations = cqc.equilibrium_input_equations_from_pdfs(pdf_center_vector)
+            result.append(eq_equations.all_assignments)
+
+        # sample velocity which will be used in the wall stress calculation
+        if self.mean_velocity:
+            if self.maronga_sampling_shift:
+                u_for_tau_wall = tuple(u_mean_i.get_shifted(
+                    self.maronga_sampling_shift * self.normal_direction[0],
+                    self.maronga_sampling_shift * self.normal_direction[1],
+                    self.maronga_sampling_shift * self.normal_direction[2]
+                ) for u_mean_i in self.mean_velocity)
+            else:
+                u_for_tau_wall = tuple(u_mean_i.get_shifted(
+                    self.sampling_shift * self.normal_direction[0],
+                    self.sampling_shift * self.normal_direction[1],
+                    self.sampling_shift * self.normal_direction[2]
+                ) for u_mean_i in self.mean_velocity)
+
+            rho_for_tau_wall = sp.Float(1)
+        else:
+            rho_for_tau_wall = cqc.density_symbol
+            u_for_tau_wall = cqc.velocity_symbols
+
+        # calculate Maronga factor in case of correction
+        maronga_fix = sp.Symbol("maronga_fix")
+        if self.maronga_sampling_shift:
+            inst_first_cell_vel = cqc.velocity_symbols
+            mean_first_cell_vel = tuple(u_mean_i.get_shifted(*self.normal_direction) for u_mean_i in self.mean_velocity)
+
+            mag_inst_vel_first_cell = sp.sqrt(sum([inst_first_cell_vel[i] ** 2 for i in self.tangential_axis]))
+            mag_mean_vel_first_cell = sp.sqrt(sum([mean_first_cell_vel[i] ** 2 for i in self.tangential_axis]))
+
+            result.append(Assignment(maronga_fix, mag_inst_vel_first_cell / mag_mean_vel_first_cell))
+        else:
+            maronga_fix = 1
+
+        # store which direction is tangential component (only those are used for the wall shear stress)
+        red_u_mag = sp.sqrt(sum([u_for_tau_wall[i]**2 for i in self.tangential_axis]))
+
+        u_mag = Assignment(u_m, red_u_mag)
+        result.append(u_mag)
+
+        wall_distance = self.maronga_sampling_shift if self.maronga_sampling_shift else self.sampling_shift
+
+        # using wall function model
+        wall_law_assignments = self.wall_function_model.shear_stress_assignments(
+            density_symbol=rho_for_tau_wall, velocity_symbol=u_m, shear_stress_symbol=tau_w,
+            wall_distance=(wall_distance - sp.Rational(1, 2) * self.dy),
+            u_tau_target=self.target_friction_velocity)
+        result.append(wall_law_assignments)
+
+        # calculate wall stress components and use them to calculate the drag
+        for i in self.tangential_axis:
+            result.append(Assignment(wall_stress[i], - u_for_tau_wall[i] / u_m * tau_w * maronga_fix))
+
+        weight, inv_weight_sq = sp.symbols("wfb_weight inverse_weight_squared")
+
+        if self.stencil.Q == 19:
+            result.append(Assignment(weight, sp.Rational(1, 2)))
+        elif self.stencil.Q == 27:
+            result.append(
+                Assignment(
+                    inv_weight_sq,
+                    sum([CastFunc(neighbor_offset[i], self.data_type)**2 for i in self.tangential_axis])
+                )
+            )
+            a, b = sp.symbols("wfb_a wfb_b")
+
+            if self.weight_method == self.WeightMethod.LATTICE_WEIGHT:
+                res_ab = sp.solve([2 * a + 4 * b - 1, a - 4 * b], [a, b])  # lattice weight scaling
+            elif self.weight_method == self.WeightMethod.GEOMETRIC_WEIGHT:
+                res_ab = sp.solve([2 * a + 4 * b - 1, a - sp.sqrt(2) * b], [a, b])  # geometric scaling
+            else:
+                raise ValueError("Unknown weighting method for the WFB D3Q27 extension. Currently, only lattice "
+                                 "weights and geometric weights are supported.")
+
+            result.append(Assignment(weight, sp.Piecewise((sp.Float(0), sp.Equality(inv_weight_sq, 0)),
+                                                          (res_ab[a], sp.Equality(inv_weight_sq, 1)),
+                                                          (res_ab[b], True))))
+
+        factor = self.dt / self.dy * weight
+        drag = sum(
+            [
+                CastFunc(neighbor_offset[i], self.data_type) * factor * wall_stress[i]
+                for i in self.tangential_axis
+            ]
+        )
+
+        result.append(Assignment(f_in.center(inv_dir[dir_symbol]), f_out[tangential_offset](mirrored_direction) - drag))
+
+        return result
+
+# end class WallFunctionBounce
 
 
 class UBB(LbBoundary):
-    """Velocity bounce back boundary condition, enforcing specified velocity at obstacle
+    r"""Velocity bounce back boundary condition, enforcing specified velocity at obstacle. Furthermore, a density
+        at the wall can be implied. The boundary condition is implemented with the following formula:
+
+    .. math ::
+        f_{\overline{i}}(\mathbf{x}_b, t + \Delta t) = f^{\star}_{i}(\mathbf{x}_b, t) -
+        2 w_{i} \rho_{w} \frac{\mathbf{c}_i \cdot \mathbf{u}_w}{c_s^2}
+
 
     Args:
-        velocity: can either be a constant, an access into a field, or a callback function.
-                  The callback functions gets a numpy record array with members, 'x','y','z', 'dir' (direction)
-                  and 'velocity' which has to be set to the desired velocity of the corresponding link
+        velocity: Prescribe the fluid velocity :math:`\mathbf{u}_w` at the wall.
+                  Can either be a constant, an access into a field, or a callback function.
+                  The callback functions gets a numpy record array with members, ``x``, ``y``, ``z``, ``dir``
+                  (direction) and ``velocity`` which has to be set to the desired velocity of the corresponding link
+        density: Prescribe the fluid density :math:`\rho_{w}` at the wall. If not prescribed the density is
+                 calculated from the PDFs at the wall. The density can only be set constant.
         adapt_velocity_to_force: adapts the velocity to the correct equilibrium when the lattice Boltzmann method holds
                                  a forcing term. If no forcing term is set and adapt_velocity_to_force is set to True
                                  it has no effect.
@@ -124,8 +794,9 @@ class UBB(LbBoundary):
         name: optional name of the boundary.
     """
 
-    def __init__(self, velocity, adapt_velocity_to_force=False, dim=None, name=None, data_type='double'):
+    def __init__(self, velocity, density=None, adapt_velocity_to_force=False, dim=None, name=None, data_type='double'):
         self._velocity = velocity
+        self._density = density
         self._adaptVelocityToForce = adapt_velocity_to_force
         if callable(self._velocity) and not dim:
             raise ValueError("When using a velocity callback the dimension has to be specified with the dim parameter")
@@ -134,7 +805,7 @@ class UBB(LbBoundary):
         self.dim = dim
         self.data_type = data_type
 
-        super(UBB, self).__init__(name)
+        super(UBB, self).__init__(name, calculate_force_on_boundary=False)
 
     @property
     def additional_data(self):
@@ -162,7 +833,7 @@ class UBB(LbBoundary):
         Returns:
             list containing LbmWeightInfo and NeighbourOffsetArrays
         """
-        return [LbmWeightInfo(lb_method), NeighbourOffsetArrays(lb_method.stencil)]
+        return [LbmWeightInfo(lb_method, self.data_type), NeighbourOffsetArrays(lb_method.stencil)]
 
     @property
     def velocity_is_callable(self):
@@ -170,15 +841,15 @@ class UBB(LbBoundary):
         This is useful if the inflow velocity should have a certain profile for instance"""
         return callable(self._velocity)
 
-    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field):
+    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector):
+        dtype = create_type(self.data_type)
         vel_from_idx_field = callable(self._velocity)
         vel = [index_field(f'vel_{i}') for i in range(self.dim)] if vel_from_idx_field else self._velocity
-        direction = dir_symbol
 
         assert self.dim == lb_method.dim, \
             f"Dimension of UBB ({self.dim}) does not match dimension of method ({lb_method.dim})"
 
-        neighbor_offset = NeighbourOffsetArrays.neighbour_offset(direction, lb_method.stencil)
+        neighbor_offset = NeighbourOffsetArrays.neighbour_offset(dir_symbol, lb_method.stencil)
 
         velocity = tuple(v_i.get_shifted(*neighbor_offset)
                          if isinstance(v_i, Field.Access) and not vel_from_idx_field
@@ -188,29 +859,38 @@ class UBB(LbBoundary):
         if self._adaptVelocityToForce:
             cqc = lb_method.conserved_quantity_computation
             shifted_vel_eqs = cqc.equilibrium_input_equations_from_init_values(velocity=velocity)
-            velocity = [eq.rhs for eq in shifted_vel_eqs.new_filtered(cqc.first_order_moment_symbols).main_assignments]
+            shifted_vel_eqs = shifted_vel_eqs.new_without_subexpressions()
+            velocity = [eq.rhs for eq in shifted_vel_eqs.new_filtered(cqc.velocity_symbols).main_assignments]
 
         c_s_sq = sp.Rational(1, 3)
-        weight_of_direction = LbmWeightInfo.weight_of_direction
-        vel_term = 2 / c_s_sq * sum([d_i * v_i for d_i, v_i in zip(neighbor_offset, velocity)]) * weight_of_direction(
-            direction, lb_method)
+        weight_info = LbmWeightInfo(lb_method, data_type=self.data_type)
+        weight_of_direction = weight_info.weight_of_direction
+        vel_term = (
+            2 / c_s_sq
+            * sum([CastFunc(d_i, dtype) * v_i for d_i, v_i in zip(neighbor_offset, velocity)]) 
+            * weight_of_direction(dir_symbol, lb_method)
+        )
 
         # Better alternative: in conserved value computation
         # rename what is currently called density to "virtual_density"
         # provide a new quantity density, which is constant in case of incompressible models
-        if not lb_method.conserved_quantity_computation.zero_centered_pdfs:
+        if lb_method.conserved_quantity_computation.compressible:
             cqc = lb_method.conserved_quantity_computation
             density_symbol = sp.Symbol("rho")
             pdf_field_accesses = [f_out(i) for i in range(len(lb_method.stencil))]
             density_equations = cqc.output_equations_from_pdfs(pdf_field_accesses, {'density': density_symbol})
-            density_symbol = lb_method.conserved_quantity_computation.defined_symbols()['density']
-            result = density_equations.all_assignments
-            result += [Assignment(f_in(inv_dir[direction]),
-                                  f_out(direction) - vel_term * density_symbol)]
+            density_symbol = lb_method.conserved_quantity_computation.density_symbol
+            if self._density:
+                result = [Assignment(density_symbol, self._density)]
+            else:
+                result = density_equations.all_assignments
+            result += [Assignment(f_in(inv_dir[dir_symbol]),
+                                  f_out(dir_symbol) - vel_term * density_symbol)]
             return result
         else:
-            return [Assignment(f_in(inv_dir[direction]),
-                               f_out(direction) - vel_term)]
+            return [Assignment(f_in(inv_dir[dir_symbol]),
+                               f_out(dir_symbol) - vel_term)]
+
 
 # end class UBB
 
@@ -238,8 +918,10 @@ class SimpleExtrapolationOutflow(LbBoundary):
         if name is None:
             name = f"Simple Outflow: {offset_to_direction_string(normal_direction)}"
 
-        self.normal_direction = normal_direction
-        super(SimpleExtrapolationOutflow, self).__init__(name)
+        self.normal_direction = tuple([int(n) for n in normal_direction])
+        assert all([n in [-1, 0, 1] for n in self.normal_direction]), \
+            "Only -1, 0 and 1 allowed for defining the normal direction"
+        super(SimpleExtrapolationOutflow, self).__init__(name, calculate_force_on_boundary=False)
 
     def get_additional_code_nodes(self, lb_method):
         """Return a list of code nodes that will be added in the generated code before the index field loop.
@@ -253,12 +935,13 @@ class SimpleExtrapolationOutflow(LbBoundary):
         """
         return [NeighbourOffsetArrays(lb_method.stencil)]
 
-    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field):
+    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector):
         neighbor_offset = NeighbourOffsetArrays.neighbour_offset(dir_symbol, lb_method.stencil)
         tangential_offset = tuple(offset - normal for offset, normal in zip(neighbor_offset, self.normal_direction))
 
         return Assignment(f_in.center(inv_dir[dir_symbol]), f_out[tangential_offset](inv_dir[dir_symbol]))
 
+
 # end class SimpleExtrapolationOutflow
 
 
@@ -278,7 +961,7 @@ class ExtrapolationOutflow(LbBoundary):
 
     Args:
         normal_direction: direction vector normal to the outflow
-        lb_method: the lattice boltzman method to be used in the simulation
+        lb_method: the lattice Boltzmann method to be used in the simulation
         dt: lattice time step size
         dx: lattice spacing distance
         name: optional name of the boundary.
@@ -304,7 +987,9 @@ class ExtrapolationOutflow(LbBoundary):
         if name is None:
             name = f"Outflow: {offset_to_direction_string(normal_direction)}"
 
-        self.normal_direction = normal_direction
+        self.normal_direction = tuple([int(n) for n in normal_direction])
+        assert all([n in [-1, 0, 1] for n in self.normal_direction]), \
+            "Only -1, 0 and 1 allowed for defining the normal direction"
         self.streaming_pattern = streaming_pattern
         self.zeroth_timestep = zeroth_timestep
         self.dx = sp.Number(dx)
@@ -329,7 +1014,7 @@ class ExtrapolationOutflow(LbBoundary):
 
             self.equilibrium_calculation = calc_eq_pdfs
 
-        super(ExtrapolationOutflow, self).__init__(name)
+        super(ExtrapolationOutflow, self).__init__(name, calculate_force_on_boundary=False)
 
     def init_callback(self, boundary_data, **_):
         dim = boundary_data.dim
@@ -382,7 +1067,7 @@ class ExtrapolationOutflow(LbBoundary):
         """
         return [NeighbourOffsetArrays(lb_method.stencil)]
 
-    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field):
+    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector):
         subexpressions = []
         boundary_assignments = []
         dtdx = sp.Rational(self.dt, self.dx)
@@ -406,11 +1091,17 @@ class ExtrapolationOutflow(LbBoundary):
 
         return AssignmentCollection(boundary_assignments, subexpressions=subexpressions)
 
+
 # end class ExtrapolationOutflow
 
 
 class FixedDensity(LbBoundary):
-    """Boundary condition that fixes the density/pressure at the obstacle.
+    r"""Boundary condition for prescribing a density at the wall. Through :math:`p = c_s^2 \rho` this boundary condition
+        can also function as a pressure boundary condition.
+
+    .. math ::
+        f_{\overline{i}}(\mathbf{x}_b, t + \Delta t) = - f^{\star}_{i}(\mathbf{x}_b, t) +
+        2 w_{i} \rho_{w} (1 + \frac{(\mathbf{c}_i \cdot \mathbf{u}_w)^2}{2c_s^4} + \frac{\mathbf{u}_w^2}{2c_s^2})
 
     Args:
         density: value of the density which should be set.
@@ -422,16 +1113,16 @@ class FixedDensity(LbBoundary):
             name = "Fixed Density " + str(density)
         self.density = density
 
-        super(FixedDensity, self).__init__(name)
+        super(FixedDensity, self).__init__(name, calculate_force_on_boundary=False)
 
-    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field):
+    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector):
         def remove_asymmetric_part_of_main_assignments(assignment_collection, degrees_of_freedom):
             new_main_assignments = [Assignment(a.lhs, get_symmetric_part(a.rhs, degrees_of_freedom))
                                     for a in assignment_collection.main_assignments]
             return assignment_collection.copy(new_main_assignments)
 
         cqc = lb_method.conserved_quantity_computation
-        velocity = cqc.defined_symbols()['velocity']
+        velocity = cqc.velocity_symbols
         symmetric_eq = remove_asymmetric_part_of_main_assignments(lb_method.get_equilibrium(),
                                                                   degrees_of_freedom=velocity)
         substitutions = {sym: f_out(i) for i, sym in enumerate(lb_method.pre_collision_pdf_symbols)}
@@ -440,42 +1131,70 @@ class FixedDensity(LbBoundary):
         simplification = create_simplification_strategy(lb_method)
         symmetric_eq = simplification(symmetric_eq)
 
-        density_symbol = cqc.defined_symbols()['density']
-
         density = self.density
         equilibrium_input = cqc.equilibrium_input_equations_from_init_values(density=density)
         equilibrium_input = equilibrium_input.new_without_subexpressions()
-        density_eq = equilibrium_input.main_assignments[0]
-        assert density_eq.lhs == density_symbol
-        transformed_density = density_eq.rhs
+        equilibrium_input = equilibrium_input.main_assignments_dict
+
+        subexpressions_dict = symmetric_eq.subexpressions_dict
+        subexpressions_dict[cqc.density_symbol] = equilibrium_input[cqc.density_symbol]
+        subexpressions_dict[cqc.density_deviation_symbol] = equilibrium_input[cqc.density_deviation_symbol]
 
         conditions = [(eq_i.rhs, sp.Equality(dir_symbol, i))
                       for i, eq_i in enumerate(symmetric_eq.main_assignments)] + [(0, True)]
         eq_component = sp.Piecewise(*conditions)
 
-        subexpressions = [Assignment(eq.lhs, transformed_density if eq.lhs == density_symbol else eq.rhs)
-                          for eq in symmetric_eq.subexpressions]
+        main_assignments = [Assignment(f_in(inv_dir[dir_symbol]), 2 * eq_component - f_out(dir_symbol))]
 
-        return subexpressions + [Assignment(f_in(inv_dir[dir_symbol]),
-                                            2 * eq_component - f_out(dir_symbol))]
+        ac = AssignmentCollection(main_assignments, subexpressions=subexpressions_dict)
+        ac = ac.new_without_unused_subexpressions()
+        ac.topological_sort()
+
+        return ac
 
-# end class FixedDensity
 
+# end class FixedDensity
 
 class DiffusionDirichlet(LbBoundary):
-    """Boundary condition for advection-diffusion problems that fixes the concentration at the obstacle.
+    """Concentration boundary which is used for concentration or thermal boundary conditions of convection-diffusion
+    equation Base on https://doi.org/10.1103/PhysRevE.85.016701.
 
     Args:
-        concentration: value of the concentration which should be set.
+        concentration: can either be a constant, an access into a field, or a callback function.
+                       The callback functions gets a numpy record array with members, ``x``, ``y``, ``z``, ``dir``
+                       (direction) and ``concentration`` which has to be set to the desired
+                       velocity of the corresponding link
+        velocity_field: if velocity field is given the boundary value is approximated by using the discrete equilibrium.
         name: optional name of the boundary.
+        data_type: data type of the concentration value. default is double
     """
 
-    def __init__(self, concentration, name=None):
+    def __init__(self, concentration, velocity_field=None, name=None, data_type='double'):
         if name is None:
-            name = "Diffusion Dirichlet " + str(concentration)
+            name = "DiffusionDirichlet"
         self.concentration = concentration
+        self._data_type = data_type
+        self.concentration_is_callable = callable(self.concentration)
+        self.velocity_field = velocity_field
 
-        super(DiffusionDirichlet, self).__init__(name)
+        super(DiffusionDirichlet, self).__init__(name, calculate_force_on_boundary=False)
+
+    @property
+    def additional_data(self):
+        """ In case of the UBB boundary additional data is a velocity vector. This vector is added to each cell to
+            realize velocity profiles for the inlet."""
+        if self.concentration_is_callable:
+            return [('concentration', create_type(self._data_type))]
+        else:
+            return []
+
+    @property
+    def additional_data_init_callback(self):
+        """Initialise additional data of the boundary. For an example see
+            `tutorial 02 <https://pycodegen.pages.i10git.cs.fau.de/lbmpy/notebooks/02_tutorial_boundary_setup.html>`_
+            or lbmpy.geometry.add_pipe_inflow_boundary"""
+        if self.concentration_is_callable:
+            return self.concentration
 
     def get_additional_code_nodes(self, lb_method):
         """Return a list of code nodes that will be added in the generated code before the index field loop.
@@ -486,12 +1205,35 @@ class DiffusionDirichlet(LbBoundary):
         Returns:
             list containing LbmWeightInfo
         """
-        return [LbmWeightInfo(lb_method)]
+        if self.velocity_field:
+            return [LbmWeightInfo(lb_method, self._data_type), NeighbourOffsetArrays(lb_method.stencil)]
+        else:
+            return [LbmWeightInfo(lb_method, self._data_type)]
+
+    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector):
+        assert lb_method.conserved_quantity_computation.zero_centered_pdfs is False, \
+            "DiffusionDirichlet only works for methods with normal pdfs storage -> set zero_centered=False"
+        weight_info = LbmWeightInfo(lb_method, self._data_type)
+        w_dir = weight_info.weight_of_direction(dir_symbol, lb_method)
+
+        if self.concentration_is_callable:
+            concentration = index_field[0]('concentration')
+        else:
+            concentration = self.concentration
+
+        if self.velocity_field:
+            neighbour_offset = NeighbourOffsetArrays.neighbour_offset(dir_symbol, lb_method.stencil)
+            u = self.velocity_field
+            cs = sp.Rational(1, 3)
+
+            equilibrium = (1 + scalar_product(neighbour_offset, u.center_vector)**2 / (2 * cs**4)
+                           - scalar_product(u.center_vector, u.center_vector) / (2 * cs**2))
+        else:
+            equilibrium = sp.Rational(1, 1)
+
+        result = [Assignment(f_in(inv_dir[dir_symbol]), 2.0 * w_dir * concentration * equilibrium - f_out(dir_symbol))]
+        return result
 
-    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field):
-        w_dir = LbmWeightInfo.weight_of_direction(dir_symbol, lb_method)
-        return [Assignment(f_in(inv_dir[dir_symbol]),
-                           2 * w_dir * self.concentration - f_out(dir_symbol))]
 
 # end class DiffusionDirichlet
 
@@ -511,11 +1253,12 @@ class NeumannByCopy(LbBoundary):
         """
         return [NeighbourOffsetArrays(lb_method.stencil)]
 
-    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field):
+    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector):
         neighbour_offset = NeighbourOffsetArrays.neighbour_offset(dir_symbol, lb_method.stencil)
         return [Assignment(f_in(inv_dir[dir_symbol]), f_out(inv_dir[dir_symbol])),
                 Assignment(f_out[neighbour_offset](dir_symbol), f_out(dir_symbol))]
 
+
 # end class NeumannByCopy
 
 
@@ -529,7 +1272,7 @@ class StreamInConstant(LbBoundary):
     """
 
     def __init__(self, constant, name=None):
-        super(StreamInConstant, self).__init__(name)
+        super(StreamInConstant, self).__init__(name, calculate_force_on_boundary=False)
         self.constant = constant
 
     def get_additional_code_nodes(self, lb_method):
@@ -543,7 +1286,7 @@ class StreamInConstant(LbBoundary):
         """
         return [NeighbourOffsetArrays(lb_method.stencil)]
 
-    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field):
+    def __call__(self, f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector):
         neighbour_offset = NeighbourOffsetArrays.neighbour_offset(dir_symbol, lb_method.stencil)
         return [Assignment(f_in(inv_dir[dir_symbol]), self.constant),
                 Assignment(f_out[neighbour_offset](dir_symbol), self.constant)]

lbmpy/boundaries/boundaryhandling.py → src/lbmpy/boundaries/boundaryhandling.py

+from dataclasses import replace
 import numpy as np
-import sympy as sp
-from lbmpy.advanced_streaming.indexing import BetweenTimestepsIndexing
-from lbmpy.advanced_streaming.utility import is_inplace, Timestep, AccessPdfValues
-from pystencils import Field, Assignment, TypedSymbol, create_indexed_kernel
-from pystencils.stencil import inverse_direction
+
+from pystencils import Assignment, CreateKernelConfig, create_kernel, Field, Target, FieldType
 from pystencils.boundaries import BoundaryHandling
 from pystencils.boundaries.createindexlist import numpy_data_type_for_boundary_object
-from pystencils.backends.cbackend import CustomCodeNode
+from pystencils.simp import add_subexpressions_for_field_reads
+from pystencils.stencil import inverse_direction
+
+from lbmpy.advanced_streaming.indexing import BetweenTimestepsIndexing
+from lbmpy.advanced_streaming.utility import is_inplace, Timestep, AccessPdfValues
+
+from .._compat import IS_PYSTENCILS_2
+
+if IS_PYSTENCILS_2:
+    from pystencils.types import PsNumericType
 
 
 class LatticeBoltzmannBoundaryHandling(BoundaryHandling):
     """
-    Enables boundary handling for LBM simulations with advanced streaming patterns. 
-    For the in-place patterns AA and EsoTwist, two kernels are generated for a boundary 
+    Enables boundary handling for LBM simulations with advanced streaming patterns.
+    For the in-place patterns AA and EsoTwist, two kernels are generated for a boundary
     object and the right one selected depending on the time step.
     """
 
     def __init__(self, lb_method, data_handling, pdf_field_name, streaming_pattern='pull',
-                 name="boundary_handling", flag_interface=None, target='cpu', openmp=True):
+                 name="boundary_handling", flag_interface=None, target=Target.CPU, openmp=False, **kwargs):
         self._lb_method = lb_method
         self._streaming_pattern = streaming_pattern
         self._inplace = is_inplace(streaming_pattern)
         self._prev_timestep = None
-        super(LatticeBoltzmannBoundaryHandling, self).__init__(data_handling, pdf_field_name, lb_method.stencil,
-                                                               name, flag_interface, target, openmp)
+        super(LatticeBoltzmannBoundaryHandling, self).__init__(
+            data_handling, pdf_field_name, lb_method.stencil,
+            name, flag_interface, target=target, openmp=openmp,
+            **kwargs
+        )
 
     #   ------------------------- Overridden methods of pystencils.BoundaryHandling -------------------------
 
@@ -37,7 +47,7 @@ class LatticeBoltzmannBoundaryHandling(BoundaryHandling):
         self._prev_timestep = None
 
     def add_fixed_steps(self, fixed_loop, **kwargs):
-        if self._inplace:   # Fixed Loop can't do timestep selection
+        if self._inplace:  # Fixed Loop can't do timestep selection
             raise NotImplementedError("Adding to fixed loop is currently not supported for inplace kernels")
         super(LatticeBoltzmannBoundaryHandling, self).add_fixed_steps(fixed_loop, **kwargs)
 
@@ -49,12 +59,14 @@ class LatticeBoltzmannBoundaryHandling(BoundaryHandling):
 
     def _add_inplace_boundary(self, boundary_obj, flag=None):
         if boundary_obj not in self._boundary_object_to_boundary_info:
-            sym_index_field = Field.create_generic('indexField', spatial_dimensions=1,
+            sym_index_field = Field.create_generic('indexField', spatial_dimensions=1, field_type=FieldType.INDEXED,
                                                    dtype=numpy_data_type_for_boundary_object(boundary_obj, self.dim))
-            kernels = [self._create_boundary_kernel(
-                self._data_handling.fields[self._field_name], sym_index_field, boundary_obj, Timestep.EVEN).compile(),
-                self._create_boundary_kernel(
-                self._data_handling.fields[self._field_name], sym_index_field, boundary_obj, Timestep.ODD).compile()]
+
+            ast_even = self._create_boundary_kernel(self._data_handling.fields[self._field_name], sym_index_field,
+                                                    boundary_obj, Timestep.EVEN)
+            ast_odd = self._create_boundary_kernel(self._data_handling.fields[self._field_name], sym_index_field,
+                                                   boundary_obj, Timestep.ODD)
+            kernels = [ast_even.compile(), ast_odd.compile()]
             if flag is None:
                 flag = self.flag_interface.reserve_next_flag()
             boundary_info = self.InplaceStreamingBoundaryInfo(self, boundary_obj, flag, kernels)
@@ -62,10 +74,15 @@ class LatticeBoltzmannBoundaryHandling(BoundaryHandling):
         return self._boundary_object_to_boundary_info[boundary_obj].flag
 
     def _create_boundary_kernel(self, symbolic_field, symbolic_index_field, boundary_obj, prev_timestep=Timestep.BOTH):
+        if IS_PYSTENCILS_2:
+            additional_args = {"default_dtype": self._default_dtype}
+        else:
+            additional_args = dict()
+            
         return create_lattice_boltzmann_boundary_kernel(
             symbolic_field, symbolic_index_field, self._lb_method, boundary_obj,
             prev_timestep=prev_timestep, streaming_pattern=self._streaming_pattern,
-            target=self._target, cpu_openmp=self._openmp)
+            target=self._target, cpu_openmp=self._openmp, **additional_args)
 
     class InplaceStreamingBoundaryInfo(object):
 
@@ -83,6 +100,7 @@ class LatticeBoltzmannBoundaryHandling(BoundaryHandling):
             self.boundary_object = boundary_obj
             self.flag = flag
             self._kernels = kernels
+
     #   end class InplaceStreamingBoundaryInfo
 
     # ------------------------------ Force On Boundary ------------------------------------------------------------
@@ -147,55 +165,85 @@ class LatticeBoltzmannBoundaryHandling(BoundaryHandling):
 
         return dh.reduce_float_sequence(list(result), 'sum')
 
+
 # end class LatticeBoltzmannBoundaryHandling
 
+def create_lattice_boltzmann_boundary_kernel(pdf_field, index_field, lb_method, boundary_functor,
+                                             prev_timestep=Timestep.BOTH, streaming_pattern='pull',
+                                             target=Target.CPU, force_vector=None, **kernel_creation_args):
+    from .._compat import IS_PYSTENCILS_2
 
-class LbmWeightInfo(CustomCodeNode):
+    indexing = BetweenTimestepsIndexing(
+        pdf_field, lb_method.stencil, prev_timestep, streaming_pattern, np.int32, np.int32)
 
-    # --------------------------- Functions to be used by boundaries --------------------------
+    dim = lb_method.stencil.D
+    f_out, f_in = indexing.proxy_fields
+    dir_symbol = indexing.dir_symbol
+    inv_dir = indexing.inverse_dir_symbol
 
-    @staticmethod
-    def weight_of_direction(dir_idx, lb_method=None):
-        if isinstance(sp.sympify(dir_idx), sp.Integer):
-            return lb_method.weights[dir_idx].evalf()
-        else:
-            return sp.IndexedBase(LbmWeightInfo.WEIGHTS_SYMBOL, shape=(1,))[dir_idx]
+    if IS_PYSTENCILS_2:
+        from pystencils.types.quick import SInt
+        config = CreateKernelConfig(
+            index_field=index_field,
+            target=target,
+            index_dtype=SInt(32),
+            skip_independence_check=True,
+            **kernel_creation_args
+        )
 
-    # ---------------------------------- Internal ---------------------------------------------
+        default_data_type: PsNumericType = config.get_option("default_dtype")
 
-    WEIGHTS_SYMBOL = TypedSymbol("weights", "double")
+        if force_vector is None:
+            force_vector_type = np.dtype([(f"F_{i}", default_data_type.numpy_dtype) for i in range(dim)], align=True)
+            force_vector = Field.create_generic('force_vector', spatial_dimensions=1,
+                                                dtype=force_vector_type, field_type=FieldType.INDEXED)
 
-    def __init__(self, lb_method):
-        weights = [str(w.evalf()) for w in lb_method.weights]
-        w_sym = LbmWeightInfo.WEIGHTS_SYMBOL
-        code = "const double %s [] = { %s };\n" % (w_sym.name, ",".join(weights))
-        super(LbmWeightInfo, self).__init__(code, symbols_read=set(), symbols_defined={w_sym})
-# end class LbmWeightInfo
+        boundary_assignments = boundary_functor(f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector)
+        boundary_assignments = indexing.substitute_proxies(boundary_assignments)
 
+        if pdf_field.dtype != default_data_type:
+            boundary_assignments = add_subexpressions_for_field_reads(boundary_assignments, data_type=default_data_type)
 
-def create_lattice_boltzmann_boundary_kernel(pdf_field, index_field, lb_method, boundary_functor,
-                                             prev_timestep=Timestep.BOTH, streaming_pattern='pull',
-                                             target='cpu', **kernel_creation_args):
+        elements: list[Assignment] = []
 
-    indexing = BetweenTimestepsIndexing(
-        pdf_field, lb_method.stencil, prev_timestep, streaming_pattern, np.int64, np.int64)
+        index_arrs_node = indexing.create_code_node()
+        elements += index_arrs_node.get_array_declarations()
+        
+        for node in boundary_functor.get_additional_code_nodes(lb_method)[::-1]:
+            elements += node.get_array_declarations()
+            
+        elements += [Assignment(dir_symbol, index_field[0]('dir'))]
+        elements += boundary_assignments.all_assignments
 
-    f_out, f_in = indexing.proxy_fields
-    dir_symbol = indexing.dir_symbol
-    inv_dir = indexing.inverse_dir_symbol
+        kernel = create_kernel(elements, config=config)
+        return kernel
+    else:
+        config = CreateKernelConfig(index_fields=[index_field], target=target, default_number_int="int32",
+                                    skip_independence_check=True, **kernel_creation_args)
+
+        default_data_type = config.data_type.default_factory()
+
+        if force_vector is None:
+            force_vector_type = np.dtype([(f"F_{i}", default_data_type.c_name) for i in range(dim)], align=True)
+            force_vector = Field.create_generic('force_vector', spatial_dimensions=1,
+                                                dtype=force_vector_type, field_type=FieldType.INDEXED)
+
+        config = replace(config, index_fields=[index_field, force_vector])
+
+        boundary_assignments = boundary_functor(f_out, f_in, dir_symbol, inv_dir, lb_method, index_field, force_vector)
+        boundary_assignments = indexing.substitute_proxies(boundary_assignments)
 
-    boundary_assignments = boundary_functor(f_out, f_in, dir_symbol, inv_dir, lb_method, index_field)
-    boundary_assignments = indexing.substitute_proxies(boundary_assignments)
+        if pdf_field.dtype != default_data_type:
+            boundary_assignments = add_subexpressions_for_field_reads(boundary_assignments, data_type=default_data_type)
 
-    #   Code Elements inside the loop
-    elements = [Assignment(dir_symbol, index_field[0]('dir'))]
-    elements += boundary_assignments.all_assignments
+        elements = [Assignment(dir_symbol, index_field[0]('dir'))]
+        elements += boundary_assignments.all_assignments
 
-    kernel = create_indexed_kernel(elements, [index_field], target=target, **kernel_creation_args)
+        kernel = create_kernel(elements, config=config)
 
-    #   Code Elements ahead of the loop
-    index_arrs_node = indexing.create_code_node()
-    for node in boundary_functor.get_additional_code_nodes(lb_method)[::-1]:
-        kernel.body.insert_front(node)
-    kernel.body.insert_front(index_arrs_node)
-    return kernel
+        #   Code Elements ahead of the loop
+        index_arrs_node = indexing.create_code_node()
+        for node in boundary_functor.get_additional_code_nodes(lb_method)[::-1]:
+            kernel.body.insert_front(node)
+        kernel.body.insert_front(index_arrs_node)
+        return kernel

src/lbmpy/boundaries/wall_function_models.py

+import sympy as sp
+from abc import ABC, abstractmethod
+
+from pystencils import Assignment
+
+
+class WallFunctionModel(ABC):
+    def __init__(self, name):
+        self._name = name
+
+    @abstractmethod
+    def shear_stress_assignments(self, density_symbol: sp.Symbol, shear_stress_symbol: sp.Symbol,
+                                 velocity_symbol: sp.Symbol, wall_distance, u_tau_target):
+        """
+        Computes a symbolic representation for the log law.
+
+        Args:
+            density_symbol: symbol density, should be provided by the LB method's conserved quantity computation
+            shear_stress_symbol: symbolic wall shear stress to which the calculated shear stress will be assigned
+            velocity_symbol: symbolic velocity that is taken as a reference in the wall functions
+            wall_distance: distance to the wall, equals to 0.5 in standard cell-centered LBM
+            u_tau_target: in implicit wall functions, a target friction velocity can be provided which will be used as
+                          initial guess in the Newton iteration. This target friction velocity can be obtained, e.g.,
+                          from the target friction Reynolds number
+        """
+        pass
+
+
+# end class WallFunctionModel
+
+
+class ExplicitWallFunctionModel(WallFunctionModel, ABC):
+    """
+    Abstract base class for explicit wall functions that can be solved directly for the wall shear stress.
+    """
+
+    def __init__(self, name):
+        super(ExplicitWallFunctionModel, self).__init__(name=name)
+
+
+class MoninObukhovSimilarityTheory(ExplicitWallFunctionModel):
+    def __init__(self, z0, kappa=0.41, phi=0, name="MOST"):
+        self.z0 = z0
+        self.kappa = kappa
+        self.phi = phi
+
+        super(MoninObukhovSimilarityTheory, self).__init__(name=name)
+
+    def shear_stress_assignments(self, density_symbol: sp.Symbol, shear_stress_symbol: sp.Symbol,
+                                 velocity_symbol: sp.Symbol, wall_distance, u_tau_target=None):
+        u_tau = velocity_symbol * self.kappa / sp.ln(wall_distance / self.z0 + self.phi)
+        return [Assignment(shear_stress_symbol, u_tau ** 2 * density_symbol)]
+
+
+class ImplicitWallFunctionModel(WallFunctionModel, ABC):
+    """
+    Abstract base class for implicit wall functions that require a Newton procedure to solve for the wall shear stress.
+    """
+
+    def __init__(self, name, newton_steps, viscosity):
+        self.newton_steps = newton_steps
+        self.u_tau = sp.symbols(f"wall_function_u_tau_:{self.newton_steps + 1}")
+        self.delta = sp.symbols(f"wall_function_delta_:{self.newton_steps}")
+
+        self.viscosity = viscosity
+
+        super(ImplicitWallFunctionModel, self).__init__(name=name)
+
+    def newton_iteration(self, wall_law):
+        m = -wall_law / wall_law.diff(self.u_tau[0])
+
+        assignments = []
+        for i in range(self.newton_steps):
+            assignments.append(Assignment(self.delta[i], m.subs({self.u_tau[0]: self.u_tau[i]})))
+            assignments.append(Assignment(self.u_tau[i + 1], self.u_tau[i] + self.delta[i]))
+
+        return assignments
+
+
+class LogLaw(ImplicitWallFunctionModel):
+    """
+    Analytical model for the velocity profile inside the boundary layer, obtained from the mean velocity gradient.
+    Only valid in the log-law region.
+    """
+
+    def __init__(self, viscosity, newton_steps=5, kappa=0.41, b=5.2, name="LogLaw"):
+        self.kappa = kappa
+        self.b = b
+
+        super(LogLaw, self).__init__(name=name, newton_steps=newton_steps, viscosity=viscosity)
+
+    def shear_stress_assignments(self, density_symbol: sp.Symbol, shear_stress_symbol: sp.Symbol,
+                                 velocity_symbol: sp.Symbol, wall_distance, u_tau_target=None):
+        def law(u_p, y_p):
+            return 1 / self.kappa * sp.ln(y_p) + self.b - u_p
+
+        u_plus = velocity_symbol / self.u_tau[0]
+        y_plus = (wall_distance * self.u_tau[0]) / self.viscosity
+
+        u_tau_init = u_tau_target if u_tau_target else velocity_symbol / sp.Float(100)
+
+        wall_law = law(u_plus, y_plus)
+        assignments = [Assignment(self.u_tau[0], u_tau_init),  # initial guess
+                       *self.newton_iteration(wall_law),  # newton iterations
+                       Assignment(shear_stress_symbol, self.u_tau[-1] ** 2 * density_symbol)]  # final result
+
+        return assignments
+
+
+class SpaldingsLaw(ImplicitWallFunctionModel):
+    """
+    Single formula for the velocity profile inside the boundary layer, proposed by Spalding :cite:`spalding1961`.
+    Valid in the inner and the outer layer.
+    """
+
+    def __init__(self, viscosity, newton_steps=5, kappa=0.41, b=5.5, name="Spalding"):
+        self.kappa = kappa
+        self.b = b
+
+        super(SpaldingsLaw, self).__init__(name=name, newton_steps=newton_steps, viscosity=viscosity)
+
+    def shear_stress_assignments(self, density_symbol: sp.Symbol, shear_stress_symbol: sp.Symbol,
+                                 velocity_symbol: sp.Symbol, wall_distance, u_tau_target=None):
+        def law(u_p, y_p):
+            k_times_u = self.kappa * u_p
+            frac_1 = (k_times_u ** 2) / sp.Float(2)
+            frac_2 = (k_times_u ** 3) / sp.Float(6)
+            return (u_p + sp.exp(-self.kappa * self.b) * (sp.exp(k_times_u) - sp.Float(1) - k_times_u - frac_1 - frac_2)
+                    - y_p)
+
+        u_plus = velocity_symbol / self.u_tau[0]
+        y_plus = (wall_distance * self.u_tau[0]) / self.viscosity
+
+        u_tau_init = u_tau_target if u_tau_target else velocity_symbol / sp.Float(100)
+
+        wall_law = law(u_plus, y_plus)
+        assignments = [Assignment(self.u_tau[0], u_tau_init),  # initial guess
+                       *self.newton_iteration(wall_law),  # newton iterations
+                       Assignment(shear_stress_symbol, self.u_tau[-1] ** 2 * density_symbol)]  # final result
+
+        return assignments
+
+
+class MuskerLaw(ImplicitWallFunctionModel):
+    """
+    Quasi-analytical model for the velocity profile inside the boundary layer, proposed by Musker. Valid in the inner
+    and the outer layer.
+    Formulation taken from :cite:`malaspinas2015`, Equation (59).
+    """
+
+    def __init__(self, viscosity, newton_steps=5, name="Musker"):
+
+        super(MuskerLaw, self).__init__(name=name, newton_steps=newton_steps, viscosity=viscosity)
+
+    def shear_stress_assignments(self, density_symbol: sp.Symbol, shear_stress_symbol: sp.Symbol,
+                                 velocity_symbol: sp.Symbol, wall_distance, u_tau_target=None):
+        def law(u_p, y_p):
+            arctan = sp.Float(5.424) * sp.atan(sp.Float(0.119760479041916168) * y_p - sp.Float(0.488023952095808383))
+            logarithm = (sp.Float(0.434) * sp.log((y_p + sp.Float(10.6)) ** sp.Float(9.6)
+                                                  / (y_p ** 2 - sp.Float(8.15) * y_p + sp.Float(86)) ** 2))
+            return (arctan + logarithm - sp.Float(3.50727901936264842)) - u_p
+
+        u_plus = velocity_symbol / self.u_tau[0]
+        y_plus = (wall_distance * self.u_tau[0]) / self.viscosity
+
+        u_tau_init = u_tau_target if u_tau_target else velocity_symbol / sp.Float(100)
+
+        wall_law = law(u_plus, y_plus)
+        assignments = [Assignment(self.u_tau[0], u_tau_init),  # initial guess
+                       *self.newton_iteration(wall_law),  # newton iterations
+                       Assignment(shear_stress_symbol, self.u_tau[-1] ** 2 * density_symbol)]  # final result
+
+        return assignments

lbmpy/chapman_enskog/chapman_enskog.py → src/lbmpy/chapman_enskog/chapman_enskog.py

@@ -21,8 +21,8 @@ class ChapmanEnskogAnalysis:
         cqc = method.conserved_quantity_computation
         self._method = method
         self._moment_cache = LbMethodEqMoments(method)
-        self.rho = cqc.defined_symbols(order=0)[1]
-        self.u = cqc.defined_symbols(order=1)[1]
+        self.rho = cqc.density_symbol
+        self.u = cqc.velocity_symbols
         self.t = sp.Symbol("t")
         self.epsilon = sp.Symbol("epsilon")
 
@@ -370,7 +370,7 @@ def take_moments(eqn, pdf_to_moment_name=(('f', '\\Pi'), ('\\Omega f', '\\Upsilo
                 if new_f_index is None:
                     rest *= factor
                 else:
-                    assert not(new_f_index and f_index)
+                    assert not (new_f_index and f_index)
                     f_index = new_f_index
 
         moment_tuple = [0] * len(velocity_terms)

lbmpy/chapman_enskog/chapman_enskog_steady_state.py → src/lbmpy/chapman_enskog/chapman_enskog_steady_state.py

 import functools
 
+import numpy as np
 import sympy as sp
 
 from lbmpy.chapman_enskog.chapman_enskog import (
@@ -150,7 +151,8 @@ class SteadyStateChapmanEnskogAnalysis:
 
                 have_shape = hasattr(arg, 'shape') and hasattr(new_prod, 'shape')
                 if have_shape and arg.shape == new_prod.shape and arg.shape[1] == 1:
-                    new_prod = sp.matrix_multiply_elementwise(new_prod, arg)
+                    # since sympy 1.9 sp.matrix_multiply_elementwise does not work anymore in this case
+                    new_prod = sp.Matrix(np.multiply(new_prod, arg))
                 else:
                     new_prod = arg * new_prod
                 if new_prod == 0:

lbmpy/continuous_distribution_measures.py → src/lbmpy/continuous_distribution_measures.py

@@ -6,21 +6,23 @@ import sympy as sp
 
 from lbmpy.moments import polynomial_to_exponent_representation
 from pystencils.cache import disk_cache, memorycache
-from pystencils.sympyextensions import complete_the_squares_in_exp
+from pystencils.sympyextensions import complete_the_squares_in_exp, scalar_product
 
 
 @memorycache()
-def moment_generating_function(generating_function, symbols, symbols_in_result):
+def moment_generating_function(generating_function, symbols, symbols_in_result, velocity=None):
     r"""
     Computes the moment generating function of a probability distribution. It is defined as:
 
     .. math ::
-        F[f(\mathbf{x})](\mathbf{t}) = \int e^{<\mathbf{x}, \mathbf{t}>} f(x)\; dx
+        F[f(\mathbf{x})](t) = \int e^{<\mathbf{x}, t>} f(\mathbf{x})\; dx
 
     Args:
         generating_function: sympy expression
-        symbols: a sequence of symbols forming the vector x
+        symbols: a sequence of symbols forming the vector :math:`\mathbf{x}`
         symbols_in_result: a sequence forming the vector t
+        velocity: if the generating function generates central moments, the velocity needs to be substracted. Thus the
+                  velocity symbols need to be passed. All generating functions need to have the same parameters.
 
     Returns:
         transformation result F: an expression that depends now on symbols_in_result
@@ -55,9 +57,27 @@ def moment_generating_function(generating_function, symbols, symbols_in_result):
     return sp.simplify(result)
 
 
-def cumulant_generating_function(func, symbols, symbols_in_result):
+def central_moment_generating_function(func, symbols, symbols_in_result, velocity=sp.symbols("u_:3")):
+    r"""
+    Computes central moment generating func, which is defined as:
+
+    .. math ::
+        K( \mathbf{\Xi} ) = \exp ( - \mathbf{\Xi} \cdot \mathbf{u} ) M( \mathbf{\Xi} ).
+
+    For parameter description see :func:`moment_generating_function`.
     """
-    Computes cumulant generating func, which is the logarithm of the moment generating func.
+    argument = - scalar_product(symbols_in_result, velocity)
+
+    return sp.exp(argument) * moment_generating_function(func, symbols, symbols_in_result)
+
+
+def cumulant_generating_function(func, symbols, symbols_in_result, velocity=None):
+    r"""
+    Computes cumulant generating func, which is the logarithm of the moment generating func:
+
+    .. math ::
+        C(\mathbf{\Xi}) = \log M(\mathbf{\Xi})
+
     For parameter description see :func:`moment_generating_function`.
     """
     return sp.ln(moment_generating_function(func, symbols, symbols_in_result))
@@ -93,16 +113,16 @@ def multi_differentiation(generating_function, index, symbols):
 
 
 @memorycache(maxsize=512)
-def __continuous_moment_or_cumulant(func, moment, symbols, generating_function):
+def __continuous_moment_or_cumulant(func, moment, symbols, generating_function, velocity=sp.symbols("u_:3")):
     if type(moment) is tuple and not symbols:
         symbols = sp.symbols("xvar yvar zvar")
 
     dim = len(moment) if type(moment) is tuple else len(symbols)
 
     # not using sp.Dummy here - since it prohibits caching
-    t = tuple([sp.Symbol("tmpvar_%d" % i, ) for i in range(dim)])
+    t = sp.symbols(f"tmpvar_:{dim}")
     symbols = symbols[:dim]
-    generating_function = generating_function(func, symbols, t)
+    generating_function = generating_function(func, symbols, t, velocity=velocity)
 
     if type(moment) is tuple:
         return multi_differentiation(generating_function, moment, t)
@@ -128,6 +148,18 @@ def continuous_moment(func, moment, symbols=None):
     return __continuous_moment_or_cumulant(func, moment, symbols, moment_generating_function)
 
 
+def continuous_central_moment(func, moment, symbols=None, velocity=sp.symbols("u_:3")):
+    """Computes central moment of given function.
+
+    Args:
+        func: function to compute moments of
+        moment: tuple or polynomial describing the moment
+        symbols: if moment is given as polynomial, pass the moment symbols, i.e. the dof of the polynomial
+    """
+    return __continuous_moment_or_cumulant(func, moment, symbols, central_moment_generating_function,
+                                           velocity=velocity)
+
+
 def continuous_cumulant(func, moment, symbols=None):
     """Computes cumulant of continuous function.
 

src/lbmpy/creationfunctions.py

+r"""
+Creating LBM kernels and Parameter Specifications
+-------------------------------------------------
+
+Kernel functions are created in four/five steps represented by five
+python functions: `create_lb_method`, *create_lb_collision_rule/create_lb_update_rule*, `create_lb_ast` and
+`create_lb_function` Each of those functions is configured with three data classes.
+
+One dataclass defines the lattice Boltzmann method itself. This class is called `LBMConfig`. It defines, for example,
+which collision space or LB stencil should be used.
+
+The second one determines optimisations that are specific to the LBM. Optimisations like the
+common subexpression elimination. Most of these optimisations act on the assignment level.
+This means they only manipulate the assignments. The config class is called `LBMOptimisation`.
+
+The third data class determines hardware optimisation. This means that contrary to the `LBMOptimisation` class,
+it acts on the level of the abstract syntax tree. Thus, it is independent of the assignments and the LBM
+and belongs to pystencils, not lbmpy. This can be found in the pystencils module as
+'pystencils.kernelcreation.CreateKernelConfig'. With this class, for example, the target (CPU, GPU etc.)
+of the generated code is specified.
+
+1. *Method*:
+         the method defines the collision process. Currently, there are two big categories:
+         moment and cumulant based methods. A method defines how each moment or cumulant is relaxed by
+         storing the equilibrium value and the relaxation rate for each moment/cumulant.
+2. *Collision/Update Rule*:
+         Methods can generate a "collision rule" which is an equation collection that define the
+         post collision values as a function of the pre-collision values. On these equation collection
+         simplifications are applied to reduce the number of floating point operations.
+         At this stage an entropic optimisation step can also be added to determine one relaxation rate by an
+         entropy condition.
+         Then a streaming rule is added which transforms the collision rule into an update rule.
+         The streaming step depends on the pdf storage (source/destination, AABB pattern, EsoTwist).
+         Currently only the simple source/destination  pattern is supported.
+3. *AST*:
+        The abstract syntax tree describes the structure of the kernel, including loops and conditionals.
+        The ast can be modified, e.g., to add OpenMP pragmas, reorder loops or apply other optimisations.
+4. *Function*:
+        This step compiles the AST into an executable function, either for CPU or GPUs. This function
+        behaves like a normal Python function and runs one LBM time step.
+
+Each stage (apart from *Function*) also adds its result to the given `LBMConfig` object. The `LBMConfig`
+thus coalesces all information defining the LBM kernel.
+
+The function :func:`create_lb_function` runs the whole pipeline, the other functions in this module
+execute this pipeline only up to a certain step. Each function optionally also takes the result of the previous step.
+
+For example, to modify the AST one can run::
+
+    ast = create_lb_ast(...)
+    # modify ast here
+    func = create_lb_function(ast=ast, ...)
+
+"""
+import copy
+
+from dataclasses import dataclass, field, replace
+from typing import Union, List, Tuple, Any, Type, Iterable
+from warnings import warn, filterwarnings
+
+from ._compat import IS_PYSTENCILS_2
+
+import sympy as sp
+
+from lbmpy.enums import Stencil, Method, ForceModel, CollisionSpace, SubgridScaleModel
+import lbmpy.forcemodels as forcemodels
+from lbmpy.fieldaccess import CollideOnlyInplaceAccessor, PdfFieldAccessor, PeriodicTwoFieldsAccessor
+from lbmpy.fluctuatinglb import add_fluctuations_to_collision_rule
+from lbmpy.partially_saturated_cells import (replace_by_psm_collision_rule, PSMConfig,
+                                             add_psm_solid_collision_to_collision_rule)
+from lbmpy.non_newtonian_models import add_cassons_model, CassonsParameters
+from lbmpy.methods import (create_mrt_orthogonal, create_mrt_raw, create_central_moment,
+                           create_srt, create_trt, create_trt_kbc)
+from lbmpy.methods.creationfunctions import CollisionSpaceInfo
+from lbmpy.methods.creationfunctions import (
+    create_with_monomial_cumulants, create_cumulant, create_with_default_polynomial_cumulants)
+from lbmpy.methods.momentbased.entropic import add_entropy_condition, add_iterative_entropy_condition
+from lbmpy.relaxationrates import relaxation_rate_from_magic_number
+from lbmpy.simplificationfactory import create_simplification_strategy
+from lbmpy.stencils import LBStencil
+from lbmpy.turbulence_models import add_sgs_model
+from lbmpy.updatekernels import create_lbm_kernel, create_stream_pull_with_output_kernel
+from lbmpy.advanced_streaming.utility import Timestep, get_accessor
+from .forcemodels import AbstractForceModel
+
+import pystencils
+from pystencils import CreateKernelConfig, create_kernel
+from pystencils.cache import disk_cache_no_fallback
+from pystencils.field import Field
+from pystencils.simp import sympy_cse, SimplificationStrategy
+# needed for the docstring
+from lbmpy.methods.abstractlbmethod import LbmCollisionRule, AbstractLbMethod
+from lbmpy.methods.cumulantbased import CumulantBasedLbMethod
+
+if IS_PYSTENCILS_2:
+    from pystencils import Kernel as KernelFunction
+else:
+    from pystencils.astnodes import KernelFunction
+
+# Filter out JobLib warnings. They are not useful for use:
+# https://github.com/joblib/joblib/issues/683
+filterwarnings("ignore", message="Persisting input arguments took")
+
+
+@dataclass
+class LBMConfig:
+    """
+    **Below all parameters for the LBMConfig are explained**
+    """
+    stencil: LBStencil = LBStencil(Stencil.D2Q9)
+    """
+    All stencils are defined in :class:`lbmpy.enums.Stencil`. From that :class:`lbmpy.stencils.LBStencil` 
+    class will be created
+    """
+    method: Method = Method.SRT
+    """
+    Name of lattice Boltzmann method. Defined by :class:`lbmpy.enums.Method`.
+    This determines the selection and relaxation pattern of moments/cumulants, i.e. which moment/cumulant basis is
+    chosen, and which of the basis vectors are relaxed together
+    """
+    relaxation_rates: Iterable = None
+    """
+    Sequence of relaxation rates, number depends on selected method. If you specify more rates than
+    method needs, the additional rates are ignored.
+
+    If no relaxation rates are specified, the parameter `relaxation_rate` will be consulted.
+    """
+    relaxation_rate: Union[int, float, Type[sp.Symbol]] = None
+    """
+    The method's primary relaxation rate. In most cases, this is the relaxation rate governing shear viscosity.
+    For SRT, this is the only relaxation rate.
+    For TRT, the second relaxation rate is then determined via magic number.
+    In the case of raw moment, central moment, and cumulant-based MRT methods, all other relaxation rates will be
+    set to unity.
+
+    If neither `relaxation_rate` nor `relaxation_rates` is specified, the behaviour is as if 
+    `relaxation_rate=sp.Symbol('omega')` was set.
+    """
+    compressible: bool = False
+    """
+    Affects the selection of equilibrium moments. Both options approximate the *incompressible*
+    Navier Stokes Equations. However when chosen as False, the approximation is better, the standard LBM derivation is
+    compressible.
+    """
+    zero_centered: bool = True
+    """
+    Governs the storage format of populations. If `False`, the discrete particle distribution vector is stored in its
+    absolute form. If `True`, instead, only the distribution's deviation from its rest state (typically given by the
+    lattice weights) is stored.
+    """
+    delta_equilibrium: bool = None
+    """
+    Determines whether or not the (continuous or discrete, see `continuous_equilibrium`) Maxwellian equilibrium is
+    expressed in its absolute form, or only by its deviation from the rest state (typically given by the reference
+    density and zero velocity). This parameter is only effective if `zero_centered` is set to `True`. Then, if
+    `delta_equilibrium` is `False`, the rest state must be reintroduced to the populations during collision. Otherwise,
+    if `delta_equilibrium` is `True`, the collision equations can be derived using only the deviations from the rest 
+    state.
+
+    If `None` is passed to `delta_equilibrium`, its value will be chosen automatically, depending on the value of
+    `zero_centered` and the chosen `method`.
+    """
+    equilibrium_order: int = 2
+    """
+    Order in velocity, at which the equilibrium moment approximation is
+    truncated. Order 2 is sufficient to approximate Navier-Stokes. This parameter has no effect on cumulant-based
+    methods, whose equilibrium terms have no contributions above order one.
+    """
+    c_s_sq: sp.Expr = sp.Rational(1, 3)
+    """
+    The squared lattice speed of sound used to derive the LB method. It is very uncommon to use a value different 
+    to 1 / 3.
+    """
+    weighted: bool = True
+    """
+    Affects only orthogonal MRT methods. If set to True a weighted Gram-Schmidt procedure is used to orthogonalise 
+    the moments.
+    """
+    nested_moments: List[List] = None
+    """
+    A list of lists of modes, grouped by common relaxation times. This is usually used in
+    conjunction with `lbmpy.methods.default_moment_sets.mrt_orthogonal_modes_literature`.
+    If this argument is not provided, Gram-Schmidt orthogonalisation of the default modes is performed.
+    """
+
+    force_model: Union[AbstractForceModel, ForceModel] = None
+    """
+    Force model to determine how forcing terms enter the collision rule.
+    Possibilities are defined in :class: `lbmpy.enums.ForceModel`
+    """
+    force: Union[Tuple, Field] = (0, 0, 0)
+    """
+    Either constant force or a symbolic expression depending on field value
+    """
+    continuous_equilibrium: bool = True
+    """
+    Way to compute equilibrium moments/cumulants, if False the standard discretised LBM equilibrium is used,
+    otherwise the equilibrium moments are computed from the continuous Maxwellian. This makes only a 
+    difference if sparse stencils are used e.g. D2Q9 and D3Q27 are not affected, D319 and DQ15 are affected.
+    """
+    maxwellian_moments: bool = None
+    """
+    Deprecated and due for removal by version 0.5; use `continuous_equilibrium` instead.
+    """
+    initial_velocity: Tuple = None,
+    """
+    Initial velocity in domain, can either be a tuple (x,y,z) velocity to set a constant
+    velocity everywhere, or a numpy array with the same size of the domain, with a last coordinate of shape dim to set
+    velocities on cell level
+    """
+    galilean_correction: bool = False
+    """
+    Special correction for D3Q27 cumulant LBMs. For Details see
+    :mod:`lbmpy.methods.cumulantbased.galilean_correction`
+    """
+    fourth_order_correction: Union[float, bool] = False
+    """
+    Special correction for rendering D3Q27 cumulant LBMs fourth-order accurate in diffusion. For Details see
+    :mod:`lbmpy.methods.cumulantbased.fourth_order_correction`. If set to `True`, the fourth-order correction is 
+    employed without limiters (or more precisely with a very high limiter, practically disabling the limiters). If this 
+    variable is set to a number, the latter is used for the limiters (uniformly for omega_3, omega_4 and omega_5). 
+    """
+    collision_space_info: CollisionSpaceInfo = None
+    """
+    Information about the LB method's collision space (see :class:`lbmpy.methods.creationfunctions.CollisionSpaceInfo`)
+    including the classes defining how populations are transformed to these spaces.
+    If left at `None`, it will be inferred according to the value of `method`.
+    If an instance of the :class:`lbmpy.enums.CollisionSpace` enum is passed, a
+    :class:`lbmpy.method.CollisionSpaceInfo` instance with the space's default setup is created.
+    Otherwise, the selected collision space must be in accord with the chosen :class:`lbmpy.enum.Method`.
+    """
+    entropic: bool = False
+    """
+    In case there are two distinct relaxation rate in a method, one of them (usually the one, not
+    determining the viscosity) can be automatically chosen w.r.t an entropy condition. For details see
+    :mod:`lbmpy.methods.momentbased.entropic`
+    """
+    entropic_newton_iterations: int = None
+    """
+    For moment methods the entropy optimum can be calculated in closed form.
+    For cumulant methods this is not possible, in that case it is computed using Newton iterations.
+    This parameter can be used to force Newton iterations and specify how many should be done
+    """
+    omega_output_field: Field = None
+    """
+    A pystencils Field can be passed here, where the calculated free relaxation rate of
+    an entropic or subgrid-scale method is written to
+    """
+    eddy_viscosity_field: Field = None
+    """
+    A pystencils Field can be passed here, where the eddy-viscosity of a subgrid-scale model is written.
+    """
+    subgrid_scale_model: Union[SubgridScaleModel, tuple[SubgridScaleModel, float], tuple[SubgridScaleModel, int]] = None
+    """
+    Choose a subgrid-scale model (SGS) for large-eddy simulations. ``omega_output_field`` can be set to
+    write out adapted relaxation rates. Either provide just the SGS and use the default model constants or provide a
+    tuple of the SGS and its corresponding model constant.
+    """
+    cassons: CassonsParameters = False
+    """
+    Adds the Cassons model according to https://doi.org/10.1007/s10955-005-8415-x
+    The parameters are set with the ``CassonsParameters`` dataclass.
+    """
+    fluctuating: dict = False
+    """
+    Enables fluctuating lattice Boltzmann by randomizing collision process.
+    Pass dictionary with parameters to  ``lbmpy.fluctuatinglb.add_fluctuations_to_collision_rule``.
+    Can only be used for weighed MRT collision operators.
+    """
+    temperature: Any = None
+    """
+    Temperature for fluctuating lattice Boltzmann methods.
+    """
+
+    psm_config: PSMConfig = None
+    """
+    If a PSM config is specified, (1 - fractionField) is added to the relaxation rates of the collision 
+    and to the potential force term, and a solid collision is build and added to the main assignments.
+    """
+
+    output: dict = field(default_factory=dict)
+    """
+    A dictionary mapping macroscopic quantites e.g. the strings 'density' and 'velocity' to pystencils
+    fields. In each timestep the corresponding quantities are written to the given fields. Possible input would be:
+    {'density': density_field, 'velocity': velocity_field}
+    """
+    velocity_input: Field = None
+    """
+    Symbolic field where the velocities are read from. If `None` is given the velocity is calculated inplace from
+    with first order moments.
+    """
+    density_input: Field = None
+    """
+    Symbolic field where the density is read from. If `None` is given the density is calculated inplace from
+    with zeroth order moment.
+    """
+    conserved_moments: bool = True
+    """
+    If lower order moments are conserved or not. If velocity or density input is set the lower order moments are not 
+    conserved anymore.
+    """
+
+    kernel_type: Union[str, Type[PdfFieldAccessor]] = 'default_stream_collide'
+    """
+    Supported values: ``'default_stream_collide'`` (default), ``'collide_only'``, ``'stream_pull_only'``. 
+    With ``'default_stream_collide'``, streaming pattern and even/odd time-step (for in-place patterns) can be specified
+    by the ``streaming_pattern`` and ``timestep`` arguments. For backwards compatibility, ``kernel_type`` also accepts
+    ``'stream_pull_collide'``, ``'collide_stream_push'``, ``'esotwist_even'``, ``'esotwist_odd'``, ``'aa_even'``
+    and ``'aa_odd'`` for selection of the streaming pattern. 
+    """
+    streaming_pattern: str = 'pull'
+    """
+    The streaming pattern to be used with a ``'default_stream_collide'`` kernel. Accepted values are
+    ``'pull'``, ``'push'``, ``'aa'`` and ``'esotwist'``.
+    """
+    timestep: Timestep = Timestep.BOTH
+    """
+    Timestep modulus for the streaming pattern. For two-fields patterns, this argument is irrelevant and
+    by default set to ``Timestep.BOTH``. For in-place patterns, ``Timestep.EVEN`` or ``Timestep.ODD`` must be specified.
+    """
+
+    field_name: str = 'src'
+    """
+    Name of the PDF field.
+    """
+    temporary_field_name: str = 'dst'
+    """
+    Name of the temporary PDF field.
+    """
+
+    lb_method: Type[AbstractLbMethod] = None
+    """
+    Instance of `lbmpy.methods.abstractlbmethod.AbstractLbMethod`. If this parameter is `None`, the lb_method is derived
+    via `create_lb_method`. 
+    """
+    collision_rule: LbmCollisionRule = None
+    """
+    Instance of :class:`lbmpy.methods.LbmCollisionRule`. If this parameter is `None`,
+    the collision rule is derived via *create_lb_collision_rule*.
+    """
+    update_rule: LbmCollisionRule = None
+    """
+    Instance of :class:`lbmpy.methods.LbmCollisionRule`. If this parameter is `None`,
+    the update rule is derived via *create_lb_update_rule*.
+    """
+    ast: KernelFunction = None
+    """
+    Instance of *pystencils.KernelFunction*. If this parameter is `None`,
+    the ast is derived via `create_lb_ast`.
+    """
+
+    def __post_init__(self):
+        if isinstance(self.method, str):
+            new_method = Method[self.method.upper()]
+            warn(f'Method "{self.method}" as str is deprecated. Use {new_method} instead')
+            self.method = new_method
+
+        if self.maxwellian_moments is not None:
+            warn("Argument 'maxwellian_moments' is deprecated and will be removed by version 0.5."
+                 "Use `continuous_equilibrium` instead.")
+            self.continuous_equilibrium = self.maxwellian_moments
+
+        if not isinstance(self.stencil, LBStencil):
+            self.stencil = LBStencil(self.stencil)
+
+        if self.relaxation_rates is None:
+            #   Fall back to regularized method
+            if self.relaxation_rate is None:
+                self.relaxation_rate = sp.Symbol("omega")
+
+        # if only a single relaxation rate is defined,
+        # it is internally treated as a list with one element and just sets the relaxation_rates parameter
+        if self.relaxation_rate is not None:
+            if self.method in [Method.TRT, Method.TRT_KBC_N1, Method.TRT_KBC_N2, Method.TRT_KBC_N3, Method.TRT_KBC_N4]:
+                self.relaxation_rates = [self.relaxation_rate,
+                                         relaxation_rate_from_magic_number(self.relaxation_rate)]
+            else:
+                self.relaxation_rates = [self.relaxation_rate]
+
+        #   Incompressible cumulant method is not available
+        if not self.compressible and self.method in (Method.MONOMIAL_CUMULANT, Method.CUMULANT):
+            raise ValueError("Incompressible cumulant-based methods are not supported (yet).")
+
+        if self.zero_centered and self.entropic:
+            raise ValueError("Entropic methods can only be created with `zero_centered=False`.")
+
+        #   Check or infer delta-equilibrium
+        if self.delta_equilibrium is not None:
+            #   Must be zero-centered
+            if self.delta_equilibrium:
+                if not self.zero_centered:
+                    raise ValueError("`delta_equilibrium=True` requires `zero_centered=True`!")
+                #   Must not be a cumulant-method
+                if self.method in (Method.MONOMIAL_CUMULANT, Method.CUMULANT):
+                    raise ValueError("Cannot create a cumulant-based method from a delta-equilibrium!")
+        else:
+            if self.zero_centered:
+                if self.method in (Method.CENTRAL_MOMENT, Method.MONOMIAL_CUMULANT, Method.CUMULANT):
+                    self.delta_equilibrium = False
+                else:
+                    self.delta_equilibrium = True
+            else:
+                self.delta_equilibrium = False
+
+        #   Check or infer collision space
+        if isinstance(self.collision_space_info, CollisionSpace):
+            self.collision_space_info = CollisionSpaceInfo(self.collision_space_info)
+
+        if self.collision_space_info is not None:
+            if (self.entropic or self.fluctuating) \
+                    and self.collision_space_info.collision_space != CollisionSpace.POPULATIONS:
+                #   Workaround until entropic method supports relaxation in subexpressions
+                #   and the problem with RNGs in the assignment collection has been solved
+                raise ValueError("Entropic and Fluctuating methods are only available in population space.")
+            elif not self.collision_space_info.collision_space.compatible(self.method):
+                raise ValueError("Given method is not compatible with given collision space.")
+        else:
+            if self.method in {Method.SRT, Method.TRT,
+                               Method.TRT_KBC_N1, Method.TRT_KBC_N2, Method.TRT_KBC_N3, Method.TRT_KBC_N4}:
+                self.collision_space_info = CollisionSpaceInfo(CollisionSpace.POPULATIONS)
+            elif self.entropic or self.fluctuating:
+                self.collision_space_info = CollisionSpaceInfo(CollisionSpace.POPULATIONS)
+            elif self.method in {Method.MRT_RAW, Method.MRT}:
+                self.collision_space_info = CollisionSpaceInfo(CollisionSpace.RAW_MOMENTS)
+            elif self.method in {Method.CENTRAL_MOMENT}:
+                self.collision_space_info = CollisionSpaceInfo(CollisionSpace.CENTRAL_MOMENTS)
+            elif self.method in {Method.MONOMIAL_CUMULANT, Method.CUMULANT}:
+                self.collision_space_info = CollisionSpaceInfo(CollisionSpace.CUMULANTS)
+            else:
+                raise Exception(f"No default collision space is given for method {self.method}."
+                                "This is a bug; please report it to the developers.")
+
+        #   for backwards compatibility
+        kernel_type_to_streaming_pattern = {
+            'stream_pull_collide': ('pull', Timestep.BOTH),
+            'collide_stream_push': ('push', Timestep.BOTH),
+            'aa_even': ('aa', Timestep.EVEN),
+            'aa_odd': ('aa', Timestep.ODD),
+            'esotwist_even': ('esotwist', Timestep.EVEN),
+            'esotwist_odd': ('esotwist', Timestep.ODD)
+        }
+
+        if self.kernel_type in kernel_type_to_streaming_pattern.keys():
+            self.streaming_pattern, self.timestep = kernel_type_to_streaming_pattern[self.kernel_type]
+            self.kernel_type = 'default_stream_collide'
+
+        if isinstance(self.force, Field):
+            self.force = tuple([self.force(i) for i in range(self.stencil.D)])
+
+        force_not_zero = False
+        for f_i in self.force:
+            if f_i != 0:
+                force_not_zero = True
+
+        if self.force_model is None and force_not_zero:
+            if self.method == Method.CUMULANT:
+                self.force_model = forcemodels.CentralMoment(self.force[:self.stencil.D])
+            else:
+                self.force_model = forcemodels.Guo(self.force[:self.stencil.D])
+
+        force_model_dict = {
+            'simple': forcemodels.Simple,
+            'luo': forcemodels.Luo,
+            'guo': forcemodels.Guo,
+            'schiller': forcemodels.Guo,
+            'buick': forcemodels.Buick,
+            'silva': forcemodels.Buick,
+            'edm': forcemodels.EDM,
+            'kupershtokh': forcemodels.EDM,
+            'he': forcemodels.He,
+            'shanchen': forcemodels.ShanChen,
+            'centralmoment': forcemodels.CentralMoment
+        }
+
+        if self.psm_config is not None and self.psm_config.fraction_field is not None:
+            self.force = [(1.0 - self.psm_config.fraction_field_symbol) * f for f in self.force]
+
+        if isinstance(self.force_model, str):
+            new_force_model = ForceModel[self.force_model.upper()]
+            warn(f'ForceModel "{self.force_model}" as str is deprecated. Use {new_force_model} instead or '
+                 f'provide a class of type AbstractForceModel', category=DeprecationWarning)
+            force_model_class = force_model_dict[new_force_model.name.lower()]
+            self.force_model = force_model_class(force=self.force[:self.stencil.D])
+        elif isinstance(self.force_model, ForceModel):
+            force_model_class = force_model_dict[self.force_model.name.lower()]
+            self.force_model = force_model_class(force=self.force[:self.stencil.D])
+
+        if self.density_input or self.velocity_input:
+            self.conserved_moments = False
+
+
+@dataclass
+class LBMOptimisation:
+    """
+    **Below all parameters for the LBMOptimisation are explained**
+    """
+    cse_pdfs: bool = False
+    """
+    Run common subexpression elimination for opposing stencil directions.
+    """
+    cse_global: bool = False
+    """
+    Run common subexpression elimination after all other simplifications have been executed.
+    """
+    simplification: Union[str, bool, SimplificationStrategy] = 'auto'
+    """
+    Simplifications applied during the derivation of the collision rule. If ``True`` or ``'auto'``,
+    a default simplification strategy is selected according to the type of the method;
+    see :func:`lbmpy.simplificationfactory.create_simplification_strategy`.
+    If ``False``, no simplification is applied.
+    Otherwise, the given simplification strategy will be applied.
+    """
+    pre_simplification: bool = True
+    """
+    Simplifications applied during the derivation of the collision rule for cumulant LBMs.
+    For details see :mod:`lbmpy.moment_transforms`.
+    """
+    split: bool = False
+    """
+    Split innermost loop, to handle only two directions per loop. This reduces the number of parallel
+    load/store streams and thus speeds up the kernel on most architectures.
+    """
+    field_size: Any = None
+    """
+    Create kernel for fixed field size. 
+    """
+    field_layout: str = 'fzyx'
+    """
+    ``'c'`` or ``'numpy'`` for standard numpy layout, ``'reverse_numpy'`` or ``'f'`` for fortran
+    layout, this does not apply when pdf_arr was given, then the same layout as pdf_arr is used.
+    """
+    symbolic_field: pystencils.field.Field = None
+    """
+    Pystencils field for source (pdf field that is read)
+    """
+    symbolic_temporary_field: pystencils.field.Field = None
+    """
+    Pystencils field for temporary (pdf field that is written in stream, or stream-collide)
+    """
+    builtin_periodicity: Tuple[bool] = (False, False, False)
+    """
+    Instead of handling periodicity by copying ghost layers, the periodicity
+    is built into the kernel. This parameters specifies if the domain is periodic in (x,y,z) direction. Even if the
+    periodicity is built into the kernel, the fields have one ghost layer to be consistent with other functions.
+    """
+
+
+def create_lb_function(ast=None, lbm_config=None, lbm_optimisation=None, config=None, optimization=None, **kwargs):
+    """Creates a Python function for the LB method"""
+    lbm_config, lbm_optimisation, config = update_with_default_parameters(kwargs, optimization,
+                                                                          lbm_config, lbm_optimisation, config)
+    if lbm_config.ast is not None:
+        ast = lbm_config.ast
+
+    if ast is None:
+        ast = create_lb_ast(lbm_config.update_rule, lbm_config=lbm_config,
+                            lbm_optimisation=lbm_optimisation, config=config)
+
+    res = ast.compile()
+
+    res.method = ast.method
+    res.update_rule = ast.update_rule
+    return res
+
+
+def create_lb_ast(update_rule=None, lbm_config=None, lbm_optimisation=None, config=None, optimization=None, **kwargs):
+    """Creates a pystencils AST for the LB method"""
+    lbm_config, lbm_optimisation, config = update_with_default_parameters(kwargs, optimization,
+                                                                          lbm_config, lbm_optimisation, config)
+
+    if lbm_config.update_rule is not None:
+        update_rule = lbm_config.update_rule
+
+    if update_rule is None:
+        update_rule = create_lb_update_rule(lbm_config.collision_rule, lbm_config=lbm_config,
+                                            lbm_optimisation=lbm_optimisation, config=config)
+
+    config = replace(config, ghost_layers=1)
+    ast = create_kernel(update_rule, config=config)
+
+    ast.method = update_rule.method
+    ast.update_rule = update_rule
+    lbm_config.ast = ast
+    return ast
+
+
+@disk_cache_no_fallback
+def create_lb_update_rule(collision_rule=None, lbm_config=None, lbm_optimisation=None, config=None,
+                          optimization=None, **kwargs):
+    """Creates an update rule (list of Assignments) for a LB method that describe a full sweep"""
+    lbm_config, lbm_optimisation, config = update_with_default_parameters(kwargs, optimization,
+                                                                          lbm_config, lbm_optimisation, config)
+
+    if lbm_config.collision_rule is not None:
+        collision_rule = lbm_config.collision_rule
+
+    if collision_rule is None:
+        collision_rule = create_lb_collision_rule(lbm_config.lb_method, lbm_config=lbm_config,
+                                                  lbm_optimisation=lbm_optimisation,
+                                                  config=config)
+
+    lb_method = collision_rule.method
+
+    if IS_PYSTENCILS_2:
+        fallback_field_data_type = config.get_option("default_dtype")
+    else:
+        fallback_field_data_type = config.data_type[lbm_config.field_name].numpy_dtype
+
+    q = collision_rule.method.stencil.Q
+
+    if lbm_optimisation.symbolic_field is not None:
+        src_field = lbm_optimisation.symbolic_field
+    elif lbm_optimisation.field_size:
+        field_size = tuple([s + 2 for s in lbm_optimisation.field_size] + [q])
+        src_field = Field.create_fixed_size(lbm_config.field_name, field_size, index_dimensions=1,
+                                            layout=lbm_optimisation.field_layout, dtype=fallback_field_data_type)
+    else:
+        src_field = Field.create_generic(lbm_config.field_name, spatial_dimensions=collision_rule.method.dim,
+                                         index_shape=(q,), layout=lbm_optimisation.field_layout,
+                                         dtype=fallback_field_data_type)
+
+    if lbm_optimisation.symbolic_temporary_field is not None:
+        dst_field = lbm_optimisation.symbolic_temporary_field
+    else:
+        dst_field = src_field.new_field_with_different_name(lbm_config.temporary_field_name)
+   
+    kernel_type = lbm_config.kernel_type
+    if kernel_type == 'stream_pull_only':
+        update_rule = create_stream_pull_with_output_kernel(lb_method, src_field, dst_field, lbm_config.output)
+    else:
+        if kernel_type == 'default_stream_collide':
+            if lbm_config.streaming_pattern == 'pull' and any(lbm_optimisation.builtin_periodicity):
+                accessor = PeriodicTwoFieldsAccessor(lbm_optimisation.builtin_periodicity, ghost_layers=1)
+            else:
+                accessor = get_accessor(lbm_config.streaming_pattern, lbm_config.timestep)
+        elif kernel_type == 'collide_only':
+            accessor = CollideOnlyInplaceAccessor
+        elif isinstance(kernel_type, PdfFieldAccessor):
+            accessor = kernel_type
+        else:
+            raise ValueError("Invalid value of parameter 'kernel_type'", lbm_config.kernel_type)
+        update_rule = create_lbm_kernel(collision_rule, src_field, dst_field, accessor)
+
+    lbm_config.update_rule = update_rule
+    return update_rule
+
+
+@disk_cache_no_fallback
+def create_lb_collision_rule(lb_method=None, lbm_config=None, lbm_optimisation=None, config=None,
+                             optimization=None, **kwargs):
+    """Creates a collision rule (list of Assignments) for a LB method describing the collision operator (no stream)"""
+    lbm_config, lbm_optimisation, config = update_with_default_parameters(kwargs, optimization,
+                                                                          lbm_config, lbm_optimisation, config)
+
+    if lbm_config.lb_method is not None:
+        lb_method = lbm_config.lb_method
+
+    if lb_method is None:
+        lb_method = create_lb_method(lbm_config)
+
+    cqc = lb_method.conserved_quantity_computation
+    rho_in = lbm_config.density_input
+    u_in = lbm_config.velocity_input
+
+    if u_in is not None and isinstance(u_in, Field):
+        u_in = u_in.center_vector
+    if rho_in is not None and isinstance(rho_in, Field):
+        rho_in = rho_in.center
+
+    pre_simplification = lbm_optimisation.pre_simplification
+    if rho_in is not None or u_in is not None:
+        cqe = cqc.equilibrium_input_equations_from_pdfs(lb_method.pre_collision_pdf_symbols)
+        cqe_main_assignments = cqe.main_assignments_dict
+
+        if rho_in is not None:
+            if u_in is None:
+                raise ValueError("When setting 'density_input' parameter, "
+                                 "'velocity_input' has to be specified as well.")
+            cqe_main_assignments[cqc.density_symbol] = rho_in
+            cqe_main_assignments[cqc.density_deviation_symbol] = rho_in - cqc.background_density
+
+        if u_in is not None:
+            for u_sym, u in zip(cqc.velocity_symbols, u_in):
+                cqe_main_assignments[u_sym] = u
+
+        cqe.set_main_assignments_from_dict(cqe_main_assignments)
+        cqe = cqe.new_without_unused_subexpressions()
+
+        collision_rule = lb_method.get_collision_rule(conserved_quantity_equations=cqe,
+                                                      pre_simplification=pre_simplification)
+    else:
+        collision_rule = lb_method.get_collision_rule(pre_simplification=pre_simplification)
+
+    if lbm_config.galilean_correction:
+        from lbmpy.methods.cumulantbased import add_galilean_correction
+        collision_rule = add_galilean_correction(collision_rule)
+
+    if lbm_config.fourth_order_correction:
+        from lbmpy.methods.cumulantbased import add_fourth_order_correction
+
+        # must provide a second relaxation rate in implementation; defaults to 1
+        if len(lbm_config.relaxation_rates) == 1:
+            lbm_config.relaxation_rates.append(1)
+
+        cumulant_limiter = 1e6 if lbm_config.fourth_order_correction is True else lbm_config.fourth_order_correction
+        collision_rule = add_fourth_order_correction(collision_rule=collision_rule,
+                                                     shear_relaxation_rate=lbm_config.relaxation_rates[0],
+                                                     bulk_relaxation_rate=lbm_config.relaxation_rates[1],
+                                                     limiter=cumulant_limiter)
+
+    if lbm_config.psm_config is not None:
+        if lbm_config.psm_config.fraction_field is None or lbm_config.psm_config.object_velocity_field is None:
+            raise ValueError("Specify a fraction and object velocity field in the PSM Config")
+        collision_rule = replace_by_psm_collision_rule(collision_rule, lbm_config.psm_config)
+
+    if lbm_config.entropic:
+        if lbm_config.subgrid_scale_model or lbm_config.cassons:
+            raise ValueError("Choose either entropic, subgrid-scale or cassons")
+        if lbm_config.entropic_newton_iterations:
+            if isinstance(lbm_config.entropic_newton_iterations, bool):
+                iterations = 3
+            else:
+                iterations = lbm_config.entropic_newton_iterations
+            collision_rule = add_iterative_entropy_condition(collision_rule, newton_iterations=iterations,
+                                                             omega_output_field=lbm_config.omega_output_field)
+        else:
+            collision_rule = add_entropy_condition(collision_rule, omega_output_field=lbm_config.omega_output_field)
+
+    elif lbm_config.subgrid_scale_model:
+        if lbm_config.cassons:
+            raise ValueError("Cassons model can not be combined with a subgrid-scale model")
+
+        model_constant = None
+        sgs_model = lbm_config.subgrid_scale_model
+
+        if isinstance(lbm_config.subgrid_scale_model, tuple):
+            sgs_model = lbm_config.subgrid_scale_model[0]
+            model_constant = lbm_config.subgrid_scale_model[1]
+
+        collision_rule = add_sgs_model(collision_rule=collision_rule, subgrid_scale_model=sgs_model,
+                                       model_constant=model_constant, omega_output_field=lbm_config.omega_output_field,
+                                       eddy_viscosity_field=lbm_config.eddy_viscosity_field)
+
+    if 'split_groups' in collision_rule.simplification_hints:
+        collision_rule.simplification_hints['split_groups'][0].append(sp.Symbol("sgs_omega"))
+
+    elif lbm_config.cassons:
+        collision_rule = add_cassons_model(collision_rule, parameter=lbm_config.cassons,
+                                           omega_output_field=lbm_config.omega_output_field)
+
+    if lbm_config.output:
+        output_eqs = cqc.output_equations_from_pdfs(lb_method.pre_collision_pdf_symbols, lbm_config.output)
+        collision_rule = collision_rule.new_merged(output_eqs)
+
+    if lbm_optimisation.simplification is True or lbm_optimisation.simplification == 'auto':
+        simplification = create_simplification_strategy(lb_method, split_inner_loop=lbm_optimisation.split)
+    elif callable(lbm_optimisation.simplification):
+        simplification = lbm_optimisation.simplification
+    else:
+        simplification = SimplificationStrategy()
+    collision_rule = simplification(collision_rule)
+
+    if isinstance(collision_rule.method, CumulantBasedLbMethod):
+        from lbmpy.methods.cumulantbased.cumulant_simplifications import check_for_logarithms
+        check_for_logarithms(collision_rule)
+
+    if lbm_config.fluctuating:
+        add_fluctuations_to_collision_rule(collision_rule, **lbm_config.fluctuating)
+
+    if lbm_optimisation.cse_pdfs:
+        from lbmpy.methods.momentbased.momentbasedsimplifications import cse_in_opposing_directions
+        collision_rule = cse_in_opposing_directions(collision_rule)
+    if lbm_optimisation.cse_global:
+        collision_rule = sympy_cse(collision_rule)
+
+    lbm_config.collision_rule = collision_rule
+    return collision_rule
+
+
+def create_lb_method(lbm_config=None, **params):
+    """Creates a LB method, defined by moments/cumulants for collision space, equilibrium and relaxation rates."""
+    lbm_config, _, _ = update_with_default_parameters(params, lbm_config=lbm_config)
+
+    relaxation_rates = lbm_config.relaxation_rates
+    dim = lbm_config.stencil.D
+
+    if isinstance(lbm_config.force, Field):
+        lbm_config.force = tuple(lbm_config.force(i) for i in range(dim))
+
+    if lbm_config.psm_config is None:
+        fraction_field = None
+    else:
+        fraction_field = lbm_config.psm_config.fraction_field_symbol
+
+    common_params = {
+        'compressible': lbm_config.compressible,
+        'zero_centered': lbm_config.zero_centered,
+        'delta_equilibrium': lbm_config.delta_equilibrium,
+        'equilibrium_order': lbm_config.equilibrium_order,
+        'force_model': lbm_config.force_model,
+        'continuous_equilibrium': lbm_config.continuous_equilibrium,
+        'c_s_sq': lbm_config.c_s_sq,
+        'collision_space_info': lbm_config.collision_space_info,
+        'fraction_field': fraction_field,
+    }
+
+    cumulant_params = {
+        'zero_centered': lbm_config.zero_centered,
+        'force_model': lbm_config.force_model,
+        'c_s_sq': lbm_config.c_s_sq,
+        'collision_space_info': lbm_config.collision_space_info,
+        'fraction_field': fraction_field,
+    }
+
+    if lbm_config.method == Method.SRT:
+        assert len(relaxation_rates) >= 1, "Not enough relaxation rates"
+        method = create_srt(lbm_config.stencil, relaxation_rates[0], **common_params)
+    elif lbm_config.method == Method.TRT:
+        assert len(relaxation_rates) >= 2, "Not enough relaxation rates"
+        method = create_trt(lbm_config.stencil, relaxation_rates[0], relaxation_rates[1], **common_params)
+    elif lbm_config.method == Method.MRT:
+        method = create_mrt_orthogonal(lbm_config.stencil, relaxation_rates, weighted=lbm_config.weighted,
+                                       nested_moments=lbm_config.nested_moments,
+                                       conserved_moments=lbm_config.conserved_moments, **common_params)
+    elif lbm_config.method == Method.CENTRAL_MOMENT:
+        method = create_central_moment(lbm_config.stencil, relaxation_rates,
+                                       nested_moments=lbm_config.nested_moments,
+                                       conserved_moments=lbm_config.conserved_moments, **common_params)
+    elif lbm_config.method == Method.MRT_RAW:
+        method = create_mrt_raw(lbm_config.stencil, relaxation_rates,
+                                conserved_moments=lbm_config.conserved_moments, **common_params)
+    elif lbm_config.method in (Method.TRT_KBC_N1, Method.TRT_KBC_N2, Method.TRT_KBC_N3, Method.TRT_KBC_N4):
+        if lbm_config.stencil.D == 2 and lbm_config.stencil.Q == 9:
+            dim = 2
+        elif lbm_config.stencil.D == 3 and lbm_config.stencil.Q == 27:
+            dim = 3
+        else:
+            raise NotImplementedError("KBC type TRT methods can only be constructed for D2Q9 and D3Q27 stencils")
+        method_nr = lbm_config.method.name[-1]
+        method = create_trt_kbc(dim, relaxation_rates[0], relaxation_rates[1], 'KBC-N' + method_nr, **common_params)
+    elif lbm_config.method == Method.CUMULANT:
+        if lbm_config.fourth_order_correction:
+            if lbm_config.stencil.D != 3 and lbm_config.stencil.Q != 27:
+                raise ValueError("Fourth-order correction can only be applied to D3Q27 cumulant methods.")
+
+            assert len(relaxation_rates) <= 2, "Optimal parametrisation for fourth-order cumulants needs either one " \
+                                               "or two relaxation rates, associated with the shear (and bulk) " \
+                                               "viscosity. All other relaxation rates are automatically chosen " \
+                                               "optimally"
+
+            # define method in terms of symbolic relaxation rates and assign optimal values later
+            from lbmpy.methods.cumulantbased.fourth_order_correction import FOURTH_ORDER_RELAXATION_RATE_SYMBOLS
+            relaxation_rates = FOURTH_ORDER_RELAXATION_RATE_SYMBOLS
+
+        if lbm_config.nested_moments is not None:
+            method = create_cumulant(lbm_config.stencil, relaxation_rates, lbm_config.nested_moments,
+                                     conserved_moments=lbm_config.conserved_moments, **cumulant_params)
+        else:
+            method = create_with_default_polynomial_cumulants(lbm_config.stencil, relaxation_rates, **cumulant_params)
+
+    elif lbm_config.method == Method.MONOMIAL_CUMULANT:
+        method = create_with_monomial_cumulants(lbm_config.stencil, relaxation_rates,
+                                                conserved_moments=lbm_config.conserved_moments, **cumulant_params)
+    else:
+        raise ValueError("Failed to create LB method. Please use lbmpy.enums.Method for the creation")
+
+    # >>Entropic methods can only be created for methods with two relaxation rates One free relaxation rate
+    # determining the viscosity and one to be determined by the entropy condition<<
+    # Thus we fix the conserved quantities to one of the relaxation rates because zero would be recognised as
+    # a third relaxation rate here.
+    if lbm_config.entropic:
+        method.set_conserved_moments_relaxation_rate(relaxation_rates[0])
+
+    lbm_config.lb_method = method
+    return method
+
+
+def create_psm_update_rule(lbm_config, lbm_optimisation):
+    if IS_PYSTENCILS_2:
+        raise NotImplementedError(
+            "`create_psm_update_rule` is not yet available when using pystencils 2.0. "
+            "To instead derive a (potentially less efficient) PSM kernel without branches, "
+            "use `create_lb_update_rule` with a `PsmConfig` object instead."
+        )
+    
+    from pystencils.astnodes import Conditional, Block
+    from pystencils.node_collection import NodeCollection
+
+    if lbm_config.psm_config is None:
+        raise ValueError("Specify a PSM Config in the LBM Config, when creating a psm update rule")
+
+    config_without_particles = copy.deepcopy(lbm_config)
+    config_without_particles.psm_config.max_particles_per_cell = 0
+
+    lb_update_rule = create_lb_update_rule(
+        lbm_config=config_without_particles, lbm_optimisation=lbm_optimisation)
+
+    node_collection = lb_update_rule.all_assignments
+
+    if lbm_config.psm_config.individual_fraction_field is None:
+        assert lbm_config.psm_config.max_particles_per_cell == 1
+        fraction_field = lbm_config.psm_config.fraction_field
+    else:
+        fraction_field = lbm_config.psm_config.individual_fraction_field
+
+    for p in range(lbm_config.psm_config.max_particles_per_cell):
+
+        psm_solid_collision = add_psm_solid_collision_to_collision_rule(lb_update_rule, lbm_config, p)
+        psm_update_rule = create_lb_update_rule(
+            collision_rule=psm_solid_collision, lbm_config=lbm_config, lbm_optimisation=lbm_optimisation)
+
+        node_collection.append(
+            Conditional(
+                fraction_field.center(p) > 0.0,
+                Block(psm_update_rule.all_assignments),
+            )
+        )
+
+    return NodeCollection(node_collection)
+
+
+# ----------------------------------------------------------------------------------------------------------------------
+def update_with_default_parameters(params, opt_params=None, lbm_config=None, lbm_optimisation=None, config=None):
+    # Fix CreateKernelConfig params
+    pystencils_config_params = ['target', 'backend', 'cpu_openmp', 'double_precision', 'gpu_indexing',
+                                'gpu_indexing_params', 'cpu_vectorize_info']
+    if opt_params is not None:
+        config_params = {k: v for k, v in opt_params.items() if k in pystencils_config_params}
+    else:
+        config_params = {}
+
+    if 'double_precision' in config_params:
+        if config_params['double_precision']:
+            config_params['data_type'] = 'float64'
+        else:
+            config_params['data_type'] = 'float32'
+        del config_params['double_precision']
+
+    if not config:
+        config = CreateKernelConfig(**config_params)
+    else:
+        for k, v in config_params.items():
+            if not hasattr(config, k):
+                raise KeyError(f'{v} is not a valid kwarg. Please look in CreateKernelConfig for valid settings')
+        config = replace(config, **config_params)
+
+    lbm_opt_params = ['cse_pdfs', 'cse_global', 'simplification', 'pre_simplification', 'split', 'field_size',
+                      'field_layout', 'symbolic_field', 'symbolic_temporary_field', 'builtin_periodicity']
+
+    if opt_params is not None:
+        opt_params_dict = {k: v for k, v in opt_params.items() if k in lbm_opt_params}
+    else:
+        opt_params_dict = {}
+
+    if not lbm_optimisation:
+        lbm_optimisation = LBMOptimisation(**opt_params_dict)
+    else:
+        for k, v in opt_params_dict.items():
+            if not hasattr(lbm_optimisation, k):
+                raise KeyError(f'{v} is not a valid kwarg. Please look in LBMOptimisation for valid settings')
+        lbm_optimisation = replace(lbm_optimisation, **opt_params_dict)
+
+    if params is None:
+        params = {}
+
+    if not lbm_config:
+        lbm_config = LBMConfig(**params)
+    else:
+        for k, v in params.items():
+            if not hasattr(lbm_config, k):
+                raise KeyError(f'{v} is not a valid kwarg. Please look in LBMConfig for valid settings')
+        lbm_config = replace(lbm_config, **params)
+
+    return lbm_config, lbm_optimisation, config

lbmpy/cumulants.py → src/lbmpy/cumulants.py

@@ -102,7 +102,7 @@ def __cumulant_raw_moment_transform(index, dependent_var_dict, outer_function, d
 
 @memorycache(maxsize=16)
 def __get_discrete_cumulant_generating_function(func, stencil, wave_numbers):
-    assert len(stencil) == len(func)
+    assert stencil.Q == len(func)
 
     laplace_transformation = sum([factor * sp.exp(scalar_product(wave_numbers, e)) for factor, e in zip(func, stencil)])
     return sp.ln(laplace_transformation)
@@ -121,10 +121,10 @@ def discrete_cumulant(func, cumulant, stencil):
                   (similar to moment description)
         stencil: sequence of directions
     """
-    assert len(stencil) == len(func)
+    assert stencil.Q == len(func)
 
     dim = len(stencil[0])
-    wave_numbers = tuple([sp.Symbol("Xi_%d" % (i,)) for i in range(dim)])
+    wave_numbers = sp.symbols(f"Xi_:{dim}")
 
     generating_function = __get_discrete_cumulant_generating_function(func, stencil, wave_numbers)
     if type(cumulant) is tuple:
@@ -157,9 +157,9 @@ def cumulants_from_pdfs(stencil, cumulant_indices=None, pdf_symbols=None):
     dim = len(stencil[0])
     if cumulant_indices is None:
         cumulant_indices = moments_up_to_component_order(2, dim=dim)
-    assert len(stencil) == len(cumulant_indices), "Stencil has to have same length as cumulant_indices sequence"
+    assert stencil.Q == len(cumulant_indices), "Stencil has to have same length as cumulant_indices sequence"
     if pdf_symbols is None:
-        pdf_symbols = __get_indexed_symbols(pdf_symbols, "f", range(len(stencil)))
+        pdf_symbols = __get_indexed_symbols(pdf_symbols, "f", range(stencil.Q))
     return {idx: discrete_cumulant(tuple(pdf_symbols), idx, stencil) for idx in cumulant_indices}
 
 

src/lbmpy/custom_code_nodes.py

+import numpy as np
+import sympy as sp
+
+from ._compat import IS_PYSTENCILS_2
+
+if IS_PYSTENCILS_2:
+    raise ImportError("`lbmpy.custom_code_nodes` is only available when running with pystencils 1.x")
+
+from pystencils.typing import TypedSymbol, create_type
+from pystencils.backends.cbackend import CustomCodeNode
+
+
+class NeighbourOffsetArrays(CustomCodeNode):
+
+    @staticmethod
+    def neighbour_offset(dir_idx, stencil):
+        if isinstance(sp.sympify(dir_idx), sp.Integer):
+            return stencil[dir_idx]
+        else:
+            return tuple([sp.IndexedBase(symbol, shape=(1,))[dir_idx]
+                         for symbol in NeighbourOffsetArrays._offset_symbols(len(stencil[0]))])
+
+    @staticmethod
+    def _offset_symbols(dim):
+        return [TypedSymbol(f"neighbour_offset_{d}", create_type('int32')) for d in ['x', 'y', 'z'][:dim]]
+
+    def __init__(self, stencil, offsets_dtype=np.int32):
+        offsets_dtype = create_type(offsets_dtype)
+        dim = len(stencil[0])
+
+        array_symbols = NeighbourOffsetArrays._offset_symbols(dim)
+        code = "\n"
+        for i, arrsymb in enumerate(array_symbols):
+            code += _array_pattern(offsets_dtype, arrsymb.name, (d[i] for d in stencil))
+
+        offset_symbols = NeighbourOffsetArrays._offset_symbols(dim)
+        super(NeighbourOffsetArrays, self).__init__(code, symbols_read=set(),
+                                                    symbols_defined=set(offset_symbols))
+
+
+class MirroredStencilDirections(CustomCodeNode):
+
+    @staticmethod
+    def mirror_stencil(direction, mirror_axis):
+        assert mirror_axis <= len(direction), f"only {len(direction)} axis available for mirage"
+        direction = list(direction)
+        direction[mirror_axis] = -direction[mirror_axis]
+
+        return tuple(direction)
+
+    @staticmethod
+    def _mirrored_symbol(mirror_axis, _stencil):
+        axis = ['x', 'y', 'z']
+        return TypedSymbol(f"{axis[mirror_axis]}_axis_mirrored_stencil_dir", create_type('int32'))
+
+    def __init__(self, stencil, mirror_axis, dtype=np.int32):
+        offsets_dtype = create_type(dtype)
+
+        mirrored_stencil_symbol = MirroredStencilDirections._mirrored_symbol(mirror_axis, stencil)
+        mirrored_directions = [stencil.index(MirroredStencilDirections.mirror_stencil(direction, mirror_axis))
+                               for direction in stencil]
+        code = "\n"
+        code += _array_pattern(offsets_dtype, mirrored_stencil_symbol.name, mirrored_directions)
+
+        super(MirroredStencilDirections, self).__init__(code, symbols_read=set(),
+                                                        symbols_defined={mirrored_stencil_symbol})
+
+
+class LbmWeightInfo(CustomCodeNode):
+    def __init__(self, lb_method, data_type='double'):
+        self.weights_symbol = TypedSymbol("weights", data_type)
+
+        weights = [f"(({self.weights_symbol.dtype.c_name})({str(w.evalf(17))}))" for w in lb_method.weights]
+        weights = ", ".join(weights)
+        w_sym = self.weights_symbol
+        code = f"const {self.weights_symbol.dtype.c_name} {w_sym.name} [] = {{{weights}}};\n"
+        super(LbmWeightInfo, self).__init__(code, symbols_read=set(), symbols_defined={w_sym})
+
+    def weight_of_direction(self, dir_idx, lb_method=None):
+        if isinstance(sp.sympify(dir_idx), sp.Integer):
+            return lb_method.weights[dir_idx].evalf(17)
+        else:
+            return sp.IndexedBase(self.weights_symbol, shape=(1,))[dir_idx]
+
+
+class TranslationArraysNode(CustomCodeNode):
+
+    def __init__(self, array_content, symbols_defined):
+        code = ''
+        for content in array_content:
+            code += _array_pattern(*content)
+        super(TranslationArraysNode, self).__init__(code, symbols_read=set(), symbols_defined=symbols_defined)
+
+    def __str__(self):
+        return "Variable PDF Access Translation Arrays"
+
+    def __repr__(self):
+        return "Variable PDF Access Translation Arrays"
+
+
+def _array_pattern(dtype, name, content):
+    return f"const {str(dtype)} {name} [] = {{ {','.join(str(c) for c in content)} }}; \n"

src/lbmpy/db.py

+import json
+import six
+import inspect
+
+from pystencils.runhelper.db import PystencilsJsonEncoder
+from pystencils.simp import SimplificationStrategy
+from lbmpy import LBStencil, Method, CollisionSpace, SubgridScaleModel
+from lbmpy.creationfunctions import LBMConfig, LBMOptimisation
+from lbmpy.methods import CollisionSpaceInfo
+from lbmpy.forcemodels import AbstractForceModel
+from lbmpy.non_newtonian_models import CassonsParameters
+
+
+class LbmpyJsonEncoder(PystencilsJsonEncoder):
+
+    def default(self, obj):
+        if isinstance(obj, (LBMConfig, LBMOptimisation, CollisionSpaceInfo, CassonsParameters)):
+            return obj.__dict__
+        if isinstance(obj, (LBStencil, Method, CollisionSpace, SubgridScaleModel)):
+            return obj.name
+        if isinstance(obj, AbstractForceModel):
+            return obj.__class__.__name__
+        if isinstance(obj, SimplificationStrategy):
+            return obj.__str__()
+        if inspect.isclass(obj):
+            return obj.__name__
+        return PystencilsJsonEncoder.default(self, obj)
+
+
+class LbmpyJsonSerializer(object):
+
+    @classmethod
+    def serialize(cls, data):
+        if six.PY3:
+            if isinstance(data, bytes):
+                return json.dumps(data.decode('utf-8'), cls=LbmpyJsonEncoder, ensure_ascii=False).encode('utf-8')
+            else:
+                return json.dumps(data, cls=LbmpyJsonEncoder, ensure_ascii=False).encode('utf-8')
+        else:
+            return json.dumps(data, cls=LbmpyJsonEncoder, ensure_ascii=False).encode('utf-8')
+
+    @classmethod
+    def deserialize(cls, data):
+        if six.PY3:
+            return json.loads(data.decode('utf-8'))
+        else:
+            return json.loads(data.decode('utf-8'))

src/lbmpy/enums.py

+from enum import Enum, auto
+
+
+class Stencil(Enum):
+    """
+    The Stencil enumeration represents all possible lattice Boltzmann stencils that are available in lbmpy.
+    It should be passed to :class:`lbmpy.stencils.LBStenil`. This class then creates a stencils representation
+    containing the concrete neighbour directions as a tuple of tuples.
+
+    The number of spatial dimensions *d* and the number of discrete velocities *q* are stated in the DdQq notation
+    """
+    D2Q9 = auto()
+    """
+    A two dimensional stencil using 9 discrete velocities.
+    """
+    D2V17 = auto()
+    """
+    A two dimensional stencil using 17 discrete velocities. (long range stencil).
+    """
+    D2V37 = auto()
+    """
+    A two dimensional stencil using 37 discrete velocities. (long range stencil).
+    """
+    D3Q7 = auto()
+    """
+    A three dimensional stencil using 7 discrete velocities.
+    """
+    D3Q15 = auto()
+    """
+    A three dimensional stencil using 15 discrete velocities.
+    """
+    D3Q19 = auto()
+    """
+    A three dimensional stencil using 19 discrete velocities.
+    """
+    D3Q27 = auto()
+    """
+    A three dimensional stencil using 27 discrete velocities.
+    """
+
+
+class Method(Enum):
+    """
+    The Method enumeration represents all possible lattice Boltzmann collision operators that are available in lbmpy.
+    It should be passed to :class:`lbmpy.creationfunctions.LBMConfig`. The LBM configuration *dataclass* then derives
+    the respective collision equations when passed to the creations functions in the `lbmpy.creationfunctions`
+    module of lbmpy.
+
+    Note here, when using a specific enumeration to derive a particular LBM collision operator,
+    different parameters of the :class:`lbmpy.creationfunctions.LBMConfig` might become necessary.
+    For example, it does not make sense to define *relaxation_rates* for a single relaxation rate method, which
+    is essential for multiple relaxation rate methods. Important specific parameters are listed below to the enum value.
+    A specific creation function is stated for each case which explains these parameters in detail.
+    """
+    SRT = auto()
+    """
+    See :func:`lbmpy.methods.create_srt`, 
+    Single relaxation time method
+    """
+    TRT = auto()
+    """
+    See :func:`lbmpy.methods.create_trt`, 
+    Two relaxation time, the first relaxation rate is for even moments and determines the
+    viscosity (as in SRT). The second relaxation rate is used for relaxing odd moments and controls the
+    bulk viscosity. For details in the TRT collision operator see :cite:`TRT`
+    """
+    MRT_RAW = auto()
+    """
+    See :func:`lbmpy.methods.create_mrt_raw`, 
+    Non-orthogonal MRT where all relaxation rates can be specified independently, i.e. there are as many relaxation 
+    rates as stencil entries. Look at the generated method in Jupyter to see which moment<->relaxation rate mapping.
+    Originally defined in :cite:`raw_moments`
+    """
+    MRT = auto()
+    """
+    See :func:`lbmpy.methods.create_mrt_orthogonal`
+    Orthogonal multi relaxation time model, relaxation rates are used in this order for *shear modes*, *bulk modes*,
+    *third-order modes*, *fourth-order modes*, etc. Requires also a parameter *weighted* that should be `True` if the
+    moments should be orthogonal w.r.t. weighted scalar product using the lattice weights. If `False`, the normal
+    scalar product is used. For custom definition of the method, a *nested_moments* can be passed.
+    For example: [ [1, x, y], [x*y, x**2, y**2], ... ] that groups all moments together that should be relaxed
+    at the same rate. Literature values of this list can be obtained through 
+    :func:`lbmpy.methods.creationfunctions.mrt_orthogonal_modes_literature`.
+    WMRT collision operators are reported to be numerically more stable and more accurate, 
+    whilst also having a lower computational cos :cite:`FAKHARI201722`
+    """
+    CENTRAL_MOMENT = auto()
+    """
+    See :func:`lbmpy.methods.create_central_moment`
+    Creates moment based LB method where the collision takes place in the central moment space. By default, 
+    a raw-moment set is used where the bulk and the shear viscosity are separated. An original derivation can be 
+    found in :cite:`Geier2006`
+    """
+    TRT_KBC_N1 = auto()
+    """
+    See :func:`lbmpy.methods.create_trt_kbc`
+    Particular two-relaxation rate method. This is not the entropic method yet, only the relaxation pattern. 
+    To get the entropic method also *entropic* needs to be set to `True`. 
+    There are four KBC methods available in lbmpy. The naming is according to :cite:`karlin2015entropic`
+    """
+    TRT_KBC_N2 = auto()
+    """
+    See :func:`lbmpy.methods.create_trt_kbc`
+    Particular two-relaxation rate method. This is not the entropic method yet, only the relaxation pattern. 
+    To get the entropic method also *entropic* needs to be set to `True`. 
+    There are four KBC methods available in lbmpy. The naming is according to :cite:`karlin2015entropic`
+    """
+    TRT_KBC_N3 = auto()
+    """
+    See :func:`lbmpy.methods.create_trt_kbc`
+    Particular two-relaxation rate method. This is not the entropic method yet, only the relaxation pattern. 
+    To get the entropic method also *entropic* needs to be set to `True`. 
+    There are four KBC methods available in lbmpy. The naming is according to :cite:`karlin2015entropic`
+    """
+    TRT_KBC_N4 = auto()
+    """
+    See :func:`lbmpy.methods.create_trt_kbc`
+    Particular two-relaxation rate method. This is not the entropic method yet, only the relaxation pattern. 
+    To get the entropic method also *entropic* needs to be set to `True`. 
+    There are four KBC methods available in lbmpy. The naming is according to :cite:`karlin2015entropic`
+    """
+    CUMULANT = auto()
+    """
+    See :func:`lbmpy.methods.create_with_default_polynomial_cumulants`
+    Cumulant-based LB method which relaxes groups of polynomial cumulants chosen to optimize rotational invariance.
+    For details on the method see :cite:`geier2015`
+    """
+    MONOMIAL_CUMULANT = auto()
+    """
+    See :func:`lbmpy.methods.create_with_monomial_cumulants`
+    Cumulant-based LB method which relaxes monomial cumulants.
+    For details on the method see :cite:`geier2015` and :cite:`Coreixas2019`
+    """
+
+
+class CollisionSpace(Enum):
+    """
+    The CollisionSpace enumeration lists all possible spaces for collision to occur in.
+    """
+    POPULATIONS = auto()
+    """
+    Population space, meaning post-collision populations are obtained directly by relaxation of linear combinations of
+    pre-collision populations. Default for `lbmpy.enums.Method.SRT` and `lbmpy.enums.Method.TRT`.
+    Results in the creation of an instance of :class:`lbmpy.methods.momentbased.MomentBasedLbMethod`.
+    """
+    RAW_MOMENTS = auto()
+    """
+    Raw moment space, meaning relaxation is applied to a set of linearly independent, polynomial raw moments of the 
+    discrete population vector. Default for `lbmpy.enums.Method.MRT`.
+    Results in the creation of an instance of :class:`lbmpy.methods.momentbased.MomentBasedLbMethod`.
+    """
+    CENTRAL_MOMENTS = auto()
+    """
+    Central moment space, meaning relaxation is applied to a set of linearly independent, polynomial central moments 
+    of the discrete population vector. Default for `lbmpy.enums.Method.CENTRAL_MOMENT`.
+    Results in the creation of an instance of :class:`lbmpy.methods.momentbased.CentralMomentBasedLbMethod`.
+    """
+    CUMULANTS = auto()
+    """
+    Cumulant space, meaning relaxation is applied to a set of linearly independent, polynomial cumulants of the
+    discrete population vector. Default for `lbmpy.enums.Method.CUMULANT` and `lbmpy.enums.Method.MONOMIAL_CUMULANT`.
+    Results in the creation of an instance of :class:`lbmpy.methods.cumulantbased.CumulantBasedLbMethod`.
+    """
+
+    def compatible(self, method: Method):
+        """Determines if the given `lbmpy.enums.Method` is compatible with this collision space."""
+        compat_dict = {
+            CollisionSpace.POPULATIONS: {Method.SRT, Method.TRT, Method.MRT_RAW, Method.MRT,
+                                         Method.TRT_KBC_N1, Method.TRT_KBC_N2, Method.TRT_KBC_N3, Method.TRT_KBC_N4},
+            CollisionSpace.RAW_MOMENTS: {Method.SRT, Method.TRT, Method.MRT_RAW, Method.MRT},
+            CollisionSpace.CENTRAL_MOMENTS: {Method.CENTRAL_MOMENT},
+            CollisionSpace.CUMULANTS: {Method.MONOMIAL_CUMULANT, Method.CUMULANT}
+        }
+
+        return method in compat_dict[self]
+
+
+class ForceModel(Enum):
+    """
+    The ForceModel enumeration defines which force model is used to introduce forcing terms in the collision operator
+    of the lattice Boltzmann method. A short summary of the theory behind is shown in `lbmpy.forcemodels`.
+    More precise definitions are given in Chapter 6 and 10 of :cite:`lbm_book`
+    """
+    SIMPLE = auto()
+    """
+    See :class:`lbmpy.forcemodels.Simple`
+    """
+    LUO = auto()
+    """
+    See :class:`lbmpy.forcemodels.Luo`
+    """
+    GUO = auto()
+    """
+    See :class:`lbmpy.forcemodels.Guo`
+    """
+    BUICK = auto()
+    """
+    See :class:`lbmpy.forcemodels.Buick`
+    """
+    SILVA = auto()
+    """
+    See :class:`lbmpy.forcemodels.Buick`
+    """
+    EDM = auto()
+    """
+    See :class:`lbmpy.forcemodels.EDM`
+    """
+    KUPERSHTOKH = auto()
+    """
+    See :class:`lbmpy.forcemodels.EDM`
+    """
+    HE = auto()
+    """
+    See :class:`lbmpy.forcemodels.He`
+    """
+    SHANCHEN = auto()
+    """
+    See :class:`lbmpy.forcemodels.ShanChen`
+    """
+    CENTRALMOMENT = auto()
+    """
+    See :class:`lbmpy.forcemodels.CentralMoment`
+    """
+
+
+class SubgridScaleModel(Enum):
+    """
+    The SubgridScaleModel enumeration defines which subgrid-scale model (SGS) is used to perform
+    Large-Eddy-Simulations (LES).
+    """
+    SMAGORINSKY = auto()
+    """
+    See :func:`lbmpy.turbulence_models.add_smagorinsky_model`
+    """
+    QR = auto()
+    """
+    See :func:`lbmpy.turbulence_models.add_qr_model`
+    """

src/lbmpy/equilibrium/__init__.py

+r"""
+This module contains various classes encapsulating equilibrium distributions used in the lattice Boltzmann
+method. These include both the continuous and the discretized variants of the Maxwellian equilibrium of
+hydrodynamics. Furthermore, a lightweight wrapper class for custom discrete equilibria is provided.
+Custom equilibria may also be implemented by manually overriding the abstract base class 
+:class:`lbmpy.equilibrium.AbstractEquilibrium`.
+"""
+
+from .abstract_equilibrium import AbstractEquilibrium
+from .continuous_hydro_maxwellian import ContinuousHydrodynamicMaxwellian, default_background_distribution
+from .generic_discrete_equilibrium import GenericDiscreteEquilibrium, discrete_equilibrium_from_matching_moments
+from .discrete_hydro_maxwellian import DiscreteHydrodynamicMaxwellian
+
+__all__ = [
+    "AbstractEquilibrium",
+    "ContinuousHydrodynamicMaxwellian", "default_background_distribution",
+    "GenericDiscreteEquilibrium", "discrete_equilibrium_from_matching_moments",
+    "DiscreteHydrodynamicMaxwellian"
+]

src/lbmpy/equilibrium/abstract_equilibrium.py

+from abc import ABC, abstractmethod
+import sympy as sp
+from pystencils.cache import sharedmethodcache
+
+from lbmpy.moments import polynomial_to_exponent_representation
+
+
+class AbstractEquilibrium(ABC):
+    """
+    Abstract Base Class for description of equilibrium distribution functions used in lattice
+    Boltzmann methods.
+
+    **Equilibrium Representation:**
+
+    This class provides the common interface for describing equilibrium distribution functions,
+    which is then used by the various method classes in the derivation of collision equations.
+    An equilibrium distribution is defined by either its continuous equation (see :attr:`continuous_equation`)
+    or a set of discrete populations
+    (see :attr:`discrete_populations` and :class:`lbmpy.equilibrium.GenericDiscreteEquilibrium`).
+    The distribution function may be given either in its regular, absolute form; or only as its
+    deviation from the rest state, represented by the background distribution (see :attr:`background_distribution`).
+
+    **Computation of Statistical Modes:**
+
+    The major computational task of an equilbrium class is the computation of the distribution's
+    statistical modes. For discrete distributions, the subclass :class:`lbmpy.equilibrium.GenericDiscreteEquilibrium`
+    provides a generic implementation for their computation. For continuous distributions, computation
+    of raw moments, central moments, and cumulants is more complicated, but may also be simplified using special
+    tricks.
+
+    As the computation of statistical modes is a time-consuming process, the abstract base class provides caching
+    functionality to avoid recomputing quantities that are already known.
+
+    **Instructions to Override:**
+
+    If you wish to model a simple custom discrete distribution, just using the class
+    :class:`lbmpy.equilibrium.GenericDiscreteEquilibrium` might already be sufficient.
+    If, however, you need to implement more specific functionality, custom properties, 
+    a background distribution, etc., or if you wish to model a continuous distribution,
+    you will have to set up a custom subclass of :class:`AbstractEquilibrium`.
+
+    A subclass must implement all abstract properties according to their docstrings.
+    For computation of statistical modes, a large part of the infrastructure is already given in the abstract base
+    class. The public interface for computing e.g. raw moments reduces the computation of polynomial moments to their
+    contained monomials (for details on how moments are represented in *lbmpy*, see :mod:`lbmpy.moments`). The values
+    of both polynomial and monomial moments, once computed, will be cached per instance of the equilibrium class.
+    To take full advantage of the caching functionality, you will have to override only :func:`_monomial_raw_moment`
+    and its central moment and cumulant counterparts. These methods will be called only once for each monomial quantity
+    when it is required for the first time. Afterward, the cached value will be used.
+    """
+
+    def __init__(self, dim=3):
+        self._dim = dim
+
+    @property
+    def dim(self):
+        """This distribution's spatial dimensionality."""
+        return self._dim
+
+    #   -------------- Abstract Properties, to be overridden in subclass ----------------------------------------------
+
+    @property
+    @abstractmethod
+    def deviation_only(self):
+        """Whether or not this equilibrium distribution is represented only by its deviation
+        from the background distribution."""
+        raise NotImplementedError("'deviation_only' must be provided by subclass.")
+
+    @property
+    @abstractmethod
+    def continuous_equation(self):
+        """Returns the continuous equation defining this equilibrium distribution,
+        or `None` if no such equation is available."""
+        raise NotImplementedError("'continuous_equation' must be provided by subclass.")
+
+    @property
+    @abstractmethod
+    def discrete_populations(self):
+        """Returns the discrete populations of this equilibrium distribution as a tuple, 
+        or `None` if none are available."""
+        raise NotImplementedError("'discrete_populations' must be provided by subclass.")
+
+    @property
+    @abstractmethod
+    def background_distribution(self):
+        """Returns this equilibrium distribution's background distribution, which is
+        the distribution the discrete populations are centered around in the case of
+        zero-centered storage. If no background distribution is available, `None` must be 
+        returned."""
+        raise NotImplementedError("'background_distribution' must be provided by subclass.")
+
+    @property
+    @abstractmethod
+    def zeroth_order_moment_symbol(self):
+        """Returns a symbol referring to the zeroth-order moment of this distribution,
+        which is the area under it's curve."""
+        raise NotImplementedError("'zeroth_order_moment' must be provided by subclass.")
+
+    @property
+    @abstractmethod
+    def first_order_moment_symbols(self):
+        """Returns a vector of symbols referring to the first-order moment of this distribution,
+        which is its mean value."""
+        raise NotImplementedError("'first_order_moments' must be provided by subclass.")
+
+    #   -------------- Statistical Modes Interface --------------------------------------------------------------------
+
+    @sharedmethodcache("_moment_cache")
+    def moment(self, exponent_tuple_or_polynomial):
+        """Returns this equilibrium distribution's moment specified by ``exponent_tuple_or_polynomial``.
+
+        Args:
+            exponent_tuple_or_polynomial: Moment specification, see :mod:`lbmpy.moments`.
+        """
+        monomials = []
+        if isinstance(exponent_tuple_or_polynomial, tuple):
+            monomials = [(1, exponent_tuple_or_polynomial)]
+        else:
+            monomials = polynomial_to_exponent_representation(exponent_tuple_or_polynomial, dim=self._dim)
+
+        moment_value = sp.Integer(0)
+        for coeff, moment in monomials:
+            moment_value += coeff * self._cached_monomial_raw_moment(moment)
+
+        return moment_value.expand()
+
+    def moments(self, exponent_tuples_or_polynomials):
+        """Returns a tuple of this equilibrium distribution's moments specified by 'exponent_tuple_or_polynomial'.
+
+        Args:
+            exponent_tuples_or_polynomials: Sequence of moment specifications, see :mod:`lbmpy.moments`.
+        """
+        return tuple(self.moment(m) for m in exponent_tuples_or_polynomials)
+
+    @sharedmethodcache("_central_moment_cache")
+    def central_moment(self, exponent_tuple_or_polynomial, frame_of_reference):
+        """Returns this equilibrium distribution's central moment specified by
+        ``exponent_tuple_or_polynomial``, computed according to the given ``frame_of_reference``.
+
+        Args:
+            exponent_tuple_or_polynomial: Moment specification, see :mod:`lbmpy.moments`.
+            frame_of_reference: The frame of reference with respect to which the central moment should be computed.
+        """
+        monomials = []
+        if isinstance(exponent_tuple_or_polynomial, tuple):
+            monomials = [(1, exponent_tuple_or_polynomial)]
+        else:
+            monomials = polynomial_to_exponent_representation(exponent_tuple_or_polynomial, dim=self._dim)
+
+        moment_value = sp.Integer(0)
+        for coeff, moment in monomials:
+            moment_value += coeff * self._cached_monomial_central_moment(moment, frame_of_reference)
+
+        return moment_value.expand()
+
+    def central_moments(self, exponent_tuples_or_polynomials, frame_of_reference):
+        """Returns a list this equilibrium distribution's central moments specified by
+        ``exponent_tuples_or_polynomials``, computed according to the given ``frame_of_reference``.
+
+        Args:
+            exponent_tuples_or_polynomials: Sequence of moment specifications, see :mod:`lbmpy.moments`.
+            frame_of_reference: The frame of reference with respect to which the central moment should be computed.
+        """
+        return tuple(self.central_moment(m, frame_of_reference) for m in exponent_tuples_or_polynomials)
+
+    @sharedmethodcache("_cumulant_cache")
+    def cumulant(self, exponent_tuple_or_polynomial, rescale=True):
+        """Returns this equilibrium distribution's cumulant specified by ``exponent_tuple_or_polynomial``.
+
+        Args:
+            exponent_tuple_or_polynomial: Moment specification, see :mod:`lbmpy.moments`.
+            rescale: If ``True``, the cumulant value should be multiplied by the zeroth-order moment.
+        """
+        monomials = []
+        if isinstance(exponent_tuple_or_polynomial, tuple):
+            monomials = [(1, exponent_tuple_or_polynomial)]
+        else:
+            monomials = polynomial_to_exponent_representation(exponent_tuple_or_polynomial, dim=self._dim)
+
+        cumulant_value = sp.Integer(0)
+        for coeff, moment in monomials:
+            cumulant_value += coeff * self._cached_monomial_cumulant(moment, rescale=rescale)
+
+        return cumulant_value.expand()
+
+    def cumulants(self, exponent_tuples_or_polynomials, rescale=True):
+        """Returns a list of this equilibrium distribution's cumulants specified by ``exponent_tuples_or_polynomial``.
+
+        Args:
+            exponent_tuples_or_polynomials: Sequence of moment specifications, see :mod:`lbmpy.moments`.
+            rescale: If ``True``, the cumulant value should be multiplied by the zeroth-order moment.
+        """
+        return tuple(self.cumulant(m, rescale) for m in exponent_tuples_or_polynomials)
+
+    #   -------------- Monomial moment computation, to be overridden in subclass --------------------------------------
+
+    @abstractmethod
+    def _monomial_raw_moment(self, exponents):
+        """See :func:`lbmpy.equilibrium.AbstractEquilibrium.moment`."""
+        raise NotImplementedError("'_monomial_raw_moment' must be implemented by a subclass.")
+
+    @abstractmethod
+    def _monomial_central_moment(self, exponents, frame_of_reference):
+        """See :func:`lbmpy.equilibrium.AbstractEquilibrium.central_moment`."""
+        raise NotImplementedError("'_monomial_central_moment' must be implemented by a subclass.")
+
+    @abstractmethod
+    def _monomial_cumulant(self, exponents, rescale):
+        """See :func:`lbmpy.equilibrium.AbstractEquilibrium.cumulant`."""
+        raise NotImplementedError("'_monomial_cumulant' must be implemented by a subclass.")
+
+    #   -------------- Cached monomial moment computation methods -----------------------------------------------------
+
+    @sharedmethodcache("_moment_cache")
+    def _cached_monomial_raw_moment(self, exponents):
+        return self._monomial_raw_moment(exponents)
+
+    @sharedmethodcache("_central_moment_cache")
+    def _cached_monomial_central_moment(self, exponents, frame_of_reference):
+        return self._monomial_central_moment(exponents, frame_of_reference)
+
+    @sharedmethodcache("_cumulant_cache")
+    def _cached_monomial_cumulant(self, exponents, rescale):
+        return self._monomial_cumulant(exponents, rescale)
+
+    #   -------------- HTML Representation ----------------------------------------------------------------------------
+
+    def _repr_html_(self):
+        html = f"""
+        <table style="border:none; width: 100%">
+            <tr>
+                <th colspan="2" style="text-align: left">
+                    Instance of {self.__class__.__name__}
+                </th>
+            </tr>
+        """
+
+        cont_eq = self.continuous_equation
+        if cont_eq is not None:
+            html += f"""
+                <tr>
+                    <td> Continuous Equation: </td>
+                    <td style="text-align: center">
+                        ${sp.latex(self.continuous_equation)}$
+                    </td>
+                </tr>
+            """
+        else:
+            pdfs = self.discrete_populations
+            if pdfs is not None:
+                html += """
+                    <tr>
+                        <td colspan="2" style="text-align: right;"> Discrete Populations: </td>
+                    </tr>
+                    """
+                for f, eq in zip(sp.symbols(f"f_:{len(pdfs)}"), pdfs):
+                    html += f'<tr><td colspan="2" style="text-align: left;"> ${f} = {sp.latex(eq)}$ </td></tr>'
+
+        html += "</table>"
+
+        return html
+
+#   end class AbstractEquilibrium

src/lbmpy/equilibrium/continuous_hydro_maxwellian.py

+import sympy as sp
+
+from .abstract_equilibrium import AbstractEquilibrium
+
+from lbmpy.moments import contained_moments
+from lbmpy.maxwellian_equilibrium import continuous_maxwellian_equilibrium
+from lbmpy.continuous_distribution_measures import continuous_moment, continuous_cumulant
+
+from pystencils.sympyextensions import remove_higher_order_terms, simplify_by_equality
+
+
+def default_background_distribution(dim):
+    return ContinuousHydrodynamicMaxwellian(dim=dim, compressible=True, deviation_only=False,
+                                            rho=sp.Integer(1), delta_rho=0, u=(0,) * dim,
+                                            c_s_sq=sp.Rational(1, 3))
+
+
+class ContinuousHydrodynamicMaxwellian(AbstractEquilibrium):
+    r"""
+        The standard continuous Maxwellian equilibrium distribution for hydrodynamics.
+
+        This class represents the Maxwellian equilibrium distribution of hydrodynamics in its continuous form
+        in :math:`d` dimensions :cite:`lbm_book`:
+        
+        .. math::
+
+            \Psi \left( \rho, \mathbf{u}, \mathbf{\xi} \right)
+             = \rho \left( \frac{1}{2 \pi c_s^2} \right)^{d/2} 
+                    \exp \left( \frac{- (\mathbf{\xi} - \mathbf{u})^2 }{2 c_s^2} \right)
+
+        Beyond this classic, 'compressible' form of the equilibrium, an alternative form known as the
+        incompressible equilibrium of the LBM can be obtained by setting the flag ``compressible=False``.
+        The continuous incompressible equilibrium can be expressed as
+        (:cite:`HeIncompressible,GruszczynskiCascadedPhaseFieldModel`):
+
+        .. math::
+
+            \Psi^{\mathrm{incomp}} \left( \rho, \mathbf{u}, \mathbf{\xi} \right)
+                = \Psi \left( \rho_0, \mathbf{u}, \mathbf{\xi} \right) 
+                  + \Psi \left( \delta\rho, \mathbf{0}, \mathbf{\xi} \right)
+
+        Here, :math:`\rho_0` (typically :math:`\rho_0 = 1`) denotes the background density, and :math:`\delta\rho` is
+        the density deviation, such that the total fluid density amounts to :math:`\rho = \rho_0 + \delta\rho`.
+
+        To simplify computations when the zero-centered storage format is used for PDFs, both equilibrium variants can
+        also be expressed in a *deviation-only* or *delta-equilibrium* form, which is obtained by subtracting the
+        constant background distribution :math:`\Psi (\rho_0, \mathbf{0})`. The delta form expresses the equilibrium
+        distribution only by its deviation from the rest state:
+
+        .. math::
+
+            \delta\Psi \left( \rho, \mathbf{u}, \mathbf{\xi} \right)
+                &=   \Psi \left( \rho, \mathbf{u}, \mathbf{\xi} \right) 
+                    - \Psi \left( \rho_0, \mathbf{0}, \mathbf{\xi} \right) \\
+            \delta\Psi^{\mathrm{incomp}} \left( \rho, \mathbf{u}, \mathbf{\xi} \right)
+                &=   \Psi^{\mathrm{incomp}} \left( \rho, \mathbf{u}, \mathbf{\xi} \right) 
+                    - \Psi \left( \rho_0, \mathbf{0}, \mathbf{\xi} \right)
+
+        Parameters:
+            dim: Spatial dimensionality
+            compressible: If `False`, the incompressible equilibrium is created
+            deviation_only: If `True`, the delta-equilibrium is created
+            order: The discretization order in velocity to which computed statistical modes should be truncated
+            rho: Symbol or value for the density
+            rho_background: Symbol or value for the background density
+            delta_rho: Symbol or value for the density deviation
+            u: Sequence of symbols for the macroscopic velocity
+            v: Sequence of symbols for the particle velocity :math:`\xi`
+            c_s_sq: Symbol or value for the squared speed of sound
+    """
+
+    def __init__(self, dim=3, compressible=True, deviation_only=False,
+                 order=None,
+                 rho=sp.Symbol("rho"),
+                 rho_background=sp.Integer(1),
+                 delta_rho=sp.Symbol("delta_rho"),
+                 u=sp.symbols("u_:3"),
+                 v=sp.symbols("v_:3"),
+                 c_s_sq=sp.Symbol("c_s") ** 2):
+        super().__init__(dim=dim)
+
+        self._order = order
+        self._compressible = compressible
+        self._deviation_only = deviation_only
+        self._rho = rho
+        self._rho_background = rho_background
+        self._delta_rho = delta_rho
+        self._u = u[:dim]
+        self._v = v[:dim]
+
+        # trick to speed up sympy integration (otherwise it takes multiple minutes, or aborts):
+        # use a positive, real symbol to represent c_s_sq -> then replace this symbol afterwards with the real c_s_sq
+        # (see maxwellian_equilibrium.py)
+        self._c_s_sq = c_s_sq
+        self._c_s_sq_helper = sp.Symbol("csq_helper", positive=True, real=True)
+
+        def psi(rho, u):
+            return continuous_maxwellian_equilibrium(dim=self._dim,
+                                                     rho=rho,
+                                                     u=u,
+                                                     v=self._v,
+                                                     c_s_sq=self._c_s_sq_helper)
+
+        zeroth_moment_arg = self._rho if self._compressible else self._rho_background
+        self._base_equation = psi(zeroth_moment_arg, self._u)
+
+        self._corrections = []
+        
+        if not self._compressible:
+            zeroth_order_correction = psi(self._delta_rho, (sp.Integer(0), ) * self._dim)
+            self._corrections.append((sp.Integer(1), zeroth_order_correction))
+
+        if self._deviation_only:
+            rest_state = psi(self._rho_background, (sp.Integer(0), ) * self._dim)
+            self._corrections.append((sp.Integer(-1), rest_state))
+
+    @property
+    def order(self):
+        return self._order
+
+    @property
+    def deviation_only(self):
+        return self._deviation_only
+
+    @property
+    def compressible(self):
+        return self._compressible
+
+    @property
+    def density(self):
+        return self._rho
+
+    @property
+    def background_density(self):
+        return self._rho_background
+
+    @property
+    def density_deviation(self):
+        return self._delta_rho
+
+    @property
+    def velocity(self):
+        return self._u
+
+    @property
+    def continuous_equation(self):
+        eq = self._base_equation + sum(f * e for f, e in self._corrections)
+        eq = eq.subs(self._c_s_sq_helper, self._c_s_sq)
+        return eq
+
+    @property
+    def zeroth_order_moment_symbol(self):
+        return self._delta_rho if self._deviation_only else self._rho
+
+    @property
+    def first_order_moment_symbols(self):
+        return self._u
+
+    @property
+    def background_distribution(self):
+        return ContinuousHydrodynamicMaxwellian(dim=self.dim, compressible=True, deviation_only=False,
+                                                order=self._order, rho=self._rho_background,
+                                                rho_background=self._rho_background,
+                                                delta_rho=0, u=(0,) * self.dim, v=self._v,
+                                                c_s_sq=self._c_s_sq)
+
+    @property
+    def discrete_populations(self):
+        return None
+
+    def central_moment(self, exponent_tuple_or_polynomial, velocity=None):
+        if velocity is None:
+            velocity = self._u
+
+        return super().central_moment(exponent_tuple_or_polynomial, velocity)
+
+    def central_moments(self, exponent_tuples_or_polynomials, velocity=None):
+        if velocity is None:
+            velocity = self._u
+
+        return super().central_moments(exponent_tuples_or_polynomials, velocity)
+
+    def cumulant(self, exponent_tuple_or_polynomial, rescale=True):
+        if not self._compressible or self._deviation_only:
+            raise Exception("Cumulants can only be computed for the compressible, "
+                            "non-deviation maxwellian equilibrium!")
+
+        return super().cumulant(exponent_tuple_or_polynomial, rescale=rescale)
+
+    #   ------------------ Overridden Moment Computation ------------------------------------------
+
+    def _monomial_raw_moment(self, exponents):
+        moment_value = continuous_moment(self._base_equation, exponents, self._v)
+
+        for coeff, corr in self._corrections:
+            moment_value += coeff * continuous_moment(corr, exponents, self._v)
+
+        moment_value = self._correct_order_and_cssq(moment_value)
+        moment_value = self._simplify_moment(moment_value)
+        return moment_value
+
+    def _monomial_central_moment(self, cm_exponents, velocity):
+        #   Setting up the central moment-generating function using SymPy integration
+        #   will take unfeasibly long at times
+        #   So we compute the central moments by binomial expansion in raw moments
+
+        cm_order = sum(cm_exponents)
+        contained_raw_moments = contained_moments(cm_exponents, exclude_original=False)
+        moment_value = sp.Integer(0)
+
+        for rm_exponents in contained_raw_moments:
+            rm_order = sum(rm_exponents)
+            factor = (-1)**(cm_order - rm_order)
+            factor *= sp.Mul(*(u**(c - i) * sp.binomial(c, i)
+                               for u, c, i in zip(velocity, cm_exponents, rm_exponents)))
+            rm_value = self._cached_monomial_raw_moment(rm_exponents)
+            moment_value += factor * rm_value
+
+        moment_value = self._correct_order_and_cssq(moment_value)
+        moment_value = self._simplify_moment(moment_value)
+        return moment_value
+
+    def _monomial_cumulant(self, c_exponents, rescale):
+        #   this implementation works only for the compressible, non-deviation equilibrium
+        cumulant_value = continuous_cumulant(self._base_equation, c_exponents, self._v)
+        cumulant_value = self._correct_order_and_cssq(cumulant_value)
+        if rescale:
+            cumulant_value = self._rho * cumulant_value
+        return cumulant_value
+
+    def _correct_order_and_cssq(self, term):
+        term = term.subs(self._c_s_sq_helper, self._c_s_sq)
+        term = term.expand()
+        if self._order is not None:
+            return remove_higher_order_terms(term, order=self._order, symbols=self._u)
+        else:
+            return term
+
+    def _simplify_moment(self, moment_value):
+        if (self.deviation_only or not self.compressible) \
+                and isinstance(self.density, sp.Symbol) and isinstance(self.density_deviation, sp.Symbol):
+            moment_value = simplify_by_equality(moment_value, self.density,
+                                                self.density_deviation, self.background_density)
+        return moment_value
+
+    #   ------------------ Utility ----------------------------------------------------------------
+
+    def __repr__(self):
+        return f"ContinuousHydrodynamicMaxwellian({self.dim}D, " \
+               f"compressible={self.compressible}, deviation_only:{self.deviation_only}" \
+               f"order={self.order})"
+
+    def _repr_html_(self):
+        def stylized_bool(b):
+            return "✓" if b else "✗"
+
+        html = f"""
+        <table style="border:none; width: 100%">
+            <tr>
+                <th colspan="3" style="text-align: left">
+                    Continuous Hydrodynamic Maxwellian Equilibrium
+                </th>
+                <td rowspan="2" style="width: 50%; text-align: center">
+                    $f ({sp.latex(self._rho)}, {sp.latex(self._u)}, {sp.latex(self._v)}) 
+                        = {sp.latex(self.continuous_equation)}$
+                </td>
+            </tr>
+            <tr>
+                <td>Compressible: {stylized_bool(self._compressible)}</td>
+                <td>Deviation Only: {stylized_bool(self._deviation_only)}</td>
+                <td>Order: {"&#8734;" if self._order is None else self._order}</td>
+            </tr>
+        </table>
+        """
+
+        return html
+
+#   end class ContinuousHydrodynamicMaxwellian

src/lbmpy/equilibrium/discrete_hydro_maxwellian.py

+import sympy as sp
+
+from pystencils.sympyextensions import simplify_by_equality
+
+from lbmpy.maxwellian_equilibrium import discrete_maxwellian_equilibrium
+
+from .generic_discrete_equilibrium import GenericDiscreteEquilibrium
+
+
+class DiscreteHydrodynamicMaxwellian(GenericDiscreteEquilibrium):
+    r"""
+        The textbook discretization of the Maxwellian equilibrium distribution of hydrodynamics.
+
+        This class represents the default discretization of the Maxwellian in velocity space,
+        computed from the distribution's expansion in Hermite polynomials (cf. :cite:`lbm_book`).
+        In :math:`d` dimensions, its populations :math:`f_i` on a given stencil 
+        :math:`(\mathbf{c}_i)_{i=0,\dots,q-1}` are given by
+
+        .. math::
+
+            f_i (\rho, \mathbf{u}) 
+                = w_i \rho \left( 
+                    1 + \frac{\mathbf{c}_i \cdot \mathbf{u}}{c_s^2}
+                    + \frac{(\mathbf{c}_i \cdot \mathbf{u})^2}{2 c_s^4}
+                    - \frac{\mathbf{u} \cdot \mathbf{u}}{2 c_s^2}
+                  \right).
+
+        Here :math:`w_i` denote the Hermite integration weights, also called lattice weights.
+        The incompressible variant of this distribution :cite:`HeIncompressible` can be written as
+
+        .. math::
+            f_i^{\mathrm{incomp}} (\rho, \mathbf{u})
+                = w_i \rho + w_i \rho_0 \left(
+                    \frac{\mathbf{c}_i \cdot \mathbf{u}}{c_s^2}
+                    + \frac{(\mathbf{c}_i \cdot \mathbf{u})^2}{2 c_s^4}
+                    - \frac{\mathbf{u} \cdot \mathbf{u}}{2 c_s^2}
+                  \right).
+
+        Again, for usage with zero-centered PDF storage, both distributions may be expressed in a delta-form
+        by subtracting their values at the background rest state at :math:`\rho = \rho_0`,
+        :math:`\mathbf{u} = \mathbf{0}`, which are exactly the lattice weights:
+
+        .. math::
+            \delta f_i &= f_i - w_i \\
+            \delta f_i^{\mathrm{incomp}} &= f_i^{\mathrm{incomp}} - w_i \\
+
+        Parameters:
+            stencil: Discrete velocity set for the discretization, see :class:`lbmpy.stencils.LBStencil`
+            compressible: If `False`, the incompressible equilibrium is created
+            deviation_only: If `True`, the delta-equilibrium is created
+            order: The discretization order in velocity to which computed statistical modes should be truncated
+            rho: Symbol or value for the density
+            delta_rho: Symbol or value for the density deviation
+            u: Sequence of symbols for the macroscopic velocity
+            c_s_sq: Symbol or value for the squared speed of sound
+    """
+
+    def __init__(self, stencil, compressible=True, deviation_only=False,
+                 order=2,
+                 rho=sp.Symbol("rho"),
+                 delta_rho=sp.Symbol("delta_rho"),
+                 u=sp.symbols("u_:3"),
+                 c_s_sq=sp.Symbol("c_s") ** 2):
+        dim = stencil.D
+
+        if order is None:
+            order = 4
+
+        self._order = order
+        self._compressible = compressible
+        self._deviation_only = deviation_only
+        self._rho = rho
+        self._rho_background = sp.Integer(1)
+        self._delta_rho = delta_rho
+        self._u = u[:dim]
+        self._c_s_sq = c_s_sq
+
+        pdfs = discrete_maxwellian_equilibrium(stencil, rho=rho, u=u,
+                                               order=order, c_s_sq=c_s_sq,
+                                               compressible=compressible)
+        if deviation_only:
+            shift = discrete_maxwellian_equilibrium(stencil, rho=self._rho_background, u=(0,) * dim,
+                                                    order=0, c_s_sq=c_s_sq, compressible=False)
+            pdfs = tuple(simplify_by_equality(f - s, rho, delta_rho, self._rho_background) for f, s in zip(pdfs, shift))
+
+        zeroth_order_moment = delta_rho if deviation_only else rho
+        super().__init__(stencil, pdfs, zeroth_order_moment, u, deviation_only)
+
+    @property
+    def order(self):
+        return self._order
+
+    @property
+    def deviation_only(self):
+        return self._deviation_only
+
+    @property
+    def compressible(self):
+        return self._compressible
+
+    @property
+    def density(self):
+        return self._rho
+
+    @property
+    def background_density(self):
+        return self._rho_background
+
+    @property
+    def density_deviation(self):
+        return self._delta_rho
+
+    @property
+    def velocity(self):
+        return self._u
+
+    @property
+    def background_distribution(self):
+        """Returns the discrete Maxwellian background distribution, which amounts exactly to the
+        lattice weights."""
+        return DiscreteHydrodynamicMaxwellian(self._stencil, compressible=True, deviation_only=False,
+                                              order=self._order, rho=self._rho_background,
+                                              delta_rho=0, u=(0,) * self.dim, c_s_sq=self._c_s_sq)
+
+    def central_moment(self, exponent_tuple_or_polynomial, velocity=None):
+        if velocity is None:
+            velocity = self._u
+
+        return super().central_moment(exponent_tuple_or_polynomial, velocity)
+
+    def central_moments(self, exponent_tuples_or_polynomials, velocity=None):
+        if velocity is None:
+            velocity = self._u
+
+        return super().central_moments(exponent_tuples_or_polynomials, velocity)
+
+    def cumulant(self, exponent_tuple_or_polynomial, rescale=True):
+        if not self._compressible or self._deviation_only:
+            raise Exception("Cumulants can only be computed for the compressible, "
+                            "non-deviation maxwellian equilibrium!")
+
+        return super().cumulant(exponent_tuple_or_polynomial, rescale=rescale)
+
+    #   ------------------ Utility ----------------------------------------------------------------
+
+    def __repr__(self):
+        return f"DiscreteHydrodynamicMaxwellian({self.stencil}, " \
+               f"compressible={self.compressible}, deviation_only:{self.deviation_only}" \
+               f"order={self.order})"
+
+    def _repr_html_(self):
+        def stylized_bool(b):
+            return "✓" if b else "✗"
+
+        html = f"""
+        <div style="max-height: 150pt; overflow-y: auto;">
+        <table style="border:none; width: 100%">
+            <tr>
+                <th colspan="3" style="text-align: left">
+                    Discrete Hydrodynamic Maxwellian Equilibrium
+                </th>
+                <td>Compressible: {stylized_bool(self._compressible)}</td>
+                <td>Deviation Only: {stylized_bool(self._deviation_only)}</td>
+                <td>Order: {"&#8734;" if self._order is None else self._order}</td>
+            </tr>
+        """
+
+        pdfs = self.discrete_populations
+        for f, eq in zip(sp.symbols(f"f_:{len(pdfs)}"), pdfs):
+            html += f'<tr><td colspan="6" style="text-align: left;"> ${f} = {sp.latex(eq)}$ </td></tr>'
+        
+        html += "</table></div>"
+
+        return html
+
+# end class DiscreteHydrodynamicMaxwellian

src/lbmpy/equilibrium/generic_discrete_equilibrium.py

+import sympy as sp
+
+from .abstract_equilibrium import AbstractEquilibrium
+
+from lbmpy.moments import discrete_moment, moment_matrix
+from lbmpy.cumulants import discrete_cumulant
+
+
+def discrete_equilibrium_from_matching_moments(stencil, moment_constraints,
+                                               zeroth_order_moment_symbol,
+                                               first_order_moment_symbols,
+                                               deviation_only=False):
+    assert len(moment_constraints) == stencil.Q
+    moments = tuple(moment_constraints.keys())
+    mm = moment_matrix(moments, stencil)
+    try:
+        pdfs = mm.inv() * sp.Matrix(list(moment_constraints.values()))
+        pdfs = pdfs.expand()
+        return GenericDiscreteEquilibrium(stencil, pdfs, zeroth_order_moment_symbol,
+                                          first_order_moment_symbols, deviation_only=deviation_only)
+    except sp.matrices.inverse.NonInvertibleMatrixError as e:
+        raise ValueError("Could not construct equilibrium from given moment constraints.") from e
+
+
+class GenericDiscreteEquilibrium(AbstractEquilibrium):
+    """
+        Class for encapsulating arbitrary discrete equilibria, given by their equilibrium populations.
+
+        This class takes both a stencil and a sequence of populations modelling a discrete distribution function
+        and provides basic functionality for computing and caching that distribution's statistical modes.
+
+        Parameters:
+            stencil: Discrete velocity set, see :class:`lbmpy.stencils.LBStencil`.
+            equilibrium_pdfs: List of q populations, describing the particle distribution on the discrete velocity
+                              set given by the stencil.
+            zeroth_order_moment_symbol: Symbol corresponding to the distribution's zeroth-order moment, the area under
+                                        it's curve (see :attr:`zeroth_order_moment_symbol`).
+            first_order_moment_symbols: Sequence of symbols corresponding to the distribution's first-order moment, the
+                                        vector of its mean values (see :attr:`first_order_moment_symbols`).
+            deviation_only: Set to `True` if the given populations model only the deviation from a rest state, to be
+                            used in junction with the zero-centered storage format.
+    """
+
+    def __init__(self, stencil, equilibrium_pdfs,
+                 zeroth_order_moment_symbol,
+                 first_order_moment_symbols,
+                 deviation_only=False):
+        super().__init__(dim=stencil.D)
+
+        if len(equilibrium_pdfs) != stencil.Q:
+            raise ValueError(f"Wrong number of PDFs."
+                             f"On the {stencil} stencil, exactly {stencil.Q} populations must be passed!")
+
+        self._stencil = stencil
+        self._pdfs = tuple(equilibrium_pdfs)
+        self._zeroth_order_moment_symbol = zeroth_order_moment_symbol
+        self._first_order_moment_symbols = first_order_moment_symbols
+        self._deviation_only = deviation_only
+
+    @property
+    def stencil(self):
+        return self._stencil
+
+    @property
+    def deviation_only(self):
+        return self._deviation_only
+
+    @property
+    def continuous_equation(self):
+        """Always returns `None`."""
+        return None
+
+    @property
+    def discrete_populations(self):
+        return self._pdfs
+
+    @property
+    def background_distribution(self):
+        """Always returns `None`. To specify a background distribution, override this class."""
+        return None
+
+    @property
+    def zeroth_order_moment_symbol(self):
+        return self._zeroth_order_moment_symbol
+
+    @property
+    def first_order_moment_symbols(self):
+        return self._first_order_moment_symbols
+
+    #   Moment Computation
+
+    def _monomial_raw_moment(self, exponents):
+        return discrete_moment(self._pdfs, exponents, self._stencil)
+
+    def _monomial_central_moment(self, exponents, frame_of_reference):
+        return discrete_moment(self._pdfs, exponents, self._stencil, shift_velocity=frame_of_reference)
+
+    def _monomial_cumulant(self, exponents, rescale):
+        value = discrete_cumulant(self._pdfs, exponents, self._stencil)
+        if rescale:
+            value = self.zeroth_order_moment_symbol * value
+        return value