centered_cumulants.py 4.13 KB
 Frederik Hennig committed Feb 02, 2021 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ``````from lbmpy.stencils import get_stencil import sympy as sp from pystencils.stencil import have_same_entries from lbmpy.moments import MOMENT_SYMBOLS, moment_sort_key, exponent_to_polynomial_representation def statistical_quantity_symbol(name, exponents): return sp.Symbol(f'{name}_{"".join(str(i) for i in exponents)}') def exponent_tuple_sort_key(x): return moment_sort_key(exponent_to_polynomial_representation(x)) def get_default_polynomial_cumulants_for_stencil(stencil): """ Returns default groups of cumulants to be relaxed with common relaxation rates as stated in literature. Groups are ordered like this: `````` Markus Holzer committed May 12, 2021 21 22 23 24 25 26 27 28 29 `````` - First group is density - Second group are the momentum modes - Third group are the shear modes - Fourth group is the bulk mode - Remaining groups do not govern hydrodynamic properties Args: stencil: can be D2Q9, D2Q19 or D3Q27 `````` Frederik Hennig committed Feb 02, 2021 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 `````` """ x, y, z = MOMENT_SYMBOLS if have_same_entries(stencil, get_stencil("D2Q9")): # Cumulants of the D2Q9 stencil up to third order are equal to # the central moments; only the fourth-order cumulant x**2 * y**2 # has a more complicated form. They can be arranged into groups # for the preservation of rotational invariance as described by # Martin Geier in his dissertation. # # Reference: Martin Geier. Ab inito derivation of the cascaded Lattice Boltzmann # Automaton. Dissertation. University of Freiburg. 2006. return [ [sp.sympify(1)], # density is conserved [x, y], # momentum is relaxed for cumulant forcing [x * y, x**2 - y**2], # shear [x**2 + y**2], # bulk [x**2 * y, x * y**2], [x**2 * y**2] ] elif have_same_entries(stencil, get_stencil("D3Q19")): # D3Q19 cumulants are obtained by pruning the D3Q27 cumulant set as # described by Coreixas, 2019. return [ [sp.sympify(1)], # density is conserved [x, y, z], # momentum might be affected by forcing [x * y, x * z, y * z, x ** 2 - y ** 2, x ** 2 - z ** 2], # shear [x ** 2 + y ** 2 + z ** 2], # bulk [x * y ** 2 + x * z ** 2, x ** 2 * y + y * z ** 2, x ** 2 * z + y ** 2 * z], [x * y ** 2 - x * z ** 2, x ** 2 * y - y * z ** 2, x ** 2 * z - y ** 2 * z], [x ** 2 * y ** 2 - 2 * x ** 2 * z ** 2 + y ** 2 * z ** 2, x ** 2 * y ** 2 + x ** 2 * z ** 2 - 2 * y ** 2 * z ** 2], [x ** 2 * y ** 2 + x ** 2 * z ** 2 + y ** 2 * z ** 2] ] elif have_same_entries(stencil, get_stencil("D3Q27")): # Cumulants grouped to preserve rotational invariance as described by Geier et al, 2015 return [ [sp.sympify(1)], # density is conserved [x, y, z], # momentum might be affected by forcing [x * y, x * z, y * z, x ** 2 - y ** 2, x ** 2 - z ** 2], # shear [x ** 2 + y ** 2 + z ** 2], # bulk [x * y ** 2 + x * z ** 2, x ** 2 * y + y * z ** 2, x ** 2 * z + y ** 2 * z], [x * y ** 2 - x * z ** 2, x ** 2 * y - y * z ** 2, x ** 2 * z - y ** 2 * z], [x * y * z], [x ** 2 * y ** 2 - 2 * x ** 2 * z ** 2 + y ** 2 * z ** 2, x ** 2 * y ** 2 + x ** 2 * z ** 2 - 2 * y ** 2 * z ** 2], [x ** 2 * y ** 2 + x ** 2 * z ** 2 + y ** 2 * z ** 2], [x ** 2 * y * z, x * y ** 2 * z, x * y * z ** 2], [x ** 2 * y ** 2 * z, x ** 2 * y * z ** 2, x * y ** 2 * z ** 2], [x ** 2 * y ** 2 * z ** 2] ] else: raise ValueError("No default set of cumulants is available for this stencil. " "Please specify your own set of polynomial cumulants.")``````