Spaces:
Sleeping
Sleeping
File size: 6,360 Bytes
66c9c8a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 | """
This example solves a 2D Stokes flow problem
-nu D(u) + grad p = 0
Div u = 0
with (soft) velocity-Dirichlet boundary conditions
"""
import argparse
import warp as wp
import numpy as np
import warp.fem as fem
try:
from .plot_utils import Plot
from .bsr_utils import bsr_to_scipy
from .mesh_utils import gen_trimesh, gen_quadmesh
except ImportError:
from plot_utils import Plot
from bsr_utils import bsr_to_scipy
from mesh_utils import gen_trimesh, gen_quadmesh
# Need to solve a saddle-point system, use scipy for simplicity
from scipy.sparse import bmat
from scipy.sparse.linalg import spsolve
@fem.integrand
def constant_form(val: wp.vec2):
return val
@fem.integrand
def viscosity_form(s: fem.Sample, u: fem.Field, v: fem.Field, nu: float):
return nu * wp.ddot(fem.D(u, s), fem.D(v, s))
@fem.integrand
def top_mass_form(
s: fem.Sample,
domain: fem.Domain,
u: fem.Field,
v: fem.Field,
):
# non zero on top boundary of domain only
nor = fem.normal(domain, s)
return wp.dot(u(s), v(s)) * wp.max(0.0, nor[1])
@fem.integrand
def mass_form(
s: fem.Sample,
u: fem.Field,
v: fem.Field,
):
return wp.dot(u(s), v(s))
@fem.integrand
def div_form(
s: fem.Sample,
u: fem.Field,
q: fem.Field,
):
return q(s) * fem.div(u, s)
class Example:
parser = argparse.ArgumentParser()
parser.add_argument("--resolution", type=int, default=50)
parser.add_argument("--degree", type=int, default=2)
parser.add_argument("--top_velocity", type=float, default=1.0)
parser.add_argument("--viscosity", type=float, default=1.0)
parser.add_argument("--boundary_strength", type=float, default=100.0)
parser.add_argument("--mesh", choices=("grid", "tri", "quad"), default="grid", help="Mesh type")
parser.add_argument(
"--nonconforming_pressures", action="store_true", help="For grid, use non-conforming pressure (Q_d/P_{d-1})"
)
def __init__(self, stage=None, quiet=False, args=None, **kwargs):
if args is None:
# Read args from kwargs, add default arg values from parser
args = argparse.Namespace(**kwargs)
args = Example.parser.parse_args(args=[], namespace=args)
self._args = args
self._quiet = quiet
# Grid or triangle mesh geometry
if args.mesh == "tri":
positions, tri_vidx = gen_trimesh(res=wp.vec2i(args.resolution))
geo = fem.Trimesh2D(tri_vertex_indices=tri_vidx, positions=positions)
elif args.mesh == "quad":
positions, quad_vidx = gen_quadmesh(res=wp.vec2i(args.resolution))
geo = fem.Quadmesh2D(quad_vertex_indices=quad_vidx, positions=positions)
else:
geo = fem.Grid2D(res=wp.vec2i(args.resolution))
# Function spaces -- Q_d for vel, P_{d-1} for pressure
u_space = fem.make_polynomial_space(geo, degree=args.degree, dtype=wp.vec2)
if args.mesh != "tri" and args.nonconforming_pressures:
p_space = fem.make_polynomial_space(
geo, degree=args.degree - 1, element_basis=fem.ElementBasis.NONCONFORMING_POLYNOMIAL
)
else:
p_space = fem.make_polynomial_space(geo, degree=args.degree - 1)
# Vector and scalar fields
self._u_field = u_space.make_field()
self._p_field = p_space.make_field()
# Interpolate initial condition on boundary (for example purposes)
self._bd_field = u_space.make_field()
f_boundary = fem.make_restriction(self._bd_field, domain=fem.BoundarySides(geo))
top_velocity = wp.vec2(args.top_velocity, 0.0)
fem.interpolate(constant_form, dest=f_boundary, values={"val": top_velocity})
self.renderer = Plot(stage)
def update(self):
args = self._args
u_space = self._u_field.space
p_space = self._p_field.space
geo = u_space.geometry
domain = fem.Cells(geometry=geo)
boundary = fem.BoundarySides(geo)
# Viscosity
u_test = fem.make_test(space=u_space, domain=domain)
u_trial = fem.make_trial(space=u_space, domain=domain)
u_visc_matrix = fem.integrate(
viscosity_form,
fields={"u": u_trial, "v": u_test},
values={"nu": args.viscosity},
)
# Weak velocity boundary conditions
u_bd_test = fem.make_test(space=u_space, domain=boundary)
u_bd_trial = fem.make_trial(space=u_space, domain=boundary)
u_rhs = fem.integrate(top_mass_form, fields={"u": self._bd_field.trace(), "v": u_bd_test})
u_bd_matrix = fem.integrate(mass_form, fields={"u": u_bd_trial, "v": u_bd_test})
# Pressure-velocity coupling
p_test = fem.make_test(space=p_space, domain=domain)
div_matrix = fem.integrate(div_form, fields={"u": u_trial, "q": p_test})
# Solve with scipy
# Assemble saddle-point system with velocity, pressure, and zero-average-pressure constraint
u_rhs = u_rhs.numpy() * args.boundary_strength
u_matrix = bsr_to_scipy(u_visc_matrix) + args.boundary_strength * bsr_to_scipy(u_bd_matrix)
div_matrix = bsr_to_scipy(div_matrix)
ones = np.ones(shape=(p_space.node_count(), 1), dtype=float)
saddle_system = bmat(
[
[u_matrix, div_matrix.transpose(), None],
[div_matrix, None, ones],
[None, ones.transpose(), None],
],
format="csr",
)
saddle_rhs = np.zeros(saddle_system.shape[0])
u_slice = slice(0, 2 * u_space.node_count())
p_slice = slice(2 * u_space.node_count(), 2 * u_space.node_count() + p_space.node_count())
saddle_rhs[u_slice] = u_rhs.flatten()
x = spsolve(saddle_system, saddle_rhs)
# Extract result
self._u_field.dof_values = x[u_slice].reshape((-1, 2))
self._p_field.dof_values = x[p_slice]
def render(self):
self.renderer.add_surface("pressure", self._p_field)
self.renderer.add_surface_vector("velocity", self._u_field)
if __name__ == "__main__":
wp.init()
wp.set_module_options({"enable_backward": False})
args = Example.parser.parse_args()
example = Example(args=args)
example.update()
example.render()
example.renderer.plot()
|