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This example computes a 2D weakly-compressible Stokes flow around a moving object, including:
- defining active cells from a mask, and restricting the computation domain to those
- utilizing the PicQuadrature to integrate over unstructured particles
"""
import math
import warp as wp
import numpy as np
import warp.fem as fem
from warp.utils import array_cast
from warp.fem.utils import array_axpy
from warp.sparse import bsr_transposed, bsr_mm, bsr_axpy, bsr_mv
# Import example utilities
# Make sure that works both when imported as module and run as standalone file
try:
from .bsr_utils import bsr_cg
from .plot_utils import Plot
except ImportError:
from bsr_utils import bsr_cg
from plot_utils import Plot
@fem.integrand
def vel_from_particles_form(s: fem.Sample, particle_vel: wp.array(dtype=wp.vec2), v: fem.Field):
vel = particle_vel[s.qp_index]
return wp.dot(vel, v(s))
@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 mass_form(
s: fem.Sample,
u: fem.Field,
v: fem.Field,
):
return wp.dot(u(s), v(s))
@fem.integrand
def scalar_mass_form(
s: fem.Sample,
p: fem.Field,
q: fem.Field,
):
return p(s) * q(s)
@fem.integrand
def div_form(
s: fem.Sample,
u: fem.Field,
q: fem.Field,
):
return q(s) * fem.div(u, s)
@fem.integrand
def cell_activity(s: fem.Sample, domain: fem.Domain, c1: wp.vec2, c2: wp.vec2, radius: float):
pos = domain(s)
if wp.length(pos - c1) < radius:
return 0.0
if wp.length(pos - c2) < radius:
return 0.0
return 1.0
@wp.kernel
def inverse_array_kernel(m: wp.array(dtype=wp.float64)):
m[wp.tid()] = wp.float64(1.0) / m[wp.tid()]
class Example:
def __init__(self, stage=None, quiet=False):
self._quiet = quiet
self.res = 50
self.cell_size = 1.0 / self.res
self.vel = 1.0
self.viscosity = 100.0
self.compliance = 0.01
self.bd_strength = 100000.0
geo = fem.Grid2D(res=wp.vec2i(self.res))
# Displacement boundary conditions are defined by two circles going in opposite directions
# Sample particles along those
circle_radius = 0.15
c1_center = wp.vec2(0.25, 0.5)
c2_center = wp.vec2(0.75, 0.5)
particles, particle_areas, particle_velocities = self._gen_particles(circle_radius, c1_center, c2_center)
# Disable cells that are interior to the circles
cell_space = fem.make_polynomial_space(geo, degree=0)
activity = cell_space.make_field()
fem.interpolate(
cell_activity,
dest=activity,
values={"c1": c1_center, "c2": c2_center, "radius": circle_radius - self.cell_size},
)
# Explicitly define the active geometry partition from those cells
self._active_partition = fem.ExplicitGeometryPartition(geo, wp.array(activity.dof_values.numpy(), dtype=int))
if not self._quiet:
print("Active cells:", self._active_partition.cell_count())
# Function spaces -- Q1 for vel, Q0 for pressure
u_space = fem.make_polynomial_space(geo, degree=1, dtype=wp.vec2)
p_space = fem.make_polynomial_space(geo, degree=0)
self._active_space_partition = fem.make_space_partition(
space=u_space, geometry_partition=self._active_partition
)
self._active_p_space_partition = fem.make_space_partition(
space=p_space, geometry_partition=self._active_partition
)
self._u_field = u_space.make_field()
self._p_field = p_space.make_field()
# Particle-based quadrature rule over active cells
domain = fem.Cells(geometry=self._active_partition)
self._pic_quadrature = fem.PicQuadrature(domain, particles, particle_areas)
self._particle_velocities = particle_velocities
self.renderer = Plot(stage)
def update(self):
u_space = self._u_field.space
p_space = self._p_field.space
# Weakly-enforced boundary condition on particles
u_test = fem.make_test(space=u_space, space_partition=self._active_space_partition)
u_trial = fem.make_trial(space=u_space, space_partition=self._active_space_partition)
u_rhs = fem.integrate(
vel_from_particles_form,
quadrature=self._pic_quadrature,
fields={"v": u_test},
values={"particle_vel": self._particle_velocities},
output_dtype=wp.vec2d,
)
u_bd_matrix = fem.integrate(mass_form, quadrature=self._pic_quadrature, fields={"u": u_trial, "v": u_test})
# Viscosity
u_visc_matrix = fem.integrate(
viscosity_form,
fields={"u": u_trial, "v": u_test},
values={"nu": self.viscosity},
)
# Pressure-velocity coupling
p_test = fem.make_test(space=p_space, space_partition=self._active_p_space_partition)
p_trial = fem.make_trial(space=p_space, space_partition=self._active_p_space_partition)
div_matrix = fem.integrate(div_form, fields={"u": u_trial, "q": p_test})
inv_p_mass_matrix = fem.integrate(scalar_mass_form, fields={"p": p_trial, "q": p_test})
wp.launch(
kernel=inverse_array_kernel,
dim=inv_p_mass_matrix.values.shape,
device=inv_p_mass_matrix.values.device,
inputs=[inv_p_mass_matrix.values],
)
# Assemble linear system
u_matrix = u_visc_matrix
bsr_axpy(u_bd_matrix, u_matrix, alpha=self.bd_strength)
div_matrix_t = bsr_transposed(div_matrix)
gradient_matrix = bsr_mm(div_matrix_t, inv_p_mass_matrix)
bsr_mm(gradient_matrix, div_matrix, u_matrix, alpha=1.0 / self.compliance, beta=1.0)
array_axpy(u_rhs, u_rhs, alpha=0.0, beta=self.bd_strength)
# Solve for displacement
u_res = wp.zeros_like(u_rhs)
bsr_cg(u_matrix, x=u_res, b=u_rhs, quiet=self._quiet)
# Compute pressure from displacement
div_u = bsr_mv(A=div_matrix, x=u_res)
p_res = bsr_mv(A=inv_p_mass_matrix, x=div_u, alpha=-1)
# Copy to fields
u_nodes = wp.indexedarray(self._u_field.dof_values, indices=self._active_space_partition.space_node_indices())
p_nodes = wp.indexedarray(self._p_field.dof_values, indices=self._active_p_space_partition.space_node_indices())
array_cast(in_array=u_res, out_array=u_nodes)
array_cast(in_array=p_res, out_array=p_nodes)
def render(self):
self.renderer.add_surface("pressure", self._p_field)
self.renderer.add_surface_vector("velocity", self._u_field)
def _gen_particles(self, circle_radius, c1_center, c2_center):
"""Generate some particles along two circles defining velocity boundary conditions"""
# Generate particles defining the transfer displacement
particles_per_circle = int(2.0 * math.pi * circle_radius * self.res)
angles = np.linspace(0, 2.0 * math.pi, particles_per_circle, endpoint=False)
n_particles = 2 * particles_per_circle
particles = np.empty((n_particles, 2), dtype=float)
particles[:particles_per_circle, 0] = c1_center[0] + circle_radius * np.cos(angles)
particles[:particles_per_circle, 1] = c1_center[1] + circle_radius * np.sin(angles)
particles[particles_per_circle:, 0] = c2_center[0] + circle_radius * np.cos(angles)
particles[particles_per_circle:, 1] = c2_center[1] + circle_radius * np.sin(angles)
particle_areas = np.ones(n_particles) * self.cell_size**2
particle_velocities = np.zeros_like(particles)
particle_velocities[:particles_per_circle, 0] = self.vel
particle_velocities[particles_per_circle:, 0] = -self.vel
particles = wp.array(particles, dtype=wp.vec2)
particle_areas = wp.array(particle_areas, dtype=float)
particle_velocities = wp.array(particle_velocities, dtype=wp.vec2)
return particles, particle_areas, particle_velocities
if __name__ == "__main__":
wp.init()
wp.set_module_options({"enable_backward": False})
example = Example()
example.update()
example.render()
example.renderer.plot(streamlines=["velocity"])
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