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import warp as wp
import numpy as np
from warp.fem.geometry import Grid3D
from warp.fem.polynomial import Polynomial, quadrature_1d, lagrange_scales, is_closed
from warp.fem.types import Coords
from warp.fem import cache
from .tet_shape_function import TetrahedronPolynomialShapeFunctions
_CUBE_EDGE_INDICES = wp.constant(
wp.mat(shape=(3, 4), dtype=int)(
[
[0, 4, 2, 6],
[3, 1, 7, 5],
[8, 11, 9, 10],
]
)
)
class CubeTripolynomialShapeFunctions:
VERTEX = 0
EDGE = 1
FACE = 2
INTERIOR = 3
def __init__(self, degree: int, family: Polynomial):
self.family = family
self.ORDER = wp.constant(degree)
self.NODES_PER_ELEMENT = wp.constant((degree + 1) ** 3)
self.NODES_PER_EDGE = wp.constant(degree + 1)
lobatto_coords, lobatto_weight = quadrature_1d(point_count=degree + 1, family=family)
lagrange_scale = lagrange_scales(lobatto_coords)
NodeVec = wp.types.vector(length=degree + 1, dtype=wp.float32)
self.LOBATTO_COORDS = wp.constant(NodeVec(lobatto_coords))
self.LOBATTO_WEIGHT = wp.constant(NodeVec(lobatto_weight))
self.LAGRANGE_SCALE = wp.constant(NodeVec(lagrange_scale))
self.ORDER_PLUS_ONE = wp.constant(self.ORDER + 1)
self._node_ijk = self._make_node_ijk()
self.node_type_and_type_index = self._make_node_type_and_type_index()
@property
def name(self) -> str:
return f"Cube_Q{self.ORDER}_{self.family}"
@wp.func
def _vertex_coords_f(vidx_in_cell: int):
x = vidx_in_cell // 4
y = (vidx_in_cell - 4 * x) // 2
z = vidx_in_cell - 4 * x - 2 * y
return wp.vec3(float(x), float(y), float(z))
def _make_node_ijk(self):
ORDER_PLUS_ONE = self.ORDER_PLUS_ONE
def node_ijk(
node_index_in_elt: int,
):
node_i = node_index_in_elt // (ORDER_PLUS_ONE * ORDER_PLUS_ONE)
node_jk = node_index_in_elt - ORDER_PLUS_ONE * ORDER_PLUS_ONE * node_i
node_j = node_jk // ORDER_PLUS_ONE
node_k = node_jk - ORDER_PLUS_ONE * node_j
return node_i, node_j, node_k
return cache.get_func(node_ijk, self.name)
def _make_node_type_and_type_index(self):
ORDER = self.ORDER
@cache.dynamic_func(suffix=self.name)
def node_type_and_type_index(
node_index_in_elt: int,
):
i, j, k = self._node_ijk(node_index_in_elt)
zi = wp.select(i == 0, 0, 1)
zj = wp.select(j == 0, 0, 1)
zk = wp.select(k == 0, 0, 1)
mi = wp.select(i == ORDER, 0, 1)
mj = wp.select(j == ORDER, 0, 1)
mk = wp.select(k == ORDER, 0, 1)
if zi + mi == 1:
if zj + mj == 1:
if zk + mk == 1:
# vertex
type_instance = mi * 4 + mj * 2 + mk
return CubeTripolynomialShapeFunctions.VERTEX, type_instance, 0
# z edge
type_instance = _CUBE_EDGE_INDICES[2, mi * 2 + mj]
type_index = k - 1
return CubeTripolynomialShapeFunctions.EDGE, type_instance, type_index
if zk + mk == 1:
# y edge
type_instance = _CUBE_EDGE_INDICES[1, mk * 2 + mi]
type_index = j - 1
return CubeTripolynomialShapeFunctions.EDGE, type_instance, type_index
# x face
type_instance = mi
type_index = wp.select(mi == 1, (j - 1) * (ORDER - 1) + k - 1, (k - 1) * (ORDER - 1) + j - 1)
return CubeTripolynomialShapeFunctions.FACE, type_instance, type_index
if zj + mj == 1:
if zk + mk == 1:
# x edge
type_instance = _CUBE_EDGE_INDICES[0, mj * 2 + mk]
type_index = i - 1
return CubeTripolynomialShapeFunctions.EDGE, type_instance, type_index
# y face
type_instance = 2 + mj
type_index = wp.select(mj == 1, (i - 1) * (ORDER - 1) + k - 1, (k - 1) * (ORDER - 1) + i - 1)
return CubeTripolynomialShapeFunctions.FACE, type_instance, type_index
if zk + mk == 1:
# z face
type_instance = 4 + mk
type_index = wp.select(mk == 1, (j - 1) * (ORDER - 1) + i - 1, (i - 1) * (ORDER - 1) + j - 1)
return CubeTripolynomialShapeFunctions.FACE, type_instance, type_index
type_index = ((i - 1) * (ORDER - 1) + (j - 1)) * (ORDER - 1) + k - 1
return CubeTripolynomialShapeFunctions.INTERIOR, 0, type_index
return node_type_and_type_index
def make_node_coords_in_element(self):
LOBATTO_COORDS = self.LOBATTO_COORDS
@cache.dynamic_func(suffix=self.name)
def node_coords_in_element(
node_index_in_elt: int,
):
node_i, node_j, node_k = self._node_ijk(node_index_in_elt)
return Coords(LOBATTO_COORDS[node_i], LOBATTO_COORDS[node_j], LOBATTO_COORDS[node_k])
return node_coords_in_element
def make_node_quadrature_weight(self):
ORDER = self.ORDER
LOBATTO_WEIGHT = self.LOBATTO_WEIGHT
def node_quadrature_weight(
node_index_in_elt: int,
):
node_i, node_j, node_k = self._node_ijk(node_index_in_elt)
return LOBATTO_WEIGHT[node_i] * LOBATTO_WEIGHT[node_j] * LOBATTO_WEIGHT[node_k]
def node_quadrature_weight_linear(
node_index_in_elt: int,
):
return 0.125
if ORDER == 1:
return cache.get_func(node_quadrature_weight_linear, self.name)
return cache.get_func(node_quadrature_weight, self.name)
def make_trace_node_quadrature_weight(self):
ORDER = self.ORDER
LOBATTO_WEIGHT = self.LOBATTO_WEIGHT
def trace_node_quadrature_weight(
node_index_in_elt: int,
):
# We're either on a side interior or at a vertex
# If we find one index at extremum, pick the two other
node_i, node_j, node_k = self._node_ijk(node_index_in_elt)
if node_i == 0 or node_i == ORDER:
return LOBATTO_WEIGHT[node_j] * LOBATTO_WEIGHT[node_k]
if node_j == 0 or node_j == ORDER:
return LOBATTO_WEIGHT[node_i] * LOBATTO_WEIGHT[node_k]
return LOBATTO_WEIGHT[node_i] * LOBATTO_WEIGHT[node_j]
def trace_node_quadrature_weight_linear(
node_index_in_elt: int,
):
return 0.25
def trace_node_quadrature_weight_open(
node_index_in_elt: int,
):
return 0.0
if not is_closed(self.family):
return cache.get_func(trace_node_quadrature_weight_open, self.name)
if ORDER == 1:
return cache.get_func(trace_node_quadrature_weight_linear, self.name)
return cache.get_func(trace_node_quadrature_weight, self.name)
def make_element_inner_weight(self):
ORDER_PLUS_ONE = self.ORDER_PLUS_ONE
LOBATTO_COORDS = self.LOBATTO_COORDS
LAGRANGE_SCALE = self.LAGRANGE_SCALE
def element_inner_weight(
coords: Coords,
node_index_in_elt: int,
):
node_i, node_j, node_k = self._node_ijk(node_index_in_elt)
w = float(1.0)
for k in range(ORDER_PLUS_ONE):
if k != node_i:
w *= coords[0] - LOBATTO_COORDS[k]
if k != node_j:
w *= coords[1] - LOBATTO_COORDS[k]
if k != node_k:
w *= coords[2] - LOBATTO_COORDS[k]
w *= LAGRANGE_SCALE[node_i] * LAGRANGE_SCALE[node_j] * LAGRANGE_SCALE[node_k]
return w
def element_inner_weight_linear(
coords: Coords,
node_index_in_elt: int,
):
v = CubeTripolynomialShapeFunctions._vertex_coords_f(node_index_in_elt)
wx = (1.0 - coords[0]) * (1.0 - v[0]) + v[0] * coords[0]
wy = (1.0 - coords[1]) * (1.0 - v[1]) + v[1] * coords[1]
wz = (1.0 - coords[2]) * (1.0 - v[2]) + v[2] * coords[2]
return wx * wy * wz
if self.ORDER == 1 and is_closed(self.family):
return cache.get_func(element_inner_weight_linear, self.name)
return cache.get_func(element_inner_weight, self.name)
def make_element_inner_weight_gradient(self):
ORDER_PLUS_ONE = self.ORDER_PLUS_ONE
LOBATTO_COORDS = self.LOBATTO_COORDS
LAGRANGE_SCALE = self.LAGRANGE_SCALE
def element_inner_weight_gradient(
coords: Coords,
node_index_in_elt: int,
):
node_i, node_j, node_k = self._node_ijk(node_index_in_elt)
prefix_xy = float(1.0)
prefix_yz = float(1.0)
prefix_zx = float(1.0)
for k in range(ORDER_PLUS_ONE):
if k != node_i:
prefix_yz *= coords[0] - LOBATTO_COORDS[k]
if k != node_j:
prefix_zx *= coords[1] - LOBATTO_COORDS[k]
if k != node_k:
prefix_xy *= coords[2] - LOBATTO_COORDS[k]
prefix_x = prefix_zx * prefix_xy
prefix_y = prefix_yz * prefix_xy
prefix_z = prefix_zx * prefix_yz
grad_x = float(0.0)
grad_y = float(0.0)
grad_z = float(0.0)
for k in range(ORDER_PLUS_ONE):
if k != node_i:
delta_x = coords[0] - LOBATTO_COORDS[k]
grad_x = grad_x * delta_x + prefix_x
prefix_x *= delta_x
if k != node_j:
delta_y = coords[1] - LOBATTO_COORDS[k]
grad_y = grad_y * delta_y + prefix_y
prefix_y *= delta_y
if k != node_k:
delta_z = coords[2] - LOBATTO_COORDS[k]
grad_z = grad_z * delta_z + prefix_z
prefix_z *= delta_z
grad = (
LAGRANGE_SCALE[node_i]
* LAGRANGE_SCALE[node_j]
* LAGRANGE_SCALE[node_k]
* wp.vec3(
grad_x,
grad_y,
grad_z,
)
)
return grad
def element_inner_weight_gradient_linear(
coords: Coords,
node_index_in_elt: int,
):
v = CubeTripolynomialShapeFunctions._vertex_coords_f(node_index_in_elt)
wx = (1.0 - coords[0]) * (1.0 - v[0]) + v[0] * coords[0]
wy = (1.0 - coords[1]) * (1.0 - v[1]) + v[1] * coords[1]
wz = (1.0 - coords[2]) * (1.0 - v[2]) + v[2] * coords[2]
dx = 2.0 * v[0] - 1.0
dy = 2.0 * v[1] - 1.0
dz = 2.0 * v[2] - 1.0
return wp.vec3(dx * wy * wz, dy * wz * wx, dz * wx * wy)
if self.ORDER == 1 and is_closed(self.family):
return cache.get_func(element_inner_weight_gradient_linear, self.name)
return cache.get_func(element_inner_weight_gradient, self.name)
def element_node_hexes(self):
from warp.fem.utils import grid_to_hexes
return grid_to_hexes(self.ORDER, self.ORDER, self.ORDER)
def element_node_tets(self):
from warp.fem.utils import grid_to_tets
return grid_to_tets(self.ORDER, self.ORDER, self.ORDER)
class CubeSerendipityShapeFunctions:
"""
Serendipity element ~ tensor product space without interior nodes
Edge shape functions are usual Lagrange shape functions times a bilinear function in the normal directions
Corner shape functions are trilinear shape functions times a function of (x^{d-1} + y^{d-1})
"""
# Node categories
VERTEX = wp.constant(0)
EDGE_X = wp.constant(1)
EDGE_Y = wp.constant(2)
def __init__(self, degree: int, family: Polynomial):
if not is_closed(family):
raise ValueError("A closed polynomial family is required to define serendipity elements")
if degree not in [2, 3]:
raise NotImplementedError("Serendipity element only implemented for order 2 or 3")
self.family = family
self.ORDER = wp.constant(degree)
self.NODES_PER_ELEMENT = wp.constant(8 + 12 * (degree - 1))
self.NODES_PER_EDGE = wp.constant(degree + 1)
lobatto_coords, lobatto_weight = quadrature_1d(point_count=degree + 1, family=family)
lagrange_scale = lagrange_scales(lobatto_coords)
NodeVec = wp.types.vector(length=degree + 1, dtype=wp.float32)
self.LOBATTO_COORDS = wp.constant(NodeVec(lobatto_coords))
self.LOBATTO_WEIGHT = wp.constant(NodeVec(lobatto_weight))
self.LAGRANGE_SCALE = wp.constant(NodeVec(lagrange_scale))
self.ORDER_PLUS_ONE = wp.constant(self.ORDER + 1)
self.node_type_and_type_index = self._get_node_type_and_type_index()
self._node_lobatto_indices = self._get_node_lobatto_indices()
@property
def name(self) -> str:
return f"Cube_S{self.ORDER}_{self.family}"
def _get_node_type_and_type_index(self):
@cache.dynamic_func(suffix=self.name)
def node_type_and_index(
node_index_in_elt: int,
):
if node_index_in_elt < 8:
return CubeSerendipityShapeFunctions.VERTEX, node_index_in_elt
type_index = (node_index_in_elt - 8) // 3
side = node_index_in_elt - 8 - 3 * type_index
return CubeSerendipityShapeFunctions.EDGE_X + side, type_index
return node_type_and_index
@wp.func
def _vertex_coords(vidx_in_cell: int):
x = vidx_in_cell // 4
y = (vidx_in_cell - 4 * x) // 2
z = vidx_in_cell - 4 * x - 2 * y
return wp.vec3i(x, y, z)
@wp.func
def _edge_coords(type_index: int):
index_in_side = type_index // 4
side_offset = type_index - 4 * index_in_side
return (wp.vec3i(index_in_side + 1, side_offset // 2, side_offset % 2),)
@wp.func
def _edge_axis(node_type: int):
return node_type - CubeSerendipityShapeFunctions.EDGE_X
@wp.func
def _cube_edge_index(node_type: int, type_index: int):
index_in_side = type_index // 4
side_offset = type_index - 4 * index_in_side
return _CUBE_EDGE_INDICES[node_type - CubeSerendipityShapeFunctions.EDGE_X, side_offset], index_in_side
def _get_node_lobatto_indices(self):
ORDER = self.ORDER
@cache.dynamic_func(suffix=self.name)
def node_lobatto_indices(node_type: int, type_index: int):
if node_type == CubeSerendipityShapeFunctions.VERTEX:
return CubeSerendipityShapeFunctions._vertex_coords(type_index) * ORDER
axis = CubeSerendipityShapeFunctions._edge_axis(node_type)
local_coords = CubeSerendipityShapeFunctions._edge_coords(type_index)
local_indices = wp.vec3i(local_coords[0], local_coords[1] * ORDER, local_coords[2] * ORDER)
return Grid3D._local_to_world(axis, local_indices)
return node_lobatto_indices
def make_node_coords_in_element(self):
LOBATTO_COORDS = self.LOBATTO_COORDS
@cache.dynamic_func(suffix=self.name)
def node_coords_in_element(
node_index_in_elt: int,
):
node_type, type_index = self.node_type_and_type_index(node_index_in_elt)
node_coords = self._node_lobatto_indices(node_type, type_index)
return Coords(
LOBATTO_COORDS[node_coords[0]], LOBATTO_COORDS[node_coords[1]], LOBATTO_COORDS[node_coords[2]]
)
return node_coords_in_element
def make_node_quadrature_weight(self):
ORDER = self.ORDER
@cache.dynamic_func(suffix=self.name)
def node_quadrature_weight(
node_index_in_elt: int,
):
node_type, type_index = self.node_type_and_type_index(node_index_in_elt)
if node_type == CubeSerendipityShapeFunctions.VERTEX:
return 1.0 / float(8 * ORDER * ORDER * ORDER)
return (1.0 - 1.0 / float(ORDER * ORDER * ORDER)) / float(12 * (ORDER - 1))
return node_quadrature_weight
def make_trace_node_quadrature_weight(self):
ORDER = self.ORDER
@cache.dynamic_func(suffix=self.name)
def trace_node_quadrature_weight(
node_index_in_elt: int,
):
node_type, type_index = self.node_type_and_type_index(node_index_in_elt)
if node_type == CubeSerendipityShapeFunctions.VERTEX:
return 0.25 / float(ORDER * ORDER)
return (0.25 - 0.25 / float(ORDER * ORDER)) / float(ORDER - 1)
return trace_node_quadrature_weight
def make_element_inner_weight(self):
ORDER = self.ORDER
ORDER_PLUS_ONE = self.ORDER_PLUS_ONE
LOBATTO_COORDS = self.LOBATTO_COORDS
LAGRANGE_SCALE = self.LAGRANGE_SCALE
DEGREE_3_SPHERE_RAD = wp.constant(2 * 0.5**2 + (0.5 - LOBATTO_COORDS[1]) ** 2)
DEGREE_3_SPHERE_SCALE = 1.0 / (0.75 - DEGREE_3_SPHERE_RAD)
@cache.dynamic_func(suffix=self.name)
def element_inner_weight(
coords: Coords,
node_index_in_elt: int,
):
node_type, type_index = self.node_type_and_type_index(node_index_in_elt)
if node_type == CubeSerendipityShapeFunctions.VERTEX:
node_ijk = CubeSerendipityShapeFunctions._vertex_coords(type_index)
cx = wp.select(node_ijk[0] == 0, coords[0], 1.0 - coords[0])
cy = wp.select(node_ijk[1] == 0, coords[1], 1.0 - coords[1])
cz = wp.select(node_ijk[2] == 0, coords[2], 1.0 - coords[2])
w = cx * cy * cz
if ORDER == 2:
w *= cx + cy + cz - 3.0 + LOBATTO_COORDS[1]
return w * LAGRANGE_SCALE[0]
if ORDER == 3:
w *= (
(cx - 0.5) * (cx - 0.5)
+ (cy - 0.5) * (cy - 0.5)
+ (cz - 0.5) * (cz - 0.5)
- DEGREE_3_SPHERE_RAD
)
return w * DEGREE_3_SPHERE_SCALE
axis = CubeSerendipityShapeFunctions._edge_axis(node_type)
node_all = CubeSerendipityShapeFunctions._edge_coords(type_index)
local_coords = Grid3D._world_to_local(axis, coords)
w = float(1.0)
w *= wp.select(node_all[1] == 0, local_coords[1], 1.0 - local_coords[1])
w *= wp.select(node_all[2] == 0, local_coords[2], 1.0 - local_coords[2])
for k in range(ORDER_PLUS_ONE):
if k != node_all[0]:
w *= local_coords[0] - LOBATTO_COORDS[k]
w *= LAGRANGE_SCALE[node_all[0]]
return w
return element_inner_weight
def make_element_inner_weight_gradient(self):
ORDER = self.ORDER
ORDER_PLUS_ONE = self.ORDER_PLUS_ONE
LOBATTO_COORDS = self.LOBATTO_COORDS
LAGRANGE_SCALE = self.LAGRANGE_SCALE
DEGREE_3_SPHERE_RAD = wp.constant(2 * 0.5**2 + (0.5 - LOBATTO_COORDS[1]) ** 2)
DEGREE_3_SPHERE_SCALE = 1.0 / (0.75 - DEGREE_3_SPHERE_RAD)
@cache.dynamic_func(suffix=self.name)
def element_inner_weight_gradient(
coords: Coords,
node_index_in_elt: int,
):
node_type, type_index = self.node_type_and_type_index(node_index_in_elt)
if node_type == CubeSerendipityShapeFunctions.VERTEX:
node_ijk = CubeSerendipityShapeFunctions._vertex_coords(type_index)
cx = wp.select(node_ijk[0] == 0, coords[0], 1.0 - coords[0])
cy = wp.select(node_ijk[1] == 0, coords[1], 1.0 - coords[1])
cz = wp.select(node_ijk[2] == 0, coords[2], 1.0 - coords[2])
gx = wp.select(node_ijk[0] == 0, 1.0, -1.0)
gy = wp.select(node_ijk[1] == 0, 1.0, -1.0)
gz = wp.select(node_ijk[2] == 0, 1.0, -1.0)
if ORDER == 2:
w = cx + cy + cz - 3.0 + LOBATTO_COORDS[1]
grad_x = cy * cz * gx * (w + cx)
grad_y = cz * cx * gy * (w + cy)
grad_z = cx * cy * gz * (w + cz)
return wp.vec3(grad_x, grad_y, grad_z) * LAGRANGE_SCALE[0]
if ORDER == 3:
w = (
(cx - 0.5) * (cx - 0.5)
+ (cy - 0.5) * (cy - 0.5)
+ (cz - 0.5) * (cz - 0.5)
- DEGREE_3_SPHERE_RAD
)
dw_dcx = 2.0 * cx - 1.0
dw_dcy = 2.0 * cy - 1.0
dw_dcz = 2.0 * cz - 1.0
grad_x = cy * cz * gx * (w + dw_dcx * cx)
grad_y = cz * cx * gy * (w + dw_dcy * cy)
grad_z = cx * cy * gz * (w + dw_dcz * cz)
return wp.vec3(grad_x, grad_y, grad_z) * DEGREE_3_SPHERE_SCALE
axis = CubeSerendipityShapeFunctions._edge_axis(node_type)
node_all = CubeSerendipityShapeFunctions._edge_coords(type_index)
local_coords = Grid3D._world_to_local(axis, coords)
w_long = wp.select(node_all[1] == 0, local_coords[1], 1.0 - local_coords[1])
w_lat = wp.select(node_all[2] == 0, local_coords[2], 1.0 - local_coords[2])
g_long = wp.select(node_all[1] == 0, 1.0, -1.0)
g_lat = wp.select(node_all[2] == 0, 1.0, -1.0)
w_alt = LAGRANGE_SCALE[node_all[0]]
g_alt = float(0.0)
prefix_alt = LAGRANGE_SCALE[node_all[0]]
for k in range(ORDER_PLUS_ONE):
if k != node_all[0]:
delta_alt = local_coords[0] - LOBATTO_COORDS[k]
w_alt *= delta_alt
g_alt = g_alt * delta_alt + prefix_alt
prefix_alt *= delta_alt
local_grad = wp.vec3(g_alt * w_long * w_lat, w_alt * g_long * w_lat, w_alt * w_long * g_lat)
return Grid3D._local_to_world(axis, local_grad)
return element_inner_weight_gradient
def element_node_tets(self):
from warp.fem.utils import grid_to_tets
if self.ORDER == 2:
element_tets = np.array(
[
[0, 8, 9, 10],
[1, 11, 10, 15],
[2, 9, 14, 13],
[3, 15, 13, 17],
[4, 12, 8, 16],
[5, 18, 16, 11],
[6, 14, 12, 19],
[7, 19, 18, 17],
[16, 12, 18, 11],
[8, 16, 12, 11],
[12, 19, 18, 14],
[14, 19, 17, 18],
[10, 9, 15, 8],
[10, 8, 11, 15],
[9, 13, 15, 14],
[13, 14, 17, 15],
]
)
middle_hex = np.array([8, 11, 9, 15, 12, 18, 14, 17])
middle_tets = middle_hex[grid_to_tets(1, 1, 1)]
return np.concatenate((element_tets, middle_tets))
raise NotImplementedError()
class CubeNonConformingPolynomialShapeFunctions:
# embeds the largest regular tet centered at (0.5, 0.5, 0.5) into the reference cube
_tet_height = 2.0 / 3.0
_tet_side = math.sqrt(3.0 / 2.0) * _tet_height
_tet_face_height = math.sqrt(3.0) / 2.0 * _tet_side
_tet_to_cube = np.array(
[
[_tet_side, _tet_side / 2.0, _tet_side / 2.0],
[0.0, _tet_face_height, _tet_face_height / 3.0],
[0.0, 0.0, _tet_height],
]
)
_TET_OFFSET = wp.constant(wp.vec3(0.5 - 0.5 * _tet_side, 0.5 - _tet_face_height / 3.0, 0.5 - 0.25 * _tet_height))
def __init__(self, degree: int):
self._tet_shape = TetrahedronPolynomialShapeFunctions(degree=degree)
self.ORDER = self._tet_shape.ORDER
self.NODES_PER_ELEMENT = self._tet_shape.NODES_PER_ELEMENT
self.element_node_tets = self._tet_shape.element_node_tets
@property
def name(self) -> str:
return f"Cube_P{self.ORDER}d"
def make_node_coords_in_element(self):
node_coords_in_tet = self._tet_shape.make_node_coords_in_element()
TET_TO_CUBE = wp.constant(wp.mat33(self._tet_to_cube))
@cache.dynamic_func(suffix=self.name)
def node_coords_in_element(
node_index_in_elt: int,
):
tet_coords = node_coords_in_tet(node_index_in_elt)
return TET_TO_CUBE * tet_coords + CubeNonConformingPolynomialShapeFunctions._TET_OFFSET
return node_coords_in_element
def make_node_quadrature_weight(self):
NODES_PER_ELEMENT = self.NODES_PER_ELEMENT
@cache.dynamic_func(suffix=self.name)
def node_uniform_quadrature_weight(
node_index_in_elt: int,
):
return 1.0 / float(NODES_PER_ELEMENT)
return node_uniform_quadrature_weight
def make_trace_node_quadrature_weight(self):
# Non-conforming, zero measure on sides
@wp.func
def zero(node_index_in_elt: int):
return 0.0
return zero
def make_element_inner_weight(self):
tet_inner_weight = self._tet_shape.make_element_inner_weight()
CUBE_TO_TET = wp.constant(wp.mat33(np.linalg.inv(self._tet_to_cube)))
@cache.dynamic_func(suffix=self.name)
def element_inner_weight(
coords: Coords,
node_index_in_elt: int,
):
tet_coords = CUBE_TO_TET * (coords - CubeNonConformingPolynomialShapeFunctions._TET_OFFSET)
return tet_inner_weight(tet_coords, node_index_in_elt)
return element_inner_weight
def make_element_inner_weight_gradient(self):
tet_inner_weight_gradient = self._tet_shape.make_element_inner_weight_gradient()
CUBE_TO_TET = wp.constant(wp.mat33(np.linalg.inv(self._tet_to_cube)))
@cache.dynamic_func(suffix=self.name)
def element_inner_weight_gradient(
coords: Coords,
node_index_in_elt: int,
):
tet_coords = CUBE_TO_TET * (coords - CubeNonConformingPolynomialShapeFunctions._TET_OFFSET)
grad = tet_inner_weight_gradient(tet_coords, node_index_in_elt)
return wp.transpose(CUBE_TO_TET) * grad
return element_inner_weight_gradient
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