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import warp as wp
import numpy as np
from warp.fem.polynomial import Polynomial, quadrature_1d, lagrange_scales, is_closed
from warp.fem.types import Coords
from warp.fem import cache
from .triangle_shape_function import Triangle2DPolynomialShapeFunctions
class SquareBipolynomialShapeFunctions:
def __init__(self, degree: int, family: Polynomial):
self.family = family
self.ORDER = wp.constant(degree)
self.NODES_PER_ELEMENT = wp.constant((degree + 1) * (degree + 1))
self.NODES_PER_SIDE = 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)
@property
def name(self) -> str:
return f"Square_Q{self.ORDER}_{self.family}"
def make_node_coords_in_element(self):
ORDER = self.ORDER
LOBATTO_COORDS = self.LOBATTO_COORDS
@cache.dynamic_func(suffix=self.name)
def node_coords_in_element(
node_index_in_elt: int,
):
node_i = node_index_in_elt // (ORDER + 1)
node_j = node_index_in_elt - (ORDER + 1) * node_i
return Coords(LOBATTO_COORDS[node_i], LOBATTO_COORDS[node_j], 0.0)
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_index_in_elt // (ORDER + 1)
node_j = node_index_in_elt - (ORDER + 1) * node_i
return LOBATTO_WEIGHT[node_i] * LOBATTO_WEIGHT[node_j]
def node_quadrature_weight_linear(
node_index_in_elt: int,
):
return 0.25
if ORDER == 1:
return cache.get_func(node_quadrature_weight_linear, self.name)
return cache.get_func(node_quadrature_weight, self.name)
@wp.func
def _vertex_coords_f(vidx_in_cell: int):
x = vidx_in_cell // 2
y = vidx_in_cell - 2 * x
return wp.vec2(float(x), float(y))
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
# I.e., either both indices are at extrema, or only one is
# Pick the interior one if possible, if both are at extrema pick any one
node_i = node_index_in_elt // (ORDER + 1)
if node_i > 0 and node_i < ORDER:
return LOBATTO_WEIGHT[node_i]
node_j = node_index_in_elt - (ORDER + 1) * node_i
return LOBATTO_WEIGHT[node_j]
def trace_node_quadrature_weight_linear(
node_index_in_elt: int,
):
return 0.5
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_index_in_elt // ORDER_PLUS_ONE
node_j = node_index_in_elt - ORDER_PLUS_ONE * node_i
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]
w *= LAGRANGE_SCALE[node_i] * LAGRANGE_SCALE[node_j]
return w
def element_inner_weight_linear(
coords: Coords,
node_index_in_elt: int,
):
v = SquareBipolynomialShapeFunctions._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]
return wx * wy
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_index_in_elt // ORDER_PLUS_ONE
node_j = node_index_in_elt - ORDER_PLUS_ONE * node_i
prefix_x = float(1.0)
prefix_y = float(1.0)
for k in range(ORDER_PLUS_ONE):
if k != node_i:
prefix_y *= coords[0] - LOBATTO_COORDS[k]
if k != node_j:
prefix_x *= coords[1] - LOBATTO_COORDS[k]
grad_x = float(0.0)
grad_y = 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
grad = LAGRANGE_SCALE[node_i] * LAGRANGE_SCALE[node_j] * wp.vec2(grad_x, grad_y)
return grad
def element_inner_weight_gradient_linear(
coords: Coords,
node_index_in_elt: int,
):
v = SquareBipolynomialShapeFunctions._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]
dx = 2.0 * v[0] - 1.0
dy = 2.0 * v[1] - 1.0
return wp.vec2(dx * wy, dy * wx)
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_triangulation(self):
from warp.fem.utils import grid_to_tris
return grid_to_tris(self.ORDER, self.ORDER)
class SquareSerendipityShapeFunctions:
"""
Serendipity element ~ tensor product space without interior nodes
Side shape functions are usual Lagrange shape functions times a linear function in the normal direction
Corner shape functions are bilinear 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(4 * degree)
self.NODES_PER_SIDE = 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"Square_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 < 4:
return SquareSerendipityShapeFunctions.VERTEX, node_index_in_elt
type_index = (node_index_in_elt - 4) // 2
side = node_index_in_elt - 4 - 2 * type_index
return SquareSerendipityShapeFunctions.EDGE_X + side, type_index
return node_type_and_index
@wp.func
def side_offset_and_index(type_index: int):
index_in_side = type_index // 2
side_offset = type_index - 2 * index_in_side
return 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 == SquareSerendipityShapeFunctions.VERTEX:
node_i = type_index // 2
node_j = type_index - 2 * node_i
return node_i * ORDER, node_j * ORDER
side_offset, index_in_side = SquareSerendipityShapeFunctions.side_offset_and_index(type_index)
if node_type == SquareSerendipityShapeFunctions.EDGE_X:
node_i = 1 + index_in_side
node_j = side_offset * ORDER
else:
node_j = 1 + index_in_side
node_i = side_offset * ORDER
return node_i, node_j
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_i, node_j = self._node_lobatto_indices(node_type, type_index)
return Coords(LOBATTO_COORDS[node_i], LOBATTO_COORDS[node_j], 0.0)
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 == SquareSerendipityShapeFunctions.VERTEX:
return 0.25 / float(ORDER * ORDER)
return (0.25 - 0.25 / float(ORDER * ORDER)) / float(ORDER - 1)
return node_quadrature_weight
def make_trace_node_quadrature_weight(self):
LOBATTO_WEIGHT = self.LOBATTO_WEIGHT
@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 == SquareSerendipityShapeFunctions.VERTEX:
return LOBATTO_WEIGHT[0]
side_offset, index_in_side = SquareSerendipityShapeFunctions.side_offset_and_index(type_index)
return LOBATTO_WEIGHT[1 + index_in_side]
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_CIRCLE_RAD = wp.constant(0.5**2 + (0.5 - LOBATTO_COORDS[1]) ** 2)
DEGREE_3_CIRCLE_SCALE = 1.0 / (0.5 - DEGREE_3_CIRCLE_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)
node_i, node_j = self._node_lobatto_indices(node_type, type_index)
if node_type == SquareSerendipityShapeFunctions.VERTEX:
cx = wp.select(node_i == 0, coords[0], 1.0 - coords[0])
cy = wp.select(node_j == 0, coords[1], 1.0 - coords[1])
w = cx * cy
if ORDER == 2:
w *= cx + cy - 2.0 + LOBATTO_COORDS[1]
return w * LAGRANGE_SCALE[0]
if ORDER == 3:
w *= (cx - 0.5) * (cx - 0.5) + (cy - 0.5) * (cy - 0.5) - DEGREE_3_CIRCLE_RAD
return w * DEGREE_3_CIRCLE_SCALE
w = float(1.0)
if node_type == SquareSerendipityShapeFunctions.EDGE_Y:
w *= wp.select(node_i == 0, coords[0], 1.0 - coords[0])
else:
for k in range(ORDER_PLUS_ONE):
if k != node_i:
w *= coords[0] - LOBATTO_COORDS[k]
w *= LAGRANGE_SCALE[node_i]
if node_type == SquareSerendipityShapeFunctions.EDGE_X:
w *= wp.select(node_j == 0, coords[1], 1.0 - coords[1])
else:
for k in range(ORDER_PLUS_ONE):
if k != node_j:
w *= coords[1] - LOBATTO_COORDS[k]
w *= LAGRANGE_SCALE[node_j]
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_CIRCLE_RAD = wp.constant(0.5**2 + (0.5 - LOBATTO_COORDS[1]) ** 2)
DEGREE_3_CIRCLE_SCALE = 1.0 / (0.5 - DEGREE_3_CIRCLE_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)
node_i, node_j = self._node_lobatto_indices(node_type, type_index)
if node_type == SquareSerendipityShapeFunctions.VERTEX:
cx = wp.select(node_i == 0, coords[0], 1.0 - coords[0])
cy = wp.select(node_j == 0, coords[1], 1.0 - coords[1])
gx = wp.select(node_i == 0, 1.0, -1.0)
gy = wp.select(node_j == 0, 1.0, -1.0)
if ORDER == 2:
w = cx + cy - 2.0 + LOBATTO_COORDS[1]
grad_x = cy * gx * (w + cx)
grad_y = cx * gy * (w + cy)
return wp.vec2(grad_x, grad_y) * LAGRANGE_SCALE[0]
if ORDER == 3:
w = (cx - 0.5) * (cx - 0.5) + (cy - 0.5) * (cy - 0.5) - DEGREE_3_CIRCLE_RAD
dw_dcx = 2.0 * cx - 1.0
dw_dcy = 2.0 * cy - 1.0
grad_x = cy * gx * (w + cx * dw_dcx)
grad_y = cx * gy * (w + cy * dw_dcy)
return wp.vec2(grad_x, grad_y) * DEGREE_3_CIRCLE_SCALE
if node_type == SquareSerendipityShapeFunctions.EDGE_X:
prefix_x = wp.select(node_j == 0, coords[1], 1.0 - coords[1])
else:
prefix_x = LAGRANGE_SCALE[node_j]
for k in range(ORDER_PLUS_ONE):
if k != node_j:
prefix_x *= coords[1] - LOBATTO_COORDS[k]
if node_type == SquareSerendipityShapeFunctions.EDGE_Y:
prefix_y = wp.select(node_i == 0, coords[0], 1.0 - coords[0])
else:
prefix_y = LAGRANGE_SCALE[node_i]
for k in range(ORDER_PLUS_ONE):
if k != node_i:
prefix_y *= coords[0] - LOBATTO_COORDS[k]
if node_type == SquareSerendipityShapeFunctions.EDGE_X:
grad_y = wp.select(node_j == 0, 1.0, -1.0) * prefix_y
else:
prefix_y *= LAGRANGE_SCALE[node_j]
grad_y = float(0.0)
for k in range(ORDER_PLUS_ONE):
if k != node_j:
delta_y = coords[1] - LOBATTO_COORDS[k]
grad_y = grad_y * delta_y + prefix_y
prefix_y *= delta_y
if node_type == SquareSerendipityShapeFunctions.EDGE_Y:
grad_x = wp.select(node_i == 0, 1.0, -1.0) * prefix_x
else:
prefix_x *= LAGRANGE_SCALE[node_i]
grad_x = 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
grad = wp.vec2(grad_x, grad_y)
return grad
return element_inner_weight_gradient
def element_node_triangulation(self):
if self.ORDER == 2:
element_triangles = [
[0, 4, 5],
[5, 4, 6],
[5, 6, 1],
[4, 2, 7],
[4, 7, 6],
[6, 7, 3],
]
else:
element_triangles = [
[0, 4, 5],
[2, 7, 8],
[3, 10, 11],
[1, 9, 6],
[5, 6, 9],
[5, 4, 6],
[8, 11, 10],
[8, 7, 11],
[4, 8, 10],
[4, 10, 6],
]
return element_triangles
class SquareNonConformingPolynomialShapeFunctions:
# embeds the largest equilateral triangle centered at (0.5, 0.5) into the reference square
_tri_height = 0.75
_tri_side = 2.0 / math.sqrt(3.0) * _tri_height
_tri_to_square = np.array([[_tri_side, _tri_side / 2.0], [0.0, _tri_height]])
_TRI_OFFSET = wp.constant(wp.vec2(0.5 - 0.5 * _tri_side, 0.5 - _tri_height / 3.0))
def __init__(self, degree: int):
self._tri_shape = Triangle2DPolynomialShapeFunctions(degree=degree)
self.ORDER = self._tri_shape.ORDER
self.NODES_PER_ELEMENT = self._tri_shape.NODES_PER_ELEMENT
self.element_node_triangulation = self._tri_shape.element_node_triangulation
@property
def name(self) -> str:
return f"Square_P{self.ORDER}d"
def make_node_coords_in_element(self):
node_coords_in_tet = self._tri_shape.make_node_coords_in_element()
TRI_TO_SQUARE = wp.constant(wp.mat22(self._tri_to_square))
@cache.dynamic_func(suffix=self.name)
def node_coords_in_element(
node_index_in_elt: int,
):
tri_coords = node_coords_in_tet(node_index_in_elt)
coords = (
TRI_TO_SQUARE * wp.vec2(tri_coords[1], tri_coords[2])
) + SquareNonConformingPolynomialShapeFunctions._TRI_OFFSET
return Coords(coords[0], coords[1], 0.0)
return node_coords_in_element
def make_node_quadrature_weight(self):
NODES_PER_ELEMENT = self.NODES_PER_ELEMENT
if self.ORDER == 2:
# Intrinsic quadrature (order 2)
@cache.dynamic_func(suffix=self.name)
def node_quadrature_weight_quadratic(
node_index_in_elt: int,
):
node_type, type_index = self._tri_shape.node_type_and_type_index(node_index_in_elt)
if node_type == Triangle2DPolynomialShapeFunctions.VERTEX:
return 0.18518521
return 0.14814811
return node_quadrature_weight_quadratic
@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):
tri_inner_weight = self._tri_shape.make_element_inner_weight()
SQUARE_TO_TRI = wp.constant(wp.mat22(np.linalg.inv(self._tri_to_square)))
@cache.dynamic_func(suffix=self.name)
def element_inner_weight(
coords: Coords,
node_index_in_elt: int,
):
tri_param = SQUARE_TO_TRI * (
wp.vec2(coords[0], coords[1]) - SquareNonConformingPolynomialShapeFunctions._TRI_OFFSET
)
tri_coords = Coords(1.0 - tri_param[0] - tri_param[1], tri_param[0], tri_param[1])
return tri_inner_weight(tri_coords, node_index_in_elt)
return element_inner_weight
def make_element_inner_weight_gradient(self):
tri_inner_weight_gradient = self._tri_shape.make_element_inner_weight_gradient()
SQUARE_TO_TRI = wp.constant(wp.mat22(np.linalg.inv(self._tri_to_square)))
@cache.dynamic_func(suffix=self.name)
def element_inner_weight_gradient(
coords: Coords,
node_index_in_elt: int,
):
tri_param = SQUARE_TO_TRI * (
wp.vec2(coords[0], coords[1]) - SquareNonConformingPolynomialShapeFunctions._TRI_OFFSET
)
tri_coords = Coords(1.0 - tri_param[0] - tri_param[1], tri_param[0], tri_param[1])
grad = tri_inner_weight_gradient(tri_coords, node_index_in_elt)
return wp.transpose(SQUARE_TO_TRI) * grad
return element_inner_weight_gradient
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