In [4]:
plt.figure(figsize=(7,7))
plt.gca().set_aspect("equal")
plt.triplot(X, Y, E)
Out[4]:
[<matplotlib.lines.Line2D at 0x7f17d53aae00>, <matplotlib.lines.Line2D at 0x7f17d53aaf20>]
Copyright (C) 2010-2020 Luke Olson
Copyright (C) 2020 Andreas Kloeckner
In this note, we look at constructing a finite element approximation to $$ \begin{align*} {}- \nabla\cdot \kappa(x,y) \nabla &u = f(x,y)\qquad((x,y)\in\Omega),\\ u &= g(x,y)\qquad ((x,y)\in \partial \Omega). \end{align*} $$ We define $\kappa$, $f$, and $g$ in a bit.
import numpy as np
import scipy.linalg as la
import scipy.sparse as sparse
import scipy.sparse.linalg as sla
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
import meshpy.triangle as triangle
def round_trip_connect(start, end):
return [(i, i+1) for i in range(start, end)] + [(end, start)]
def make_mesh():
points = [(-1, -1), (1, -1), (1, 1), (-1, 1)]
facets = round_trip_connect(0, len(points)-1)
circ_start = len(points)
points.extend(
(0.25 * np.cos(angle), 0.25 * np.sin(angle))
for angle in np.linspace(0, 2*np.pi, 30, endpoint=False))
facets.extend(round_trip_connect(circ_start, len(points)-1))
def needs_refinement(vertices, area):
bary = np.sum(np.array(vertices), axis=0)/3
max_area = 0.01 + la.norm(bary, np.inf)*0.01
return bool(area > max_area)
info = triangle.MeshInfo()
info.set_points(points)
info.set_facets(facets)
built_mesh = triangle.build(info, refinement_func=needs_refinement)
return np.array(built_mesh.points), np.array(built_mesh.elements)
V, E = make_mesh()
nv = len(V)
ne = len(E)
print(V.shape)
print(E.shape)
print(E.max())
X, Y = V[:, 0], V[:, 1]
(237, 2) (438, 3) 236
plt.figure(figsize=(7,7))
plt.gca().set_aspect("equal")
plt.triplot(X, Y, E)
[<matplotlib.lines.Line2D at 0x7f17d53aae00>, <matplotlib.lines.Line2D at 0x7f17d53aaf20>]
Map the reference triangle to the triangle given by these vertices:
v1 = np.array([1.0, 1.0])
v2 = np.array([3.0, 1.0])
v3 = np.array([2.0, 2.0])
Come up with the matrix TA and vector Tb of the affine mapping.
TA = np.array([v2-v1, v3-v1]).T
Tb = v1
Test the mapping.
# make random points in the reference triangle
r = np.random.rand(1000, 2)
r = r[r[:, 0]+r[:, 1] < 1]
x = np.einsum("ij,pj->pi", TA, r) + Tb
plt.plot(x[:, 0], x[:, 1], "o")
plt.plot(r[:, 0], r[:, 1], "o")
plt.plot(v1[0], v1[1], "o", label="v1")
plt.plot(v2[0], v2[1], "o", label="v2")
plt.plot(v3[0], v3[1], "o", label="v3")
plt.legend()
<matplotlib.legend.Legend at 0x7f17d316c0d0>
Define $\kappa$, $f$, and $g$.
def kappa(xvec):
x, y = xvec
if (x**2 + y**2)**0.5 <= 0.25:
return 25.0
else:
return 1.0
def f(xvec):
x, y = xvec
if (x**2 + y**2)**0.5 <= 0.25:
return 100.0
else:
return 0.0
def g(xvec):
x, y = xvec
return 1 * (1 - x**2)
class MatrixBuilder:
def __init__(self):
self.rows = []
self.cols = []
self.vals = []
def add(self, rows, cols, submat):
for i, ri in enumerate(rows):
for j, cj in enumerate(cols):
self.rows.append(ri)
self.cols.append(cj)
self.vals.append(submat[i, j])
def coo_matrix(self):
return sparse.coo_matrix((self.vals, (self.rows, self.cols)))
Recall the nodal linear basis:
Create a $2\times N_p$ array containing $\nabla_{\boldsymbol r} \varphi_i$.
dbasis = np.array([
[-1, 1, 0],
[-1, 0, 1]])
Assemble the matrix. Use a MatrixBuilder a_builder. Recall (from the notes):
$$
\let\b=\boldsymbol
\int_{E} \kappa(\b{x}) \nabla \varphi_i ( \b{x} )^T \nabla \varphi_j ( \b{x} ) d\b{x}
= ( J_T^{-T} \nabla_{\b r} \varphi_i )^T ( J_T^{-T} \nabla_{\b r} \varphi_j ) | J_T | \int_{\hat E} \kappa( T( \b{r} ) ) d\b{r}
$$
Using a 1-point Gauss Quadrature rule: $\int_{\hat E} f \approx \frac 12 f(\bar{\boldsymbol x})$, where $\bar{\boldsymbol x}$ is the element centroid.
a_builder = MatrixBuilder()
for ei in range(0, ne):
vert_indices = E[ei, :]
x0, x1, x2 = el_verts = V[vert_indices]
centroid = np.mean(el_verts, axis=0)
J = np.array([x1-x0, x2-x0]).T
invJT = la.inv(J.T)
detJ = la.det(J)
dphi = invJT @ dbasis
Aelem = kappa(centroid) * (detJ / 2.0) * dphi.T @ dphi
a_builder.add(vert_indices, vert_indices, Aelem)
Eliminate duplicate entries in the COO-form sparse matrix:
A = a_builder.coo_matrix().tocsr().tocoo()
Compute the right-hand side b using a 1-point Gauss Quadrature rule:
$$
\int_{E_i} f(\boldsymbol x) \phi_i\,d\boldsymbol x
= |J| \int_{E} f(T_i(\boldsymbol r)) \phi_i(\alpha)\,d\boldsymbol r\\
\approx \frac 12 |J| f(\bar{\boldsymbol x}) \phi_i(\boldsymbol r)
= \frac 16 |J| f(\bar{\boldsymbol x}),
$$
where $\bar{\boldsymbol x}$ is the element centroid.
b = np.zeros(nv)
for ei in range(0, ne):
vert_indices = E[ei, :]
x0, x1, x2 = el_verts = V[vert_indices]
centroid = np.mean(el_verts, axis=0)
J = np.array([x1-x0, x2-x0]).T
detJ = la.det(J)
belem = f(centroid) * (detJ / 6.0) * np.ones((3,))
for i, vi in enumerate(vert_indices):
b[vi] += belem[i]
Create flags for the boundary vertices/DoFs:
tol = 1e-12
is_boundary = (
(np.abs(X+1) < tol)
| (np.abs(X-1) < tol)
| (np.abs(Y+1) < tol)
| (np.abs(Y-1) < tol))
is_g_boundary = np.abs(Y+1) < tol
Next, construct the 'volume-lifted' boundary condition $u^0$.
u0 = np.zeros(nv)
u0[is_g_boundary] = g(V[is_g_boundary].T)
Compute the "post-lifting" right hand side rhs.
Note: The Riesz representer of rhs needs to be in $H^1_0$. (I.e. what should its values for the boundary DoFs be?)
rhs = b - A @ u0
rhs[is_boundary] = 0.0
Next, set the rows corresponding to boundary DoFs to be identity rows:
for k in range(A.nnz):
i = A.row[k]
j = A.col[k]
if is_boundary[i]:
A.data[k] = 1 if i == j else 0
Now solve and correct for lifting:
uhat = sla.spsolve(A.tocsr(), rhs)
u = uhat + u0
And plot:
fig = plt.figure(figsize=(8,8))
ax = plt.gca(projection='3d')
ax.plot_trisurf(X, Y, u, triangles=E, cmap=plt.cm.jet, linewidth=0.2)
plt.show()
/tmp/ipykernel_39743/2828084510.py:2: MatplotlibDeprecationWarning: Calling gca() with keyword arguments was deprecated in Matplotlib 3.4. Starting two minor releases later, gca() will take no keyword arguments. The gca() function should only be used to get the current axes, or if no axes exist, create new axes with default keyword arguments. To create a new axes with non-default arguments, use plt.axes() or plt.subplot(). ax = plt.gca(projection='3d')
