Developing FEM in 2D¶
Copyright (C) 2010-2020 Luke Olson
Copyright (C) 2020 Andreas Kloeckner
MIT License
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.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]
plt.figure(figsize=(7,7))
plt.gca().set_aspect("equal")
plt.triplot(X, Y, E)
Explore Connectivity¶
Compute the vertex-to-edge connections as V2E.
print('V shape: ', V.shape)
print('E shape: ', E.shape)
element_ids = np.empty((ne, 3), dtype=np.intp)
element_ids[:] = np.arange(ne).reshape(-1, 1)
V2E = sparse.coo_matrix(
(np.ones((ne*3,), dtype=np.intp),
(E.ravel(),
element_ids.ravel(),)))
print('V2E shape: ', V2E.shape)
Compute
- the element-to-element connections
E2E, and - the vertex-to-vertex connections
V2V.
E2E = V2E.T @ V2E
V2V = V2E @ V2E.T
print('V2V shape: ', V2V.shape)
print('E2E shape: ', E2E.shape)
Plot the vertex degrees.
plt.scatter(X, Y, c=V2V.diagonal(), clip_on=False)
plt.colorbar()
plt.show()
Explain this:
E2E.diagonal()
Plot the number of neighbors of each element.
E2E.data[:] = 1
num_neighbors = np.array(E2E.sum(axis=0)).ravel()
plt.tripcolor(X, Y, triangles=E, facecolors=num_neighbors)
plt.colorbar()
plt.show()
Constructing Element Mappings¶
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()
Problem Data¶
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)
Assembly Helper¶
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)))
Assembly¶
Recall the nodal linear basis:
- $\varphi_1(r,s) =1-r-s$
- $\varphi_2(r,s) =r$
- $\varphi_3(r,s) =s$
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]
Boundary Conditions¶
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
Solve and Plot¶
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()
