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December 8, 2023 23:47
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1D Finite Element Analysis in 100 Lines of Python Code
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import numpy as np | |
from types import SimpleNamespace as SN | |
from scipy.sparse import csr_array as sparray | |
from scipy.sparse.linalg import spsolve | |
import matplotlib.pyplot as plt | |
def Mesh(npoints): | |
points = np.linspace(0, 1, npoints) | |
cells = np.arange(len(points)).repeat(2)[1:-1].reshape(-1, 2) | |
return SN(points=points, cells=cells) | |
def Element(points): | |
function = np.array([1 - points, 1 + points]) / 2 | |
gradient = np.array([-np.ones_like(points), np.ones_like(points)]) / 2 | |
return SN(function=function, gradient=gradient) | |
def Region(mesh, Element, quadrature): | |
element = Element(quadrature.points) | |
gradient = mesh.points[mesh.cells] @ element.gradient | |
return SN( | |
mesh=mesh, | |
element=element, | |
gradient=element.gradient[..., None, :] / gradient, | |
dx=quadrature.weights * gradient, | |
) | |
def Field(region, values): | |
return SN(values=values, region=region) | |
def SolidBody(field, E, A): | |
def gradient(field): | |
region = field.region | |
return (field.values[region.mesh.cells] * region.gradient.T).sum(-1).T | |
stretch = 1 + gradient(field) | |
return SN( | |
force=E * A * (stretch - 1 / stretch**2), | |
stiffness=E * A * (1 + 2 / stretch**3), | |
field=field, | |
) | |
def Assemble(solid): | |
dx = solid.field.region.dx | |
gradv = solid.field.region.gradient[:, None] | |
gradu = solid.field.region.gradient[None, :] | |
vector = gradv * solid.force * dx | |
matrix = gradv * solid.stiffness * gradu * dx | |
vidx = (mesh.cells.T.ravel(), np.zeros(mesh.cells.size, dtype=int)) | |
midx = ( | |
np.tile(mesh.cells.T.ravel(), len(mesh.cells.T)), | |
np.repeat(mesh.cells.T.ravel(), len(mesh.cells.T)), | |
) | |
return SN( | |
vector=sparray((vector.sum(axis=-1).ravel(), vidx)).toarray().ravel(), | |
matrix=sparray((matrix.sum(axis=-1).ravel(), midx)), | |
) | |
mesh = Mesh(npoints=10001) | |
quadrature = SN(points=np.array([-1, 1]) / np.sqrt(3), weights=np.ones([1, 1])) | |
region = Region(mesh, Element, quadrature) | |
field = Field(region, values=np.zeros_like(mesh.points)) | |
dof = SN(fixed=np.arange(1), active=np.arange(len(mesh.points))[1:]) | |
extforce = np.append(np.zeros(len(mesh.points) - 1), 1) | |
for iteration in range(8): | |
system = Assemble(SolidBody(field, E=1, A=1)) | |
A = system.matrix[dof.active, :][:, dof.active] | |
b = extforce[dof.active] - system.vector[dof.active] | |
norm = np.linalg.norm(b) | |
print(f"Iteration {iteration + 1} | norm(force)={norm:1.2e}") | |
if norm < 1e-3: | |
break | |
x = spsolve(A, b) | |
field.values[dof.active] += x | |
plt.plot(field.values + mesh.points, np.zeros_like(mesh.points), "C0o-") |
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