Git reference: Benchmark 03-linear-elasticity.
This problem is a coupled system of two equations with a mixed derivative in the coupling term (Lame equations); the context of the problem comes from the subject of linear elasticity.
Equation solved: Coupled system of two equations
-E \frac{1-\nu^2}{1-2\nu} \frac{\partial^{2} u}{\partial x^{2}} - E\frac{1-\nu^2}{2-2\nu} \frac{\partial^{2} u}{\partial y^{2}} -E \frac{1-\nu^2}{(1-2\nu)(2-2\nu)} \frac{\partial^{2} v}{\partial x \partial y} - F_{x} = 0.
-E \frac{1-\nu^2}{2-2\nu} \frac{\partial^{2} v}{\partial x^{2}} - E\frac{1-\nu^2}{1-2\nu} \frac{\partial^{2} v}{\partial y^{2}} -E \frac{1-\nu^2}{(1-2\nu)(2-2\nu)} \frac{\partial^{2} u}{\partial x \partial y} - F_{y} = 0.
where F_{x} = F_{y} = 0, u and v are the x and y displacements, E is Young’s Modulus, and \nu is Poisson’s ratio.
Domain of interest: (-1, 1)^2 with a slit from (0, 0) to (1, 0).
Boundary conditions: Dirichlet, given by exact solution.
Known exact solution for mode 1:
u(x, y) = \frac{1}{2G} r^{\lambda}[(k - Q(\lambda + 1))cos(\lambda \theta) - \lambda cos((\lambda - 2) \theta)].
v(x, y) = \frac{1}{2G} r^{\lambda}[(k + Q(\lambda + 1))sin(\lambda \theta) + \lambda sin((\lambda - 2) \theta)].
here lambda = 0.5444837367825, and Q = 0.5430755788367.
Known exact solution for mode 2:
u(x, y) = \frac{1}{2G} r^{\lambda}[(k - Q(\lambda + 1))sin(\lambda \theta) - \lambda sin((\lambda - 2) \theta)].
v(x, y) = -\frac{1}{2G} r^{\lambda}[(k + Q(\lambda + 1))cos(\lambda \theta) + \lambda cos((\lambda - 2) \theta)].
here lambda = 0.9085291898461, and Q = -0.2189232362488. Both in mode 1 and mode 2, k = 3 - 4 \nu, and G = E / (2(1 + \nu)).
Final mesh (h-FEM, p=1, anisotropic refinements):
Final mesh (h-FEM, p=2, anisotropic refinements):
Final mesh (hp-FEM, h-anisotropic refinements):
DOF convergence graphs:
CPU convergence graphs:
Final mesh (hp-FEM, h-anisotropic refinements):
Final mesh (hp-FEM, hp-anisotropic refinements):
DOF convergence graphs:
CPU convergence graphs: