A 2-D plane-stress elasticity eigenproblem

This page treats a plane-stress linear-elasticity eigenproblem with per-face Lamé coefficients. It is the only vector-valued PDE among the examples — useful as evidence that the greedy machinery is not specialised to scalar diffusion.

The model

2D plane-stress linear elasticity on \Omega = [0,1]^2, fully Dirichlet:

-\nabla \cdot \sigma(u) \;=\; \lambda\,\rho\,u \quad \text{in } \Omega, \qquad u = 0 \text{ on } \partial\Omega,

with the linear-elastic stress

\sigma(u) \;=\; \lambda_L(\mu)\,(\nabla\!\cdot u)\,I \;+\; 2\,\mu_L(\mu)\,\varepsilon(u), \quad \varepsilon(u) = \tfrac{1}{2}\bigl(\nabla u + (\nabla u)^\top\bigr).

The Lamé parameters \bigl(\lambda_L,\,\mu_L\bigr) are piecewise-constant on faces; the parameter mapping converts each face’s (E_f, \nu_f) to its Lamé pair (\lambda_L^{(f)}, \mu_L^{(f)}).

Discretisation

Quantity	Value
Element	Vector P1
Default geometry	unit square split into N vertical strips
Affine decomposition	K(\mu) = \sum_f \lambda_L^{(f)}\,K_{\text{lam}}^{(f)} + \mu_L^{(f)}\,K_{\mu}^{(f)}

The stiffness matrix is affine in the per-face Lamé parameters, so each face’s two component matrices are assembled once and recombined per parameter point.

Parameter mappings

Two parameter mappings are available:

Mapping	d_{\text{param}}	Use
5D — (E_1, \nu_1, E_2, \nu_2, E_3) with \nu_3 = 0.1 fixed	5	one Poisson ratio held constant so the box edges line up
6D — all (E_f, \nu_f) free	6	full elasticity sweep with every face’s pair free

This page uses the 5D mapping. The figures here are sized for a fast page render; a deeper 6D exploration is deferred to a follow-up.

A 5-D run

Three vertical strips (\texttt{mesh\_n}=16, three faces, n_{\text{dof}}=450), 5D box

(E_1, \nu_1, E_2, \nu_2, E_3) \in [0.8, 2.4] \times [0.22, 0.34] \times [1.2, 2.4] \times [0.22, 0.34] \times [1.0, 2.0],

with n_{\text{eigs}}=4, two refinement levels, t_\pi=0.35, and t_\lambda=0.05.

Spectrum along E_1

First four eigenvalues over E_1 \in [0.8, 2.4] with the other four parameters fixed at the box midpoint. Smooth, well-separated, monotone in E_1 — what plane-stress elasticity is expected to look like in this regime.

Adaptive grid (5D, projected to E_1 \times E_2)

Adaptive-greedy node set on the 5D box, projected onto the (E_1, E_2) plane. Colour = insertion level. The non-trivial L=1 refinement (vs L=0 termination on the two-parameter diffusion sweep) shows the algorithm exercising bisection on a vector-valued elasticity problem, not just on the scalar diffusion family the other pages cover.

--- title: "A 2-D plane-stress elasticity eigenproblem" --- This page treats a plane-stress linear-elasticity eigenproblem with per-face Lamé coefficients. It is the only vector-valued PDE among the examples — useful as evidence that the greedy machinery is not specialised to scalar diffusion. ## The model 2D plane-stress linear elasticity on $\Omega = [0,1]^2$, fully Dirichlet: $$ -\nabla \cdot \sigma(u) \;=\; \lambda\,\rho\,u \quad \text{in } \Omega, \qquad u = 0 \text{ on } \partial\Omega, $$ with the linear-elastic stress $$ \sigma(u) \;=\; \lambda_L(\mu)\,(\nabla\!\cdot u)\,I \;+\; 2\,\mu_L(\mu)\,\varepsilon(u), \quad \varepsilon(u) = \tfrac{1}{2}\bigl(\nabla u + (\nabla u)^\top\bigr). $$ The Lamé parameters $\bigl(\lambda_L,\,\mu_L\bigr)$ are piecewise-constant on faces; the parameter mapping converts each face's $(E_f, \nu_f)$ to its Lamé pair $(\lambda_L^{(f)}, \mu_L^{(f)})$. ## Discretisation | Quantity | Value | |---|---| | Element | Vector P1 | | Default geometry | unit square split into $N$ vertical strips | | Affine decomposition | $K(\mu) = \sum_f \lambda_L^{(f)}\,K_{\text{lam}}^{(f)} + \mu_L^{(f)}\,K_{\mu}^{(f)}$ | The stiffness matrix is affine in the per-face Lamé parameters, so each face's two component matrices are assembled once and recombined per parameter point. ## Parameter mappings Two parameter mappings are available: | Mapping | $d_{\text{param}}$ | Use | |---|---|---| | 5D — $(E_1, \nu_1, E_2, \nu_2, E_3)$ with $\nu_3 = 0.1$ fixed | 5 | one Poisson ratio held constant so the box edges line up | | 6D — all $(E_f, \nu_f)$ free | 6 | full elasticity sweep with every face's pair free | This page uses the **5D mapping**. The figures here are sized for a fast page render; a deeper 6D exploration is deferred to a follow-up. ## A 5-D run Three vertical strips ($\texttt{mesh\_n}=16$, three faces, $n_{\text{dof}}=450$), 5D box $$ (E_1, \nu_1, E_2, \nu_2, E_3) \in [0.8, 2.4] \times [0.22, 0.34] \times [1.2, 2.4] \times [0.22, 0.34] \times [1.0, 2.0], $$ with $n_{\text{eigs}}=4$, two refinement levels, $t_\pi=0.35$, and $t_\lambda=0.05$. ### Spectrum along $E_1$ ![First four eigenvalues over $E_1 \in [0.8, 2.4]$ with the other four parameters fixed at the box midpoint. Smooth, well-separated, monotone in $E_1$ — what plane-stress elasticity is expected to look like in this regime.](../figures/elasticity-2d/spectrum.png) ### Adaptive grid (5D, projected to $E_1 \times E_2$) ![Adaptive-greedy node set on the 5D box, projected onto the $(E_1, E_2)$ plane. Colour = insertion level. The non-trivial $L=1$ refinement (vs $L=0$ termination on the [two-parameter diffusion sweep](diffusion-3-3-3.qmd)) shows the algorithm exercising bisection on a vector-valued elasticity problem, not just on the scalar diffusion family the other pages cover.](../figures/elasticity-2d/greedy.png)