Lecture 5: FEM, cont'd

Linear Elasticity

• Consider a domain $\Omega$ as shown here: We assume it is a piece of elastic material (similar to a beam) of this shape. Fixed at the left end and the force $P$ is applied to its right end.

• Boundary:

  • leftmost: $\Gamma_1$
  • rightmost: $\Gamma_3$
  • rest: $\Gamma_2$

• Displacement: $(u,v)$ (in the $x$ and $y$ coord)

• Elasticity tensors: $$ \begin{align*} C_{11} &= \begin{pmatrix} \frac{E (1-\nu)}{(1-2\nu)(1+\nu)} & 0 \\ 0 & \frac{E}{2(1+\nu)} \end{pmatrix}, &\qquad C_{12} &= \begin{pmatrix} 0 & \frac{E \nu}{(1-2\nu)(1+\nu)} \\ \frac{E}{2(1+\nu)} & 0 \end{pmatrix}, \\ C_{21} &= C_{12}^{\rm T}, &\qquad C_{22} &= \begin{pmatrix} \frac{E}{2(1+\nu)} & 0 \\ 0 & \frac{E (1-\nu)}{(1-2\nu)(1+\nu)} \end{pmatrix} \end{align*} $$ Here $E$ is called the modulus of elasticity and $\nu$ is called the Poisson ratio. (In fact, $C$ is a four-dimensional tensor, denoted by $C$, most likely, after Cauchy.)

• Equations: $$ \begin{align*} -{\rm div} (C_{11} \nabla u + C_{12} \nabla v) &= 0 \qquad\text{in }\Omega \\ -{\rm div} (C_{21} \nabla u + C_{22} \nabla v) &= 0 \qquad\text{in }\Omega \\ u = v &= 0 \qquad\text{on }\Gamma_1 \\ C_{11} u_n + C_{12} v_n = C_{21} u_n + C_{22} v_n &= 0 \qquad\text{on }\Gamma_2 \\ C_{11} u_n + C_{12} v_n &= 0 \qquad\text{on }\Gamma_3 \\ C_{21} u_n + C_{22} v_n &= \frac{P}{H W} \qquad\text{on }\Gamma_3, \end{align*} $$ where $u_n$ and $v_n$ are normal derivatives of the solution.

• Variational formulation:

  • test functions $p=p(x,y)$ and $q = q(x,y)$
  • Dirichet BCs: $p=q=0$ on $\Gamma_1$
  • Variational form: $$ \int_{\Omega} \big[ (C_{11} \nabla u + C_{12} \nabla v)\cdot \nabla p + (C_{21} \nabla u + C_{22} \nabla v)\cdot \nabla q \big] = \frac{P}{H W} \int_{\Gamma_3} q \qquad\forall p, q $$

• A good way to go about this problem is to think that your space consists of vector-valued functions $(u,v)\in\mathbb R^2$. Thus, at each node you have two basis functions: such that $(u,v) = (1,0)$ for that node, and $(u,v) = (0,1)$ for that node.

More advanced stuff

Error analysis

• If $u$ is the exact solution and $u_h$ is the approximate solution then one can prove that $$ \|\nabla u_h - \nabla u\|_{L^2} \leq C h \|\nabla^2 u\|_{L^2}, $$ where $h$ is the maximal size of the triangle and $\nabla^2 u$ is the Hessian matrix.

• Comes from a truly beautiful argument:

  • One can show that $u_h$ is the minimizer of $$ \int \nabla (v_h-u) \cdot \nabla (v_h-u) = \|\nabla v_h - \nabla u\|_{L^2}^2 $$ over all $v_h\in \mathcal P_1(\mathcal T)$ (let's prove it).
  • It remains to invoke a geometry-related argument that given $u$ there exists $v_h$ such that $\|\nabla v_h - \nabla u\| \leq h \|\nabla^2 u\|$. This is achieved with, e.g., the Clément interpolation (warning: google for that on your own risk!)
  • Exercise: learn the difference in pronounciation between Clement and Clément
  • Know this guy?

Mesh refinement / Adaptivity

• Often done through the a posteriori error estimate, e.g., $$ \|\nabla u_h - \nabla u\|^2 \leq \sum_{{\rm edge}} h_{{\rm edge}} \Big[\frac{\nabla u_h}{h_{{\rm edge}}}\Big]^2_{{\rm edge}} $$ where $[\bullet]_{\rm edge}$ is the jump across the edge (recall that $\nabla u_h$ is piecewise constant)

Mesh quality

• Mesh should not have very sharp angles (otherwise the approximation will be bad)!

Collocation

(this is an alternative to Galerkin)

• If we have non-constant coefficients, we use the quadrature rules $$ \int_T k(x,y) \nabla u\cdot\nabla v \approx \sum_{i} w_i k(x_i,y_i) \nabla u(x_i,y_i)\cdot\nabla v(x_i,y_i) $$

• must be careful to take the quadrature rule of sufficiently high order

Quad(rilateral) elements:

High-order elements

• $\mathcal P^p(\mathcal T)$: polynomials of power $p$
• $hp$-refinement

A good source for these is
C. Schwab, "p- and hp- Finite Element Methods: Theory and Applications to Solid and Fluid Mechanics", Clarendon Press, 1998.

More stuff:

• Discontinous Galerkin, $\mathcal P^0$ elements
• Other bases: wavelets, radial basis functions
• mixed FEM, generalized FEM, extended FEM, what-not FEM...

Adv/Disadv of FEM

• Good for simple physics, solid mechanics
• Good energy (and other) conservation properties

But...

• Not easy for fluid flow (must work hard for stability)
• sometimes non-monotonicity

Questions?