Weighted Residual Methods

How to find element stiffness matrix and equations:

• Direct equilibrium method: Good for simple problems (1D elements)
• Energy method: Good for all elements
  • Principle of minimum potential energy: elastic materials
  • Castigliano's theorem: elastic materials
  • Principle of virtual work: any material
• Variational method
• Method of weighted residuals

Variational Method

Find a functional \(\Pi_p\) equivalent to the PDE.

• Structural stress analysis:

  \[ \Pi_p = \frac{1}{2} \int_V \sigma_{ij} \varepsilon_{ij} dV - \int_V f_i u_i dV - \int_S T_i u_i dS \]

Sometimes we can't find a functional!

Weighted Residual Method

We want to find an approximate solution to any PDE, as long as we've obtained the governing equations and boundary conditions.

Linear Elasticity

\[ \begin{aligned} \varepsilon_{ij} &= \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \\ \sigma_{ij} &= 2 G \varepsilon_{ij} + \lambda \varepsilon_{kk} \delta_{ij} \\ \frac{\partial \sigma_{ij}}{\partial x_j} + f_i &= 0 \end{aligned} \]

15 equations + BC \(\implies\) 15 unknowns, but the PDEs are very difficult to solve. Because the solution has to satisfy the PDEs at every material point, and this is a very strong requirement (Strong form).

Instead of finding a precise solution

\[ \frac{\partial \sigma_{ij}(u_1, u_2, u_3)}{\partial x_j} + f_i = 0 \]

we ask the above equations to be