Introduction¶

Basic Concepts¶

graph LR
    A[<b>Engineering Problems</b>] --> B[<b>Physical Models</b><br>Governing Eqs + BCs]
    B --> C[<b>FEM</b><br>KU = F]

Vertical machining center

Identify the geometry of the problem
Select a displacement function
Axial Tension
- choose \(u(x) = a_0 + a_1 x\) for the displacement function
  
  you can't choose \(u(x) = a_0 + a_2 x^2\), because it doesn't satisfy the convergence condition
- determine \(a_0, a_1\) by BC: \(u(0) = u_i, u(L) = u_j\)
  
  \[ \begin{aligned} a_0 &= u_i \\ a_1 &= \frac{u_j - u_i}{L} \end{aligned} \]
- rewrite \(u(x)\) in terms of nodal displacements \(u_i, u_j\):
  
  \[ \begin{aligned} u(x) &= u_i + \frac{x}{L} (u_j - u_i) \\ &= \Big(1 - \frac{x}{L}\Big) u_i + \frac{x}{L} u_j \\ :&= N_i u_i + N_j u_j \end{aligned} \]
  - \(N_i, N_j\): shape functions
Define the strain/displacement and stress/strain relationships

\[ \boldsymbol{\varepsilon} = \nabla \mathbf{u}, \quad \boldsymbol{\sigma} = \mathbf{D} \boldsymbol{\varepsilon} \]
Derive the element stiffness matrix and equations

\[ \mathbf{k}_e \mathbf{u}_e = \mathbf{f}_e \]
- 直接平衡法
- 能量法
  - 最小势能原理（弹性材料）
  - Castigliano 定理（弹性材料）
  - 虚功原理（任何材料）
- 最小残差法（weighted residual method）
  - Galerkin method
- Assemble the element equations to obtain the global equations, introduce BCs
\[ \mathbf{k}_e \mathbf{u}_e = \mathbf{f}_e \xrightarrow{\text{Direct stiffness method}} \mathbf{KU} = \mathbf{F} \]

Commercial FEM Software Packages

FEM only obtains approximate solutions with inherent errors. The errors can be reduced by:

History of FEM