Isoparametric Formulation

Isoparametric Formula for 2D Elements

Finite Element for Plane Problems

Element types:

• Triangular
  • Constant Strain Triangle (CST, T3)
  • Linear Strain Triangle (LST, T6)
• Rectangular
  • Bilinear rectangle (Q4)
  • Quadratic rectangle (Q8)

In reality, both cannot meet our needs (Triangular: low accuracy; Rectangular: simple geometry only). So we need to use isoparametric elements:

• high accuracy
• arbitrary geometry
• simple computer programming

We will first consider 4 node isoparametric 2D plane elements.

Isoparametric Formula

\[ \mathbf{k}^e = \int_V \mathbf{B}^T \mathbf{D} \mathbf{B} \, \mathrm{d} V \]

用全局坐标求解 \(\mathbf{k}^e\) 的话，\(\mathbf{B}\) 不是常数，积分非常麻烦！

全局坐标系 \((x, y)\) 与自然（局部）坐标系 \((s, t)\) 之间存在一一映射关系。设映射为

\[ \begin{aligned} x &= a_1 + a_2 s + a_3 t + a_4 s t \\ y &= b_1 + b_2 s + b_3 t + b_4 s t \end{aligned} \]

代入四个节点的坐标，当然可以得到各个系数。不过更简单的做法是从形函数出发。因为形函数仅在节点处取值为 \(1\)，所以可以直接构造

\[ \begin{aligned} N_1 &= \frac{1}{4} (1 - s) (1 - t), \quad N_2 = \frac{1}{4} (1 + s) (1 - t) \\ N_3 &= \frac{1}{4} (1 + s) (1 + t), \quad N_4 = \frac{1}{4} (1 - s) (1 + t) \end{aligned} \]

Shape and Displacement Interpolation

Shape:

\[ x = \sum_{i = 1}^4 N_i(s, t) x_i, \quad y = \sum_{i = 1}^4 N_i(s, t) y_i \]

Displacement:

\[ u = \sum_{i = 1}^4 N_i(s, t) u_i, \quad v = \sum_{i = 1}^4 N_i(s, t) v_i \]

Here, the shape and displacements are interpolated by the same shape functions, which is the reason why this formulation is called "isoparametric".

Strain matrix

\[ \boldsymbol{\varepsilon} = \begin{bmatrix} \frac{\partial}{\partial x} & 0 \\ 0 & \frac{\partial}{\partial y} \\ \frac{\partial}{\partial y} & \frac{\partial}{\partial x} \end{bmatrix} \begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} \frac{\partial}{\partial x} & 0 \\ 0 & \frac{\partial}{\partial y} \\ \frac{\partial}{\partial y} & \frac{\partial}{\partial x} \end{bmatrix} \mathbf{N} \mathbf{d} = \mathbf{B} \mathbf{d} \]

we need to solve the derivatives \(\frac{\partial N_i}{\partial x}\) and \(\frac{\partial N_i}{\partial y}\). By the chain rule,

\[ \begin{aligned} \frac{\partial N_i}{\partial s} &= \frac{\partial N_i}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial N_i}{\partial y} \frac{\partial y}{\partial s} \\ \frac{\partial N_i}{\partial t} &= \frac{\partial N_i}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial N_i}{\partial y} \frac{\partial y}{\partial t} \end{aligned} \]