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Chap1 Introduction

Basic concept of the FEM

FEM

Finite Element Method: A numerical technique for obtaining approximate solutions to differential equations

  • FEM discretizes a physical domain into many elements

  • Then reconnect elements at nodes as if nodes were pins or drops of glue that hold elements together

  • This process results in a set of simultaneous algebraic equations( 一组联立的代数方程组 )

example

Engineering problems( 工程问题 ): structural analysis( 结构分析 )、thermal analysis( 热分析 )、fluid flow( 流体流动 )、electromagnetics( 电磁学 )、buckling( 变形 )

\[\downarrow\]

Physical model( 物理模型 ): governing equation( 控制方程 )\(L(\phi)+f=0\)、boundary conditions( 边界条件 )\(B(\phi)+g=0\)

\[\downarrow\]

FEM: a set of simultaneous algebraic equations

\[KU=F\]

concepts summary

  • A set of simultaneous algebraic equations at nodes
\[KU=F\]

  • K-property U-Behavior F-action

  • FEM uses the concept of piecewise polynomial( 分段多项式 ) interpolation( 插值 )

  • By connecting elements together, the field quantity becomes interpolated over the entire structure in piecewise fashion

General steps of the FEM

Step 1: Discretize( 离散化 ) and select the element type( 选择单元类型 )

wfxs

Step 2: Select a displacement function( 位移函数 )

Step 3: Define the strain( 应变 )/displacement and stress( 应力 )/strain relationships

对于一维线弹性情况，小应变的应变 \(\varepsilon_x\) 和位移 \(u\) 的关系为

\[\varepsilon_x=\frac{du}{dx}\]

应力和应变通过本构关系关联，胡克定律为最简单的应力 / 应变定律

\[\sigma_x=E\varepsilon_x\]

Step 4: Derive( 推导 ) the element stiffness matrix( 单元刚度矩阵 ) and equations

Step 5: Assemble( 组合 ) the element equations( 单元方程 ) to obtain the global equations( 总体方程 ) and introduce boundary conditions

组合方程或总体方程的矩阵形式如下

\[\{F\}=[K]\{d\}\]

其中 \(\{F\}\) 为总体节点力矢量，\([K]\) 为结构总体刚度矩阵，\(\{d\}\) 是结构已知和结构未知的节点的自由度或广义位移

Step 6: Solve for the unknown degrees of freedom( 未知自由度 )

写为扩展的矩阵形式

\[\left\{\begin{aligned}F_1 \\ F_2 \\ ... \\ F_n \end{aligned}\right\}=\begin{bmatrix}K_{11 } & K_{12} & ... & K_{1n} \\ K_{21} & K_{22} & ... & K_{2n} \\ ... \\ K_{n1} & K_{n2} & ... & K_{nn}\end{bmatrix}\left\{\begin{aligned}d_1 \\ d_2 \\ ... \\ d_n \end{aligned}\right\}\]

其中 n 是结构未知节点自由度的总数

方程可用消元法或迭代法求解，得出所有 d 值；d 值是利用刚度（或位移）有限元方法获得的第一组量，因此称为初始未知量

Step 7: Solve for the element strains and stresses

Step 8: Interpret the results

Applications of the FEM

Application in Engineering

  • Structural problems
    • Stress analysis(static/dynamic, linear/nonlinear)
    • Buckling( 屈曲 )
    • Vibration( 振动 ) analysis
  • Nonstructural problems
    • Heat transfer analysis(steady/transient( 瞬时 ))
    • Fluid flow
    • Distribution of electric or magnetic petential( 电势或磁势的分布 ))

Errors of the FEM

FEM only obtains approximate solutions with inherent errors( 固有误差 )

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