有限元笔记4-二维单元形函数(2D-4,9Node)

在前面一节，我们有了一维的形函数，可以对一维问题进行有限元计算了，这一节我们来看一下二维单元的形函数。

这里先说明一下，二维单元的类型最常用的主要有三角形单元和四边形单元，这两种单元类型各有各的好处。因为在AsFem的网格生成模块中主要考虑二维的矩形区域，所以本章就优先介绍四边形单元中的4点线性单元和9节点二次单元。其他单元的结构非常类似，随便google一下都能找到具体的表达式，就不在多说了。

先回顾下一维情形下的一阶Lagrange插值多项式：

N 1 = 1 - ξ 2, N 2 = 1 + ξ 2

$N_{1}=\frac{1-\xi}{2},N_{2}=\frac{1+\xi}{2}$
现在我们假设在y(也就是

η $\eta$ )方向也有一个一阶的Lagrange插值多项式：

M 1 = 1 - η 2, M 2 = 1 + η 2

$M_{1}=\frac{1-\eta}{2},M_{2}=\frac{1+\eta}{2}$
做一个向量乘法就可以得到二维一阶4节点单元的形函数为：

[M 2 M 1] \cdot [N 1 N 2] = [M 2 N 1 M 1 N 1 M 2 N 2 M 1 N 2]

$\begin{bmatrix}M_{2}\\{M_{1}}\end{bmatrix}\cdot\begin{bmatrix}N_{1}&N_{2}\end{bmatrix}=\begin{bmatrix}M_{2}N_{1}&M_{2}N_{2}\\M_{1}N_{1}&M_{1}N_{2}\end{bmatrix}$
为了方便，还是用

N $N$ 来表示形函数，注意去上面符号的区别。

所以在二维4节点线性单元中，4个节点对应的形函数为：

N 1 = 1 - η 2 1 - ξ 2 = ( 1 - η ) ( 1 - ξ ) 4

$N_{1}=\frac{1-\eta}{2}\frac{1-\xi}{2}=\frac{(1-\eta)(1-\xi)}{4}$

N 2 = 1 - η 2 1 + ξ 2 = ( 1 - η ) ( 1 + ξ ) 4

$N_{2}=\frac{1-\eta}{2}\frac{1+\xi}{2}=\frac{(1-\eta)(1+\xi)}{4}$

N 3 = 1 + η 2 1 + ξ 2 = ( 1 + η ) ( 1 + ξ ) 4

$N_{3}=\frac{1+\eta}{2}\frac{1+\xi}{2}=\frac{(1+\eta)(1+\xi)}{4}$

N 4 = 1 + η 2 1 - ξ 2 = ( 1 + η ) ( 1 - ξ ) 4

$N_{4}=\frac{1+\eta}{2}\frac{1-\xi}{2}=\frac{(1+\eta)(1-\xi)}{4}$

同理，二维9节点二次单元的形函数就可以由一下公式计算得到：

N 9 n o d e = ⎡ ⎣ ⎢ ⎢ M 3 M 2 M 1 ⎤ ⎦ ⎥ ⎥ \cdot [N 1 N 2 N 3]

$N_{9node}=\begin{bmatrix}M_{3}\\M_{2}\\M_{1}\end{bmatrix}\cdot\begin{bmatrix}N_{1}&N_{2}&N_{3}\end{bmatrix}$

N i j = ⎡ ⎣ ⎢ ⎢ M 3 N 1 M 2 N 1 M 1 N 1 M 3 N 2 M 2 N 2 M 1 N 2 M 3 N 3 M 2 N 3 M 1 N 3 ⎤ ⎦ ⎥ ⎥

$N_{ij}=\begin{bmatrix} M_{3}N_{1}&M_{3}N_{2}&M_{3}N_{3}\\ M_{2}N_{1}&M_{2}N_{2}&M_{2}N_{3}\\ M_{1}N_{1}&M_{1}N_{2}&M_{1}N_{3} \end{bmatrix}$
具体表达式如下：

N 1 = η ( η - 1 ) 2 ξ ( ξ - 1 ) 2 = 1 4 η (η - 1) ξ (ξ - 1)

$N_{1}=\frac{\eta(\eta-1)}{2}\frac{\xi(\xi-1)}{2}=\frac{1}{4}\eta(\eta-1)\xi(\xi-1)$

N 2 = η ( η - 1 ) 2 ξ ( ξ + 1 ) 2 = 1 4 η (η - 1) ξ (ξ + 1)

$N_{2}=\frac{\eta(\eta-1)}{2}\frac{\xi(\xi+1)}{2}=\frac{1}{4}\eta(\eta-1)\xi(\xi+1)$

N 3 = η ( η + 1 ) 2 ξ ( ξ + 1 ) 2 = 1 4 η (η + 1) ξ (ξ + 1)

$N_{3}=\frac{\eta(\eta+1)}{2}\frac{\xi(\xi+1)}{2}=\frac{1}{4}\eta(\eta+1)\xi(\xi+1)$

N 4 = η ( η + 1 ) 2 ξ ( ξ - 1 ) 2 = 1 4 η (η + 1) ξ (ξ - 1)

$N_{4}=\frac{\eta(\eta+1)}{2}\frac{\xi(\xi-1)}{2}=\frac{1}{4}\eta(\eta+1)\xi(\xi-1)$

N 5 = η ( η - 1 ) 2 (1 + ξ) (1 - ξ) = 1 2 η (η - 1) (1 - ξ 2)

$N_{5}=\frac{\eta(\eta-1)}{2}(1+\xi)(1-\xi)=\frac{1}{2}\eta(\eta-1)(1-\xi^{2})$

N 6 = (1 + η) (1 - η) ξ ( 1 + ξ ) 2 = 1 2 (1 - η 2) ξ (1 + ξ)

$N_{6}=(1+\eta)(1-\eta)\frac{\xi(1+\xi)}{2}=\frac{1}{2}(1-\eta^{2})\xi(1+\xi)$

N 7 = η ( η + 1 ) 2 (1 + ξ) (1 - ξ) = 1 2 η (η + 1) (1 - ξ 2)

$N_{7}=\frac{\eta(\eta+1)}{2}(1+\xi)(1-\xi)=\frac{1}{2}\eta(\eta+1)(1-\xi^{2})$

N 8 = (1 + η) (1 - η) ξ ( 1 - ξ ) 2 = 1 2 (1 - η 2) ξ (1 - ξ)

$N_{8}=(1+\eta)(1-\eta)\frac{\xi(1-\xi)}{2}=\frac{1}{2}(1-\eta^{2})\xi(1-\xi)$

N 9 = (1 + η) (1 - η) (1 + ξ) (1 - ξ) = (1 - η 2) (1 - ξ 2)

$N_{9}=(1+\eta)(1-\eta)(1+\xi)(1-\xi)=(1-\eta^{2})(1-\xi^{2})$

以上是针对4,9节点取巧的计算方法，真正的形函数推导应该先根据单元的节点数构造出对应的位移模式：

u = α 1 + α 2 x + α 3 y + α 4 x y

$u=\alpha_{1}+\alpha_{2}x+\alpha_{3}y+\alpha_{4}xy$
然后带入4个点的(

ξi $\xi_{i}$ ,

ηj $\eta_{j}$ )值来求解

u=A∗α $u=A*{\alpha}$ 方程得到

A $A$ ，进而得到各个点对应的形函数。

我们可以画一下对应的形函数曲面，代码如下：

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import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from matplotlib.ticker import LinearLocator, FormatStrFormatter
from mpl_toolkits.mplot3d import Axes3D

xi=np.linspace(-1.0,1.0,100)
eta=np.linspace(-1.0,1.0,100)

Xi,Eta=np.meshgrid(xi,eta)
N1=Eta*(1-Eta*Eta)*(1-Xi*Xi)
#N1=(1-Eta*Eta)*Xi*(1+Xi)/2.0

fig=plt.figure(1)
ax=fig.gca(projection='3d')
surf=ax.plot_surface(Xi,Eta,N1,cmap=cm.coolwarm,linewidth=0, antialiased=False)

# Customize the z axis.
ax.set_zlim(-1.01, 1.01)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f'))
plt.title('$N_{9}$ of 2D-9Node element')
plt.xlim([-1.0,1.0])
plt.ylim([-1.0,1.0])
plt.xlabel("$\\xi$")
plt.ylabel('$\eta$')
# Add a color bar which maps values to colors.
fig.colorbar(surf, shrink=0.5, aspect=5)
plt.savefig('N9-9Node.png',dpi=500)
plt.show()

结果如下：

这里写图片描述

同上一节一样，有了形函数后我们还需要知道其对自然坐标的导数才能计算 $\nabla$ 和 $\Delta$ 等算符，计算也很简单，做一下二维情形的Jacobian变换就可以了。
比如：

\partial N i ( ξ , η ) \partial x = \partial N i \partial ξ \cdot \partial ξ \partial x + \partial N i \partial η \cdot \partial η \partial x

$\frac{\partial N_{i}(\xi,\eta)}{\partial x}=\frac{\partial N_{i}}{\partial\xi}\cdot\frac{\partial\xi}{\partial x}+\frac{\partial N_{i}}{\partial\eta}\cdot\frac{\partial\eta}{\partial x}$

\partial N i ( ξ , η ) \partial y = \partial N i \partial ξ \cdot \partial ξ \partial y + \partial N i \partial η \cdot \partial η \partial y

$\frac{\partial N_{i}(\xi,\eta)}{\partial y}=\frac{\partial N_{i}}{\partial\xi}\cdot\frac{\partial\xi}{\partial y}+\frac{\partial N_{i}}{\partial\eta}\cdot\frac{\partial\eta}{\partial y}$
写成矩阵形式：

⎡ ⎣ ⎢ ⎢ ⎢ \partial N i \partial x \partial N i \partial x ⎤ ⎦ ⎥ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎢ \partial ξ \partial x \partial ξ \partial y \partial η \partial x \partial η \partial y ⎤ ⎦ ⎥ ⎥ ⎥ \cdot ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ \partial N i \partial ξ \partial N i \partial η ⎤ ⎦ ⎥ ⎥ ⎥ ⎥

$\begin{bmatrix}\frac{\partial N_{i}}{\partial x}\\\frac{\partial N_{i}}{\partial x}\end{bmatrix} =\begin{bmatrix} \frac{\partial\xi}{\partial x}&\frac{\partial\eta}{\partial x}\\ \frac{\partial\xi}{\partial y}&\frac{\partial\eta}{\partial y} \end{bmatrix}\cdot \begin{bmatrix}\frac{\partial N_{i}}{\partial\xi}\\\frac{\partial N_{i}}{\partial\eta}\end{bmatrix}$
这是数学上最直观的表达，但是在程序中，为了方便计算我们反过来做：

\partial N i \partial ξ = \partial N i \partial x \cdot \partial x \partial ξ + \partial N i \partial y \cdot \partial y \partial ξ

$\frac{\partial N_{i}}{\partial\xi}=\frac{\partial N_{i}}{\partial x}\cdot\frac{\partial x}{\partial\xi}+\frac{\partial N_{i}}{\partial y}\cdot\frac{\partial y}{\partial\xi}$

\partial N i \partial η = \partial N i \partial x \cdot \partial x \partial η + \partial N i \partial y \cdot \partial y \partial η

$\frac{\partial N_{i}}{\partial\eta}=\frac{\partial N_{i}}{\partial x}\cdot\frac{\partial x}{\partial\eta}+\frac{\partial N_{i}}{\partial y}\cdot\frac{\partial y}{\partial\eta}$
其矩阵形式为:

⎡ ⎣ ⎢ ⎢ ⎢ ⎢ \partial N i \partial ξ \partial N i \partial η ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ \partial x \partial ξ \partial x \partial η \partial y \partial ξ \partial y \partial η ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ \cdot ⎡ ⎣ ⎢ ⎢ ⎢ \partial N i \partial x \partial N i \partial x ⎤ ⎦ ⎥ ⎥ ⎥

$\begin{bmatrix}\frac{\partial N_{i}}{\partial\xi}\\\frac{\partial N_{i}}{\partial\eta}\end{bmatrix} =\begin{bmatrix} \frac{\partial x}{\partial\xi}&\frac{\partial y}{\partial\xi}\\ \frac{\partial x}{\partial\eta}&\frac{\partial y}{\partial\eta} \end{bmatrix}\cdot \begin{bmatrix}\frac{\partial N_{i}}{\partial x}\\\frac{\partial N_{i}}{\partial x}\end{bmatrix}$
Jacobian变换矩阵就是：

J = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ \partial x \partial ξ \partial x \partial η \partial y \partial ξ \partial y \partial η ⎤ ⎦ ⎥ ⎥ ⎥ ⎥

$J=\begin{bmatrix}\frac{\partial x}{\partial\xi}&\frac{\partial y}{\partial\xi}\\ \frac{\partial x}{\partial\eta}&\frac{\partial y}{\partial\eta}\end{bmatrix}$
其中

\partial x \partial ξ = \sum i = 1 n \partial N i \partial ξ x i, \partial x \partial η = \sum i = 1 n \partial N i \partial η x i

$\frac{\partial x}{\partial\xi}=\sum_{i=1}^{n}\frac{\partial N_{i}}{\partial\xi}x_{i},\frac{\partial x}{\partial\eta}=\sum_{i=1}^{n}\frac{\partial N_{i}}{\partial\eta}x_{i}$

\partial y \partial ξ = \sum i = 1 n \partial N i \partial ξ y i, \partial y \partial η = \sum i = 1 n \partial N i \partial η y i

$\frac{\partial y}{\partial\xi}=\sum_{i=1}^{n}\frac{\partial N_{i}}{\partial\xi}y_{i},\frac{\partial y}{\partial\eta}=\sum_{i=1}^{n}\frac{\partial N_{i}}{\partial\eta}y_{i}$
最后，形函数对自然坐标的偏导就可以写为：

⎡ ⎣ ⎢ ⎢ ⎢ \partial N i \partial x \partial N i \partial y ⎤ ⎦ ⎥ ⎥ ⎥ = J - 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ \partial N i \partial ξ \partial N i \partial η ⎤ ⎦ ⎥ ⎥ ⎥ ⎥

$\begin{bmatrix}\frac{\partial N_{i}}{\partial x}\\\frac{\partial N_{i}}{\partial y}\end{bmatrix} =J^{-1}\begin{bmatrix}\frac{\partial N_{i}}{\partial\xi}\\\frac{\partial N_{i}}{\partial\eta}\end{bmatrix}$
这里

J−1 $J^{-1}$ 为Jacobian矩阵的逆矩阵。

AsFem对应的计算代码如下:

void Shp2D(const int &nDims,const double &xi,const double &eta,const MatrixXd &Coords, MatrixXd &shp, double &DetJac)
{
    int nNodes;
    int i, j;
    double dxdxi, dydxi, dzdxi;
    double dxdeta, dydeta, dzdeta;
    double temp;

    nNodes = int(Coords.rows());


    if (nDims<2||nDims>3)
    {
        cout << "Wrong dimension!!!" << endl;
        abort();
    }

    if (nNodes<3 || nNodes>16)
    {
        cout << "Wrong nodes on current element!!!" << endl;
        cout<<"AsFem only support 2D-4,8,9 nodes element!!!"<<endl;
        abort();
    }

    MatrixXd Jac(2,2);
    Matrix2d XJac;
    shp.setZero();

    if(nNodes==4)
    {
        // 2D-4Nodes rectangle element
        shp(0,0)=(1.0-xi)*(1.0-eta)/4.0;
        shp(1,0)=(1.0+xi)*(1.0-eta)/4.0;
        shp(2,0)=(1.0+xi)*(1.0+eta)/4.0;
        shp(3,0)=(1.0-xi)*(1.0+eta)/4.0;

        shp(0,1)= (eta-1.0)/4.0;
        shp(0,2)= (xi -1.0)/4.0;

        shp(1,1)= (1.0-eta)/4.0;
        shp(1,2)=-(1.0+xi )/4.0;

        shp(2,1)= (1.0+eta)/4.0;
        shp(2,2)= (1.0+xi )/4.0;

        shp(3,1)=-(1.0+eta)/4.0;
        shp(3,2)= (1.0-xi )/4.0;
    }
    else if(nNodes==8)
    {
        // 2D-8Nodes rectangle element
        // Go to AsFem for details
    }
    else if(nNodes==9)
    {
        // 2D-9Nodes rectangle element
        shp(0,0)=(xi*xi-xi )*(eta*eta-eta)/4.0;
        shp(1,0)=(xi*xi+xi )*(eta*eta-eta)/4.0;
        shp(2,0)=(xi*xi+xi )*(eta*eta+eta)/4.0;
        shp(3,0)=(xi*xi-xi )*(eta*eta+eta)/4.0;
        shp(4,0)=(1.0-xi*xi)*(eta*eta-eta)/2.0;
        shp(5,0)=(xi*xi+xi )*(1.0-eta*eta)/2.0;
        shp(6,0)=(1.0-xi*xi)*(eta*eta+eta)/2.0;
        shp(7,0)=(xi*xi-xi )*(1.0-eta*eta)/2.0;
        shp(8,0)=(1.0-xi*xi)*(1.0-eta*eta);

        shp(0,1)=(2.0*xi-1.0)*(eta*eta-eta)/4.0;
        shp(0,2)=(xi*xi-xi  )*(2.0*eta-1.0)/4.0;

        shp(1,1)=(2.0*xi+1.0)*(eta*eta-eta)/4.0;
        shp(1,2)=(xi*xi+xi  )*(2.0*eta-1.0)/4.0;

        // Go to AsFem for details
    }


    // compute jacob transform matrix
    if(nDims==2)
    {
        dxdxi=0.0;dxdeta=0.0;
        dydxi=0.0;dydeta=0.0;
        for(i=0;i<nNodes;i++)
        {
            dxdxi+=Coords(i,0)*shp(i,1);
            dxdeta+=Coords(i,0)*shp(i,2);

            dydxi+=Coords(i,1)*shp(i,1);
            dydeta+=Coords(i,1)*shp(i,2);
        }
        Jac(0,0)=dxdxi;Jac(0,1)=dydxi;
        Jac(1,0)=dxdeta;Jac(1,1)=dydeta;

        DetJac=Jac(0,0)*Jac(1,1)-Jac(0,1)*Jac(1,0);
        if(DetJac==0.0)
        {
            cout<<"Singular element!!!"<<endl;
            abort();
        }
        XJac=Jac.inverse();
    }
    else if(nDims==3)
    {
        // Go to AsFem for details
    }

    for(i=0;i<nNodes;i++)
    {
        temp=XJac(0,0)*shp(i,1)+XJac(0,1)*shp(i,2);
        shp(i,2)=XJac(1,0)*shp(i,1)+XJac(1,1)*shp(i,2);
        shp(i,1)=temp;
    }
}

有限元笔记4-二维单元形函数(2D-4,9Node)

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