Chapter 4 Approximation of trial solutions, weight functions and Gauss quadrature for one-dimensional problems
Continuity + Completeness -----> Convergence of FEM
Continuity : The trial solutions and weight functions are suffciently smooth
Based on the order of derivatives that appear in the weak form
Completeness : The capability of a series of functions to approximate a given smooth function with arbitary accuracy
Approximation of trial solutions in element
Two-node linear element
Linear ploynomial function :
\(\theta^e(x)=\alpha_0^e+\alpha_1^ex=\pmb{P}(x)\pmb{\alpha}^e\)
where \(\pmb{P}(x)=\left[ \begin{matrix} 1 & x \end{matrix} \right]\), \(\pmb{\alpha}^e=\left[ \begin{matrix} \alpha_0^e & \alpha_1^e \end{matrix} \right]^T\)
Express \(\pmb{\alpha}^e\) terms of the nodal values \(\theta^e(x_i^e)\equiv \theta_i^e\)
\(\left[ \begin{matrix} \theta_1^e \\ \theta_2^e \end{matrix} \right]=\left[ \begin{matrix} 1 & x_1^e\\ 1 & x_2^e \end{matrix} \right] \left[ \begin{matrix} \alpha_0^e \\ \alpha_1^e \end{matrix} \right]\)
\(\pmb{d}^e=\pmb{M}^e\pmb{\alpha}^e\)
\(\pmb{\alpha}^e=(\pmb{M}^e)^{-1}\pmb{d}^e\)
\(\theta^e(x)=\pmb{P}(x)\pmb{\alpha}^e=\pmb{P}(x)(\pmb{M}^e)^{-1}\pmb{d}^e=\pmb{N}^e(x)\pmb{d}^e=\sum_{I=1}^{n_{en}}N_I^e(x)\theta_I^e\)
where \(\pmb{N}^e(x)= \left[ \begin{matrix} N_1^e & N_2^e \end{matrix} \right]\) is the element shape function matrix with the interpolation property : \(N_I^e(x_J^e)=\delta_{IJ}\)
For the derivatives in the weak form
\(\frac{d\theta^e}{dx}=\frac{d}{dx}(\pmb{N}^e\pmb{d}^e)=\frac{d\pmb{N}^e}{dx}\pmb{d}^e= \left[ \begin{matrix} \frac{dN_1^e}{dx} & \frac{dN_2^e}{dx} \end{matrix} \right]\pmb{d}^e=\pmb{B}^e\pmb{d}^e\)
Quadratic one-dimensional element
Three nodes in an element ...
In general, the third node placed at the center of the element perform better than others
Direct construction of shape functions in one dimension
Lagrange interpolants ...
Approximation of the weight functions
Galerkin FEM : use the same interpolants for the weight functions and trial solutions
\(w^e(x)=\pmb{N}^e(x)\pmb{w}^e\)
\(\frac{dw^e}{dx}=\pmb{B}^e\pmb{w}^e\)
Gloabl approximation and continuity
Global approximation of trial solutions and weight functions:
\(\theta^h=\pmb{N}\pmb{d}=\sum_{I=1}^{n_{np}}N_Id_I\)
\(w^h=\pmb{N}\pmb{w}=\sum_{I=1}^{n_{np}}N_Iw_I\)
\(N_I\in H^1\) and satisfies the Kronecker delta property
Global shape functions are \(C^0\) continuous ---> \(\theta^h\) and \(w^h\) are also \(C^0\) continuous (Linear combination)
Guass quadrature
To find the following integral
\(I=\int_a^bf(x)dx\)
Use the a linear mapping to map the parent domian of Guass quadrature [\(-1,1\)] to physical domain [\(a,b\)]
\(x=\frac{b-a}{1-(-1)}(\xi-1)+b=\frac{1-\xi}{2}a+\frac{\xi+1}{2}b=N_1(\xi)x_1+N_2(\xi)x_2\)
\(dx=\frac{b-a}{2}d\xi=Jd\xi\) , where \(J=\frac{b-a}{2}\) is the Jacobi
The integral becomes
\(I=\int_a^bf(x)dx=J\int_{-1}^1f(\xi)d\xi=J\hat{I}\)
Approximate the integral \(\hat{I}\) by
\(\hat{I}=W_1f(\xi_1)+W_2f(\xi_2)+...=\pmb{W}^T\pmb{f}\)
The function \(f(\xi)\) is approximated by a polynominal
\(f(\xi)=\alpha_1+\alpha_2\xi+...=\pmb{P}(\xi)\pmb{\alpha}\)
where \(\pmb{P}= \left[ \begin{matrix} 1 & \xi & \xi^2 & \cdots \end{matrix} \right]\)
Then \(f(\xi)\) at the integral points can be expressed as
\(\pmb{f}=\pmb{M}\pmb{\alpha}\)
where \(\pmb{M}= \left[ \begin{matrix} 1 & \xi_1 & \xi_1^2 & \cdots\\ 1 & \xi_2 & \xi_2^2 & \cdots\\ \vdots & \vdots & \vdots & \vdots\\ 1 & \xi_n & \xi_n^2 & \cdots \end{matrix} \right]\)
The guass quadrature form:
\(\hat{I}=\pmb{W}^T\pmb{M}\pmb{\alpha}\)
The polynominal integration form:
\(\hat{I}=\int_{-1}^{1}f(\xi)d\xi= \int_{-1}^{1} \left[ \begin{matrix} 1 & \xi & \xi^2 & \cdots \end{matrix} \right] \pmb{\alpha} d\xi= \int_{-1}^{1} \left[ \begin{matrix} \xi & \frac{\xi^2}{2} & \frac{\xi^3}{3} & \cdots \end{matrix} \right]_{-1}^1 \pmb{\alpha} d\xi= \left[ \begin{matrix} 2 & 0 & \frac{2}{3} & \cdots \end{matrix} \right]_{-1}^1 \pmb{\alpha}= \hat{\pmb{P}}\pmb{\alpha}\)
To give the exact numerical integral results
\(\pmb{W}^T\pmb{M}=\hat{\pmb{P}}\) -----> \(\pmb{M}^T\pmb{W}=\hat{\pmb{P}}^T\)
For \(n_{gp}\) gauss points, there are \(2n_{gp}\) adjustable parameters (weights and integral points)
\(\pmb{M}\) (integral points) and \(\pmb{W}\) (weights) are all unknown and \(\hat{\pmb{P}}\) is already known (\(P+1\) parameters for \(P\)-prder polynominal)
\(2n_{gp}\geq P+1\)
So for the \(P\)-order polynominal, the required number of intergral points is
\(n_{gp}\geq \frac{P+1}{2}\)