Finite Element Method

The elements in PeriLab are formulated using the matrix of shape functions $\mathbf{N}$ and the matrix of derivatives $\mathbf{B}$ [21], [22].

\[\mathbf{M}=\mathbf{N}^T\mathbf{N}\rho\]

To run a point wise analysis and bring it into the same structure as the PD formulation the lumped mass matrix form is used

\[m_i=\sum\limits_{j=1}^{n} M_{i,j}\]

to get a diagonal mass matrix. To compute the stiffness matrix the following form is used.

\[\mathbf{K}=\int_V\mathbf{B}^T\boldsymbol{\sigma}(\boldsymbol{\varepsilon})dV\]

The element strain is given as

\[\boldsymbol{\varepsilon}=\mathbf{B}^T\mathbf{u}\]

This formulation allows the flexible integration of material laws. Using the linear elastic material with the elasticity matrix $\mathbf{C}$

\[\mathbf{K}=\int_V \mathbf{B}^T\mathbf{CB}dV\]

Lagrange functions

Lagrange polynomials can be used to formulated finite elements [23]. These polynomials can be defined recursively for a polynomial $p$.

\[L(x) = \prod\limits_{\begin{smallmatrix}0\le m\le p\\ m\neq j\end{smallmatrix}} \frac{x-x_m}{x_j-x_m}\]

The values $x$ are defined in local coordinated $[-1\,1]$. These shape functions can be defined seperatly for each direction, also with different polynomial orders. In combination these functions are used in the matrix $\mathbf{N}$.

The derivative can be computed recursively as well.

\[L_j'(x)=L_j(x)\sum\limits_{\begin{smallmatrix}i=0\\ i\neq j\end{smallmatrix}}^{p}\frac{1}{x-x_i}\]

The number of nodes per element is depended on the degrees of freedom (dof) $(p+1)^{dof}$.