Finite Element Method

Daniel Weschke

November 24, 2016

• Bar theory
• Cross-section and derived sizes
• Rigid body elements
• Aspects of Singularities
• ANSYS
• Code_Aster
  • Appendix
  • Elements
  • Study Bar
  • Study Ax
  • Study Mirror

List of Symbols

\(\tensorI{b}\)	Acceleration
\(\tensor{f}\ti{int}, \tensor{f}\ti{ext}\)	Internal force and external force
\(h, \tensor{H}\)	Shape functions
\(k\)	Node (of Element)
\(\tensor{L}\)	Differential operator
\(\tensor{M}\)	Mass matrix
\(\tensorI{n}\)	Normal vector
\(s\)	Surface
\(\tensorI{u},\tensor{u}\)	Displacement
\(v\)	Volume
\(\tensor{Z}\)	Incidence matrix
\(\tensorI{x}, x, y, z\)	Physical coordinates and \(\tensorI{\xi}, \xi, \eta, \zeta\) natural coordinates
\(\tensor\varepsilon\)	Strain
\(\rho\)	Density
\(\tensorII\sigma\)	Cauchy stress tensor
\(\tensor\sigma\)	Cauchy stress tensor represented in Voigt’s notation as a 6-dimensional vector

1 Balance of forces

\[\tensorI\nabla\cdot\tensorII\sigma + \rho\tensorI{b} = \rho\tensorI{\ddot{u}}\]

With Voigt’s notation

\[\begin{split}\tensor{L} = \begin{bmatrix} \frac{\partial}{\partial x} & 0 & 0 \\ 0 & \frac{\partial}{\partial y} & 0 \\ 0 & 0 & \frac{\partial}{\partial z} \\ 0 & \frac{\partial}{\partial z} & \frac{\partial}{\partial y} \\ \frac{\partial}{\partial z} & 0 & \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 \end{bmatrix} , \tensor\sigma = \begin{bmatrix} \sigma\ti{xx} \\ \sigma\ti{yy} \\ \sigma\ti{zz} \\ \sigma\ti{yz} \\ \sigma\ti{xz} \\ \sigma\ti{xy} \end{bmatrix}\end{split}\]

follows

\[\tensor{L}^\T\tensor\sigma + \rho\tensorI{b} = \rho\tensorI{\ddot{u}}\]

2 Weak formulation

\[\int\delta\tensorI{u}^\T\tensor{L}^\T\tensor\sigma\dif{v} + \int\delta\tensorI{u}^\T\rho\tensorI{b}\dif{v} = \int\delta\tensorI{u}^\T\rho\tensorI{\ddot{u}}\dif{v}\]

With Gauss’s theorem / Green’s theorem

\[\int\delta\tensorI{u}^\T\tensor{L}^\T\tensor\sigma\dif{v} = \int\delta\tensorI{u}^\T(\tensorII\sigma\cdot\tensorI{n})\dif{s} - \int(\tensor{L}\delta\tensorI{u})^\T\tensor\sigma\dif{v}\]

follows with \(\tensorI{t}=\tensorII\sigma\cdot\tensorI{n}\)

\[\int\rho\,\delta\tensorI{u}^\T\tensorI{\ddot{u}}\dif{v} + \int(\tensor{L}\delta\tensorI{u})^\T\tensor\sigma\dif{v} = \int\rho\,\delta\tensorI{u}^\T\tensorI{b}\dif{v} + \int\delta\tensorI{u}^\T\tensorI{t}\dif{s}\]

\[\tensor{L}\delta\tensorI{u} = \delta\tensor\varepsilon\]

For quasistatic stress conditions follows

\[\int\delta\tensor\varepsilon^\T\tensor\sigma\dif{v} = \int\rho\,\delta\tensorI{u}^\T\tensorI{b}\dif{v} + \int\delta\tensorI{u}^\T\tensorI{t}\dif{s}\]

3 Spatial discretization by finite elements

\[v = \bigcup\limits_{e}^{n_e} v_e \qquad \Rightarrow \qquad \int_v \dif{v} = \sum_e^{n_e} \int_{v_e}\]

3D ansatz function for an element

\[\tensorI{u} \approx \tensorI{\tilde{u}} = \sum_k^n h_k(\xi,\eta,\zeta) \tensorI{u}_k = \tensor{H} \tensor{u}_e\]

\(k\): Element node \(n\): Number of nodes per element \(h\): Shape function

\[\begin{split}\tensor{H} = \begin{bmatrix} h_1 & 0 & 0 & h_2 & 0 & 0 & \dots & h_n & 0 & 0 \\ 0 & h_1 & 0 & 0 & h_2 & 0 & \dots & 0 & h_n & 0 \\ 0 & 0 & h_1 & 0 & 0 & h_2 & \dots & 0 & 0 & h_n \end{bmatrix}\end{split}\]

\[\tensor{u}_e = \tensor{Z}_e\tensor{u}\]

\(\tensor{Z}_e\): Incidence matrix of size \(3n\times N\), where \(N\) represents the global degrees of freedom

\[\int_v \rho\,\delta\tensorI{u}^\T\tensorI{\ddot{u}}\dif{v} + \int_v(\tensor{L}\delta\tensorI{u})^\T\tensor\sigma\dif{v} = \int_v\rho\,\delta\tensorI{u}^\T\tensorI{b}\dif{v} + \int_s\delta\tensorI{u}^\T\tensorI{t}\dif{s}\]

\[\begin{split}\begin{split} \sum_e^{n_e} \int\limits_{v_e} \rho(\tensor{H}\tensor{Z}_e\delta\tensor{u})^\T\tensor{H}\tensor{Z}_e\tensor{\ddot{u}}\dif{v} + \sum_e^{n_e} \int_{v_e} (\tensor{L}\tensor{H}\tensor{Z}_e\delta\tensor{u})^\T\tensor\sigma\dif{v} \\ \quad = \sum_e^{n_e} \int_{v_e} \rho(\tensor{H}\tensor{Z}_e\delta\tensor{u})^\T\tensorI{b}\dif{v} + \sum_e^{n_e} \int_{s_e} (\tensor{H}\tensor{Z}_e\delta\tensor{u})^\T\tensorI{t}\dif{s} \end{split}\end{split}\]

\[\begin{split}\begin{split} \delta\tensor{u}^\T \underbrace{\sum_e^{n_e} \tensor{Z}_e^\T \int\limits_{v_e} \rho\,\tensor{H}^\T\tensor{H}\dif{v}\, \tensor{Z}_e}_{M} \tensor{\ddot{u}} + \delta\tensor{u}^\T \underbrace{\sum_e^{n_e} \tensor{Z}_e^\T \int_{v_e} \tensor{B}^\T\tensor\sigma\dif{v}}_{\tensor{f}\ti{int}} \\ \quad = \delta\tensor{u}^\T \underbrace{\left( \sum_e^{n_e} \tensor{Z}_e^\T \int_{v_e} \rho\,\tensor{H}^\T\tensorI{b}\dif{v} + \sum_e^{n_e} \tensor{Z}_e^\T \int_{s_e} \tensor{H}^\T\tensorI{t}\dif{s} \right)}_{\tensor{f}\ti{ext}} \end{split}\end{split}\]

with \(\tensor{B}=\tensor{L}\tensor{H}\)

\[\delta\tensor{u}^\T\tensor{M}\tensor{\ddot{u}} + \delta\tensor{u}^\T \tensor{f}\ti{int} = \delta\tensor{u}^\T \tensor{f}\ti{ext}\]

\[\tensor{M}\tensor{\ddot{u}} + \tensor{f}\ti{int} = \tensor{f}\ti{ext}\]

4 Polynomial interpolation and their derivatives

\[\tensor{B}=\tensor{L}\tensor{H},~ \tensorI{\xi}=\begin{bmatrix}\xi&\eta&\zeta\end{bmatrix}^\T,~ \tensorI{x}=\begin{bmatrix}x&y&z\end{bmatrix}^\T\]

\[\frac{\partial h_k}{\partial\tensorI\xi} = \frac{\partial \tensorI{x}}{\partial\tensorI\xi} \frac{\partial h_k}{\partial\tensorI{x}} = \tensorII{J} \frac{\partial h_k}{\partial\tensorI{x}} \quad \rightarrow \quad \frac{\partial h_k}{\partial\tensorI{x}} = \tensorII{J}^{-1} \frac{\partial h_k}{\partial\tensorI\xi}\]

\[\begin{split}\begin{split} \rightarrow \quad \tensor{B}_\xi = \tensor{J}^{-1} & \left[\begin{array}{ccc} \frac{\partial h_1}{\partial\xi} & 0 & 0 \\ 0 & \frac{\partial h_1}{\partial\eta} & 0 \\ 0 & 0 & \frac{\partial h_1}{\partial\zeta} \\ 0 & \frac{\partial h_1}{\partial\zeta} & \frac{\partial h_1}{\partial\eta} \\ \frac{\partial h_1}{\partial\zeta} & 0 & \frac{\partial h_1}{\partial\xi} \\ \frac{\partial h_1}{\partial\eta} & \frac{\partial h_1}{\partial\xi} & 0 \end{array} \right. & \dots \\ &\begin{matrix} \phantom{\frac{\partial h_1}{\partial\xi}} & \,k=1 & \phantom{\frac{\partial h_1}{\partial\xi}} \end{matrix} & \dots & \qquad k=n \end{split}\end{split}\]

\[\begin{split}\tensorII{J} = \frac{\partial \tensorI{x}}{\partial\tensorI\xi} = \begin{bmatrix} \frac{\partial x}{\partial\xi} & \frac{\partial x}{\partial\eta} & \frac{\partial x}{\partial\zeta} \\ \frac{\partial y}{\partial\xi} & \frac{\partial y}{\partial\eta} & \frac{\partial y}{\partial\zeta} \\ \frac{\partial z}{\partial\xi} & \frac{\partial z}{\partial\eta} & \frac{\partial z}{\partial\zeta} \end{bmatrix}\end{split}\]

\[\dif{v} = \det\tensorII{J}\dif\xi\dif\eta\dif\zeta\]

\(\tensorI{x}\) in analogy to \(\tensorI{u}\)

\[\begin{split}\begin{split} \tensorI{x} &= \tensorI{x}(\tensorI{\xi}) \\ &= \sum_k^n h_k(\xi,\eta\zeta) \tensorI{x}_k \end{split}\end{split}\]

\[\rightarrow \quad \tensorII{J} = \frac{\partial \tensorI{x}}{\partial\tensorI\xi} = \sum_k^n \frac{\partial h_k}{\partial\tensorI\xi} \tensorI{x}_k^\T\]

\[\begin{split}\begin{split} \left( \sum_e^{n_e} \tensor{Z}_e^\T \int\limits_{v_e} \rho\,\tensor{H}^\T\tensor{H} \det\tensorII{J}\dif\xi\dif\eta\dif\zeta\, \tensor{Z}_e \right) \tensor{\ddot{u}} \\ \quad + \left( \sum_e^{n_e} \tensor{Z}_e^\T \int_{v_e} \tensor{B}_\xi^\T\tensor\sigma \det\tensorII{J}\dif\xi\dif\eta\dif\zeta \right) \\ \quad = \left( \sum_e^{n_e} \tensor{Z}_e^\T \int_{v_e} \rho\,\tensor{H}^\T\tensorI{b} \det\tensorII{J}\dif\xi\dif\eta\dif\zeta \\ \quad + \sum_e^{n_e} \tensor{Z}_e^\T \int_{s_e} \tensor{H}^\T\tensorI{t}\dif{s} \right) \end{split}\end{split}\]

Polynomial interpolation à la Lagrange. In example of a hexahedron element, with \(n=8\) and \(\xi,\eta,\zeta=[-1,1]\):

\[\begin{split}\begin{array}{ll} h_1 = \frac{1}{8}(1-\xi)(1-\eta)(1+\zeta) & h_5 = \frac{1}{8}(1-\xi)(1-\eta)(1-\zeta) \\ h_2 = \frac{1}{8}(1+\xi)(1-\eta)(1+\zeta) & h_6 = \frac{1}{8}(1+\xi)(1-\eta)(1-\zeta) \\ h_3 = \frac{1}{8}(1+\xi)(1+\eta)(1+\zeta) & h_7 = \frac{1}{8}(1+\xi)(1+\eta)(1-\zeta) \\ h_4 = \frac{1}{8}(1-\xi)(1+\eta)(1+\zeta) & h_8 = \frac{1}{8}(1-\xi)(1+\eta)(1-\zeta) \end{array}\end{split}\]

\[\begin{split}\begin{array}{ll} \frac{\partial h_1}{\partial\xi} = -\frac{1}{8}(1-\eta)(1+\zeta) \\ \frac{\partial h_1}{\partial\eta} = -\frac{1}{8}(1-\xi)(1+\zeta) \\ \frac{\partial h_1}{\partial\zeta} = \phantom{+}\frac{1}{8}(1-\xi)(1-\eta) \\[0.5em] \frac{\partial h_2}{\partial\xi} = \phantom{+}\frac{1}{8}(1-\eta)(1+\zeta) \\ \frac{\partial h_2}{\partial\eta} = -\frac{1}{8}(1+\xi)(1+\zeta) \\ \frac{\partial h_2}{\partial\zeta} = \phantom{+}\frac{1}{8}(1+\xi)(1-\eta) \\[0.5em] \frac{\partial h_3}{\partial\xi} = \phantom{+}\frac{1}{8}(1+\eta)(1+\zeta) \\ \frac{\partial h_3}{\partial\eta} = \phantom{+}\frac{1}{8}(1+\xi)(1+\zeta) \\ \frac{\partial h_3}{\partial\zeta} = \phantom{+}\frac{1}{8}(1+\xi)(1+\eta) \\[0.5em] \frac{\partial h_4}{\partial\xi} = -\frac{1}{8}(1+\eta)(1+\zeta) \\ \frac{\partial h_4}{\partial\eta} = \phantom{+}\frac{1}{8}(1-\xi)(1+\zeta) \\ \frac{\partial h_4}{\partial\zeta} = \phantom{+}\frac{1}{8}(1-\xi)(1+\eta) \end{array}~~\begin{array}{ll} \frac{\partial h_5}{\partial\xi} = -\frac{1}{8}(1-\eta)(1-\zeta) \\ \frac{\partial h_5}{\partial\eta} = -\frac{1}{8}(1-\xi)(1-\zeta) \\ \frac{\partial h_5}{\partial\zeta} = -\frac{1}{8}(1-\xi)(1-\eta) \\[0.5em] \frac{\partial h_6}{\partial\xi} = \phantom{+}\frac{1}{8}(1-\eta)(1-\zeta) \\ \frac{\partial h_6}{\partial\eta} = -\frac{1}{8}(1+\xi)(1-\zeta) \\ \frac{\partial h_6}{\partial\zeta} = -\frac{1}{8}(1+\xi)(1-\eta) \\[0.5em] \frac{\partial h_7}{\partial\xi} = \phantom{+}\frac{1}{8}(1+\eta)(1-\zeta) \\ \frac{\partial h_7}{\partial\eta} = \phantom{+}\frac{1}{8}(1+\xi)(1-\zeta) \\ \frac{\partial h_7}{\partial\zeta} = -\frac{1}{8}(1+\xi)(1+\eta) \\[0.5em] \frac{\partial h_8}{\partial\xi} = -\frac{1}{8}(1+\eta)(1-\zeta) \\ \frac{\partial h_8}{\partial\eta} = \phantom{+}\frac{1}{8}(1-\xi)(1-\zeta) \\ \frac{\partial h_8}{\partial\zeta} = -\frac{1}{8}(1-\xi)(1+\eta) \end{array}\end{split}\]