8.3 – Matrixformulierung

 

Auszug aus dem Skript der Vorlesung Finite Elemente bei Dr.-Ing Philipp Höfer an der UniBw München.

Vektor des Temperaturgradienten:

{\left\{ g \right\}^T} = \left\{{\begin{array}{*{20}{c}}{\frac{{\partial \theta }}{{\partial x}}}&{\frac{{\partial \theta }}{{\partial y}}}&{\frac{{\partial \theta }}{{\partial z}}} \end{array}} \right\} = \left[ D \right]\theta

bzw.

{\left\{{\delta g} \right\}^T} = \left\{{\begin{array}{*{20}{c}}{\frac{{\partial \delta \theta }}{{\partial x}}}&{\frac{{\partial \delta \theta }}{{\partial y}}}&{\frac{{\partial \delta \theta }}{{\partial z}}} \end{array}} \right\} = \left[ D \right]\delta \theta

mit

\left[ D \right] = \left[ {\begin{array}{*{20}{c}}{\frac{\partial }{{\partial x}}} \\ {\frac{\partial }{{\partial y}}} \\ {\frac{\partial }{{\partial z}}} \end{array}} \right]

Normalenvektor:

{\left\{ n \right\}^T} = \left\{{\begin{array}{*{20}{c}}{{n_1}}&{{n_2}}&{{n_3}} \end{array}} \right\}

\Rightarrow \quad \int\limits_V {\rho {c_V}\dot \theta \delta \theta dV} = -\int\limits_V {{{\left\{{\delta g} \right\}}^T}\lambda \left\{ g \right\}dV} +\int\limits_V {\rho r\delta \theta dV} +\int\limits_A {\sigma \delta \theta {{\left\{ g \right\}}^T}\left\{ n \right\}dA}