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The method of vertical lines in non-linear finite elements
The method of vertical lines is commonly drawn on to solve partial differential equations, where two steps have to be carried out. First, the spatial discretization – here, by using finite elements – , and, second, the time discretization. In this lecture, several issues concerning solid mechanics applications with material non-linearities are discussed. In the spatial discretization step, approaches of mixed element formulations, or high-order approximations based on integrated Legendre shape functions are compared apart from the classical Lagrange-element formulations. This leads to systems of differential-algebraic equations in quasi-static approaches. The time discretization step of such systems involves further aspects. First, how are classical constitutive models of evolutionary-type are embedded, and how classical implicit finite elements can be interpreted. Second, how do further time discretization methods work. A particular focus lies on diagonally-implicit Runge-Kutta methods implying the solution of block-structured systems of non-linear equations. In this context, a misinterpretation of the Newton-Raphson method is addressed as well. The talk discusses several extensions such as thermo-mechanics, dynamics, or a specific contact formulation. Caused by the stringent mathematical formulation, parameter identification aspects can be connected to internal and external numerical differentiation schemes as well. Thus, adantages and disadvantages of the approaches, historical aspects and future implications, and the interconnection between engineering approaches and mathematical treatment are discussed.