Topological derivatives for the optimization of crashloaded structures
Optimizing structural crash safety is an important field of research in vehicle safety. Based on crash simulation, it can be useful to apply methods for dimensioning, shape and topology optimization. However, a crash is a highly dynamic process and characterized by strong nonlinearities. This makes it difficult to determine analytical sensitivities of the objective and the constraints with respect to the design variables, which are important for many optimization procedures. In this thesis, the Topological Derivative considering material nonlinearity, large deformation and time dependency is presented. Here, the Topological Derivative indicates how a functional would be affected by introducing a cut-out. Using the adjoint method and an interpolation of the elasto-plastic material behavior, a method for the determination of this sensitivity is developed. The functionals are extended by the mechanical nonlinear transient equilibrium equations with a Lagrange multiplier (the adjoint). This is chosen in a way that the implicit derivatives, such as the derivative of the displacement with respect to the design variable, are circumvented and only the explicit derivation terms, as for example the derivation of the functional with respect to the design variable, are required. For the calculation of the adjoint, a terminal value problem has to be solved. Depending on whether first differentiation and then discretization in the time domain is performed or vice versa, a separate solution scheme for the adjoint is derived. The general development allows also the usage of the adjoint solution scheme for functionals, which contain velocities or accelerations. For the internal
energy of a structure and the displacement of a single point, the Topological Derivative is specified in the analytical form. For the calculation of the Topological Derivative, integration terms remain, whose analytical resolution is no longer possible. This integration is replaced by a material interpolation, which represents the material behavior resulting from plastic strain and isotropic hardening in the temporarily linearized form.
Shaker Verlag, ISBN: 978-3-8440-8248-7
