Parametric Reduction of FE Models with Variable Mesh Topology

Abstract

Designing whole machines or processes you may need both, an integrated dynamic simulation of all components on system level and a detailed analysis of how the macroscopic behavior of a component depends on geometry and material parameters. The former analysis is usually based on systems of differential algebraic equations representing a component by not more than a few hundred states and requires tools like Matlab-Simulink R or Dymola R . The latter analysis solves discretized partial differential equations with several 100,000 degrees of freedom using finite element software like Ansys R or Comsol R . Model reduction bridges the gap between the two worlds providing small state space models with approximately the same input-output behavior as the original large finite element models. Building systems from generic components, e.g. a gas transport network from pipeline models with variable length, or optimizing the design of a device with respect to mechanical or thermal properties, we need parametric reduced models. The idea is to reduce FE models offline for selected parameter sets and to generate models for new parameters by cheap interpolation rather than expensive reduction. The different approaches to parametric linear model reduction may be divided into three classes [1]. Interpolation of transfer functions is well suited for parabolic or highly damped hyperbolic problems. However, poles are duplicated rather than shifted, which is unacceptable for weakly damped hyperbolic problems like in mechanics. The second class of methods look for a basis of state space covering system behavior over the full range of parameters. They share the critical assumption that number and meaning of states do not change with the parameters. In terms of finite elements this means that the meshes for different design parameters are morphed variants of the same reference mesh.This may become a severe restriction in practice when automatic meshing is to be applied to complicated geometries. Therefore, we propose a method from the third class, which is based on interpolating reduced system matrices [2]. Only those parts of the mesh need to share a constant topology where nodal inputs are applied and outputs are collected. The inner mesh, however, may change for different parameters. The main challenges arise from the fact that state space representations of a system are unique only up to a change of basis and that interpolating matrices which refer to non-fitting bases may cause arbitrary errors. In the article we will show how problems like leaving and entering modes or eigenvalue crossing can be overcome by using normal forms and eigenvalue tracking in parameter space. The method, which is implemented in the Fraunhofer Model Reduction Toolbox, is applied to a parametric model of a mechanical device the eigenfrequencies of which have to be kept away from some dominant excitation frequencies.