Investigation of Accuracy, Speed and Stability of Hyper-Reduction Techniques for Nonlinear FE

Abstract

The field of Hyper-reduction for Nonlinear Finite Element Method attempts to address the large durations due to repeated evaluation and assembly of the internal force and Jacobian. Stability, accuracy and speed are three aspects of thesemethods that has been dealt with in this thesis. There are two methods that are popular within the FEMframework, these are, DEIM and it variants, and ECSW. By construction, DEIM is quite unstable and has convergence issues, as the Lagrangian structure is not preserved during hyper-reduction. A recent paper by Chaturantabut, preserves the structure while using DEIM and hence assures stability and passivity of the hyper-reduced model in the context of reducing internal forces that are scalar-valued. With this thesis the possibility of restoring the structure in the context of FEM, i.e., reduction of vector-valued internal forces is investigated. It is found that, the extension of structure preserving DEIM to FEM, did not work as expected, owing to certain characteristics of FEM. In DEIM, traditionally the degrees of freedoms (dofs) at which the internal force is evaluated is equal to the number of force modes. The effect of having more number of evaluations as compared to force modes is investigated. It is found that increase in the number of evaluations does improve accuracy and also the resulting stability, with increases in the computation time. ECSW is a recent hyper-reduction technique, and is stable as a result of the preservation of the Lagrangian structure. The properties of this method are investigated. As a conclusion to this thesis, a study is performed on the different methods across five examples of varied complexity. It is found that UDEIM with nodal collocation performs well with accuracy, speed and stability across all examples.