Efficient Interpolation Methods for Nonlinear Finite Element Problems

This thesis investigates various types of interpolation strategies to simplify nonlinear finite element problems. Two major hindrances that often plague such problems are the number of variables and the computational costs associated with the assembly of the nonlinear operators. Therefore, we define an efficient interpolation framework based on the group finite element method and model order reduction techniques to handle these difficulties. First, we develop the extended group finite element method, which interpolates nonlinear terms onto general finite element spaces. With the help of a tensor formulation, we are able to drastically simplify the assembly process. Following this, we introduce proper orthogonal decomposition and the discrete empirical interpolation method to our extended group finite element method to further reduce the problem size. Along the way, a broad selection of benchmark problems demonstrate the advantages of our proposed interpolation framework. Finally, we consider a practical medical application with the goal of identifying patient-specific parameters in real-time. Faced with additional difficulties, we apply the space-mapping method to our interpolation framework in order to derive an efficient optimization approach that warrants future research.