Reduced Basis method for closed-form affine dependent second order systems

Abstract

This thesis proposes the use of Reduced Basis (RB) methods to improve the computational efficiency of simulations in the field of elastodynamics and acoustics including poroelastic materials. RB methods are Model Order Reduction techniques used to generate parametric Reduced Order Models (ROM). The are many reasons for current researchers to focus on MOR for computational improvements. The technological development of computers and hardware has led to using brute force for calculations of large matrices projected onto simple shape-functions rather than, as it was normally done in the 60s and 70s, trying shrink the size of the matrices using special shapefunctions (i.e., specific for the different systems) [1]. The purpose of MOR techniques is to use these enormously detailed but slow (to compute and even to read) data to generate those smart shape functions. Hence, the resulting ROM contain the level of detail of those huge models, referred to as high fidelity models or full order models (FOM), offering high computational performances. These characteristics of ROM can strongly enlarge the horizons of optimization techniques enabling repeated simulation at high rate or, in some cases, allow real-time simulations paving the way for e.g. virtual sensing, haptic technology, computer graphics. A common strategy to do MOR is to use projection-based techniques that apply to semi-discretised models (e.g. finite element models). A projection transforms the basis that describes the multidimensional space of the model to be much smaller. Thus, a projection of the model into a subspace that contains all and only the dimensions necessary to describe the model will minimize the computation effort. The field of MOR includes dozens of methodologies and this thesis does not pretend to cover all of them. The focus of the work is to develop methods based on projection that are able to generate ROM with explicit parametric dependency typically indicated under the category Parametric Model Order Reduction (PMOR). Changes of the parameters configuration affect the shape of the multidimensional space. Therefore, to obtain a reduced parametric solution, a manifold of all the basis corresponding to the different parameter configurations is needed. Among the possible approaches available to do PMOR, the RB methods achieve efficient results separating the parametric dependent and parametric independent quantities in the FOM. This enable an efficient reduction and originates ROM whose operations are independent from the size of the former FOM. The research brought to a parametric approach in the frequency domain that can take into account the nonlinear frequency dependent characteristics of poroelastic materials (PEM). Also this methodology is verified using few numerical examples. In addition, a parametric approach to study elastodynamic problems of linear structures made of beams is presented and applied. The results of the study are discussed and validated with direct comparison to direct FE simulations. In addition to the original contribution, the research reported in this thesis raises some new questions that could set the start of new research projects in the field of PMOR and are discussed in the conclusion to this work.