On Error-Controlled Numerical Model Reduction for Linear Transient FE² Analysis

Sammanfattning: Multiscale modeling is of high interest in the engineering community due to its ability to capture the overall response, while still accounting for processes and structures on underlying fine scales. One standard approach to multiscale modeling is the so-called FE² procedure, where the classic constitutive relation is replaced by a boundary value problem on a Representative Volume Element (RVE) comprising the underlying microscale features. It is well realized that straight-forward use of the FE²-strategy can be computationally intractable for a fine macroscale mesh. Therefore, it is of interest to reduce the cost of solving the individual RVE-problem(s) by introducing some kind of reduced basis, here denoted Numerical Model Reduction (NMR). However, it is important to note that the richness of the reduced basis will determine the accuracy of the solution, which calls for error control. This thesis concerns numerical model reduction for linear transient problems in the FE² setting, in particular the problems of heat flow and poroelasticity. Two different reduction techniques – Spectral Decomposition and Proper Orthogonal Decomposition – are applied in order to obtain an efficient method of solving and evaluating homogenized quantities on the microscale. For the model problem of linear transient heat flow, the microscale finite element problem reduces to a set of (uncoupled) ordinary differential equations, which, obviously, can be solved more efficiently than the original fully resolved finite element problem. For the error estimation, we focus solely on the error due to the reduced basis and ignore time- and space-discretization errors. We derive guaranteed, explicit bounds on the error in (i) a constructed “energy” norm and (ii) a user-defined quantity of interest (QoI) within the realm of goal-oriented error estimation. As a “workhorse” for the error computation, we introduce an associated (non-physical) symmetrized variational problem in space-time. We obtain low cost estimators, based on the residual, which, in particular, requires no extra modes than the ones used for the reduced basis approximation. The performance of the estimator is demonstrated with numerical examples, and, for both the heat flow problem and the poroelastic problem, we overestimate the error with an order of magnitude, which is deemed acceptable given that the estimate is fully explicit and the extra cost is negligible.