On Error-Controlled Numerical Model Reduction for Linear Transient FE² Analysis
Licentiate thesis, 2019
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.
Chalmers, Industrial and Materials Science, Material and Computational Mechanics
Numerical model reduction with error control in computational homogenization of transient heat flow
Computer Methods in Applied Mechanics and Engineering,; Vol. 326(2017)p. 193-222
On error controlled numerical model reduction in FE2-analysis of transient heat flow
International Journal for Numerical Methods in Engineering,; Vol. 119(2019)p. 38-73
F. Ekre, F. Larsson, K. Runesson, and R. Jänicke. A posteriori error estimation for numerical model reduction in computational homogenization of porous media
Numerisk modellreduktion vid beräkningsbaserad homogenisering av deformation och strömning i porösa medier
Swedish Research Council (VR), 2016-01-01 -- 2019-12-31.
Chalmers University of Technology
Virtual Development Laboratory (VDL), Hörsalsvägen 7
Opponent: Pedro Díez, LaCàN – Mathematical and Computational Modeling, Universitat Politècnica de Catalunya, Barcelona, Spain