Abstract
Numerical Model Reduction (NMR) is exploited for solving the finite element problem on a Representative Volume Element (RVE) that arises from the computational homogenization of a model problem of transient heat flow. Since the problem is linear, an orthogonal basis is obtained via the classical method of spectral decomposition. A symmetrized version of the space–time variational format is adopted for estimating the error from the model reduction in (i) energy norm and in (ii) given Quantities of Interest. This technique, which was recently developed in the context of the (non-selfadjoint) stationary diffusion–convection problem, is novel in the present context of NMR. By considering the discrete, unreduced, model as exact, we are able to obtain guaranteed bounds on the error while using only the reduced basis and with minor computational effort. The performance of the error estimates is demonstrated via numerical results, where the subscale is modeled in both one and three spatial dimensions.
Highlights
FE2 is the technique of solving two-scale finite element problems using computational homogenization on Representative Volume Elements (RVEs) pertinent to each macroscale quadrature point
Quite important is the obvious fact that the richness of the reduced basis will determine the accuracy of the RVE-solution, which calls for error control. (Here, we consider the full FE-solution as the exact one.) An example of error estimation due to model reduction, not in a homogenization context and for a PGD-basis, is Ladeveze and Chamoin[3]
We aim for guaranteed, explicit bounds on the error in (i) energy norm and (ii) an arbitrary “quantity of interest” (QoI) within the realm of goal-oriented error estimation
Summary
FE2 is the technique of solving two-scale finite element problems using computational homogenization on Representative Volume Elements (RVEs) pertinent to each macroscale quadrature point. There is significant interest in reducing the cost of solving the individual RVE-problem(s) by introducing some kind of reduced basis, here denoted Numerical Model Reduction (NMR). (Here, we consider the full FE-solution as the exact one.) An example of error estimation due to model reduction, not in a homogenization context and for a PGD-basis, is Ladeveze and Chamoin[3]. We consider the transient heat flow as a model problem and choose, for simplicity, to use spectral decomposition to establish the reduced basis. For this particular choice of basis, we discuss a few strategies to estimate the “solution error” without computing additional basis functions (modes). Numerical results for a simple 1D-problem illustrate the performance and quality of the proposed error estimates
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