Numerical Model Reduction and Error Control for Computational Homogenization of Transient Problems
Doctoral thesis, 2021

Multiscale modeling is a class of methods useful for numerical simulation of mechanics, in particular, when the microstructure of a material is of importance. The main advantage is the ability to capture the overall response, and, at the same time, account for processes and structures on the underlying fine scales. The FE2 procedure, "finite element squared", is one standard multiscale approach in which the constitutive relation is replaced with a boundary value problem defined on an Representative Volume Element (RVE) which contains the microscale features. The procedure thus involves the solution of finite element problems on two scales: one macroscopic problem and multiple RVE problems, typically one for each quadrature point in the macroscale mesh. While the solution of the independent RVE problems can be trivially parallelized it can still be computationally impractical to solve the two-scale problem, in particular for fine macroscale meshes. It is, therefore, of interest to investigate methods for reducing the computational cost of solving the individual RVE problems, while still having control of the accuracy.

In this thesis the concept of Numerical Model Reduction (NMR) is applied for reducing the RVE problems by constructing a reduced spatial basis using Spectral Decomposition (SD) and Proper Orthogonal Decomposition. Computational homogenization of two different transient model problems have been studied: heat flow and consolidation. In both cases the RVE problem reduces to a system of ordinary differential equations, with dimension much smaller than of the finite element system.

With the reduced basis and decreased computational time comes also loss of accuracy. Thus, in order to assess results from a reduced computation, it is useful to quantify the error. This thesis focuses solely on estimation of the error stemming from the reduced basis by assuming the fully resolved finite element solution to be exact, thereby ignoring e.g. time- and space-discretization errors. For the linear model problems guaranteed, fully computable, bounds are derived for the error in (i) a constructed "energy" norm and (ii) a user-defined quantity of interest within the realm of goal-oriented error estimation. In the non-linear case approximate, fully computable, bounds are derived based on the linearized error equation.

In all cases an associated (non-physical) symmetrized variational problem in space-time is introduced as a "driver" for the estimate. From this residual-based estimates with low computational cost are obtained. In particular, no extra modes than the ones used for the reduced basis approximation are required. The performance of the estimator is demonstrated with numerical examples, and, for both the heat flow problem and the poroelastic problem, the error is overestimated by an order of magnitude, which is deemed acceptable given that the estimate is fully explicit and the extra cost is negligible.

model reduction

error estimation

computational homogenization

Online (link below, contact for password)
Opponent: Professor Ludovic Chamoin, LMT (Laboratory of Mechanics and Technology), ENS Paris-Saclay, France


Fredrik Ekre

Chalmers, Industrial and Materials Science, Material and Computational Mechanics

A posteriori error estimation for numerical model reduction in computational homogenization of porous media

International Journal for Numerical Methods in Engineering,; Vol. 121(2020)p. 5350-5380

Journal article

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

Journal article

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

Journal article

Efficient Two-Scale Modeling of Porous Media Using NumericalModel Reduction with Fully Computable Error Bounds

Current Trends and Open Problems in Computational Mechanics,; (2022)p. 121-129

Book chapter

Numerical Model Reduction with error estimation for computational homogenization of non-linear consolidation

Computer Methods in Applied Mechanics and Engineering,; Vol. 389(2022)

Journal article

Today computers are used in the design process for almost all products, ranging from smartphones and milk cartons to cars and bridges. Instead of testing different materials or product designs in a laboratory, it is now possible to perform virtual testing using computer simulations. Such virtual experiments have many benefits over real life testing. There is, for example, no need to manufacture test samples and no need for expensive test equipment. Usually it is also faster to perform computer experiments which means that new designs can be evaluated faster.

Even though computers become more powerful every year there are situations where it is not possible to carry out this type of simulation. One example of such a demanding application is the detailed analysis of transport processes in concrete. At a first glance concrete appears to be a homogeneous material, however, when looking closer it becomes clear that this is not the case and performing a detailed simulation will require lots of computational resources. In order to remedy this problem, the concept of multiscale modeling has been developed where the material is analyzed at different scales. In this case the original expensive simulation is split up into many, less demanding, simulations: one simulation for the macroscopic behavior, and many small simulations for the microscopic behavior.

This thesis investigates methods for further reducing the computational cost of simulating all of the microscopic problems. As point of departure, the simulations are established using the finite element method, where the sought solution fields are discretized using a finite, but large, set of degrees of freedom. In order to reduce the computation cost, numerical model reduction is employed, whereby characteristic response modes of the solution are identified. Each simulation can thereby be carried out using a much smaller number of degrees of freedom, and thus at a reduced computational cost. While such an approximation results in faster simulations it also introduces an error in the answer. An important part of this thesis has therefore been to develop methods for quantifying this error to make sure that the accuracy of the approximation is within acceptable bounds.

Numerisk modellreduktion vid beräkningsbaserad homogenisering av deformation och strömning i porösa medier

Swedish Research Council (VR) (2015-05422), 2016-01-01 -- 2019-12-31.

Numerical model reduction for computational homogenization of polycrystals

Swedish Research Council (VR) (2019-05080), 2020-01-01 -- 2023-12-31.

Subject Categories

Computational Mathematics

Control Engineering



Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4974



Online (link below, contact for password)


Opponent: Professor Ludovic Chamoin, LMT (Laboratory of Mechanics and Technology), ENS Paris-Saclay, France

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