Stephan Rave

Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing

We present a new method for the nonlinear approximation of the solution manifolds of parameterized nonlinear evolution problems, in particular in hyperbolic regimes with moving discontinuities. Given the action of a Lie group on the solution space, the original problem is reformulated as a partial differential algebraic equation system by decomposing the solution into a group component and a spatial shape component, imposing appropriate algebraic constraints on the decomposition. The system is then projected onto a reduced basis space. We show that efficient online evaluation of the scheme is possible and study a numerical example showing its strongly improved performance in comparison to a scheme without freezing.

pyMOR -- Generic Algorithms and Interfaces for Model Order Reduction

Reduced basis methods are projection-based model order reduction techniques for reducing the computational complexity of solving parametrized partial differential equation problems. In this work we discuss the design of pyMOR, a freely available software library of model order reduction algorithms, in particular reduced basis methods, implemented with the Python programming language. As its main design feature, all reduction algorithms in pyMOR are implemented generically via operations on well-defined vector array, operator and discretization interface classes. This allows for an easy integration with existing open-source high-performance partial differential equation solvers without adding any model reduction specific code to these solvers. Besides an in-depth discussion of pyMORs design philosophy and architecture, we present several benchmark results and numerical examples showing the feasibility of our approach.

ArbiLoMod, a Simulation Technique Designed for Arbitrary Local Modifications

Engineers manually optimizing a structure using finite element based simulation software often employ an iterative approach where in each iteration they change the structure slightly and resimulate. Standard finite element based simulation software is usually not well suited for this workflow, as it restarts in each iteration, even for tiny changes. In settings with complex local microstructure, where a fine mesh is required to capture the geometric detail, localized model reduction can improve this workflow. To this end, we introduce ArbiLoMod, a method which allows fast recomputation after arbitrary local modifications. It employs a domain decomposition and a localized form of the reduced basis method for model order reduction. It assumes that the reduced basis on many of the unchanged domains can be reused after a localized change. The reduced model is adapted when necessary, steered by a localized error indicator. The global error introduced by the model order reduction is controlled by a robust and e...

Model Reduction for Multiscale Lithium-Ion Battery Simulation

In this contribution we are concerned with efficient model reduction for multiscale problems arising in lithium-ion battery modeling with spatially resolved porous electrodes. We present new results on the application of the reduced basis method to the resulting instationary 3D battery model that involves strong non-linearities due to Buttler-Volmer kinetics. Empirical operator interpolation is used to efficiently deal with this issue. Furthermore, we present the localized reduced basis multiscale method for parabolic problems applied to a thermal model of batteries with resolved porous electrodes. Numerical experiments are given that demonstrate the reduction capabilities of the presented approaches for these real world applications.

A Model Reduction Framework for Efficient Simulation of Li-Ion Batteries

In order to achieve a better understanding of degradation processes in lithium-ion batteries, the modelling of cell dynamics at the mircometer scale is an important focus of current mathematical research. These models lead to large-dimensional, highly nonlinear finite volume discretizations which, due to their complexity, cannot be solved at cell scale on current hardware. Model order reduction strategies are therefore necessary to reduce the computational complexity while retaining the features of the model. The application of such strategies to specialized high performance solvers asks for new software designs allowing flexible control of the solvers by the reduction algorithms. In this contribution we discuss the reduction of microscale battery models with the reduced basis method and report on our new software approach on integrating the model order reduction software pyMOR with third-party solvers. Finally, we present numerical results for the reduction of a 3D microscale battery model with porous electrode geometry.

True Error Control for the Localized Reduced Basis Method for Parabolic Problems

We present an abstract framework for a posteriori error estimation for approximations of scalar parabolic evolution equations, based on elliptic reconstruction techniques (Makridakis and Nochetto, SIAM J. Numer. Anal. 41(4):1585–1594, 2003. doi:10.1137/S0036142902406314; Lakkis and Makridakis, Math. Comput. 75(256):1627–1658, 2006. doi:10.1090/S0025-5718-06-01858-8; Demlow et al., SIAM J. Numer. Anal. 47(3):2157–2176, 2009. doi:10.1137/070708792; Georgoulis et al., SIAM J. Numer. Anal. 49(2):427–458, 2011. doi:10.1137/080722461). In addition to its original application (to derive error estimates on the discretization error), we extend the scope of this framework to derive offline/online decomposable a posteriori estimates on the model reduction error in the context of Reduced Basis (RB) methods. In addition, we present offline/online decomposable a posteriori error estimates on the full approximation error (including discretization as well as model reduction error) in the context of the localized RB method (Ohlberger and Schindler, SIAM J. Sci. Comput. 37(6):A2865–A2895, 2015. doi:10.1137/151003660). Hence, this work generalizes the localized RB method with true error certification to parabolic problems. Numerical experiments are given to demonstrate the applicability of the approach.

ArbiLoMod: Local Solution Spaces by Random Training in Electrodynamics

The simulation method ArbiLoMod (Buhr et al., SIAM J. Sci. Comput. 2017, accepted) has the goal of providing users of Finite Element based simulation software with quick re-simulation after localized changes to the model under consideration. It generates a Reduced Order Model (ROM) for the full model without ever solving the full model. To this end, a localized variant of the Reduced Basis method is employed, solving only small localized problems in the generation of the reduced basis. The key to quick re-simulation lies in recycling most of the localized basis vectors after a localized model change. In this publication, ArbiLoMod’s local training algorithm is analyzed numerically for the non-coercive problem of time harmonic Maxwell’s equations in 2D, formulated in H(curl).

Localized Reduced Basis Approximation of a Nonlinear Finite Volume Battery Model with Resolved Electrode Geometry

In this contribution we present first results towards localized model order reduction for spatially resolved, three-dimensional lithium-ion battery models. We introduce a localized reduced basis scheme based on non-conforming local approximation spaces stemming from a finite volume discretization of the analytical model and localized empirical operator interpolation for the approximation of the model’s nonlinearities. Numerical examples are provided indicating the feasibility of our approach.

MULTIBAT: Unified workflow for fast electrochemical 3D simulations of lithium-ion cells combining virtual stochastic microstructures, electrochemical degradation models and model order reduction

We present a simulation workflow for efficient investigations of the interplay between 3D lithium-ion electrode microstructures and electrochemical performance, with emphasis on lithium plating. Our approach addresses several challenges. First, the 3D microstructures of porous electrodes are generated by a parametric stochastic model, in order to significantly reduce the necessity of tomographic imaging. Secondly, we integrate a consistent microscopic, 3D spatially-resolved physical model for the electrochemical behavior of the lithium-ion cells taking lithium plating and stripping into account. This highly non-linear mathematical model is solved numerically on the complex 3D microstructures to compute the transient cell behavior. Due to the complexity of the model and the considerable size of realistic microstructures even a single charging cycle of the battery requires several hours computing time. This renders large scale parameter studies extremely time consuming. Hence, we develop a mathematical model order reduction scheme. We demonstrate how these aspects are integrated into one unified workflow, which is a step towards computer aided engineering for the development of more efficient lithium-ion cells.