Reduced-Order Modelling and Homogenisation in Magneto-Mechanics: A Numerical Comparison of Established Hyper-Reduction Methods

The simulation of complex engineering structures built from magneto-rheological elastomers is a computationally challenging task. Using the FE 2 method, which is based on computational homogenisation, leads to the repetitive solution of micro-scale FE problems, causing excessive computational effort. In this paper, the micro-scale FE problems are replaced by POD reduced models of comparable accuracy. As these models do not deliver the required reductions in computational effort, they are combined with hyper-reduction methods like the Discrete Empirical Interpolation Method (DEIM), Gappy POD, Gauss–Newton Approximated Tensors (GNAT), Empirical Cubature (EC) and Reduced Integration Domain (RID). The goal of this work is the comparison of the aforementioned hyper-reduction techniques focusing on accuracy and robustness. For the application in the FE 2 framework, EC and RID are favourable due to their robustness, whereas Gappy POD rendered both the most accurate and efficient reduced models. The well-known DEIM is discarded for this application as it suffers from serious robustness deficiencies

A numerical study of different projection-based model reduction techniques applied to computational homogenisation

Computing the macroscopic material response of a continuum body commonly involves the formulation of a phenomenological constitutive model. However, the response is mainly influenced by the heterogeneous microstructure. Computational homogenisation can be used to determine the constitutive behaviour on the macro-scale by solving a boundary value problem at the micro-scale for every so-called macroscopic material point within a nested solution scheme. Hence, this procedure requires the repeated solution of similar microscopic boundary value problems. To reduce the computational cost, model order reduction techniques can be applied. An important aspect thereby is the robustness of the obtained reduced model. Within this study reduced-order modelling (ROM) for the geometrically nonlinear case using hyperelastic materials is applied for the boundary value problem on the micro-scale. This involves the Proper Orthogonal Decomposition (POD) for the primary unknown and hyper-reduction methods for the arising nonlinearity. Therein three methods for hyper-reduction, differing in how the nonlinearity is approximated and the subsequent projection, are compared in terms of accuracy and robustness. Introducing interpolation or Gappy-POD based approximations may not preserve the symmetry of the system tangent, rendering the widely used Galerkin projection sub-optimal. Hence, a different projection related to a Gauss-Newton scheme (Gauss-Newton with Approximated Tensors- GNAT) is favoured to obtain an optimal projection and a robust reduced model

The deal.II Library, Version 9.0

This paper provides an overview of the new features of the finite element library deal.II version 9.0

Modellreduktion für gekoppelte Multiskalen-Probleme der Mechanik

The predictive simulation of elaborate engineering structures based on magneto-active polymers is computationally demanding as the material’s microstructure, which defines the constitutive behaviour, has to be taken into account. The cost of a direct numerical simulation based on a discretisation of the microstructure within the macro body exceeds even the capabilities of modern supercomputers. The computational burden is alleviated using multi-scale techniques such as the FE^2 method. The FE^2 method, whose root is computational homogenisation, requires the repeated solution of microstructural finite element models and nonetheless causes enormous numerical cost. To cut cost, the finite element models are substituted by POD reduced models. However, these models do not provide the desired speed-ups and make the application of hyper- reduction methods inevitable. To pick an appropriate approach and obtain efficient reduced models, the discrete empirical interpolation method, Gappy POD, reduced integration domain and empirical quadrature are validated for their applicability to coupled problems with regard to accuracy and robustness. Empirical quadrature outperforms the other aforementioned methods and is employed in FE^2 simulations. A comparison with a reference finite element solution reveals that the integration of reduced models produces sufficiently small errors. The result of using reduced models are compelling speed-ups in the order of 10^2 to 10^3.Die Simulation ausgeklügelter Bauteile bestehend aus magneto-sensitiven Materialien ist mit hohem Rechenaufwand verbunden, da die für das Materialverhalten verantwortliche Mikrostruktur berücksichtigt werden muss. Der Rechenaufwand einer direkten numerischen Simulation, welche die Mikro- zusammen mit der Makrostruktur diskretisiert, überschreitet selbst die Kapazitäten heutiger Hochleistungsrechner. Die Rechenlast wird durch den Einsatz von Multiskalen-Verfahren wie der FE^2 -Methode vermindert. Die FE^2 -Methode, welche auf numerischer Homogenisierung basiert, erfordert die wiederholte Lösung von Finite-Elemente Problemen auf der Mikroskala und benötigt nichtsdestotrotz enorme Rechenkapazitäten. Die FE-Probleme auf der Mikroskala werden durch POD-reduzierte Modelle ersetzt, um den Rechenaufwand entscheidend zu senken. Diese Modelle liefern jedoch immer noch nicht die gewünschten Speed-Ups und erfordern den Einsatz sogenannter Hyperreduktions Methoden. Um effiziente reduzierte Modelle zu erstellen, muss eine geeignete Hyperreduktions Methode ausgewählt werden. Die diskrete empirische Interpolationsmethode, Gappy POD, das reduziertes Integrationsgebiet und die empirische Quadratur werden mit Berücksichtung auf deren Eignung für gekoppelte Probleme hinsichtlich Genauigkeit und Robustheit miteinander verglichen und validiert. Die empirische Quadratur übertrifft die anderen Verfahren und wird deswegen in FE^2 -Simulationen eingesetzt. Ein Vergleich mit einer Referenzlösung, die mit FEM berechnet wurde, zeigt, dass der auf die reduzierten Modelle zurückzuführende Fehler vergleichsweise gering ist. Der Einsatz reduzierter Modelle ermöglicht Speed-Ups in der Größenordnung von 10^2 bis zu 10^3

Reduced-order modelling for linear heat conduction with parametrised moving heat sources

The purpose of Reduced‐Order Modelling (ROM) is to substantially lower the computational cost of numerical simulations. As an example proper orthogonal decomposition (POD), which is optimal in a least‐square sense, is applied to compute basis functions using existing solutions, e.g. from previously computed simulations. These reduced basis functions are subsequently used for the Galerkin projection of the governing equations. There is intense research on the application of ROM to parametrised partial differential equations (pPDE). However, in the case of transient equations research on ROM is mainly focused on models with constant parameters. In this paper an approach to cope with the linear heat conduction problem with a moving heat source is introduced. Problems of such kind are encountered in the simulation of laser or electron beam melting processes in additive manufacturing. To construct the reduced basis a nested POD method is used to mitigate the com‐putational costs of large‐scale eigenproblems. To further gain computational efficiency the discrete empirical interpolation method (DEIM) is applied to deal with non‐affine parameter dependencies. Based on various numerical examples the approximation quality of the reduced models is discussed through direct comparison with large‐scale Finite Element simulations

Efficient E-Cash in Practice: NFC-Based Payments for Public Transportation Systems

Abstract. Near field communication (NFC) is a recent popular technology that will facilitate many aspects of payments with mobile tokens. In the domain of public transportation payment systems electronic payments have many benefits, including improved throughput, new capabilities (congestion-based pricing etc.) and user convenience. A common concern when using electronic payments is that a user’s privacy is sacrificed. However, cryptographic e-cash schemes pro-vide provable guarantees for both security and user privacy. Even though e-cash protocols have been proposed three decades ago, there are relatively few ac-tual implementations, since their computation complexity makes an execution on lightweight devices rather difficult. This paper presents an efficient implemen-tation of Brands [11] and ACL [4] e-cash schemes on an NFC smartphone: the BlackBerry Bold 9900. Due to their efficiency during the spending phase, when compared to other schemes, and the fact that payments can be verified offline, these schemes are especially suited for, but not limited to, use in public trans-port. Additionally, the encoding of validated attributes (e.g. a user’s age range, zip code etc.) is possible in the coins being withdrawn, which allows for addi-tional features such as variable pricing (e.g. reduced fare for senior customers) and privacy-preserving data collection. We present a subtle technique to make use of the ECDHKeyAgreement class that is available in the BlackBerry API (and in the API of other systems) and show how the schemes can be implemented efficiently to satisfy the tight timing imposed by the transportation setting.