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Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics

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A special issue of Mathematical and Computational Applications (ISSN 2297-8747). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: closed (31 March 2019) | Viewed by 44885

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Prof. Dr. Felix Fritzen

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Guest Editor

EMMA—Efficient Methods for Mechanical Analysis, Institute of Applied Mechanics (CE), University of Stuttgart, Pfaffenwaldring 7, 70569 Stuttgart, Germany
Interests: solid mechanics; material modeling; model reduction and computer-aided simulation

Prof. Dr. David Ryckelynck

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Guest Editor

Head of the Ph.D Program in Mechanical Engineering, Head of CoCas Team Affiliation: MAT-Centre des Matériaux, MINES ParisTech, PSL Research University, CNRS UMR 7633, BP 87, 91003 Evry, France
Interests: computational methods in structural mechanics (model reduction of non-linear systems, hyper-reduction, incremental proper orthogonal decomposition, contact and parallel computations); multiphysic modeling of thermomechanical processes; inverse problems

Special Issue Information

Dear Colleagues,

The solution of engineering problems using simulation tools has reached a mature state over the past few decades. With the ever-increasing features included into the simulations, the technical challenges for simulations have increased. This is particularly true as simulations are also used to explore high-dimensional parameter spaces, to optimize designs, provide means for optimal control problems and—more recently—they are carried out on low-cost devices. The replacement of dedicated simulations by data-driven methods, by low-rank approximations and by reduced modeling strategies is an active field of research that is quickly finding its way into industrial applications for obvious reasons. Articles related to the development and the properties of methods from the fields of machine learning, tensor and low-rank approximations and reduced order modeling are welcome in this special issue. Papers connecting the different disciplines and regarding error control for surrogate models are particularly welcome. Authors are invited to upload supplementary material, e.g., software, data-sets or instructive videos complementing the research.

Prof. Dr. Felix Fritzen
Prof. Dr. David Ryckelynck
Guest Editors

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Published Papers (11 papers)

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27 pages, 4128 KiB

Open AccessFeature PaperArticle

Data-Driven Microstructure Property Relations

by Julian Lißner and Felix Fritzen

Math. Comput. Appl. 2019, 24(2), 57; https://doi.org/10.3390/mca24020057 - 31 May 2019

Cited by 9 | Viewed by 4205

Abstract

An image based prediction of the effective heat conductivity for highly heterogeneous microstructured materials is presented. The synthetic materials under consideration show different inclusion morphology, orientation, volume fraction and topology. The prediction of the effective property is made exclusively based on image data with the main emphasis being put on the 2-point spatial correlation function. This task is implemented using both unsupervised and supervised machine learning methods. First, a snapshot proper orthogonal decomposition (POD) is used to analyze big sets of random microstructures and, thereafter, to compress significant characteristics of the microstructure into a low-dimensional feature vector. In order to manage the related amount of data and computations, three different incremental snapshot POD methods are proposed. In the second step, the obtained feature vector is used to predict the effective material property by using feed forward neural networks. Numerical examples regarding the incremental basis identification and the prediction accuracy of the approach are presented. A Python code illustrating the application of the surrogate is freely available. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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28 pages, 1750 KiB

Open AccessFeature PaperArticle

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

by Oliver Kunc and Felix Fritzen

Math. Comput. Appl. 2019, 24(2), 56; https://doi.org/10.3390/mca24020056 - 27 May 2019

Cited by 22 | Viewed by 4009 | Correction

Abstract

The computational homogenization of hyperelastic solids in the geometrically nonlinear context has yet to be treated with sufficient efficiency in order to allow for real-world applications in true multiscale settings. This problem is addressed by a problem-specific surrogate model founded on a reduced basis approximation of the deformation gradient on the microscale. The setup phase is based upon a snapshot POD on deformation gradient fluctuations, in contrast to the widespread displacement-based approach. In order to reduce the computational offline costs, the space of relevant macroscopic stretch tensors is sampled efficiently by employing the Hencky strain. Numerical results show speed-up factors in the order of 5–100 and significantly improved robustness while retaining good accuracy. An open-source demonstrator tool with 50 lines of code emphasizes the simplicity and efficiency of the method. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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21 pages, 21203 KiB

Open AccessFeature PaperArticle

Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations

by Nissrine Akkari, Fabien Casenave and Vincent Moureau

Math. Comput. Appl. 2019, 24(2), 45; https://doi.org/10.3390/mca24020045 - 24 Apr 2019

Cited by 14 | Viewed by 4587

Abstract

In the following paper, we consider the problem of constructing a time stable reduced order model of the 3D turbulent and incompressible Navier–Stokes equations. The lack of stability associated with the order reduction methods of the Navier–Stokes equations is a well-known problem and, in general, it is very difficult to account for different scales of a turbulent flow in the same reduced space. To remedy this problem, we propose a new stabilization technique based on an a priori enrichment of the classical proper orthogonal decomposition (POD) modes with dissipative modes associated with the gradient of the velocity fields. The main idea is to be able to do an a priori analysis of different modes in order to arrange a POD basis in a different way, which is defined by the enforcement of the energetic dissipative modes within the first orders of the reduced order basis. This enables us to model the production and the dissipation of the turbulent kinetic energy (TKE) in a separate fashion within the high ranked new velocity modes, hence to ensure good stability of the reduced order model. We show the importance of this a priori enrichment of the reduced basis, on a typical aeronautical injector with Reynolds number of 45,000. We demonstrate the capacity of this order reduction technique to recover large scale features for very long integration times (25 ms in our case). Moreover, the reduced order modeling (ROM) exhibits periodic fluctuations with a period of

2.2

ms corresponding to the time scale of the precessing vortex core (PVC) associated with this test case. We will end this paper by giving some prospects on the use of this stable reduced model in order to perform time extrapolation, that could be a strategy to study the limit cycle of the PVC. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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26 pages, 691 KiB

Open AccessFeature PaperArticle

Symplectic Model Order Reduction with Non-Orthonormal Bases

by Patrick Buchfink, Ashish Bhatt and Bernard Haasdonk

Math. Comput. Appl. 2019, 24(2), 43; https://doi.org/10.3390/mca24020043 - 21 Apr 2019

Cited by 32 | Viewed by 4488

Abstract

Parametric high-fidelity simulations are of interest for a wide range of applications. However, the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g., structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such an ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As a new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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28 pages, 6635 KiB

Open AccessFeature PaperArticle

An Error Indicator-Based Adaptive Reduced Order Model for Nonlinear Structural Mechanics—Application to High-Pressure Turbine Blades

by Fabien Casenave and Nissrine Akkari

Math. Comput. Appl. 2019, 24(2), 41; https://doi.org/10.3390/mca24020041 - 19 Apr 2019

Cited by 5 | Viewed by 3848

Abstract

The industrial application motivating this work is the fatigue computation of aircraft engines’ high-pressure turbine blades. The material model involves nonlinear elastoviscoplastic behavior laws, for which the parameters depend on the temperature. For this application, the temperature loading is not accurately known and can reach values relatively close to the creep temperature: important nonlinear effects occur and the solution strongly depends on the used thermal loading. We consider a nonlinear reduced order model able to compute, in the exploitation phase, the behavior of the blade for a new temperature field loading. The sensitivity of the solution to the temperature makes the classical unenriched proper orthogonal decomposition method fail. In this work, we propose a new error indicator, quantifying the error made by the reduced order model in computational complexity independent of the size of the high-fidelity reference model. In our framework, when the error indicator becomes larger than a given tolerance, the reduced order model is updated using one time step solution of the high-fidelity reference model. The approach is illustrated on a series of academic test cases and applied on a setting of industrial complexity involving five million degrees of freedom, where the whole procedure is computed in parallel with distributed memory. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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28 pages, 3279 KiB

Open AccessFeature PaperArticle

An Artificial Neural Network Based Solution Scheme for Periodic Computational Homogenization of Electrostatic Problems

by Felix Selim Göküzüm, Lu Trong Khiem Nguyen and Marc-André Keip

Math. Comput. Appl. 2019, 24(2), 40; https://doi.org/10.3390/mca24020040 - 17 Apr 2019

Cited by 9 | Viewed by 5210

Abstract

The present work addresses a solution algorithm for homogenization problems based on an artificial neural network (ANN) discretization. The core idea is the construction of trial functions through ANNs that fulfill a priori the periodic boundary conditions of the microscopic problem. A global potential serves as an objective function, which by construction of the trial function can be optimized without constraints. The aim of the new approach is to reduce the number of unknowns as ANNs are able to fit complicated functions with a relatively small number of internal parameters. We investigate the viability of the scheme on the basis of one-, two- and three-dimensional microstructure problems. Further, global and piecewise-defined approaches for constructing the trial function are discussed and compared to finite element (FE) and fast Fourier transform (FFT) based simulations. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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14 pages, 409 KiB

Open AccessFeature PaperArticle

Toward Optimality of Proper Generalised Decomposition Bases

by Shadi Alameddin, Amélie Fau, David Néron, Pierre Ladevèze and Udo Nackenhorst

Math. Comput. Appl. 2019, 24(1), 30; https://doi.org/10.3390/mca24010030 - 5 Mar 2019

Cited by 7 | Viewed by 3939

Abstract

The solution of structural problems with nonlinear material behaviour in a model order reduction framework is investigated in this paper. In such a framework, greedy algorithms or adaptive strategies are interesting as they adjust the reduced order basis (ROB) to the problem of interest. However, these greedy strategies may lead to an excessive increase in the size of the ROB, i.e., the solution is no more represented in its optimal low-dimensional expansion. Here, an optimised strategy is proposed to maintain, at each step of the greedy algorithm, the lowest dimension of a Proper Generalized Decomposition (PGD) basis using a randomised Singular Value Decomposition (SVD) algorithm. Comparing to conventional approaches such as Gram–Schmidt orthonormalisation or deterministic SVD, it is shown to be very efficient both in terms of numerical cost and optimality of the ROB. Examples with different mesh densities are investigated to demonstrate the numerical efficiency of the presented method. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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28 pages, 2862 KiB

Open AccessFeature PaperArticle

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

by Benjamin Brands, Denis Davydov, Julia Mergheim and Paul Steinmann

Math. Comput. Appl. 2019, 24(1), 20; https://doi.org/10.3390/mca24010020 - 1 Feb 2019

Cited by 8 | Viewed by 3894

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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25 pages, 6972 KiB

Open AccessFeature PaperArticle

Data Pruning of Tomographic Data for the Calibration of Strain Localization Models

by William Hilth, David Ryckelynck and Claire Menet

Math. Comput. Appl. 2019, 24(1), 18; https://doi.org/10.3390/mca24010018 - 28 Jan 2019

Cited by 3 | Viewed by 3539

Abstract

The development and generalization of Digital Volume Correlation (DVC) on X-ray computed tomography data highlight the issue of long-term storage. The present paper proposes a new model-free method for pruning experimental data related to DVC, while preserving the ability to identify constitutive equations (i.e., closure equations in solid mechanics) reflecting strain localizations. The size of the remaining sampled data can be user-defined, depending on the needs concerning storage space. The proposed data pruning procedure is deeply linked to hyper-reduction techniques. The DVC data of a resin-bonded sand tested in uniaxial compression is used as an illustrating example. The relevance of the pruned data was tested afterwards for model calibration. A Finite Element Model Updating (FEMU) technique coupled with an hybrid hyper-reduction method aws used to successfully calibrate a constitutive model of the resin bonded sand with the pruned data only. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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17 pages, 1073 KiB

Open AccessFeature PaperArticle

Multiple Tensor Train Approximation of Parametric Constitutive Equations in Elasto-Viscoplasticity

by Clément Olivier, David Ryckelynck and Julien Cortial

Math. Comput. Appl. 2019, 24(1), 17; https://doi.org/10.3390/mca24010017 - 28 Jan 2019

Cited by 4 | Viewed by 3743

Abstract

This work presents a novel approach to construct surrogate models of parametric differential algebraic equations based on a tensor representation of the solutions. The procedure consists of building simultaneously an approximation given in tensor-train format, for every output of the reference model. A parsimonious exploration of the parameter space coupled with a compact data representation allows alleviating the curse of dimensionality. The approach is thus appropriate when many parameters with large domains of variation are involved. The numerical results obtained for a nonlinear elasto-viscoplastic constitutive law show that the constructed surrogate model is sufficiently accurate to enable parametric studies such as the calibration of material coefficients. Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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2 pages, 196 KiB

Open AccessCorrection

Correction: Kunc, O.; Fritzen, F. Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling. Math. Comput. Appl. 2019, 24, 56

by Oliver Kunc and Felix Fritzen

Math. Comput. Appl. 2019, 24(4), 95; https://doi.org/10.3390/mca24040095 - 6 Nov 2019

Cited by 1 | Viewed by 1920

Abstract

The authors wish to make a correction to Formula (42) of the paper [...] Full article

(This article belongs to the Special Issue Machine Learning, Low-Rank Approximations and Reduced Order Modeling in Computational Mechanics)

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