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

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)

Special Issue Editors

Guest Editor
Prof. Dr. Felix Fritzen

EMMA—Efficient Methods for Mechanical Analysis, Institute of Applied Mechanics (CE), University of Stuttgart, Pfaffenwaldring 7, 70569 Stuttgart, Germany
Website | E-Mail
Interests: solid mechanics; material modeling; model reduction and computer-aided simulation
Guest Editor
Prof. Dr. David Ryckelynck

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
Website | E-Mail
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

Manuscript Submission Information

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Published Papers (10 papers)

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Research

Open AccessFeature PaperArticle
Data-Driven Microstructure Property Relations
Math. Comput. Appl. 2019, 24(2), 57; https://doi.org/10.3390/mca24020057
Received: 26 March 2019 / Revised: 24 May 2019 / Accepted: 28 May 2019 / Published: 31 May 2019
PDF Full-text (3957 KB)
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 [...] Read more.
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
Open AccessFeature PaperArticle
Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling
Math. Comput. Appl. 2019, 24(2), 56; https://doi.org/10.3390/mca24020056
Received: 31 March 2019 / Revised: 24 May 2019 / Accepted: 24 May 2019 / Published: 27 May 2019
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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 [...] Read more.
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
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Open AccessFeature PaperArticle
Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations
Math. Comput. Appl. 2019, 24(2), 45; https://doi.org/10.3390/mca24020045
Received: 6 March 2019 / Revised: 23 April 2019 / Accepted: 23 April 2019 / Published: 24 April 2019
Cited by 1 | PDF Full-text (21203 KB) | HTML Full-text | XML Full-text
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, [...] Read more.
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
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Open AccessFeature PaperArticle
Symplectic Model Order Reduction with Non-Orthonormal Bases
Math. Comput. Appl. 2019, 24(2), 43; https://doi.org/10.3390/mca24020043
Received: 26 February 2019 / Revised: 15 April 2019 / Accepted: 17 April 2019 / Published: 21 April 2019
PDF Full-text (691 KB) | HTML Full-text | XML Full-text | Supplementary Files
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, [...] Read more.
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
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Open AccessFeature PaperArticle
An Error Indicator-Based Adaptive Reduced Order Model for Nonlinear Structural Mechanics—Application to High-Pressure Turbine Blades
Math. Comput. Appl. 2019, 24(2), 41; https://doi.org/10.3390/mca24020041
Received: 26 February 2019 / Revised: 30 March 2019 / Accepted: 15 April 2019 / Published: 19 April 2019
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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 [...] Read more.
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
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Open AccessFeature PaperArticle
An Artificial Neural Network Based Solution Scheme for Periodic Computational Homogenization of Electrostatic Problems
Math. Comput. Appl. 2019, 24(2), 40; https://doi.org/10.3390/mca24020040
Received: 22 March 2019 / Revised: 9 April 2019 / Accepted: 9 April 2019 / Published: 17 April 2019
PDF Full-text (3279 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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
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Open AccessFeature PaperArticle
Toward Optimality of Proper Generalised Decomposition Bases
Math. Comput. Appl. 2019, 24(1), 30; https://doi.org/10.3390/mca24010030
Received: 31 January 2019 / Revised: 28 February 2019 / Accepted: 1 March 2019 / Published: 5 March 2019
PDF Full-text (409 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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
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Open AccessFeature PaperArticle
Reduced-Order Modelling and Homogenisation in Magneto-Mechanics: A Numerical Comparison of Established Hyper-Reduction Methods
Math. Comput. Appl. 2019, 24(1), 20; https://doi.org/10.3390/mca24010020
Received: 10 December 2018 / Revised: 29 January 2019 / Accepted: 29 January 2019 / Published: 1 February 2019
PDF Full-text (2862 KB) | HTML Full-text | XML Full-text
Abstract
The simulation of complex engineering structures built from magneto-rheological elastomers is a computationally challenging task. Using the FE2 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
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Open AccessFeature PaperArticle
Data Pruning of Tomographic Data for the Calibration of Strain Localization Models
Math. Comput. Appl. 2019, 24(1), 18; https://doi.org/10.3390/mca24010018
Received: 12 November 2018 / Revised: 9 January 2019 / Accepted: 22 January 2019 / Published: 28 January 2019
PDF Full-text (6972 KB) | HTML Full-text | XML Full-text | Supplementary Files
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 [...] Read more.
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
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Open AccessFeature PaperArticle
Multiple Tensor Train Approximation of Parametric Constitutive Equations in Elasto-Viscoplasticity
Math. Comput. Appl. 2019, 24(1), 17; https://doi.org/10.3390/mca24010017
Received: 9 November 2018 / Revised: 11 January 2019 / Accepted: 23 January 2019 / Published: 28 January 2019
PDF Full-text (1073 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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
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