Lars Beex: Unsupervised learning to select modes for reduced-order models of hyperelastoplasticity: application to RVEs

Machine Learning Seminar presentation

Topic: A Physics-guided Machine Learning Model Based on Peridynamics

Speaker: Dr Lars Beex, FSTM, University of Luxembourg

Authors: S. Vijayaraghavan (1; 2), L.A.A.Beex (1), L.Noels (2), and S.P.A.Bordas (1)

1) University of Luxembourg, Ave. de la Fonte 6, L-4364 Belval, Luxembourg.

2) University of Liege, CM3 B52, Alle de la découverte 9, B4000 Liege, Belgium

Time: Wednesday 2021.01.13, 10:00 CET

How to join: Please contact Jakub Lengiewicz


Many model order reduction approaches use solutions of a few ‘offline’ training simulations to reduce the number of degrees of freedom of the many ‘online’ simulations of interest. In proper orthogonal decomposition, singular value decomposition is applied to a matrix with the training solutions in order to capture the most essential characteristics in the first few modes – which are used as global interpolation bases.

Proper orthogonal decomposition has proven itself as an accurate reduced-order model approach for elliptical partial differential equations. In the field of solid mechanics, this means that it is accurate for (hyper)elastic material models, but not for (hyper)elastoplasticity. Based on the study of [1], the current contribution investigates how clustering of the training solutions and extracting global modes from each cluster can improve the accuracy of proper orthogonal decomposition for hyperelastoplasticity. Both centroid-based clustering (i.e. k-means clustering) and connectivity-based clustering (based on Chinese whispers) are investigated.

The approach is applied to a hyperelastoplastic representative volume element exposed to monotonic loading, quasi-monotonic loading, and quasi-random loading. In case of monotonic and quasi-monotonic loading, the components of the macroscopic deformation tensor are the variables to which clustering is applied. In case of quasi-random loading, however, not only the components of the macroscopic deformation tensor and the incremental changes of these components are the variables to which clustering is applied, but also all history variables of all integration points.

Additional material:

[1] David Amsallem, Matthew J. Zahr and Charbel Farhat. Nonlinear model order reduction based on local reduced-order bases. Int. J. Numer. Meth. Engng. VOL 92 IS-10 SN-0029-5981

Tittu Mathew: Bayesian uncertainty quantification and model selection

Machine Learning Seminar presentation

Topic: Bayesian uncertainty quantification and model selection

Speaker: Tittu Mathew, Indian Institute of Technology Madras | IIT Madras · Department of Mechanical Engineering

Time: Wednesday 2020.11.11, 10:00 CET

How to join: Please contact Jakub Lengiewicz


Adaptive Importance Sampling based Neural Network framework for Reliability and Sensitivity prediction for Variable Stiffness Composite Laminates with hybrid uncertainties

In this work, we propose to leverage the advantages of both the Artificial Neural Network (ANN) based Second-Order Reliability Method (SORM) and Importance sampling to yield an Adaptive Importance Sampling based ANN, with specific application towards failure probability and sensitivity estimates of Variable Stiffness Composite Laminate (VSCL) plates, in the presence of multiple independent geometric and material uncertainties. The performance function for the case studies is defined based on the fundamental frequency of the VSCL plate. The accuracy in both the reliability estimates and sensitivity studies using the proposed method were found to be in close agreement with that obtained using the ANN-based brute-force Monte Carlo Simulations (MCS) method, with a significant computational savings of 95%.

Moreover, the importance of taking into account the randomness in ply thickness for failure probability estimates is also highlighted quantitatively under the sensitivity studies section.

Additional material:

[1] Tittu Varghese Mathew, P. Prajith, R.O. Ruiz, E. Atroshchenko, S. Natarajan, Adaptive importance sampling based neural network framework for reliability and sensitivity prediction for variable stiffness composite laminates with hybrid uncertainties, Composite Structures, 2020




Cosmin Anitescu: Methods Based on Artificial Neural Networks for the Solution of Partial Differential Equations

Machine Learning Seminar presentation

Topic: Methods Based on Artificial Neural Networks for the Solution of Partial Differential Equations

Speaker: Dr. Cosmin Anitescu, Bauhaus-Universität Weimar

Time: Wednesday 2020.10.28, 10:00 CET

How to join: Please contact Jakub Lengiewicz


Machine learning and methods based on artificial neural networks have become increasingly common in a variety of topics for areas such as image processing, voice recognition, and object detection. The success in these areas has also led to optimized hardware and software solutions for efficiently training large neural networks and solving previously intractable problems. There is also a great deal of interest in using these techniques for solving complex engineering problems.

In this talk, I will give a brief overview of some algorithms for solving partial differential equations using artificial neural networks, particularly with regard to dealing with the boundary conditions. I will also discuss some possibilities for adaptively choosing the training points and possibilities for further improvements in the efficiency and reliability of neural network-based PDE solvers.

Additional material:

(1) An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications or



2 Ph.D. positions in Computational Mechanics in Luxembourg

The computational mechanic’s group of Prof. Stéphane Bordas (www.legato‐, ) is searching for two Ph.D. students to work on one of the following projects. Both projects focus on
phase-field damage modeling, rubbery polymers, and experimental validation. All work is carried out in close cooperation with the industrial partner SISTO Armaturen S.A. (, ).

Project 1

The project aims to analyze the formation and growth of micro-cracks in rubber due to fatigue loading. Recently published articles on fatigue failure of rubber show that micro-cracks nucleate at
initially present natural flaws. The initial distribution of flaws and the micro-crack growth is investigated with CT-scans and interrupted fatigue tests. These results are used to refine a fatigue
phase-field damage model for this scale. Key points are:

  • Measure the initial flaw size with CT-scans for various process parameter and compounds
  • Interrupted fatigue tests with CT-scans to study the micro-crack growth
  • Fatigue tests with samples with artificially introduced flaws
  • Refine a fatigue phase-field damage model for micro-scale simulations
  • Implementation of a fatigue phase-field fatigue model
  • Parameter calibration and validation of the numerical model with experimental data

Project 2

This project focuses on the thermal aging of rubber. An Arrhenius law is often applied to the test data to predict behavior at lower temperatures. However, this implies that the degradation mechanism at
the highest temperature is the same as at the lower temperatures. A new thermo-chemical model should be developed to study the degradation of material properties over the entire temperature
range. This model is combined with a recently published fatigue phase-field damage model to study the influence of aging on the failure of rubber parts. Key points are:

  • Experimental study of thermal aging taking into account the influence of temperature,
    media, and sample thickness
  • Development of a numerical model to predict the influence of aging on the material
    properties of rubber
  • Couple the new model to a fatigue phase-field damage model
  • Parameter calibration and validation of the numerical model with experimental data


Four years of funding for the Ph.D. student is provided with a competitive salary. Funds to attend conferences and summer schools are available. Each student will be employed by the university, but
she/he will also spend a significant amount of time in the company.

Candidate profile

Aspiring researchers interested in a Ph.D. topic with strong industrial relevance are encouraged to apply. Only those who hold an MSc degree or will hold one in the near future will be considered. We
are specifically looking for those who hold an MSc degree in Engineering. A background in and affinity with some of the following fields is required:

  • Material/constitutive modeling and structural mechanics
  • Finite element analysis
  • Experimental work
  • Scientific computing (numerical integration, optimization, etc.)
  • Some form of programming (MATLAB, Python, C++, FORTRAN, etc.)


Send one combined email with your application letter and CV to all of the following email addresses:

  • (Prof. Stéphane Bordas)
  • (Dr. Pascal J. Loew)

ERCIM “Alain Bensoussan” Fellowship Programme

ERCIM offers fellowships for PhD holders from all over the world. The next round is open!

Deadline for applications: 30 September 2019.

Topics cover most disciplines in Computer Science, Information Technology, and Applied Mathematics.
Fellowships are of 12-month duration spent in one ERCIM member institute.

Detailed description of the programme, application guide and application form :

Applicants must:

  • have obtained a PhD degree during the last 8 years (prior to the application year deadline) or be in the last year of the thesis work with an outstanding academic record. Nevertheless, before starting the grant, a proof of the PhD degree will be requested.
  • complete, submit the application form and send via the online system:
    • a detailed curriculum vitae
    • a list of publications
    • two scientific papers in English
    • contact details of two referees
  • start the fellowship no later than 1 May 2020.

IWR School 2019: “A Crash Course in Machine Learning with Applications in Natural- and Life Sciences (ML4Nature)”

We would like to bring to your attention our upcoming IWR School which will focus on machine
learning with applications from Natural Sciences and Life Sciences. The school will take place
from September 23 to 27, 2019, in Heidelberg, and addresses young researchers (PhD,
PostDoc, Master) from Natural and Life Sciences who want to learn more about machine
A background in Machine Learning is not required. Besides introducing the basic concepts we
teach selected topics in more depth, such as deep learning, metric learning, transfer learning,
Bayesian inverse problems, and causality. Experts from machine learning, Natural Science and
Life Science explain how these machine learning approaches are utilized to solve problems in
their respective fields of research.


Further information on the IWR School 2019 are available at:

Michael Ortiz “Model-Free Data-Driven Computing”

It was a pleasure to welcome Michael Ortiz from the California Institute of Technology to give a seminar on the topic of “Model-Free Data-Driven Computing” at the University of Luxembourg. You can watch the entire seminar below.


We develop a new computing paradigm, which we refer to as Data-Driven Computing, according to which calculations are carried out directly from experimental material data and pertinent kinematic constraints and conservation laws, such as compatibility and equilibrium, thus bypassing the empirical material modeling step of conventional computing altogether. Data-driven solvers seek to assign to each material point the state from a prespecified data set that is closest to satisfying the conservation laws. Equivalently, data-driven solvers aim to find the state satisfying the conservation laws that is closest to the data set. The resulting data-driven problem thus consists of the minimization of a distance function to the data set in phase space subject to constraints introduced by the conservation laws. We demonstrate the data-driven paradigm and investigate the performance of data-driven solvers by means of several examples of application, including statics and dynamics of nonlinear three-dimensional trusses, and linear and nonlinear elasticity. In these tests, the data-driven solvers exhibit good convergence properties both with respect to the number of data points and with regard to local data assignment, including noisy material data sets containing outliers. The variational structure of the data-driven problem also renders it amenable to analysis. We find that the classical solutions are recovered in the case of linear elasticity. We identify conditions for convergence of Data-Driven solutions corresponding to sequences of approximating material data sets. Specialization to constant material data set sequences in turn establishes an appropriate notion of relaxation. We find that relaxation within the Data-Driven framework is fundamentally different from the classical relaxation of energy functions. For instance, we show that in the Data-Driven framework the relaxation of a bistable material leads to effective material data sets that are not graphs. I will finish my presentation with highlights on work in progress, including closed-loop Data-Driven analysis and experiments, Data-Driven molecular dynamics, Data-Driven inelasticity and publicly-editable material data repositories and data management from a Data-Driven perspective.