# Lars BeexUniversity of Luxembourg · Faculty of Science Technology and Medicine

Lars Beex

PhD, MSc, BSc

## About

49

Publications

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929

Citations

Introduction

I am a Senior Research Scientist at the University of Luxembourg with the right to supervise and promote my own PhD students. I develop new computational frameworks to predict the behaviour of solid materials. My research interests include multiscale methods, probabilistic mechanics, damage mechanics, contact mechanics, and elastoplasticity. I have applied said frameworks to metals, rubbers and fibrous materials such as industrial fabrics, metal foams, paper materials and wire ropes.

Additional affiliations

July 2014 - present

January 2013 - July 2014

August 2008 - November 2012

## Publications

Publications (49)

Many materials and structures consist of numerous slender struts or fibers. Due to the manufacturing processes of different types of struts and the growth processes of natural fibers, their mechanical response frequently fluctuates from strut to strut, as well as locally within each strut. In associated mechanical models each strut is often represe...

Compared to conventional projection-based model-order-reduction, its neural-network acceleration has the advantage that the online simulations are equation-free, meaning that no system of equations needs to be solved iteratively. Consequently, no stiffness matrix needs to be constructed and the stress update needs to be computed only once per incre...

Patient-specific surgical simulations require the patient-specific identification of the constitutive parameters. The sparsity of the experimental data and the substantial noise in the data (e.g., recovered during surgery) cause considerable uncertainty in the identification. In this exploratory work, parameter uncertainty for incompressible hypere...

At every iteration or timestep of the online phase of some reduced-order modelling schemes for non-linear or time-dependent systems, large linear systems must be assembled and then projected onto a reduced order basis of small dimension. The projected small linear systems are cheap to solve, but assembly and projection become the dominant computati...

The quasicontinuum method is a concurrent multiscale approach in which lattice models are fully resolved in small regions of interest and coarse‐grained elsewhere. Since the method was originally proposed to accelerate atomistic lattice simulations, its refinement criteria – that drive refining coarse‐grained regions and/or increasing fully‐resolve...

This contribution presents the extensions of beam‐to‐beam and beam‐inside‐beam contact schemes of the same authors towards frictional interactions. Since the schemes are based on the beams’ true surfaces (instead of surfaces implicitly deduced from the beams’ centroid lines), the presented enhancements are not only able to account for frictional sl...

Phase-field damage models are able to describe crack nucleation as well as crack propagation and coalescence without additional technicalities, because cracks are treated in a continuous, spatially finite manner. Previously, we have developed a phase-field model to capture the rate-dependent failure of rubber, and we have further enhanced it to des...

The article “Non-localised contact between beams with circular and elliptical cross-sections”, written by “Marco Magliulo, Jakub Lengiewicz, Andreas Zilian and Lars A. A. Beex”, was originally published Online First without open access. After publication in volume 65, issue 5, page 1247–1266 the authors decided to opt for Open Choice and to make th...

We propose a generalized quasicontinuum method to model the mechanical response of 3D lattice structures. The method relies on the spatial coupling of fully-resolved domains and coarse-grained domains. In the fully-resolved domain, the full micro-structure is taken into account. In the coarse-grained domain, the kinematics of the micro-structure ar...

Background and Objective
This contribution presents a rapid computational framework to mechanically simulate the insertion of a slender medical instrument in a tubular structure such as an artery, the cochlea or another slender instrument. Methods
Beams are employed to rapidly simulate the mechanical behaviour of the medical instrument and the tubu...

The key novelty of this contribution is a dedicated technique to efficiently determine the distance (gap) function between parallel or almost parallel beams with circular and elliptical cross-sections. The technique consists of parametrizing the surfaces of the two beams in contact, fixing a point on the centroid line of one of the beams and search...

Slender constituents are present in many structures and materials. In associated mechanical models, each slender constituent is often described as a beam. Contact between beams is essential to incorporate in mechanical models, but associated contact frameworks are only demonstrated to work for beams with circular cross sections. Only two studies ha...

By regularizing sharp cracks within a pure continuum setting, phase-damage
models offer the ability to capture crack nucleation as well as crack propagation.
Crack branching and coalescence can furthermore be described without any ad-
ditional efforts, as geometrical descriptions of the cracks are not required. In
this contribution, we extend our p...

In this contribution, we introduce a contact formulation between beams finite elements with (hyper)elliptical cross sections. The contact scheme allows to model scenarios in which the contact area is finite or the contact area occurs along a line. Although some contact schemes are yet able to do this, they require one of the beams to have a circula...

For many mechanical models, we frequently assume that the material parameters do not vary in space, nor that they vary from one product realization to another. If the length scale of the application approaches the length scale of the micro-structure however, spatially fluctuating parameter fields (which vary from one realization to another) can be...

Strain energy densities based on the Seth–Hill strain tensors are often used to describe the hyperelastic mechanical behaviours of isotropic, transversely isotropic and orthotropic materials for relatively large deformations. Since one parameter distinguishes which strain tensor of the Seth–Hill family is used, one has in theory the possibility to...

In this paper, we propose a smoothed stable extended finite element method (S$^2$XFEM) by combining the strain smoothing with the stable extended finite element method (SXFEM) to efficiently treat inclusions and/or voids in hyperelastic matrix materials. The interface geometries are implicitly represented through level sets and a geometry based err...

We present a stochastic approach combining Bayesian Inference (BI) with homogenization theories in order to identify, on the one hand, the parameters inherent to the model assumptions and, on the other hand, the composite material constituents behaviors, including their variability. In particular, we characterize the model parameters of a Mean-Fiel...

Phase-field damage models have the ability to model crack nucleation, propagation, coalescence
and branching in a unified scheme, in contrast to approaches in which the geometrical description of the crack(s)
is required. We extend the well-established theory for fracture so that the failure under cyclic loading can be
predicted in a finite strain...

Numerous materials are essentially structures of discrete fibres, yarns or struts. Considering these materials at their discrete scale, one may distinguish two types of intrinsic randomness that affect the structural behaviours of these discrete structures: geometrical randomness and material randomness. Identifying the material randomness is an ex...

In this paper, the cell based smoothed finite element method is extended to solve stochastic partial differential equations with uncertain input parameters. The spatial field of Young's moduli and the corresponding stochastic results are represented by Karhunen-Lo\'eve expansion and polynomial chaos expansion, respectively. The Young's Modulus of s...

Phase-field models have the advantage in that no geometric descriptions of cracks are required, which means that crack coalescence and branching can be treated without additional effort. Miehe and Schänzel, 2014 introduced a rate-independent phase-field damage model for finite strains in which a viscous damage regularization was proposed. We extend...

The aim of this contribution is to explain in a straightforward manner how Bayesian inference can be used to identify material parameters of material models for solids. Bayesian approaches have already been used for this purpose, but most of the literature is not necessarily easy to understand for those new to the field. The reason for this is that...

We discuss Bayesian inference for the identification of elastoplastic material parameters. In addition to errors in the stress measurements, which are commonly considered, we furthermore consider errors in the strain measurements. Since a difference between the model and the experimental data may still be present if the data is not contaminated by...

This paper presents an equation-free, data-driven approach for reduced order modeling of a Chemical Vapor Deposition (CVD) process. The proposed approach is based on process information provided by detailed, high-fidelity models, but can also use spatio-temporal measurements. The Reduced Order Model (ROM) is built using the method-of-snapshots vari...

At every iteration or timestep of the online phase of some reduced-order modelling schemes, large linear systems must be assembled and then projected onto a reduced order basis of small dimension. The projected small linear systems are cheap to solve, but assembly and projection are now the dominant computational cost. In this paper we introduce a...

This contribution discusses Bayesian inference (BI) as an approach to identify parameters in viscoelasticity. The aims are: (i) to show that the prior has a substantial influence for viscoelasticity, (ii) to show that this influence decreases for an increasing number of measurements and (iii) to show how different types of experiments influence the...

Lattice networks with dissipative interactions can be used to describe the mechanics of discrete micro- or meso-structures of materials such as 3D-printed structures and foams, or more generally heterogeneous materials. This contribution deals with the crack initiation and propagation in such materials and focuses on an adaptive multiscale approach...

This guide provides a closer description of the MATLAB implementation accompanying our paper Rokoš et al., An Adaptive Variational Quasicontinuum Methodology for Lattice Networks with Localized Damage, Int. J. Numer. Meth. Eng., in press, which closely follows our previous work Rokoš et al., A Variational Formulation of Dissipative Quasicontinuum M...

The purpose of this guide is to provide a description of the MATLAB implementation accompanying the paper Rokoš et al., A Variational Formulation of Dissipative Quasicontinuum Methods, Int. J. Solids Struct., 102-103: 214--229, 2016. It serves to familiarize the interested reader with the structure of the implementation, with the meaning of individ...

Lattice systems and discrete networks with dissipativ are successfully employed as meso-scale models of heterogeneous solids. As the application scale generally is much larger than that of the discrete links, physically relevant simulations are computationally expensive. The QuasiContinuum (QC) method is a multiscale approach that reduces the compu...

Bayesian inference (BI) can be used for the probabilistic identification of material parameters. For inverse models for instance, BI may be considered convenient as it introduces a statistically regularisation, which is not present in alternative approaches. Understanding the concepts and the application of BI is however not trivial if one is only...

A short movie capturing deformation and damage evolution in an L-shaped plate test using variational QC.

A short movie capturing deformation and damage evolution in a four-point bending test using variational QC.

The quasicontinuum (QC) method is a numerical strategy to reduce the computational cost of direct lattice computations-in this study we achieve a speed up of a factor of 40. It has successfully been applied to (conservative) atomistic lattices in the past, but using a virtual-power statement it was recently shown that QC approaches can also be used...

Compared to most bulk materials, open-cell aluminium (Al) foams (OCAFs) are lightweight and can absorb a significant amount of energy in compression , e.g. during impact. When coated with nickel (Ni), OCAFs can absorb even more energy, making them more appropriate for impacts at higher velocities than uncoated OCAFs. When Ni-coated OCAFs experience...

The quasicontinuum (QC) method [Tadmor, E.B., Phillips, R., Ortiz, M., 1996, Mixed atomistics and continuum models of deformation in solids, Langmuir, Vol. 12, 4529-4534] is a multiscale methodology to significantly reduce the computational cost of atomistic simulations. The method ensures an accurate incorporation of small-scale atomistic effects...

The quasicontinuum (QC) method is a multiscale approach that aims to reduce the computational cost of discrete lattice computations. The method incorporates small-scale local lattice phenomena (e.g. a single lattice defect) in macroscale simulations. Since the method works directly and only on the beam lattice, QC frameworks do not require the cons...

Based on sensitivity analysis, we determine the key meso-scale uncertain input variables that influence the macro-scale mechanical response of a dry textile subjected to uni-axial and biaxial deformation. We assume a transversely isotropic fashion at the macro-scale of dry woven fabric. This paper focuses on global sensitivity analysis; i.e. regres...

Structural lattice models incorporating trusses and beams are frequently used to mechanically model fibrous materials, because they can capture (local) mesoscale phenomena. Physically relevant lattice computations are however computationally expensive. A suitable multiscale approach to reduce the computational cost of large-scale lattice computatio...

The quasicontinuum (QC) method is a multiscale method for the solution of lattice models that combines coarse-grained regions and fully resolved regions with individual lattice events. QC methodologies are mainly used to reduce the computational costs of conservative atomistic lattice computations. Recently, a virtual-power-based variant has been p...

Discrete network models and lattice models using trusses or beams can be used to mechanically model fibrous materials, since the discrete elements represent the individual fibers or yarns at the mesoscale of these materials. Consequently, local mesoscale phenomena, such as individual fiber failure and interfiber bond failure, can be incorporated. O...

Lattice models employing trusses and beams are suitable to investigate the mechanical behavior of woven fabrics. The discrete features of the mesostructures of woven fabrics are naturally incorporated by the discrete elements of lattice models. In this paper, a lattice model for woven materials is adopted which consists of a network of trusses in w...

Lattice models and discrete networks naturally describe mechanical phenomena at the mesoscale of fibrous materials. A disadvantage of lattice models is their computational cost. The quasicontinuum (QC) method is a suitable multiscale approach that reduces the computational cost of lattice models and allows the incorporation of local lattice defects...

Laminated paperboard is used as a packaging material for a wide range of products. During production of the packaging, the fold lines are first defined in a so-called creasing (or scoring) operation in order to obtain uncracked folds. During creasing as well as folding, cracking of the board is to be avoided. A mechanical model for a single fold li...

Many studies in different research fields use lattice models to investigate the mechanical behavior of materials. Full lattice calculations are often performed to determine the influence of localized microscale phenomena on large-scale responses but they are usually computationally expensive. In this study the quasicontinuum (QC) method (Phil. Mag....

a b s t r a c t Laminated paperboard is often used as a packaging material for products such as toys, tea and frozenfoods. To make the paperboard packages appealing for consumers, the fold lines must be both neat and undam-aged. The quality of the folds depends on two converting processes: the manufacture of fold lines (creas-ing) and the subsequen...

## Projects

Project (1)

Discrete materials models simulate materials behavior with that of a dense regular network of inelastic lattices or beams. In this framework, the material behavior emerges from the response of individual units, which is often easier to formulate and calibrate than the macro-scale description, but comes with the cost of computational analysis of large discrete networks.
The project goal is to reduce these costs by harnessing the principles of Quasi-Continuum (QC) methods, proposed originally for conservative atomistic systems, and extending them with dissipative effects. Exploring the variational structure of QC methods, we aim to develop adaptive, energetically consistent, and theoretically supported algorithms for distributed and localized phenomena that reduce the simulation time by one order of magnitude while exhibiting errors in quantities of interest of the order of percent.