Pierre Kerfriden | Cardiff University (original) (raw)

Papers by Pierre Kerfriden

Mechanical free vibrations with unknown field of elasticity field → • Find unknown parameters giv... more Mechanical free vibrations with unknown field of elasticity field → • Find unknown parameters given observations, Inverse problem

This paper proposes a new methodology to guarantee the accuracy of the homogenisation schemes tha... more This paper proposes a new methodology to guarantee the accuracy of the homogenisation schemes that are traditionally employed to approximate the solution of PDEs with random, fastly evolving diffusion coefficients. We typically consider linear elliptic diffusion problems in randomly packed particulate composites. Our work extends the pioneering work presented in [27, 33] in order to bound the error in the expectation and second moment of quantities of interest, without ever solving the fine-scale, intractable stochastic problem. The most attractive feature of our approach is that the error bounds are computed without any integration of the fine-scale features. Our computations are purely macroscopic, deterministic, and remain tractable even for small scale ratios. The second contribution of the paper is an alternative derivation of modelling error bounds through the Prager-Synge hypercircle theorem. We show that this approach allows us to fully characterise and optimally tighten the interval in which predicted quantities of interest are guaranteed to lie. We interpret our optimum result as an extension of Reuss-Voigt approaches, which are classically used to estimate the homogenised diffusion coefficients of composites, to the estimation of macroscopic engineering quantities of interest. Finally, we make use of these derivations to obtain an efficient procedure for multiscale model verification and adaptation.

In this paper, we present new reliable model order reduction strategies for computational microme... more In this paper, we present new reliable model order reduction strategies for computational micromechanics. The difficulties rely mainly upon the high dimensionality of the parameter space represented by any load path applied onto the representative volume element (RVE). We take special care of the challenge of selecting an exhaustive snapshot set. This is treated by first using a random sampling of energy dissipating load paths and then in a more advanced way using Bayesian optimization associated with an interlocked division of the parameter space. Results show that we can insure the selection of an exhaustive snapshot set from which a reliable reduced-order model (ROM) can be built.

Simplifying the geometry of a CAD model using defeaturing techniques enables more efficient discr... more Simplifying the geometry of a CAD model using defeaturing techniques enables more efficient discretisation and subsequent simulation for engineering analysis problems. Understanding the effect this simplification has on the solution helps to decide whether the simplification is suitable for a specific simulation problem. It can also help to understand the functional effect of a geometry feature. The effect of the simplification is quantified by a user-defined quantity of interest which is assumed to be (approximately) linear in the solution. A bound on the difference between the quantity of interest of the original and simplified solutions based on the energy norm is derived. The approach is presented in the context of electrostatics problems, but can be applied in general to a range of elliptic partial differential equations. Numerical results on the efficiency of the bound are provided for electrostatics problems with simplifications involving changes inside the problem domain as well as changes to the boundaries.

This paper proposes a new reduced basis algorithm for the metamodelling of parametrised elliptic ... more This paper proposes a new reduced basis algorithm for the metamodelling of parametrised elliptic problems. The developments rely on the Constitutive Relation Error (CRE), and the construction of separate reduced order models for the primal variable (displacement) and flux
(stress) fields. A two-field Greedy sampling strategy is proposed to construct these two fields simultaneously and efficient manner: at each iteration, one of the two fields is enriched by increasing the dimension of its reduced space in such a way that the CRE is minimised. This sampling strategy is then used as a basis to construct goal-oriented reduced order modelling. The resulting algorithm is certified and “tuning-free”: the only requirement from the engineer is the level of accuracy that is desired for each of the outputs of the surrogate. It is also one order of magnitude more efficient in terms of computational expenses than competing methodologies.

Core shearing and core/face debonding are two common failure states of sandwich beams which are m... more Core shearing and core/face debonding are two common failure states of sandwich beams which are mainly the result of excessive shear stresses in the core. Generally, the core made of homogeneous Fiber Reinforced Polymer (FRP) shows better shear resistance in comparison with that made of pure polymer. Usually, this enhancement is however somewhat limited. This paper proposes a methodology to decrease interfacial stresses by presenting the optimal distribution of reinforcing ingredients in the polymeric matrix. For this purpose, a Non-Uniform Rational B- spline (NURBS) based reinforcement distribution optimizer is developed. This technique aims at the local stress minimization within any arbitrary zone of the design domain. In our methodology, optimization and model analysis (calculation of the objective function and the design constraints) have common data sets. The quadratic NURBS basis functions smoothly define the reinforcement distribution function as a NURBS surface. The core and face sheets are modeled as multi-patches and compatibility in the displacement field is enforced by the penalty method. An adjoint sensitivity method is devised to minimize the objective function within areas of interest defined over arbitrary regions in the design domain. It is also used for efficient updating of design variables through optimization iterations. The method is verified by several examples.

In this paper, we study the class of linear elastodynamic problems with affine parameter dependen... more In this paper, we study the class of linear elastodynamic problems with affine parameter dependence using a goal-oriented approach by finite element (FE) and reduced basis (RB) methods. The main contribution of this paper is the ``goal-oriented'' proper orthogonal decomposition (POD)--Greedy sampling strategy within the RB approximation context. The proposed sampling strategy looks for the parameter points such that the output error approximation will be minimized by Greedy iterations. In estimating such output error approximation, the standard POD--Greedy algorithm is invoked to provide enriched RB approximations for the FE outputs. We propose a so-called ``cross-validation'' process to choose adaptively the dimension of the enriched RB space corresponding with the dimension of the RB space under consideration. Numerical results show that the new goal-oriented POD--Greedy sampling procedure with the cross-validation process improves significantly the space-time output computations in comparison with the ones computed by the standard POD--Greedy algorithm. The method is thus ideally suited for repeated, rapid and reliable evaluations of input-output relationships in the space-time setting.

In this paper, we propose upper and lower error bounding techniques for reduced order modelling a... more In this paper, we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites. The upper bound relies on the construction of a reduced model for the stress field. Upon ensuring that the reduced stress satisfies the equilibrium in the finite element sense, the desired bounding property is obtained. The lower bound is obtained by defining a hierarchical enriched reduced model for the displacement. We show that the sharpness of both error estimates can be seamlessly controlled by adapting the parameters of the corresponding reduced order model.

Goal-oriented error estimates (GOEE) have become popular tools to quantify and control the local ... more Goal-oriented error estimates (GOEE) have become popular tools to quantify and control the local error in quantities of interest (QoI), which are often more pertinent than local errors in energy for design purposes (e.g. the mean stress or mean displacement in a particular area, the stress intensity factor for fracture problems). These GOEE are one of the key unsolved problems of advanced engineering applications in, for example, the aerospace industry. This work presents a simple recovery-based error estimation technique for QoIs whose main characteristic is the use of an enhanced version of the Superconvergent Patch Recovery (SPR) technique previously used for error estimation in the energy norm. This enhanced SPR technique is used to recover both the primal and dual solutions. It provides a nearly statically admissible stress field that results in accurate estimations of the local contributions to the discretisation error in the QoI and, therefore, in an accurate estimation of this magnitude. This approach leads to a technique with a reasonable computational cost that could easily be implemented into already available finite element codes, or as an independent postprocessing tool.

$Research paper thumbnail of A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics$

We propose in this paper an adaptive reduced order modelling technique based on domain partitioni... more We propose in this paper an adaptive reduced order modelling technique based on domain partitioning for parametric problems of fracture. We show that coupling domain decomposition and projection-based model order reduction permits to focus the numerical effort where it is most needed: around the zones where damage propagates. No \textit{a priori} knowledge of the damage pattern is required, the extraction of the corresponding spatial regions being based solely on algebra. The efficiency of the proposed approach is demonstrated numerically with an example relevant to engineering fracture.

By using the strain smoothing technique proposed by Chen et al. [1] for meshless methods in the c... more By using the strain smoothing technique proposed by Chen et al. [1] for meshless methods in the context of the finite element method (FEM), Liu et al. [2] developed the Smoothed FEM (SFEM).

$Research paper thumbnail of Local/global model order reduction strategy for the simulation of quasi-brittle fracture$

This paper proposes a novel technique to reduce the computational burden associated with the simu... more This paper proposes a novel technique to reduce the computational burden associated with the simulation of localised failure. The proposed methodology affords the simulation of damage initiation and propagation whilst concentrating the computational effort where it is most needed, i.e. in the localisation zones. To do so, a local/global technique is devised where the global (slave) problem (far from the zones undergoing severe damage and cracking) is solved for in a reduced space computed by the classical Proper Orthogonal Decomposition, while the local (master) degrees of freedom (associated with the part of the structure where most of the damage is taking place) are fully resolved. Both domains are coupled through a local/global technique. This method circumvents the difficulties associated with model order reduction for the simulation of highly non-linear mechanical failure and offers an alternative or complementary approach to the development of multiscale fracture simulators.

This article describes a bridge between POD-based model order reduction techniques and classical ... more This article describes a bridge between POD-based model order reduction techniques and classical Newton/Krylov solvers. This bridge is used to derive an efficient algorithm to correct, ``on-the-fly", the reduced order modelling of highly nonlinear problems undergoing strong topological changes. Damage initiation problems are addressed and tackle via a corrected hyperreduction method. It is shown that the relevancy of reduced order model can be significantly improved with reasonable additional costs when using this algorithm, even when strong topological changes are involved.

In quasi-static nonlinear time-dependent analysis, the choice of the time discretization is a com... more In quasi-static nonlinear time-dependent analysis, the choice of the time discretization is a complex issue. The most basic strategy consists in determining a value of the load increment that ensures the convergence of the solution with respect to time on the base of preliminary simulations. In more advanced applications, the load increments can be controlled for instance by prescribing the number of iterations of the nonlinear resolution procedure, or by using an arc-length algorithm. These techniques usually introduce a parameter whose correct value is not easy to obtain. In this paper, an alternative procedure is proposed. It is based on the continuous control of the residual of the reference problem over time, whose measure is easy to interpret. This idea is applied in the framework of a multiscale domain decomposition strategy in order to perform 3D delamination analysis.

"We present numerical enhancements of a multiscale domain decomposition strategy based on a LaTIn... more "We present numerical enhancements of a multiscale domain decomposition strategy based on a LaTIn solver and dedicated to the computation of the debounding in laminated composites. We show that the classical scale separation is irrelevant in the process zones, which results in a drop in the convergence rate of the strategy.
We show that performing nonlinear subresolutions in the vicinity of the front of the crack at each prediction stage of the iterative solver permits to restore the effectiveness of the method."

The prediction of the quasi-static response of industrial laminate structures requires to use fin... more The prediction of the quasi-static response of industrial laminate structures requires to use fine descriptions of the material, especially when debonding is involved. Even when modeled at the mesoscale, the computation of these structures results in very large numerical problems. In this paper, the exact mesoscale solution is sought using parallel iterative solvers. The LaTIn-based mixed domain decomposition method makes it very easy to handle the complex description of the structure; moreover the provided multiscale features enable us to deal with numerical difficulties at their natural scale; we present the various enhancements we developed to ensure the scalability of the method. An extension of the method designed to handle instabilities is also presented.

In this paper, the linear free flexural vibration of cracked functionally graded material plates ... more In this paper, the linear free flexural vibration of cracked functionally graded material plates is studied using the extended finite element method. A 4-noded quadrilateral plate bending element based on field and edge consistency requirement with 20 degrees of freedom per element is used for this study. The natural frequencies and mode shapes of simply supported and clamped square and rectangular plates are computed as a function of gradient index, crack length, crack orientation and crack location.

Abstract: This paper aims to incorporate the node-based smoothed finite element method (NS-FEM) i... more Abstract: This paper aims to incorporate the node-based smoothed finite element method (NS-FEM) into the extended finite element method (XFEM) to form a novel numerical method (NS-XFEM) for analyzing fracture problems of 2D elasticity. NS-FEM uses the strain smoothing technique over the smoothing domains associated with nodes to compute the system stiffness matrix, which leads to the line integrations using directly the shape function values along the boundaries of the smoothing domains.

This paper presents a strain smoothing procedure for the extended finite element method (XFEM). T... more This paper presents a strain smoothing procedure for the extended finite element method (XFEM). The resulting “edge-based” smoothed extended finite element method (ESm-XFEM) is tailored to linear elastic fracture mechanics and, in this context, to outperform the standard XFEM. In the XFEM, the displacement-based approximation is enriched by the Heaviside and asymptotic crack tip functions using the framework of partition of unity.

$Research paper thumbnail of Statistical extraction of process zones and representative subspaces in fracture of random composites$

We propose to identify process zones in heterogeneous materials by tailored statistical tools. Th... more We propose to identify process zones in heterogeneous materials by tailored statistical tools. The process zone is redefined as the part of the structure where the random process cannot be correctly approximated in a low-dimensional deterministic space. Such a low-dimensional space is obtained by a spectral analysis performed on pre-computed solution samples. A greedy algorithm is proposed to identify both process zone and low-dimensional representative subspace for the solution in the complementary region. In addition to the novelty of the tools proposed in this paper for the analysis of localised phenomena, we show that the reduced space generated by the method is a valid basis for the construction of a reduced order model.

$Research paper thumbnail of A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics$

$Research paper thumbnail of Local/global model order reduction strategy for the simulation of quasi-brittle fracture$

$Research paper thumbnail of Statistical extraction of process zones and representative subspaces in fracture of random composites$

Proceedings of USNCCM2014, 2017

$Research paper thumbnail of Reduced order modelling and multiscale methods for simulating fracture in heterogeneous media$

.1 A two-dimensional solid containing a cohesive crack and an ad Note that Γ coh represents only ... more .1 A two-dimensional solid containing a cohesive crack and an ad Note that Γ coh represents only the portion of the cohesive crack where t M is non on the Dirichlet boundary Γ u M ⊆ Γ M . The discontinuity surface Γ posed of cohesive cracks Γ coh and adhesive cracks Γ adh . It is em that although the material is heterogeneous, the macro solid is m being homogeneous with effective properties coming from a hete micro model. Hereafter, subscripts M and m are used to indicate if belongs to the macro or micro scale, respectively.

This article describes a series of contributions in the field of real-time simulation of soft tis... more This article describes a series of contributions in the field of real-time simulation of soft tissue biomechanics. These contributions address various requirements for interactive simulation of complex surgical procedures such as surgical cuts. The contributions described in this article share a common underlying model of deformation and rely on GPU implementations to significantly improve computation times.

Length--scale: length of a typical cons$tuent of the structure whose physics can be described in ... more Length--scale: length of a typical cons$tuent of the structure whose physics can be described in a self--consistent manner M. Ortiz SIMM 12/08 MS&S Metal plasticity Lattice defects, EoS Dislocation dynamics Subgrain structures length time mm nm µm ms µs ns Polycrystals Engineering applications Quantum mechanical or atomistic Discrete or linear elastic Continuum Objective: Derive ansatz-free, physics-based, predictive models of macroscopic behavior

$Research paper thumbnail of An adaptive multiscale method for fracture based on concurrent - hierarchical hybrid modelling$

In this paper, we present a multiscale method to analyse quasi-brittle crack propagation in metal... more In this paper, we present a multiscale method to analyse quasi-brittle crack propagation in metals. The fracture model is described at the grain level by damage mechanics. In order a tractable solution to be obtained by numerical means in an engineering component, the microscale behaviour is upscaled at an engineering scale by classical computational homogenization. The lack of scale separation due to the coalescence of microscopic defects is tackled by a concurrent computation of the process zone. The paper focuses on the tools for such a method to be used in practice. In particular, we investigate the adaptive refinement of the process zone within the hybrid concurrent/homogenization-based multiscale strategy, and the time integration of the resulting multiscale problem by an arc-length method.

$Research paper thumbnail of Spatial localisation and treatment of the lack of correlation in  reduced order modelling applied to fracture mechanics.$

$Research paper thumbnail of Recent advances in local/global model order reduction  for quasi-brittle fracture$

International Journal for Uncertainty Quantification, 2020

The presented adaptive modelling approach aims to jointly control the level of refinement for eac... more The presented adaptive modelling approach aims to jointly control the level of refinement for each of the building-blocks employed in a typical chain of finite element approximations for stochas-tically parametrized systems, namely: (i) finite error approximation of the spatial fields (ii) surro-gate modelling to interpolate quantities of interest(s) in the parameter domain and (iii) Monte-Carlo sampling of associated probability distribution(s). The control strategy seeks accurate calculation of any statistical measure of the distributions at minimum cost, given an acceptable margin of error as only tunable parameter. At each stage of the greedy-based algorithm for spatial discreti-sation, the mesh is selectively refined in the subdomains with highest contribution to the error in the desired measure. The strictly incremental complexity of the surrogate model is controlled by enforcing preponderant discretisation error integrated across the parameter domain. Finally, the number of Monte-Carlo samples is chosen such that either (a) the overall precision of the chain of approximations can be ascertained with sufficient confidence, or (b) the fact that the computational model requires further mesh refinement is statistically established. The efficiency of the proposed approach is discussed for a frequency-domain vibration structural dynamics problem with forward uncertainty propagation. Results show that locally adapted finite element solutions converge faster than those obtained using uniformly refined grids.

This paper presents a novel CutFEM-LaTIn algorithm to solve multiple unilateral contact problems ... more This paper presents a novel CutFEM-LaTIn algorithm to solve multiple unilateral contact problems over geome-tries that do not conform with the finite element mesh. We show that our method is (i) stable, independently of the interface locations (ii) optimally convergent with mesh refinement and (iii) efficient from an algorithmic point of view.

Introduction Enriched and unfitted finite element methods are powerful extensions of the popular ... more Introduction Enriched and unfitted finite element methods are powerful extensions of the popular classical Finite Element Method (FEM). In FEM, piecewise polynomial functions defined over geometrically simple elements are used to represent both the geometry of the computational domain and the solution of the boundary value problem of interest. Over the years, researchers have identified limitations to this framework. (i) Meshing complex domains may be difficult, numerically costly and may lack robustness; (ii) Adaptive modeling requires numerous re-meshing operations during the solution process in order to improve the accuracy and/or stability of the numerical simulation. These re-meshing cycles can be computationally expensive and should be avoided; (iii) FE solvers for boundary value problems whose solutions exhibit singularities or discontinuities are slow to converge with mesh refinement. This is particularly true for higher-order methods, where sharp solution features may trigger instabilities. Unfitted and enriched FEMs, such as the Partition-of-Unity FEM [Babuška-1997], eXtended FEM (XFEM) [Belytschko-1999], Generalised FEM [Strouboulis-2001], the Strong Discontinuity Approach [Oliver-2002] or the Finite Cell Method [Parvizian-2007], enhance the description of the geometry and solution field to circumvent some or all of these limitations. In this paper, we showcase one particular family of unfitted FEMs that is derived from the CutFEM method

Thèse d'Habilitation à Diriger des recherches - Ecole Normale Supérieure Paris Saclay, 2018

High-fidelity modelling and simulation have profoundly transformed the area of material and struc... more High-fidelity modelling and simulation have profoundly transformed the area of material and structural design. Through advances in computer hardware and software, material failure can be reliably predicted using multiscale high-fidelity models coupled with appropriately designed discretisation strategies. Yet, such heavy numerical tasks are restricted to “one-shot” virtual experiments. Emerging applications such as real-time control or interactive design require performing thousands of repeated analyses, with potentially limited computational facilities. Models used for such applications require extreme robustness and swiftness of execution. To unleash the full potential of high-fidelity computational mechanics, we need to develop a new generation of numerical tools that will bridge the gap between, on the one hand, heavy numerical solvers and, on the other hand, computationally demanding “online” engineering tasks. This thesis introduces and summarises research contributions that aim to help bridge this gap, through the development of robust model reduction approaches to control the cost associated with multiscale and physically detailed numerical simulations, with a particular emphasis on reliability assessment for composite materials and fracture.

Course on Probabilistic Inverse Problems, 2020

Deterministic and Bayesian formalisms for PDE-based inverse problems: general characteristics (la... more Deterministic and Bayesian formalisms for PDE-based inverse problems: general characteristics (lack of existence and/or uniqueness), regularisation approaches, optimisation and sampling algorithms (e.g. gradient descent, Markov-Chain Monte-Carlo), adjoint methodology, Bayesian model selection.