arXiv subjects

Pierre Gosselet

Publications and source records attributed to Pierre Gosselet.

32 records · Page 2Linked to original sources

arXiv2013

A strict error bound with separated contributions of the discretization and of the iterative solver in non-overlapping domain decomposition methods

This paper deals with the estimation of the distance between the solution of a static linear mechanic problem and its approximation by the finite element method solved with a non-overlapping domain decomposition method (FETI or BDD). We propose a new strict upper bound of the error which separates the contribution of the iterative solver and the contribution of the discretization. Numerical assessments show that the bound is sharp and enables us to define an objective stopping criterion for the iterative solver

physics.comp-ph↗

arXiv2013

Study of the strong prolongation equation for the construction of statically admissible stress fields: implementation and optimization

This paper focuses on the construction of statically admissible stress fields (SA-fields) for a posteriori error estimation. In the context of verification, the recovery of such fields enables to provide strict upper bounds of the energy norm of the discretization error between the known finite element solution and the unavailable exact solution. The reconstruction is a difficult and decisive step insofar as the effectiveness of the estimator strongly depends on the quality of the SA-fields. This paper examines the strong prolongation hypothesis, which is the starting point of the Element Equilibration Technique (EET). We manage to characterize the full space of SA-fields satisfying the prolongation equation so that optimizing inside this space is possible. The computation exploits topological properties of the mesh so that implementation is easy and costs remain controlled. In this paper, we describe the new technique in details and compare it to the classical EET and to the flux-free technique for different 2D mechanical problems. The method is explained on first degree triangular elements, but we show how extensions to different elements and to 3D are straightforward.

math.NA↗

arXiv2013

Virtual Delamination Testing through Non-Linear Multi-Scale Computational Methods: Some Recent Progress

This paper deals with the parallel simulation of delamination problems at the meso-scale by means of multi-scale methods, the aim being the Virtual Delamination Testing of Composite parts. In the non-linear context, Domain Decomposition Methods are mainly used as a solver for the tangent problem to be solved at each iteration of a Newton-Raphson algorithm. In case of strongly nonlinear and heterogeneous problems, this procedure may lead to severe difficulties. The paper focuses on methods to circumvent these problems, which can now be expressed using a relatively general framework, even though the different ingredients of the strategy have emerged separately. We rely here on the micro-macro framework proposed in (Ladevèze, Loiseau, and Dureisseix, 2001). The method proposed in this paper introduces three additional features: (i) the adaptation of the macro-basis to situations where classical homogenization does not provide a good preconditioner, (ii) the use of non-linear relocalization to decrease the number of global problems to be solved in the case of unevenly distributed non-linearities, (iii) the adaptation of the approximation of the local Schur complement which governs the convergence of the proposed iterative technique. Computations of delamination and delamination-buckling interaction with contact on potentially large delaminated areas are used to illustrate those aspects.

math.NA↗

arXiv2013

Total and selective reuse of Krylov subspaces for the resolution of sequences of nonlinear structural problems

This paper deals with the definition and optimization of augmentation spaces for faster convergence of the conjugate gradient method in the resolution of sequences of linear systems. Using advanced convergence results from the literature, we present a procedure based on a selection of relevant approximations of the eigenspaces for extracting, selecting and reusing information from the Krylov subspaces generated by previous solutions in order to accelerate the current iteration. Assessments of the method are proposed in the cases of both linear and nonlinear structural problems.

math.NA↗

arXiv2012

A domain decomposition strategy to efficiently solve structures containing repeated patterns

This paper presents a strategy for the computation of structures with repeated patterns based on domain decomposition and block Krylov solvers. It can be seen as a special variant of the FETI method. We propose using the presence of repeated domains in the problem to compute the solution by minimizing the interface error on several directions simultaneously. The method not only drastically decreases the size of the problems to solve but also accelerates the convergence of interface problem for nearly no additional computational cost and minimizes expensive memory accesses. The numerical performances are illustrated on some thermal and elastic academic problems.

math.NA↗

arXiv2012

On a multiscale strategy and its optimization for the simulation of combined delamination and buckling

This paper investigates a computational strategy for studying the interactions between multiple through-the-width delaminations and global or local buckling in composite laminates taking into account possible contact between the delaminated surfaces. In order to achieve an accurate prediction of the quasi-static response, a very refined discretization of the structure is required, leading to the resolution of very large and highly nonlinear numerical problems. In this paper, a nonlinear finite element formulation along with a parallel iterative scheme based on a multiscale domain decomposition are used for the computation of 3D mesoscale models. Previous works by the authors already dealt with the simulation of multiscale delamination assuming small perturbations. This paper presents the formulation used to include geometric nonlinearities into this existing multiscale framework and discusses the adaptations that need to be made to the iterative process in order to ensure the rapid convergence and the scalability of the method in the presence of buckling and delamination. These various adaptations are illustrated by simulations involving large numbers of DOFs.

math.NA↗

arXiv2012

Fast estimation of discretization error for FE problems solved by domain decomposition

This paper presents a strategy for a posteriori error estimation for substructured problems solved by non-overlapping domain decomposition methods. We focus on global estimates of the discretization error obtained through the error in constitutive relation for linear mechanical problems. Our method allows to compute error estimate in a fully parallel way for both primal (BDD) and dual (FETI) approaches of non-overlapping domain decomposition whatever the state (converged or not) of the associated iterative solver. Results obtained on an academic problem show that the strategy we propose is efficient in the sense that correct estimation is obtained with fully parallel computations; they also indicate that the estimation of the discretization error reaches sufficient precision in very few iterations of the domain decomposition solver, which enables to consider highly effective adaptive computational strategies.

math.NA↗

arXiv2012

On the initial estimate of interface forces in FETI methods

The Balanced Domain Decomposition (BDD) method and the Finite Element Tearing and Interconnecting (FETI) method are two commonly used non-overlapping domain decomposition methods. Due to strong theoretical and numerical similarities, these two methods are generally considered as being equivalently efficient. However, for some particular cases, such as for structures with strong heterogeneities, FETI requires a large number of iterations to compute the solution compared to BDD. In this paper, the origin of the bad efficiency of FETI in these particular cases is traced back to poor initial estimates of the interface stresses. To improve the estimation of interface forces a novel strategy for splitting interface forces between neighboring substructures is proposed. The additional computational cost incurred is not significant. This yields a new initialization for the FETI method and restores numerical efficiency which makes FETI comparable to BDD even for problems where FETI was performing poorly. Various simple test problems are presented to discuss the efficiency of the proposed strategy and to illustrate the so-obtained numerical equivalence between the BDD and FETI solvers.

math.NA↗

arXiv2012

Non-overlapping domain decomposition methods in structural mechanics

The modern design of industrial structures leads to very complex simulations characterized by nonlinearities, high heterogeneities, tortuous geometries... Whatever the modelization may be, such an analysis leads to the solution to a family of large ill-conditioned linear systems. In this paper we study strategies to efficiently solve to linear system based on non-overlapping domain decomposition methods. We present a review of most employed approaches and their strong connections. We outline their mechanical interpretations as well as the practical issues when willing to implement and use them. Numerical properties are illustrated by various assessments from academic to industrial problems. An hybrid approach, mainly designed for multifield problems, is also introduced as it provides a general framework of such approaches.

math.NA↗

arXiv2011

Improved multiscale computational strategies for delamination

This paper presents a three-scale computational strategy for the study of composite modeled at the mesoscale so that delamination can be reliably simulated. The solver is based on a LaTIn approach so that nonlinearities can be tackled at the local scale. We show how search directions which are parameters of the method need to be updated according to the delamination status of interfaces. We also present the parallelization of the macro problem which transmits long-range phenomena and warranty the scalabilty of the method; this parallelization is based on the balancing domain decomposition method, it leads to the introduction of a third (super-macro) scale.

physics.comp-ph↗

arXiv2011

A three-scale domain decomposition method for the 3D analysis of debonding in laminates

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.

physics.comp-ph↗

arXiv2011

On the control of the load increments for a proper description of multiple delamination in a domain decomposition framework

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.

math.NA↗

arXiv2011

A relocalization technique for the multiscale computation of delamination in composite structures

In this article, we present numerical enhancements of a multiscale domain decomposition strategy based on a LaTIn solver and dedicated to the computation of the debounding in laminates. 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. The scalability is restored by performing nonlinear subresolutions in the vincinity of the front of the crack, at each prediction stage of the iterative solver.

physics.class-ph↗

arXiv2011

Bridging Proper Orthogonal Decomposition methods and augmented Newton-Krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems

This article describes a bridge between POD-based model order reduction techniques and the 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.

physics.class-ph↗