A multiscale large time increment/FAS algorithm with time‐space model reduction for frictional contact problems

A multiscale strategy using model reduction for frictional contact computation is presented. This new approach aims to improve computation time of finite element simulations involving frictional contact between linear and elastic bodies. This strategy is based on a combination between the LATIN (LArge Time INcrement) method and the FAS multigrid solver. The LATIN method is an iterative solver operating on the whole time‐space domain. Applying an a posteriori analysis on solutions of different frictional contact problems shows a great potential as far as reducibility for frictional contact problems is concerned. Time‐space vectors forming the so‐called reduced basis depict particular scales of the problem. It becomes easy to make analogies with multigrid method to take full advantage of multiscale information. Copyright © 2013 John Wiley & Sons, Ltd.     

Generation of a cokriging metamodel using a multiparametric strategy

In the course of designing structural assemblies, performing a full optimization is very expensive in terms of computation time. In order or reduce this cost, we propose a multilevel model optimization approach. This paper lays the foundations of this strategy by presenting a method for constructing an approximation of an objective function. This approach consists in coupling a multiparametric mechanical strategy based on the LATIN method with a gradient-based metamodel called a cokriging metamodel. The main difficulty is to build an accurate approximation while keeping the computation cost low. Following an introduction to multiparametric and cokriging strategies, the performance of kriging and cokriging models is studied using one- and two-dimensional analytical functions; then, the performance of metamodels built from mechanical responses provided by the multiparametric strategy is analyzed based on two mechanical test examples.     

A scalable time–space multiscale domain decomposition method: adaptive time scale separation

This paper deals with the scalability of a time–space multiscale domain decomposition method in the framework of time-dependent nonlinear problems. The strategy which is being studied is the multiscale LATIN method, whose scalability was shown in previous works when the distinction between macro and micro parts is made on the spatial level alone. The objective of this work is to propose an explanation of the loss-of-scalability phenomenon, along with a remedy which guarantees full scalability provided a suitable macro time part is chosen. This technique, which is quite general, is based on an adaptive separation of scales which is achieved by adding the most relevant functions to the temporal macrobasis automatically. When this method is used, the numerical scalability of the strategy is confirmed by the examples presented.     

Multiparametric analysis within the proper generalized decomposition framework

Optimization campaigns, which are being launched more and more often, require the execution of many parametric studies which can make the approach very costly in terms of computation time. Here, in order to reduce these computation times, we undertake to develop a multiparametric strategy using the LATIN method along with Proper Generalized Decomposition. This approach is compared to other common strategies, especially those based on POD.     

LATIN: A new view and an extension to wave propagation in nonlinear media

The LATIN (acronym of LArge Time INcrement) method was originally devised as a non‐incremental procedure for the solution of quasi‐static problems in continuum mechanics with material nonlinearity. In contrast to standard incremental methods like Newton and modified Newton, LATIN is an iterative procedure applied to the entire loading path. In each LATIN iteration, two problems are solved: a local problem, which is nonlinear but algebraic and miniature, and a global problem, which involves the entire loading process but is linear. The convergence of these iterations, which has been shown to occur for a large class of nonlinear problems, provides an approximate solution to the original problem. In this paper, the LATIN method is presented from a different viewpoint, taking advantage of the causality principle. In this new view, LATIN is an incremental method, and the LATIN iterations are performed within each load step, similarly to the way that Newton iterations are performed. The advantages of the new approach are discussed. In addition, LATIN is extended for the solution of time‐dependent wave problems. As a relatively simple model for illustrating the new formulation, lateral wave propagation in a flat membrane made of a nonlinear material is considered. Numerical examples demonstrate the performance of the scheme, in conjunction with finite element discretization in space and the Newmark trapezoidal algorithm in time. Copyright © 2017 John Wiley & Sons, Ltd.     

A computational strategy for poroelastic problems with a time interface between coupled physics

This paper deals with a computational strategy suitable for the simulation of multiphysics problems and based on the LArge Time INcrement (LATIN) method. One of the main issues in the design of advanced tools for the simulation of such problems is to take into account the different time and space scales that usually arise with the different physics. Here, we focus on using different time discretizations for each physics by introducing an interface with its own discretization. The proposed application concerns the simulation of a 2‐physics problem: the fluid–structure interaction in porous media. Copyright © 2007 John Wiley & Sons, Ltd.     

Time‐space PGD for the rapid solution of 3D nonlinear parametrized problems in the many‐query context

This work deals with the question of the resolution of nonlinear problems for many different configurations in order to build a ‘virtual chart’ of solutions. The targeted problems are three‐dimensional structures driven by Chaboche‐type elastic‐viscoplastic constitutive laws. In this context, parametric analysis can lead to highly expensive computations when using a direct treatment. As an alternative, we present a technique based on the use of the time‐space proper generalized decomposition in the framework of the LATIN method. To speed up the calculations in the parametrized context, we use the fact that at each iteration of the LATIN method, an approximation over the entire time‐space domain is available. Then, a global reduced‐order basis is generated, reused and eventually enriched, by treating, one‐by‐one, all the various parameter sets. The novelty of the current paper is to develop a strategy that uses the reduced‐order basis for any new set of parameters as an initialization for the iterative procedure. The reduced‐order basis, which has been built for a set of parameters, is reused to build a first approximation of the solution for another set of parameters. An error indicator allows adding new functions to the basis only if necessary. The gain of this strategy for studying the influence of material or loading variability reaches the order of 25 in the industrial examples that are presented. Copyright © 2015 John Wiley & Sons, Ltd.     

A computational strategy for thermo-poroelastic structures with a time–space interface coupling

This paper deals with a computational strategy suitable for the simulation of multiphysics problems, based on the large time increment (LATIN) method. The simulation of such problems must encounter the possible different time and space scales that usually arise with the different physics. Herein, we focus on using different time and space discretizations for each physics by introducing an interface with its own discretization. The feasibility of both time and space couplings is exemplified on a non-linear 3-physics strongly coupled problem: the thermal/fluid/structure interaction in a thermo-poroelastic structure. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive multi-analysis strategy for contact problems with friction

The objective of the work presented here is to develop an efficient strategy for the parametric analysis of bolted joints designed for aerospace applications. These joints are used in elastic structural assemblies with local nonlinearities (such as unilateral contact with friction) under quasi-static loading. Our approach is based on a decomposition of an assembly into substructures (representing the parts) and interfaces (representing the connections). The problem within each substructure is solved by the finite element method, while an iterative scheme based on the LATIN method (Ladevèze in Nonlinear computational structural mechanics—new approaches and non-incremental methods of calculation, 1999) is used for the global resolution. The proposed strategy consists in calculating response surfaces (Rajashekhar and Ellingwood in Struct Saf 12:205–220, 1993) such that each point of a surface is associated with a design configuration. Each design configuration corresponds to a set of values of all the variable parameters (friction coefficients, prestresses) which are introduced into the mechanical analysis. Here, instead of carrying out a full calculation for each point of the surface, we propose to use the capabilities of the LATIN method and reutilize the solution of one problem (for one set of parameters) in order to solve similar problems (for the other sets of parameters) (Boucard and Champaney in Int J Numer Methods Eng 57:1259–1281, 2003). The strategy is adaptive in the sense that it takes into account the results of the previous calculations. The method presented can be used for several types of nonlinear problems requiring multiple analyses: for example, it has already been used for structural identification (Allix and Vidal in Comput Methods Appl Mech Eng 191:2727–2758, 2001).     

A suitable computational strategy for the parametric analysis of problems with multiple contact

The aim of the present work is to develop an application of the LArge Time INcrement (LATIN) approach for the parametric analysis of static problems with multiple contacts. The methodology adopted was originally introduced to solve viscoplastic and large‐transformation problems. Here, the applications concern elastic, quasi‐static structural assemblies with local non‐linearities such as unilateral contact with friction. Our approach is based on a decomposition of the assembly into substructures and interfaces. The interfaces play the vital role of enabling the local non‐linearities, such as contact and friction, to be modelled easily and accurately. The problem on each substructure is solved by the finite element method and an iterative scheme based on the LATIN method is used for the global resolution. More specifically, the objective is to calculate a large number of design configurations. Each design configuration corresponds to a set of values of all the variable parameters (friction coefficients, prestress) which are introduced into the mechanical analysis. A full computation is needed for each set of parameters. Here we propose, as an alternative to carrying out these full computations, to use the capability of the LATIN method to re‐use the solution to a given problem (for one set of parameters) in order to solve similar problems (for the other sets of parameters). Copyright © 2003 John Wiley & Sons, Ltd.     

A local multigrid X‐FEM strategy for 3‐D crack propagation

A multiscale method for 3‐D crack propagation simulation in large structures is proposed. The method is based on the extended finite element method (X‐FEM). The asymptotic behavior of the crack front is accurately modeled using enriched elements and no remeshing is required during crack propagation. However, the different scales involved in fracture mechanics problems can differ by several orders of magnitude and industrial meshes are usually not designed to account for small cracks. Enrichments are therefore useless if the crack is too small compared with the element size. To overcome this drawback, a project combining different numerical techniques was started. The first step was the implementation of a global multigrid algorithm within the X‐FEM framework and was presented in a previous paper (Eur. J. Comput. Mech. 2007; 16 :161–182). This work emphasized the high efficiency in cpu time but highlighted that mesh refinement is required on localized areas only (cracks, inclusions, steep gradient zones). This paper aims at linking the different scales by using a local multigrid approach. The coupling of this technique with the X‐FEM is described and computational aspects dealing with intergrid operators, optimal multiscale enrichment strategy and level sets are pointed out. Examples illustrating the accuracy and efficiency of the method are given. Copyright © 2008 John Wiley & Sons, Ltd.     

A LATIN computational strategy for multiphysics problems: application to poroelasticity

Multiphysics phenomena and coupled‐field problems usually lead to analyses which are computationally intensive. Strategies to keep the cost of these problems affordable are of special interest. For coupled fluid–structure problems, for instance, partitioned procedures and staggered algorithms are often preferred to direct analysis. In this paper, we describe a new strategy for solving coupled multiphysics problems which is built upon the LArge Time INcrement (LATIN) method. The proposed application concerns the consolidation of saturated porous soil, which is a strongly coupled fluid–solid problem. The goal of this paper is to discuss the efficiency of the proposed approach, especially when using an appropriate time‐space approximation of the unknowns for the iterative resolution of the uncoupled global problem. The use of a set of radial loads as an adaptive approximation of the solution during iterations will be validated and a strategy for limiting the number of global resolutions will be tested on multiphysics problems. Copyright © 2003 John Wiley & Sons, Ltd.     

Elastoplasticity with linear tetrahedral elements: A variational multiscale method

We present a computational framework for the simulation of J₂‐elastic/plastic materials in complex geometries based on simple piecewise linear finite elements on tetrahedral grids. We avoid spurious numerical instabilities by means of a specific stabilization method of the variational multiscale kind. Specifically, we introduce the concept of subgrid‐scale displacements, velocities, and pressures, approximated as functions of the governing equation residuals. The subgrid‐scale displacements/velocities are scaled using an effective (tangent) elastoplastic shear modulus, and we demonstrate the beneficial effects of introducing a subgrid‐scale pressure in the plastic regime. We provide proofs of stability and convergence of the proposed algorithms. These methods are initially presented in the context of static computations and then extended to the case of dynamics, where we demonstrate that, in general, naïve extensions of stabilized methods developed initially for static computations seem not effective. We conclude by proposing a dynamic version of the stabilizing mechanisms, which obviates this problematic issue. In its final form, the proposed approach is simple and efficient, as it requires only minimal additional computational and storage cost with respect to a standard finite element relying on a piecewise linear approximation of the displacement field.     

Adaptive sparse grid based HOPGD: Toward a nonintrusive strategy for constructing space‐time welding computational vademecum

Simulation‐based engineering usually needs the construction of computational vademecum to take into account the multiparametric aspect. One example concerns the optimization and inverse identification problems encountered in welding processes. This paper presents a nonintrusive a posteriori strategy for constructing quasi‐optimal space‐time computational vademecum using the higher‐order proper generalized decomposition method. Contrary to conventional tensor decomposition methods, based on full grids (eg, parallel factor analysis/higher‐order singular value decomposition), the proposed method is adapted to sparse grids, which allows an efficient adaptive sampling in the multidimensional parameter space. In addition, a residual‐based accelerator is proposed to accelerate the higher‐order proper generalized decomposition procedure for the optimal aspect of computational vademecum. Based on a simplified welding model, different examples of computational vademecum of dimension up to 6, taking into account both geometry and material parameters, are presented. These vademecums lead to real‐time parametric solutions and can serve as handbook for engineers to deal with optimization, identification, or other problems related to repetitive task.     

Contact mechanics at the nanoscale,a 3D multiscale approach

We present a multiscale coupling method to address contact problems. The components of the model are a molecular dynamics engine, a finite element program and a coupling scheme. We validate the approach, first on Hertzian contact and then with a rough surface contacting a rigid body plane. Various measures are provided to highlight limitations and new opportunities in conducting large‐scale simulations of contact brought by the proposed multiscale approach. Copyright © 2009 John Wiley & Sons, Ltd.     

Stabilized space–time computation of wind-turbine rotor aerodynamics

We show how we use the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) formulation for accurate 3D computation of the aerodynamics of a wind-turbine rotor. As the test case, we use the NREL 5MW offshore baseline wind-turbine rotor. This class of computational problems are rather challenging, because they involve large Reynolds numbers and rotating turbulent flows, and computing the correct torque requires an accurate and meticulous numerical approach. We compute the problem with both the original version of the DSD/SST formulation and a recently introduced version with an advanced turbulence model. The DSD/SST formulation with the advanced turbulence model is a space–time version of the residual-based variational multiscale method. We compare our results to those reported recently, which were obtained with the residual-based variational multiscale Arbitrary Lagrangian–Eulerian method using NURBS for spatial discretization and which we take as the reference solution. While the original DSD/SST formulation yields torque values not far from the reference solution, the DSD/SST formulation with the variational multiscale turbulence model yields torque values very close to the reference solution.     

Model reduction for nonlinear multiscale parabolic problems using dynamic mode decomposition

In this article, a model reduction technique is presented to solve nonlinear multiscale parabolic problems using dynamic mode decomposition. The multiple scales and nonlinearity bring great challenges for simulating the problems. To overcome this difficulty, we develop a model reduction method for the nonlinear multiscale dynamic problems by integrating constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with dynamic mode decomposition (DMD). CEM-GMsFEM has shown great efficiency to solve linear multiscale problems in a coarse space. However, using CEM-GMsFEM to directly solve multiscale nonlinear parabolic models involves dynamically computing the residual and the Jacobian on a fine grid. This may be very computationally expensive because the evaluation of the nonlinear term is implemented in a high-dimensional fine scale space. As a data-driven method, DMD can use observation data and give an explicit expression to accurately describe the underlying nonlinear dynamic system. To efficiently compute the multiscale nonlinear parabolic problems, we propose a CEM-DMD model reduction by combing CEM-GMsFEM and DMD. The CEM-DMD reduced model is a coarsen linear model, which avoids the nonlinear solver in the fine space. It is crucial to judiciously choose observation in DMD. Only proper observation can render an accurate DMD model. In the context of CEM-DMD, we introduce two different observations: fine scale observation and coarse scale observation. In the construction of DMD model, the coarse scale observation requires much less computation than the fine scale observation. The CEM-DMD model using the coarse scale observation gives a complete coarse model for the nonlinear multiscale dynamic systems and significantly improves the computation efficiency. To show the performance of the CEM-DMD using the different observations, we present a few numerical results for the nonlinear multiscale parabolic problems in heterogeneous porous media.     

FEM×DEM multiscale modeling: Model performance enhancement from Newton strategy to element loop parallelization

This paper presents a multiscale model based on a FEM×DEM approach, a method that couples discrete elements at the microscale and finite elements at the macroscale. FEM×DEM has proven to be an effective way to treat real‐scale engineering problems by embedding constitutive laws numerically obtained using discrete elements into a standard finite element framework. This proposed paper focuses on some numerical open issues of the method. Given the nonlinearity of the problem, Newton's method is required. The standard full Newton method is modified by adopting operators different from the consistent tangent matrix and by developing ad‐hoc solution strategies. The efficiency of several existing operators is compared, and a new and original strategy is proposed, which is shown to be numerically more efficient than the existing propositions. Furthermore, a shared memory parallelization framework using OpenMP directives is introduced. The combination of these enhancements allows to overcome the FEM×DEM computational limitations, thus making the approach competitive with classical FEM in terms of stability and computational cost.     

Impact on highly compressible media in explicit dynamics using the X-FEM

Finite element simulations of impact problems on highly compressible media often lead to poor accuracy due to mesh distortion. In explicit dynamics, poorly shaped elements also reduce the stable time step. In order to have satisfactory results and an acceptable computational time, the structure has to be remeshed regularly. A remeshing process can be a burdensome task, especially for 3D problems with complex geometries. In explicit methods, remeshing can also be time consuming compared to the time required for the computation. In this article, we propose to use the extended finite element method (X-FEM) to simplify the remeshing work. This simplification relies on the fact that the X-FEM allows to remesh with meshes that do not match the shape of the deformed structure. A unique simple structured mesh can be used whenever remeshing is needed. A specific algorithm is designed in order to ensure data transfer between successive meshes in the X-FEM context. Several examples demonstrate the efficiency of the proposed method. The final part of the article is dedicated to the treatment of impact problems. It is shown that the use of the penalty method with X-FEM in explicit dynamics leads to a decrease of the stable time step. We propose a specific mass scaling strategy to overcome this issue.