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.     

A multilevel projection‐based model order reduction framework for nonlinear dynamic multiscale problems in structural and solid mechanics

A reduction/hyper reduction framework is presented for dramatically accelerating the solution of nonlinear dynamic multiscale problems in structural and solid mechanics. At each scale, the dimensionality of the governing equations is reduced using the method of snapshots for proper orthogonal decomposition, and computational efficiency is achieved for the evaluation of the nonlinear reduced‐order terms using a carefully designed configuration of the energy conserving sampling and weighting method. Periodic boundary conditions at the microscales are treated as linear multipoint constraints and reduced via projection onto the span of a basis formed from the singular value decomposition of Lagrange multiplier snapshots. Most importantly, information is efficiently transmitted between the scales without incurring high‐dimensional operations. In this proposed proper orthogonal decomposition–energy conserving sampling and weighting nonlinear model reduction framework, training is performed in two steps. First, a microscale hyper reduced‐order model is constructed in situ, or using a mesh coarsening strategy, in order to achieve significant speedups even in non‐parametric settings. Next, a classical offline–online training approach is performed to build a parametric hyper reduced‐order macroscale model, which completes the construction of a fully hyper reduced‐order parametric multiscale model capable of fast and accurate multiscale simulations. A notable feature of this computational framework is the minimization, at the macroscale level, of the cost of the offline training using the in situ or coarsely trained hyper reduced‐order microscale model to accelerate snapshot acquisition. The effectiveness of the proposed hyper reduction framework at accelerating the solution of nonlinear dynamic multiscale problems is demonstrated for two problems in structural and solid mechanics. Speedup factors as high as five orders of magnitude are shown to be achievable. Copyright © 2017 John Wiley & Sons, Ltd.     

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. 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 is used for the computation of three‐dimensional 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 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. Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive atomistic‐to‐continuum modeling of propagating defects

An adaptive atomistic‐to‐continuum method is presented for modeling the propagation of material defects. This method extends the bridging domain method to allow the atomic domain to dynamically conform to the evolving defect regions during a simulation, without introducing spurious oscillations and without requiring mesh refinement. The atomic domain expands as defects approach the bridging domain method coupling domain by fine graining nearby finite elements into equivalent atomistic subdomains. Additional algorithms coarse grain portions of the atomic domain to the continuum scale, reducing the degrees of freedom, when the atomic displacements in a subdomain can be approximated by FEM or extended FEM elements to within a certain homogeneity tolerance. The extended FEM approximations are created by fitting the broken inter‐atomic bonds of fractured surfaces and dislocation slip planes. Because atomic degrees of freedom are maintained only where needed for each timestep, the solution retains the advantages of multiscale modeling, with a reduced computational cost compared with other multiscale methods. Copyright © 2012 John Wiley & Sons, Ltd.     

An efficient coarse‐grained parallel algorithm for global–local multiscale computations on massively parallel systems

The existing global–local multiscale computational methods, using finite element discretization at both the macro‐scale and micro‐scale, are intensive both in terms of computational time and memory requirements and their parallelization using domain decomposition methods incur substantial communication overhead, limiting their application. We are interested in a class of explicit global–local multiscale methods whose architecture significantly reduces this communication overhead on massively parallel machines. However, a naïve task decomposition based on distributing individual macro‐scale integration points to a single group of processors is not optimal and leads to communication overheads and idling of processors. To overcome this problem, we have developed a novel coarse‐grained parallel algorithm in which groups of macro‐scale integration points are distributed to a layer of processors. Each processor in this layer communicates locally with a group of processors that are responsible for the micro‐scale computations. The overlapping groups of processors are shown to achieve optimal concurrency at significantly reduced communication overhead. Several example problems are presented to demonstrate the efficiency of the proposed algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

Solving large‐scale nonlinear eigenvalue problems by rational interpolation and resolvent sampling based Rayleigh–Ritz method

Numerical solution of nonlinear eigenvalue problems (NEPs) is frequently encountered in computational science and engineering. The applicability of most existing methods is limited by the matrix structures, properties of the eigen‐solutions, sizes of the problems, etc. This paper aims to remove those limitations and develop robust and universal NEP solvers for large‐scale engineering applications. The novelty lies in two aspects. First, a rational interpolation approach (RIA) is proposed based on the Keldysh theorem for holomorphic matrix functions. Comparing with the existing contour integral approach, the RIA provides the possibility to select sampling points in more general regions and has advantages in improving the accuracy and reducing the computational cost. Second, a resolvent sampling scheme using the RIA is proposed to construct reliable search spaces for the Rayleigh–Ritz procedure, based on which a robust eigen‐solver, called resolvent sampling based Rayleigh–Ritz method (RSRR), is developed for solving general NEPs. The RSRR can be easily implemented and parallelized. The advantages of the RIA and the performance of the RSRR are demonstrated by a variety of benchmark and application examples. Copyright © 2016 John Wiley & Sons, Ltd.     

A local thermomechanical model of friction for computing structures under extreme loading conditions

A multiscale thermomechanical model of friction is proposed for metallic interfaces submitted to extreme loading conditions with large sliding velocities ( [v] > 100 ms^?1) and high contact pressures (P > 1 GPa). This model decomposes the friction problem into a global multidimensional structural problem and a local interface subproblem. At global scale, the structure behavior is governed by an elastoplastic model in large strains, and thermal conduction is neglected. The local model describes the micrometric scale and the underlying thermomechanical mechanisms: mechanical hardening, frictional heating, and plastic work. Using dimensional and asymptotic analysis, the corresponding set of equations at local scale is simplified and reduces to a one‐dimensional quasistatic thermoelastoplastic friction model. It is solved locally at each global point by a finite differences subgrid model using a nonlinear time‐implicit solver. This solver is coupled to a time‐explicit Lagrangian hydrocode used at the global structural scale. The coupling strategy between the solvers is force‐based: the frictional stress is evaluated at the local scale by our multiscale friction model and transferred to the global model that can then predict local velocity corrections. This coupling strategy has been successfully tested on real‐life cases. Copyright © 2011 John Wiley & Sons, Ltd.     

夹芯梁结构有效性能的多尺度变分渐近模型

为有效分析夹芯梁结构性能,基于变分渐近法建立多尺度变分渐近模型。首先基于旋转张量分解概念建立三维夹芯梁几何非线性的能量方程;利用梁结构细长和非均质较小的特征,将三维夹芯梁结构各向异性、非均质问题严格分解为宏观层面的梁轴线的一维非线性分析和细观层面的单胞本构分析。基于最小势能原理,通过对单胞应变能泛函变分主导项最小化得到有效属性和波动函数解,代入梁的一维模型进行全局非线性响应分析。利用得到的全局响应、波动函数解重构局部场。由于变分特性,构建多尺度模型可以很容易通过有限元数值实现。通过三类夹芯梁结构算例表明:构建模型得到的全局位移和局部应力场与三维有限元具有很好的一致性,但计算成本和建模工作量明显减少,为结构设计人员在初始设计阶段对夹芯梁结构性能评估提供了一种简洁的途径。      

A new impedance accounting for short‐ and long‐range effects in mixed substructured formulations of nonlinear problems

An efficient method for solving large nonlinear problems combines Newton solvers and domain decomposition methods. In the domain decomposition method framework, the boundary conditions can be chosen to be primal, dual, or mixed. The mixed approach presents the advantage to be eligible for the research of an optimal interface parameter (often called impedance), which can increase the convergence rate. The optimal value for this parameter is usually too expensive to be computed exactly in practice: An approximate version has to be sought, along with a compromise between efficiency and computational cost. In the context of parallel algorithms for solving nonlinear structural mechanical problems, we propose a new heuristic for the impedance, which combines short‐ and long‐range effects at a low computational cost.     

Output error estimation strategies for discontinuous Galerkin discretizations of unsteady convection‐dominated flows

We study practical strategies for estimating numerical errors in scalar outputs calculated from unsteady simulations of convection‐dominated flows, including those governed by the compressible Navier–Stokes equations. The discretization is a discontinuous Galerkin finite element method in space and time on static spatial meshes. Time‐integral quantities are considered for scalar outputs and these are shown to superconverge with temporal refinement. Output error estimates are calculated using the adjoint‐weighted residual method, where the unsteady adjoint solution is obtained using a discrete approach with an iterative solver. We investigate the accuracy versus computational cost trade‐off for various approximations of the fine‐space adjoint and find that exact adjoint solutions are accurate but expensive. To reduce the cost, we propose a local temporal reconstruction that takes advantage of superconvergence properties at Radau points, and a spatial reconstruction based on nearest‐neighbor elements. This inexact adjoint yields output error estimates at a computational cost of less than 2.5 times that of the forward problem for the cases tested. The calculated error estimates account for numerical error arising from both the spatial and temporal discretizations, and we present a method for identifying the percentage contributions of each discretization to the output error. Copyright © 2011 John Wiley & Sons, Ltd.     

The finite element square reduced (FE2R) method with GPU acceleration: towards three‐dimensional two‐scale simulations

The FE² method is a renown computational multiscale simulation technique for solid materials with fine‐scale microstructure. It allows for the accurate prediction of the mechanical behavior of structures made of heterogeneous materials with nonlinear material behavior. However, the FE² method leads to excessive CPU time and storage requirements, even for academic two‐dimensional problems. In order to allow for realistic three‐dimensional two‐scale simulations, a significant reduction of the CPU and memory usage is required. For this purpose, the authors have recently proposed a reduced basis homogenization scheme based on a mixed incremental variational principle. The approach exploits the potential structure of generalized standard materials. Thereby, important speed‐ups and memory savings can be achieved. Using high‐performance GPUs, the reduced‐basis method can be further accelerated. In the present contribution, our previous works are combined and extended to form the FE²‐reduced method: the FE^2R. The FE^2R can be used to simulate three‐dimensional structural problems with consideration of the nonlinearity and microstructure of the underlying material at acceptable computational cost. Thereby, it allows for a new level of complexity in nonlinear multiscale simulations. Numerical examples illustrate the capabilities of the chosen approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture

Adaptive mesh refinement and coarsening schemes are proposed for efficient computational simulation of dynamic cohesive fracture. The adaptive mesh refinement consists of a sequence of edge‐split operators, whereas the adaptive mesh coarsening is based on a sequence of vertex‐removal (or edge‐collapse) operators. Nodal perturbation and edge‐swap operators are also employed around the crack tip region to improve crack geometry representation, and cohesive surface elements are adaptively inserted whenever and wherever they are needed by means of an extrinsic cohesive zone model approach. Such adaptive mesh modification events are maintained in conjunction with a topological data structure (TopS). The so‐called PPR potential‐based cohesive model (J. Mech. Phys. Solids 2009; 57 :891–908) is utilized for the constitutive relationship of the cohesive zone model. The examples investigated include mode I fracture, mixed‐mode fracture and crack branching problems. The computational results using mesh adaptivity (refinement and coarsening) are consistent with the results using uniform mesh refinement. The present approach significantly reduces computational cost while exhibiting a multiscale effect that captures both global macro‐crack and local micro‐cracks. Copyright © 2012 John Wiley & Sons, Ltd.     

A multiscale virtual element method for the analysis of heterogeneous media

We introduce a novel heterogeneous multiscale method for the elastic analysis of two-dimensional domains with a complex microstructure. To this end, the multiscale finite element method is revisited and originally upgraded by introducing virtual element discretizations at the microscale, hence allowing for generalized polygonal and nonconvex elements. The microscale is upscaled through the numerical evaluation of a set of multiscale basis functions. The solution of the equilibrium equations is performed at the coarse scale at a reduced computational cost. We discuss the computation of the multiscale basis functions and corresponding virtual projection operators. The performance of the method in terms of accuracy and computational efficiency is evaluated through a set of numerical examples.     

A domain decomposition technique applied to the solution of the coupled electro‐mechanical problem

A domain decomposition approach for the solution of the coupled electro‐mechanical problem in dynamics is proposed. The finite element analysis of a coupled electro‐mechanical system is frequently found, for example, in the modelling and design of microsystems and may lead to a burdensome nonlinear problem solution, particularly in the dynamic case. Two versions of the algorithm are proposed: the first one, called single‐level decomposition, exploits the natural partition of the analysis domain given by the two physics to be solved; the second one, called two‐level decomposition, adds a further subdivision of each physics into subdomains. The multilevel domain decomposition strategy here proposed is shown to accurately predict the response of microsystems subjected to electro‐mechanical coupling and to allow for a significant reduction in the computational burden. Copyright © 2012 John Wiley & Sons, Ltd.     

Efficiently solving linear bilevel programming problems using off-the-shelf optimization software

Many optimization models in engineering are formulated as bilevel problems. Bilevel optimization problems are mathematical programs where a subset of variables is constrained to be an optimal solution of another mathematical program. Due to the lack of optimization software that can directly handle and solve bilevel problems, most existing solution methods reformulate the bilevel problem as a mathematical program with complementarity conditions (MPCC) by replacing the lower-level problem with its necessary and sufficient optimality conditions. MPCCs are single-level non-convex optimization problems that do not satisfy the standard constraint qualifications and therefore, nonlinear solvers may fail to provide even local optimal solutions. In this paper we propose a method that first solves iteratively a set of regularized MPCCs using an off-the-shelf nonlinear solver to find a local optimal solution. Local optimal information is then used to reduce the computational burden of solving the Fortuny-Amat reformulation of the MPCC to global optimality using off-the-shelf mixed-integer solvers. This method is tested using a wide range of randomly generated examples. The results show that our method outperforms existing general-purpose methods in terms of computational burden and global optimality.     

A nonintrusive stochastic multiscale solver

In this paper, we describe a practical nonintrusive multiscale solver that permits consideration of uncertainties in heterogeneous materials without exhausting the available computational resources. The computational complexity of analyzing heterogeneous material systems is governed by the physical and probability spaces at multiple scales. To deal with these large spaces, we employ reduced order homogenization approach in combination with the Karhunen–Loeve expansion and stochastic collocation method based on sparse grid. The resulting nonintrusive multiscale solver, which is aimed at providing practical solutions for complex multiscale stochastic problems, has been verified against the Latin Hypercube Monte–Carlo method. Copyright © 2011 John Wiley & Sons, Ltd.     

Variational multiscale enrichment for modeling coupled mechano‐diffusion problems

In this paper, a new multiscale–multiphysics computational methodology is devised for the analysis of coupled diffusion–deformation problems. The proposed methodology is based on the variational multiscale principles. The basic premise of the approach is accurate fine‐scale representation at a small subdomain where it is known a priori that important physical phenomena are likely to occur. The response within the remainder of the problem domain is idealized on the basis of coarse‐scale representation. We apply this idea to evaluate a coupled mechano‐diffusion problem that idealizes the response of titanium structures subjected to a thermo–chemo–mechanical environment. The proposed methodology is used to devise a multiscale model in which the transport of oxygen into titanium is modeled as a diffusion process, whereas the mechanical response is idealized using concentration‐dependent elasticity equations. A coupled solution strategy based on operator split is formulated to evaluate the coupled multiphysics and multiscale problem. Numerical experiments are conducted to assess the accuracy and computational performance of the proposed methodology. Numerical simulations indicate that the variational multiscale enrichment has reasonable accuracy and is computationally efficient in modeling the coupled mechano‐diffusion response. Copyright © 2011 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.