A fast implementation of the FETI‐DP method: FETI‐DP‐RBS‐LNA and applications on large scale problems with localized non‐linearities

As parallel and distributed computing gradually becomes the computing standard for large scale problems, the domain decomposition method (DD) has received growing attention since it provides a natural basis for splitting a large problem into many small problems, which can be submitted to individual computing nodes and processed in a parallel fashion. This approach not only provides a method to solve large scale problems that are not solvable on a single computer by using direct sparse solvers but also gives a flexible solution to deal with large scale problems with localized non‐linearities. When some parts of the structure are modified, only the corresponding subdomains and the interface equation that connects all the subdomains need to be recomputed. In this paper, the dual–primal finite element tearing and interconnecting method (FETI‐DP) is carefully investigated, and a reduced back‐substitution (RBS) algorithm is proposed to accelerate the time‐consuming preconditioned conjugate gradient (PCG) iterations involved in the interface problems. Linear–non‐linear analysis (LNA) is also adopted for large scale problems with localized non‐linearities based on subdomain linear–non‐linear identification criteria. This combined approach is named as the FETI‐DP‐RBS‐LNA algorithm and demonstrated on the mechanical analyses of a welding problem. Serial CPU costs of this algorithm are measured at each solution stage and compared with that from the IBM Watson direct sparse solver and the FETI‐DP method. The results demonstrate the effectiveness of the proposed computational approach for simulating welding problems, which is representative of a large class of three‐dimensional large scale problems with localized non‐linearities. Copyright © 2005 John Wiley & Sons, Ltd.     

Coupled multi‐scale cohesive modeling of failure in heterogeneous adhesives

A multi‐scale cohesive numerical framework is proposed to simulate the failure of heterogeneous adhesively bonded systems. This multi‐scale scheme is based on Hill's variational principle of energy equivalence between the higher and lower level scales. It provides an easy way to obtain accurate homogenized macroscopic properties while capturing the physics of failure processes at the micro‐scale in sufficient detail. We use an isotropic rate‐dependent damage model to mimic the failure response of the constituents of heterogeneous adhesives. The finite element method is used to solve the equilibrium equation at each scale. A nested iterative scheme inspired by the return mapping algorithm used in computational inelasticity is implemented. We propose a computationally attractive technique to couple the macro‐ and micro‐scales for rate‐dependent constitutive laws. We introduce an adhesive patch test to study the numerical performance, including spatial and temporal convergence of the multi‐scale scheme. We compare the solution of the multi‐scale cohesive scheme with a direct numerical simulation. Finally, we solve mode I and mode II fracture problems to demonstrate failure at the macro‐scale. Copyright © 2010 John Wiley & Sons, Ltd.     

A FETI‐DP method for the parallel iterative solution of indefinite and complex‐valued solid and shell vibration problems

The dual‐primal finite element tearing and interconnecting (FETI‐DP) domain decomposition method (DDM) is extended to address the iterative solution of a class of indefinite problems of the form ( K ?σ² M ) u = f , and a class of complex problems of the form ( K ?σ² M +iσ D ) u = f , where K , M , and D are three real symmetric matrices arising from the finite element discretization of solid and shell dynamic problems, i is the imaginary complex number, and σ is a real positive number. A key component of this extension is a new coarse problem based on the free‐space solutions of Navier's equations of motion. These solutions are waves, and therefore the resulting DDM is reminiscent of the FETI‐H method. For this reason, it is named here the FETI‐DPH method. For a practically large σ range, FETI‐DPH is shown numerically to be scalable with respect to all of the problem size, substructure size, and number of substructures. The CPU performance of this iterative solver is illustrated on a 40‐processor computing system with the parallel solution, for various σ ranges, of several large‐scale, indefinite, or complex‐valued systems of equations associated with shifted eigenvalue and forced frequency response structural dynamics problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Inexact FETI‐DP methods

Inexact FETI‐DP domain decomposition methods are considered. Preconditioners based on formulations of FETI‐DP as a saddle point problem are used which allow for an inexact solution of the coarse problem. A positive definite reformulation of the preconditioned saddle point problem, which also allows for approximate solvers, is considered as well. In the formulation that iterates on the original FETI‐DP saddle point system, it is also possible to solve the local Neumann subdomain problems inexactly. Given good approximate solvers for the local and coarse problems, convergence bounds of the same quality as for the standard FETI‐DP methods are obtained. Numerical experiments which compare the convergence of the inexact methods with that of standard FETI‐DP are shown for 2D and 3D elasticity using GMRES and CG as Krylov space methods. Based on parallel computations, a comparison of one variant of the inexact FETI‐DP algorithms and the standard FETI‐DP method is carried out and similar parallel performance is achieved. Parallel scalability of the inexact variant is also demonstrated. It is shown that for a very large number of subdomains and a very large coarse problem, the inexact method can be superior. Copyright © 2006 John Wiley & Sons, Ltd.     

Multi‐scale modeling of moving interface problems with flux and field jumps: Application to oxidative degradation of ceramic matrix composites

Problems involving reaction and species diffusion involve field and flux jumps at a moving reaction front. In multi‐scale problems such as carbon fiber composite oxidation, these effects need to be tracked at the microscopic scale of individual carbon fibers. A multi‐scale model is derived in this paper for predicting species distribution in such problems using a fully coupled multi‐scale homogenization approach. The homogenized fluxes from the micro‐scale are derived using Hill's macro‐homogeneity condition accounting for both flux jumps and species density field jumps at the reacting interface in the micro‐scale unit cell. At the macro‐scale, the competition between the transport of reacting species (oxygen) and the reaction product (carbon dioxide) is modeled using homogenized mass conservation equations. The moving reaction front in carbon fibers at the micro‐scale is tracked using level set method and an adaptive meshing strategy. The macroscopic weight loss of the composite when exposed to oxygen is simulated as a function of time using a coupled finite element methodology at various locations in a validated macroscopic model. Copyright © 2010 John Wiley & Sons, Ltd.     

Exploiting semantics of temporal multi‐scale methods to optimize multi‐level mesh partitioning

Multi‐scale problems are often solved by decomposing the problem domain into multiple subdomains, solving them independently using different levels of spatial and temporal refinement, and coupling the subdomain solutions back to obtain the global solution. Most commonly, finite elements are used for spatial discretization, and finite difference time stepping is used for time integration. Given a finite element mesh for the global problem domain, the number of possible decompositions into subdomains and the possible choices for associated time steps is exponentially large, and the computational costs associated with different decompositions can vary by orders of magnitude. The problem of finding an optimal decomposition and the associated time discretization that minimizes computational costs while maintaining accuracy is nontrivial. Existing mesh partitioning tools, such as METIS, overlook the constraints posed by multi‐scale methods and lead to suboptimal partitions with a high performance penalty. We present a multi‐level mesh partitioning approach that exploits domain‐specific knowledge of multi‐scale methods to produce nearly optimal mesh partitions and associated time steps automatically. Results show that for multi‐scale problems, our approach produces decompositions that outperform those produced by state‐of‐the‐art partitioners like METIS and even those that are manually constructed by domain experts. Copyright © 2017 John Wiley & Sons, Ltd.     

A two‐level nonoverlapping Schwarz algorithm for the Stokes problem without primal pressure unknowns

A two‐level nonoverlapping Schwarz algorithm is developed for the Stokes problem. The main feature of the algorithm is that a mixed problem with both velocity and pressure unknowns is solved with a balancing domain decomposition by constraints (BDDC)‐type preconditioner, which consists of solving local Stokes problems and one global coarse problem related to only primal velocity unknowns. Our preconditioner allows to use a smaller set of primal velocity unknowns than other BDDC preconditioners without much concern on certain flux conditions on the subdomain boundaries and the inf–sup stability of the coarse problem. In the two‐dimensional case, velocity unknowns at subdomain corners are selected as the primal unknowns. In addition to them, averages of each velocity component across common faces are employed as the primal unknowns for the three‐dimensional case. By using its close connection to the Dual–primal finite element tearing and interconnecting (FETI‐DP algorithm) (SIAM J Sci Comput 2010; 32 : 3301–3322; SIAM J Numer Anal 2010; 47 : 4142–4162], it is shown that the resulting matrix of our algorithm has the same eigenvalues as the FETI‐DP algorithm except zero and one. The maximum eigenvalue is determined by H/h, the number of elements across each subdomains, and the minimum eigenvalue is bounded below by a constant, which does not depend on any mesh parameters. Convergence of the method is analyzed and numerical results are included. Copyright © 2011 John Wiley & Sons, Ltd.     

A new parallel sparse direct solver: Presentation and numerical experiments in large‐scale structural mechanics parallel computing

The main purpose of this work is to present a new parallel direct solver: Dissection solver. It is based on LU factorization of the sparse matrix of the linear system and allows to detect automatically and handle properly the zero‐energy modes, which are important when dealing with DDM. A performance evaluation and comparisons with other direct solvers (MUMPS, DSCPACK) are also given for both sequential and parallel computations. Results of numerical experiments with a two‐level parallelization of large‐scale structural analysis problems are also presented: FETI is used for the global problem parallelization and Dissection for the local multithreading. In this framework, the largest problem we have solved is of an elastic solid composed of 400 subdomains running on 400 computation nodes (3200 cores) and containing about 165 millions dof. The computation of one single iteration consumes less than 20 min of CPU time. Several comparisons to MUMPS are given for the numerical computation of large‐scale linear systems on a massively parallel cluster: performances and weaknesses of this new solver are highlighted. Copyright © 2011 John Wiley & Sons, Ltd.     

Bubble‐enhanced smoothed finite element formulation: a variational multi‐scale approach for volume‐constrained problems in two‐dimensional linear elasticity

This paper presents a bubble‐enhanced smoothed finite element formulation for the analysis of volume‐constrained problems in two‐dimensional linear elasticity. The new formulation is derived based on the variational multi‐scale approach in which unequal order displacement‐pressure pairs are used for the mixed finite element approximation and hierarchical bubble function is selected for the fine‐scale displacement approximation. An area‐weighted averaging scheme is employed for the two‐scale smoothed strain calculation under the framework of edge‐based smoothed FEM. The smoothed fine‐scale solution is shown to naturally contain the stress field jump of the smoothed coarse‐scale solution across the boundary of edge‐based smoothing domain and thus provides the possibility to stabilize the global solution for volume‐constrained problems. A global monolithic solution strategy is employed, and the fine‐scale solution is solved without the consideration of approximating the strong form of the fine‐scale equation. Several numerical examples are analyzed to demonstrate the accuracy of the present formulation. Copyright © 2014 John Wiley & Sons, Ltd.     

FETI‐DP: a dual–primal unified FETI method—part I: A faster alternative to the two‐level FETI method

The FETI method and its two‐level extension (FETI‐2) are two numerically scalable domain decomposition methods with Lagrange multipliers for the iterative solution of second‐order solid mechanics and fourth‐order beam, plate and shell structural problems, respectively.The FETI‐2 method distinguishes itself from the basic or one‐level FETI method by a second set of Lagrange multipliers that are introduced at the subdomain cross‐points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations. In this paper, we present a dual–primal formulation of the FETI‐2 concept that eliminates the need for that second set of Lagrange multipliers, and unifies all previously developed one‐level and two‐level FETI algorithms into a single dual–primal FETI‐DP method. We show that this new FETI‐DP method is numerically scalable for both second‐order and fourth‐order problems. We also show that it is more robust and more computationally efficient than existing FETI solvers, particularly when the number of subdomains and/or processors is very large. Copyright © 2001 John Wiley & Sons, Ltd.     

A dual‐primal FETI method for solving a class of fluid–structure interaction problems in the frequency domain

The dual‐primal finite element tearing and interconnecting method (FETI‐DP) is extended to systems of linear equations arising from a finite element discretization for a class of fluid–structure interaction problems in the frequency domain. A preconditioned generalized minimal residual method is used to solve the linear equations for the Lagrange multipliers introduced on the subdomain boundaries to enforce continuity of the solution. The coupling between the fluid and the structure on the fluid–structure interface requires an appropriate choice of coarse level degrees of freedom in the FETI‐DP algorithm to achieve fast convergence. Several choices are proposed and tested by numerical experiments on three‐dimensional fluid–structure interaction problems in the mid‐frequency regime that demonstrate the greatly improved performance of the proposed algorithm over the standard FETI‐DP method. Copyright © 2011 John Wiley & Sons, Ltd.     

Propagation of material and surface profile uncertainties on MEMS micro‐resonators using a stochastic second‐order computational multi‐scale approach

This paper aims at accounting for the uncertainties because of material structure and surface topology of micro‐beams in a stochastic multi‐scale model. For micro‐resonators made of anisotropic polycrystalline materials, micro‐scale uncertainties exist because of the grain size, grain orientation, and the surface profile. First, micro‐scale realizations of stochastic volume elements are obtained based on experimental measurements. To account for the surface roughness, the stochastic volume elements are defined as a volume element having the same thickness as the microelectromechanical system (MEMS), with a view to the use of a plate model at the structural scale. The uncertainties are then propagated up to an intermediate scale, the meso‐scale, through a second‐order homogenization procedure. From the meso‐scale plate‐resultant material property realizations, a spatially correlated random field of the in‐plane, out‐of‐plane, and cross‐resultant material tensors can be characterized. Owing to this characterized random field, realizations of MEMS‐scale problems can be defined on a plate finite element model. Samples of the macro‐scale quantity of interest can then be computed by relying on a Monte Carlo simulation procedure. As a case study, the resonance frequency of MEMS micro‐beams is investigated for different uncertainty cases, such as grain‐preferred orientations and surface roughness effects. Copyright © 2016 John Wiley & Sons, Ltd.     

Multi‐scale modeling of surface effects in nano‐materials with temperature‐related Cauchy‐Born hypothesis via the modified boundary Cauchy‐Born model

In nano‐structures, the influence of surface effects on the properties of material is highly important because the ratio of surface to volume at the nano‐scale level is much higher than that of the macro‐scale level. In this paper, a novel temperature‐dependent multi‐scale model is presented based on the modified boundary Cauchy‐Born (MBCB) technique to model the surface, edge, and corner effects in nano‐scale materials. The Lagrangian finite element formulation is incorporated into the heat transfer analysis to develop the thermo‐mechanical finite element model. The temperature‐related Cauchy‐Born hypothesis is implemented by using the Helmholtz free energy to evaluate the temperature effect in the atomistic level. The thermo‐mechanical multi‐scale model is applied to determine the temperature related characteristics at the nano‐scale level. The first and second derivatives of free energy density are computed using the first Piola‐Kirchhoff stress and tangential stiffness tensor at the macro‐scale level. The concept of MBCB is introduced to capture the surface, edge, and corner effects. The salient point of MBCB model is the definition of radial quadrature used at the surface, edge, and corner elements as an indicator of material behavior. The characteristics of quadrature are derived by interpolating the data from the atomic level laid in a circular support around the quadrature in a least‐square approach. Finally, numerical examples are modeled using the proposed computational algorithm, and the results are compared with the fully atomistic model to illustrate the performance of MBCB multi‐scale model in the thermo‐mechanical analysis of metallic nano‐scale devices. Copyright © 2013 John Wiley & Sons, Ltd.     

Iterative solution of large‐scale acoustic scattering problems with multiple right hand‐sides by a domain decomposition method with Lagrange multipliers

An extension of the FETI‐H method is designed for the solution of acoustic scattering problems with multiple right‐hand sides. A new local pre‐conditioning of this domain decomposition method is also presented. The potential of the resulting iterative solver is demonstrated by numerical experiments for two‐dimensional problems with high wavenumbers, as many as 2.5 million complex degrees of freedom, and a sweep on the angle of the incident wave. Preliminary results for a three‐dimensional submarine problem are also included. The FETI‐H method, whose numerical scalability with respect to the mesh and subdomain sizes was previously established, is shown here to be also numerically scalable with respect to the wavenumber. Copyright © 2001 John Wiley & Sons, Ltd.     

Analysis of three‐dimensional fracture mechanics problems: A two‐scale approach using coarse‐generalized FEM meshes

This paper presents a generalized finite element method (GFEM) based on the solution of interdependent global (structural) and local (crack)‐scale problems. The local problems focus on the resolution of fine‐scale features of the solution in the vicinity of three‐dimensional cracks, while the global problem addresses the macro‐scale structural behavior. The local solutions are embedded into the solution space for the global problem using the partition of unity method. The local problems are accurately solved using an hp‐GFEM and thus the proposed method does not rely on analytical solutions. The proposed methodology enables accurate modeling of three‐dimensional cracks on meshes with elements that are orders of magnitude larger than the process zone along crack fronts. The boundary conditions for the local problems are provided by the coarse global mesh solution and can be of Dirichlet, Neumann or Cauchy type. The effect of the type of local boundary conditions on the performance of the proposed GFEM is analyzed. Several three‐dimensional fracture mechanics problems aimed at investigating the accuracy of the method and its computational performance, both in terms of problem size and CPU time, are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

A parallel hp‐FEM infrastructure for three‐dimensional structural acoustics

This paper describes a parallel three‐dimensional numerical infrastructure for the solution of a wide range of time‐harmonic problems in structural acoustics and vibration. High accuracy and rate of error‐convergence, in the mid‐frequency regime,is achieved by the use of hp‐finite and infinite element approximations. The infrastructure supports parallel computation in both single and multi‐frequency settings. Multi‐frequency solves utilize concurrent factoring of the frequency‐dependent linear algebraic systems and are naturally scalable. Scalability of large‐scale single‐frequency problems is realized by using FETI‐DP—an iterative domain‐decomposition scheme. Numerical examples are presented to cover applications in vibratory response of fluid‐filled elastic structures as well as radiation and scattering from elastic structures submerged in an infinite acoustic medium. We demonstrate both the numerical accuracy as well as parallel scalability of the infrastructure in terms of problem parameters that include wavenumber and number of frequencies, polynomial degree of finite/infinite element approximations as well as the number of processors. Scalability and accuracy is evaluated for both single and multiple frequency sweeps on four high‐performance parallel computing platforms: SGI Altix, SGI Origin, IBM p690 SP and Linux‐cluster. Results show good performance on shared as well as distributed‐memory architecture. Copyright © 2006 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.     

Coarse mesh evolution strategies in the Galerkin multigrid method with adaptive remeshing for geometrically non‐linear problems

This paper addresses the strategies of evolving the coarse mesh configurations in the context of the Galerkin multi‐grid (GMG) method when dealing with problems involving large deformations. A new coarse mesh evolution scheme, which continuously and in a simple manner moves the coarse mesh nodal points along with the deformation of the fine mesh, is proposed and its two implementation versions aiming at further improving the efficiency of the scheme are also developed. In addition, the practical aspects of integrating the GMG with adaptive remeshing techniques are discussed. Finally, several large strain elasto‐plastic problems are presented to verify the performances of the proposed schemes and the behaviour of the combined GMG/mesh adaptivity is also illustrated. Numerical results show that up to 40 per cent reduction in the number of MG iterations has been achieved by the new coarse mesh evolution scheme. Copyright © 2000 John Wiley & Sons, Ltd.