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.     

A parallel two‐level domain decomposition based one‐shot method for shape optimization problems

A two‐level domain decomposition method is introduced for general shape optimization problems constrained by the incompressible Navier–Stokes equations. The optimization problem is first discretized with a finite element method on an unstructured moving mesh that is implicitly defined without assuming that the computational domain is known and then solved by some one‐shot Lagrange–Newton–Krylov–Schwarz algorithms. In this approach, the shape of the domain, its corresponding finite element mesh, the flow fields and their corresponding Lagrange multipliers are all obtained computationally in a single solve of a nonlinear system of equations. Highly scalable parallel algorithms are absolutely necessary to solve such an expensive system. The one‐level domain decomposition method works reasonably well when the number of processors is not large. Aiming for machines with a large number of processors and robust nonlinear convergence, we introduce a two‐level inexact Newton method with a hybrid two‐level overlapping Schwarz preconditioner. As applications, we consider the shape optimization of a cannula problem and an artery bypass problem in 2D. Numerical experiments show that our algorithm performs well on a supercomputer with over 1000 processors for problems with millions of unknowns. Copyright © 2014 John Wiley & Sons, Ltd.     

Finite element analysis of time‐dependent semi‐infinite wave‐guides with high‐order boundary treatment

A new finite element (FE) scheme is proposed for the solution of time‐dependent semi‐infinite wave‐guide problems, in dispersive or non‐dispersive media. The semi‐infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC), based on the Higdon non‐reflecting operators, is developed and applied on ??. The new NRBC does not involve any high derivatives beyond second order, but its order of accuracy is as high as one desires. It involves some parameters which are chosen automatically as a pre‐process. A C⁰ semi‐discrete FE formulation incorporating this NRBC is constructed for the problem in the finite domain bounded by ??. Augmented and split versions of this FE formulation are proposed. The semi‐discrete system of equations is solved by the Newmark time‐integration scheme. Numerical examples concerning dispersive waves in a semi‐infinite wave guide are used to demonstrate the performance of the new method. Copyright © 2003 John Wiley & Sons, Ltd.     

Basis mapping methods for forward and inverse problems

This paper describes a novel method for mapping between basis representation of a field variable over a domain in the context of numerical modelling and inverse problems. In the numerical solution of inverse problems, a continuous scalar or vector field over a domain may be represented in different finite‐dimensional basis approximations, such as an unstructured mesh basis for the numerical solution of the forward problem, and a regular grid basis for the representation of the solution of the inverse problem. Mapping between the basis representations is generally lossy, and the objective of the mapping procedure is to minimise the errors incurred. We present in this paper a novel mapping mechanism that is based on a minimisation of the L² or H¹ norm of the difference between the two basis representations. We provide examples of mapping in 2D and 3D problems, between an unstructured mesh basis representative of an FEM approximation, and different types of structured basis including piecewise constant and linear pixel basis, and blob basis as a representation of the inverse basis. A comparison with results from a simple sampling‐based mapping algorithm shows the superior performance of the method proposed here. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.     

A finite element approach combining a reduced‐order system,Padé approximants,and an adaptive frequency windowing for fast multi‐frequency solution of poro‐acoustic problems

In this work, a solution strategy is investigated for the resolution of multi‐frequency structural‐acoustic problems including 3D modeling of poroelastic materials. The finite element method is used, together with a combination of a modal‐based reduction of the poroelastic domain and a Padé‐based reconstruction approach. It thus takes advantage of the reduced‐size of the problem while further improving the computational efficiency by limiting the number of frequency resolutions of the full‐sized problem. An adaptive procedure is proposed for the discretization of the frequency range into frequency intervals of reconstructed solution. The validation is presented on a 3D poro‐acoustic example. Copyright © 2013 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.     

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.     

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.     

The Direct Solution of Nonconvex Nonlinear FIR Filter Design Problems by a SIP Method

FIR filter design problems in the frequency domain are nonlinear (semi-infinite) optimization problems. In practice, however, these almost always have not been approached directly, but been solved in a simplified form and/or only under restricting assumptions. In this paper, quite general mathematical formulations of the four main design approximation problems in the frequency domain are presented, which enable the derivation of theoretical results (collected here from R. Reemtsen, 2000b, 2000c) and the application of general-purpose optimization procedures to their direct solution. For the actual solution, a nonlinear semi-infinite programming method from the thesis (S. Görner, Ph.D. Thesis, Technische Universität Berlin, Berlin, 1997) of the first author is discussed and applied to several specific design problems. In some cases, the computed solution of the nonlinear problem is compared with that of a convex approximation of the problem.     

A hybrid boundary node method

A new variational formulation for boundary node method (BNM) using a hybrid displacement functional is presented here. The formulation is expressed in terms of domain and boundary variables, and the domain variables are interpolated by classical fundamental solution; while the boundary variables are interpolated by moving least squares (MLS). The main idea is to retain the dimensionality advantages of the BNM, and get a truly meshless method, which does not require a ‘boundary element mesh’, either for the purpose of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regular shaped domains (in general, semi‐sphere in the 3‐D problem) and their boundaries. Numerical examples presented in this paper for the solution of Laplace's equation in 2‐D show that high rates of convergence with mesh refinement are achievable, and the computational results for unknown variables are most accurate. No further integrations are required to compute the unknown variables inside the domain as in the conventional BEM and BNM. Copyright © 2001 John Wiley & Sons, Ltd.     

A two‐scale approximation of the Schur complement and its use for non‐intrusive coupling

This paper presents a two‐scale approximation of the Schur complement of a subdomain's stiffness matrix, obtained by combining local (i.e. element strips) and global (i.e. homogenized) contributions. This approximation is used in the context of a coupling strategy that is designed to embed local plasticity and geometric details into a small region of a large linear elastic structure; the strategy consists in creating a local model that contains the desired features of the concerned region and then substituting it into the global problem by the means of a non‐intrusive solver coupling technique adapted from domain decomposition methods. Using the two‐scale approximation of the Schur complement as a Robin condition on the local model enables to reach high efficiency. Examples include a large 3D problem provided by our industrial partner Snecma. Copyright © 2011 John Wiley & Sons, Ltd.     

Coupled adjoint‐based sensitivities in large‐displacement fluid‐structure interaction using algorithmic differentiation

A methodology for the calculation of gradients with respect to design parameters in general fluid‐structure interaction problems is presented. It is based on fixed‐point iterations on the adjoint variables of the coupled system using algorithmic differentiation. This removes the need for the construction of the analytic Jacobian for the coupled physical problem, which is the usual limitation for the computation of adjoints in most realistic applications. The formulation is shown to be amenable to partitioned solution methods for the adjoint equations. It also poses no restrictions to the nonlinear physics in either the fluid or structural field, other than the existence of a converged solution to the primal problem from which to compute the adjoints. We demonstrate the applicability of this procedure and the accuracy of the computed gradients on coupled problems involving viscous flows with geometrical and material nonlinearities in the structural domain.     

A semi‐analytic collocation technique for steady‐state strongly nonlinear advection‐diffusion‐reaction equations with variable coefficients

The aim of this paper is to present a new semi‐analytic numerical method for strongly nonlinear steady‐state advection‐diffusion‐reaction equation (ADRE) in arbitrary 2‐D domains. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. Each basis function used in the algorithm is a sum of an analytic basis function and a special correcting function which is chosen to satisfy the homogeneous boundary conditions of the problem. The polynomials, trigonometric functions, conical radial basis functions, and the multiquadric radial basis functions are used in approximation of the ADRE. This allows us to seek an approximate solution in the analytic form which satisfies the boundary conditions of the initial problem with any choice of free parameters. As a result, we separate the approximation of the boundary conditions and the approximation of the ADRE inside the solution domain. The numerical examples confirm the high accuracy and efficiency of the proposed method in solving strongly nonlinear equations in an arbitrary domain.     

An advanced boundary element method for solving 2D and 3D static problems in Mindlin's strain‐gradient theory of elasticity

An advanced boundary element method (BEM) for solving two‐ (2D) and three‐dimensional (3D) problems in materials with microstructural effects is presented. The analysis is performed in the context of Mindlin's Form‐II gradient elastic theory. The fundamental solution of the equilibrium partial differential equation is explicitly derived. The integral representation of the problem, consisting of two boundary integral equations, one for displacements and the other for its normal derivative, is developed. The global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2D and 3D problems, respectively. Representative 2D and 3D numerical examples are presented to illustrate the method, demonstrate its accuracy and efficiency and assess the gradient effect on the response. The importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2D problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Meshless analysis of potential problems in three dimensions with the hybrid boundary node method

Combining a modified functional with the moving least‐squares (MLS) approximation, the hybrid boundary node method (Hybrid BNM) is a truly meshless, boundary‐only method. The method may have advantages from the meshless local boundary integral equation (MLBIE) method and also the boundary node method (BNM). In fact, the Hybrid BNN requires only the discrete nodes located on the surface of the domain. The Hybrid BNM has been applied to solve 2D potential problems. In this paper, the Hybrid BNM is extended to solve potential problems in three dimensions. Formulations of the Hybrid BNM for 3D potential problems and the MLS approximation on a generic surface are developed. A general computer code of the Hybrid BNM is implemented in C++. The main drawback of the ‘boundary layer effect’ in the Hybrid BNM in the 2D case is circumvented by an adaptive face integration scheme. The parameters that influence the performance of this method are studied through three different geometries and known analytical fields. Numerical results for the solution of the 3D Laplace's equation show that high convergence rates with mesh refinement and high accuracy are achievable. Copyright © 2004 John Wiley & Sons, Ltd.     

A variational‐inequality approach to stochastic boundary value problems with inequality constraints and its application to contact and elastoplasticity

This paper is concerned with stochastic boundary value problems (SBVPs) whose formulation involves inequality constraints. A class of stochastic variational inequalities (SVIs) is defined, which is well adapted to characterize the solution of specified inequality‐constrained SBVPs. A methodology for solving such SVIs is proposed, which involves their discretization by projection onto polynomial chaos and collocation of the inequality constraints, followed by the solution of a finite‐dimensional constrained optimization problem. Simulation studies in contact and elastoplasticity are provided to demonstrate the proposed framework. Copyright © 2011 John Wiley & Sons, Ltd.     

An edge‐based smoothed finite element method for 3D analysis of solid mechanics problems

The edge‐based smoothed finite element method (ES‐FEM) was proposed recently in Liu, Nguyen‐Thoi, and Lam to improve the accuracy of the FEM for 2D problems. This method belongs to the wider family of the smoothed FEM for which smoothing cells are defined to perform the numerical integration over the domain. Later, the face‐based smoothed FEM (FS‐FEM) was proposed to generalize the ES‐FEM to 3D problems. According to this method, the smoothing cells are centered along the faces of the tetrahedrons of the mesh. In the present paper, an alternative method for the extension of the ES‐FEM to 3D is investigated. This method is based on an underlying mesh composed of tetrahedrons, and the approximation of the field variables is associated with the tetrahedral elements; however, in contrast to the FS‐FEM, the smoothing cells of the proposed ES‐FEM are centered along the edges of the tetrahedrons of the mesh. From selected numerical benchmark problems, it is observed that the ES‐FEM is characterized by a higher accuracy and improved computational efficiency as compared with linear tetrahedral elements and to the FS‐FEM for a given number of degrees of freedom. Copyright © 2013 John Wiley & Sons, Ltd.     

Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC‐PIM)

It is well known that the displacement‐based fully compatible finite element method (FEM) provides a lower bound in energy norm for the exact solution to elasticity problems. It is, however, much more difficult to bound the solution from above for general problems in elasticity, and it has been a dream of many decades to find a systematical way to obtain such an upper bound. This paper presents a very important and unique property of the linearly conforming point interpolation method (LC‐PIM): it provides a general means to obtain an upper bound solution in energy norm for elasticity problems. This paper conducts first a thorough theoretical study on the LC‐PIM: we derive its weak form based on variational principles, study a number of properties of the LC‐PIM, and prove that LC‐PIM is variationally consistent and that it produces upper bound solutions. We then demonstrate these properties through intensive numerical studies with many examples of 1D, 2D, and 3D problems. Using the LC‐PIM together with the FEM, we now have a systematical way to numerically obtain both upper and lower bounds of the exact solution to elasticity problems, as shown in these example problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Efficient numerical methods for the large‐scale,parallel solution of elastoplastic contact problems

Quasi‐static elastoplastic contact problems are ubiquitous in many industrial processes and other contexts, and their numerical simulation is consequently of great interest in accurately describing and optimizing production processes. The key component in these simulations is the solution of a single load step of a time iteration. From a mathematical perspective, the problems to be solved in each time step are characterized by the difficulties of variational inequalities for both the plastic behavior and the contact problem. Computationally, they also often lead to very large problems. In this paper, we present and evaluate a complete set of methods that are (1) designed to work well together and (2) allow for the efficient solution of such problems. In particular, we use adaptive finite element meshes with linear and quadratic elements, a Newton linearization of the plasticity, active set methods for the contact problem, and multigrid‐preconditioned linear solvers. Through a sequence of numerical experiments, we show the performance of these methods. This includes highly accurate solutions of a three‐dimensional benchmark problem and scaling our methods in parallel to 1024 cores and more than a billion unknowns. Copyright © 2015 John Wiley & Sons, Ltd.     

A face‐based smoothed finite element method (FS‐FEM) for 3D linear and geometrically non‐linear solid mechanics problems using 4‐node tetrahedral elements

This paper presents a novel face‐based smoothed finite element method (FS‐FEM) to improve the accuracy of the finite element method (FEM) for three‐dimensional (3D) problems. The FS‐FEM uses 4‐node tetrahedral elements that can be generated automatically for complicated domains. In the FS‐FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements. The results demonstrated that the FS‐FEM is significantly more accurate than the FEM using tetrahedral elements for both linear and geometrically non‐linear solid mechanics problems. In addition, a novel domain‐based selective scheme is proposed leading to a combined FS/NS‐FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The implementation of the FS‐FEM is straightforward and no penalty parameters or additional degrees of freedom are used. The computational efficiency of the FS‐FEM is found better than that of the FEM. Copyright © 2008 John Wiley & Sons, Ltd.