Theoretically supported scalable BETI method for variational inequalities

The boundary element tearing and interconnecting (BETI) methods have recently been introduced as boundary element counterparts of the well‐established finite element tearing and interconnecting (FETI) methods. In this paper we present inexact data‐sparse versions of the BETI methods which avoid the elimination of the primal unknowns and dense matrices. However, instead of symmetric and positive definite systems, we finally have to solve twofold saddle point problems. The proposed iterative solvers and preconditioners result in almost optimal solvers whose complexity is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor. Moreover, the solvers are robust with respect to large coefficient jumps.

A domain decomposition method with Lagrange multipliers for the Stokes problem is developed and analysed. A common approach to solve the Stokes problem, termed the Uzawa algorithm, is to decouple the velocity and the pressure. This approach yields the Schur complement system for the pressure Lagrange multiplier which is solved with an iterative solver. Each outer iteration of the Uzawa procedure involves the inversion of a Laplacian in each spatial direction. The objective of this paper is to effectively solve this inner system (the vector Laplacian system) by applying the finite‐element tearing and interconnecting (FETI) method. Previously calculated search directions for the FETI solver are reused in subsequent outer Uzawa iterations. The advantage of the approach proposed in this paper is that pressure is continuous across the entire computational domain. Numerical tests are performed by solving the driven cavity problem. An analysis of the number of outer Uzawa iterations and inner FETI iterations is reported. Results show that the total number of inner iterations is almost numerically scalable since it grows asymptotically with the mesh size and the number of subdomains. Copyright © 2003 John Wiley & Sons, Ltd.

A class of finite element tearing and interconnecting (FETI) methods for the edge element approximation of vector field problems in two dimensions is introduced and analyzed. First, an abstract framework is presented for the analysis of a class of FETI methods where a natural coarse problem, associated with the substructures, is lacking. Then, a family of FETI methods for edge element approximations is proposed. It is shown that the condition number of the corresponding method is independent of the number of substructures and grows only polylogarithmically with the number of unknowns associated with individual substructures. The estimate is also independent of the jumps of both of the coefficients of the original problem. Numerical results validating our theoretical bounds are given. The method and its analysis can be easily generalized to Raviart--Thomas element approximations in two and three dimensions.

This paper presents a novel approach, based on the domain decomposition method in conjunction with finite element tearing and interconnecting (FETI) algorithm, to model complex EMC problems. FETI algorithm results in the computation of numerical Green's function, which can be compressed efficiently via an efficient rank-revealing SVD algorithm. The numerical Green's function, once obtained, can be readily combined with other domains as a matrix-vector multiplication in the DDM solution process. Few examples are shown to validate the present approach.

Despite their merits, composites have weakness in transverse strength making them susceptible to failures such as delamination. In order to understand the failure mechanism an accurate evaluation of the interlaminar stresses is required. An e‐cient procedure for accurate analysis of multi-layer composite structures is presented, with particular interest in the interlaminar stresses. Parallelization of sequential simulations approach together with the use of domain decomposition methods signiflcantly broadens the range of applications of numerical simulations for computationally demanding problems. In the present paper, attention will be restricted to the Finite Element Tearing and Interconnecting (FETI) method, an e‐cient solution scheme based on the domain decomposition of the structure and enforcing the compatibility conditions between the difierent sub-domains by means of Lagrange multipliers. With this approach, connecting forces between the difierent sub-domains at the interfacial nodes can be obtained and, as a consequence, an accurate post-processing recovery procedure of the interlaminar stresses based on equilibrium has been formulated. In addition, a consistent variational recovery procedure has been implemented to completely recover stress flelds. The reliability of the approach has been tested with problems involving composite plates of various geometries subjected to difierent mechanical loadings.

A new domain decomposition method with Lagrange multipliers for elliptic problems is introduced. It is based on a reformulation of the well-known finite element tearing and interconnecting (FETI) method as a saddle point problem with both primal and dual variables as unknowns. The resulting linear system is solved with block-structured preconditioners combined with a suitable Krylov subspace method. This approach allows the use of inexact subdomain solvers for the positive definite subproblems. It is shown that the condition number of the preconditioned saddle point problem is bounded independently of the number of subregions and depends only polylogarithmically on the number of degrees of freedom of individual local subproblems. Numerical results are presented for a plane stress cantilever membrane problem.

We focus on the analysis of fracture in quasi-brittle materials by exploiting the potential of domain decomposition techniques. More specifically, we restrict our attention to the FETI (Finite Element Tearing and Interconnecting) method which is used as a solver in our non-linear solution scheme. We develop criteria to anticipate the linear/non-linear character of different regions in the structure according to a damage constitutive model. A first application of our scheme focuses on the efficiency increase of a monoscale analysis by simplifying the computations in those areas that remain linear. The second application treats the problem in a multiscale fashion where the resolution of the non-linear domains is increased in order to describe failure phenomena with a higher degree of accuracy. Both applications represent a significant improvement from a computational standpoint when the main non-linear regions are small compared to the size of the whole specimen. This is often the case for brittle and quasi-brittle materials where strain localization is expected to occur upon failure.

In this paper, the transient electrical-thermal co-simulation is implemented with the finite-element method. The capability of the co-simulation is extended to solve large-scale problems by incorporating a domain decomposition scheme called the finite element tearing and interconnecting (FETI). With the high parallel efficiency of FETI for computing with multiple processors, a significant reduction in computation time is achieved. The transient electrical-thermal behaviors of largescale power distribution networks (PDNs), including on-chip power grids, solder bump arrays, and TSV-based PDN are simulated and analyzed with the proposed method. The impacts on the design of the PDNs from different types of input pulses, power maps, and via pitches are also investigated.

This paper gives new existence results for elliptic and evolutionary variational and quasi-variational inequalities. Specifically, we give an existence theorem for evolutionary variational inequalities involving different types of pseudo-monotone operators. Another existence result embarks on elliptic variational inequalities driven by maximal monotone operators. We propose a new recessivity assumption that extends all the classical coercivity conditions. We also obtain criteria for solvability of general quasi-variational inequalities treating in a unifying way elliptic and evolutionary problems. Two of the given existence results for evolutionary quasi-variational inequalities rely on Mosco-type continuity properties and Kluge's fixed point theorem for set-valued maps. We also focus on the case of compact constraints in the evolutionary quasi-variational inequalities. Here a relevant feature is that the underlying space is the domain of a linear, maximal monotone operator endowed with the graph norm. Applications are also given.

Variational inequalities form an important family of nonlinear problems. Some of the more complex physical processes are described by variational inequalities. Several comprehensive monographs can be consulted for the theory and numerical solution of variational inequalities, e.g., [45, 54, 58, 59, 95, 96, 128]. We study standard elliptic variational inequalities (EVIs) in this chapter. We give a brief introduction to some well-known results on the existence, uniqueness and stability of solutions to elliptic variational inequalities. We also discuss numerical approximations of EVIs and their error analysis.

We consider nonlinear elliptic second-order variational inequalities with degenerate (with respect to the spatial variable) and anisotropic coefficients and L1-data. We study the cases where the set of constraints belongs to a certain anisotropic weighted Sobolev space and to a larger function class. In the first case, some new properties of T-solutions and shift T-solutions of the investigated variational inequalities are established. Moreover, the notion of W1,1-regular T-solution is introduced, and a theorem of existence and uniqueness of such a solution is proved. In the second case, we introduce the notion of T-solution of the variational inequalities under consideration and establish conditions of existence and uniqueness of such a solution.

Asymptotic Solutions in Free Boundary Problems of Singularly Perturbed Elliptic Variational Inequalities

The paper is concerned with variational inequalities arising from an optimal-stopping problem for some stochastic partial differential equations. For linear equations, the associated elliptic variational equation and inequality are formulated and studied in an L2-setting. By introducing appropriate Gaussian Sobolev type of Hilbert spaces, existence and uniqueness theorems are proved. Variational formulation for nonlinear equations is also discussed briefly.

This paper deals with state-constrained optimal control problems governed by semilinear elliptic equations or variational inequalities. By using Ekeland's principle, a minimum principle of Pontryagin's type under some stability conditions of the optimal cost with respect to the state constraints is derived.

A geometric inverse problem identification procedure for detection of cavities