A substructure‐based iterative inner solver coupled with Uzawa's algorithm for the Stokes problem

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.

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.

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.

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.

We consider the computation of a few eigenvectors and corresponding eigenvalues of a large sparse nonsymmetric matrix using shift-invert Arnoldi's method with and without implicit restarts. For the inner iterations we use preconditioned GMRES as the inexact iterative solver. The costs of the solves are measured by the number of inner iterations needed by the iterative solver at each outer step of the algorithm. We first extend the relaxation strategy developed by Simoncini [SIAM J. Numer. Anal., 43 (2005), pp. 1155–1174] to implicitly restarted Arnoldi's method, which yields an improvement in the overall costs of the method. Secondly, we apply a new preconditioning strategy to the inner solver. We show that small rank changes to the preconditioner can produce significant savings in the total number of iterations. The combination of the new preconditioner with the relaxation strategy in implicitly restarted Arnoldi produces enhancement in the overall costs of around 50 percent in the examples considered here. Numerical experiments illustrate the theory throughout the paper.

The two-stage iterative methods for symmetric positive definite matrices

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.

The Boundary Element Tearing and Interconnecting (BETI) methods were recently introduced as boundary element counterparts of the well established Finite Element Tearing and Interconnecting (FETI) methods. Here we combine the BETI method preconditioned by the projector to the “natural coarse grid” with recently proposed optimal algorithms for the solution of bound and equality constrained quadratic programming problems in order to develop a theoretically supported scalable solver for elliptic multidomain boundary variational inequalities such as those describing the equilibrium of a system of bodies in mutual contact. The key observation is that the “natural coarse grid” defines a subspace that contains the solution, so that the preconditioning affects also the non-linear steps. The results are validated by numerical experiments.

This work presents an efficient technique for simultaneously reconstructing an image of a moving object from X-ray projections along with estimating the motion parameters. Current state-of-the-art iterative methods rely on objective functions that contain the solution of an iterative procedure as one of its terms. This complicates the reconstruction process as it leads to nested iterations and makes analytic differentiation impractical. The presented technique relies on an objective function that simultaneously depends on the image and the motion parameters. The derivatives of this objective function towards the image and the motion parameters are known exactly and implemented matrix free and in parallel. Moreover, the stepsizes of both the iterative reconstruction and motion parameter estimation schemes are chosen following a mathematical formulation that guarantees a fast convergence speed. The result is a method that not only needs less iterations, but also removes the need to tune the finite differences stepsize and the number of inner iterations, which allows for efficient, scalable reconstruction using modern optimizers, without nested iterations.

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.

Effect of time-step size on flow around generic car models

In this paper, we apply the projective integration method to moment models of the Boltzmann-BGK equation and investigate the numerical properties of the resulting scheme. Projective integration is an explicit, asymptotic-preserving scheme that is tailored to problems with a large spectral gap between fast and slow eigenvalues of the model. A spectral analysis of the moment model shows a clear spectral gap and reveals the multi-scale nature of the model. The new scheme overcomes the severe time step constraint of standard explicit schemes like the forward Euler scheme by performing a number of inner iterations and then extrapolating the solution forward in time. The projective integration scheme is non-intrusive and yields fast and accurate solutions, as demonstrated using a 1D shock tube test case. These observations open up many possibilities for further use of the scheme for high-resolution discretizations and different collision models.

The monotonicity of two-stage iterative methods

Convergence study of DGFEM SN based 2D/1D coupling method for solving neutron transport k-eigenvalue problems with Fourier analysis