An iterative domain decomposition method for the solution of a class of indefinite problems in computational structural dynamics

Abstract

The FETI-DP domain decomposition method (DDM) is extended to address the iterative solution of a class of indefinite problems of the form (A − σM)x = b, where A and M are two real symmetric positive semi-definite matrices arising from the finite element discretization of second-order elastodynamic problems, and σ is a positive number. A key component of this extension is a new coarse problem based on the free-space solutions of Navier's homogeneous displacement 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 given σ, this method is numerically shown to be scalable with respect to all of the problem size, subdomain size, and number of subdomains. Its intrinsic CPU performance is illustrated for various ranges of σ with the solution on an Origin 3800 parallel processor of several large-scale structural dynamics problems.

A balancing domain decomposition method by constraints for advection-diffusion problems

“…Cai and Widlund [5; 6; 7] studied overlapping Schwarz methods for such problems, using a perturbation approach in their analysis, and established that the convergence rates of the two-level overlapping Schwarz methods are independent of the mesh size if the coarse mesh is fine enough. Motivated by the FETI-DPH method proposed by Farhat and Li [16] for solving symmetric indefinite problems, the authors [23] studied a BDDC algorithm for solving Helmholtz equations and estimated its convergence rate using a similar perturbation approach. For some other results using the perturbation approach for domain decomposition methods, see [40; 38; 19].…”

A FETI‐preconditioned conjugate gradient method for large‐scale stochastic finite element problems

“…Solving these problems by the FETI-DP method requires first transforming them into the interface problems (for example, see [8][9][10][11][12][13])…”

“…In this paper, the scalable domain decomposition (DD)-based finite element tearing and interconnecting dual-primal (FETI-DP) method [8][9][10][11][12][13], which is an enhanced variant of the ubiquitous iterative FETI solver [10,[14][15][16][17][18][19][20][21][22][23][24][25][26][27], is proposed as an incomplete block-diagonal preconditioning solver for Equation (1). It is equipped with the Krylov reusage technique first proposed in [15,16] for accelerating its convergence for systems with multiple and repeated right-hand sides.…”

A damping preconditioner for time-harmonic wave equations in fluid and elastic material

“…Domain decomposition methods have been proposed for Helmholtz problems in [7,8,9,10,11,12], and for elastic problems in [13,14,15,16]. Controllability methods have been proposed for both Helmholtz and Navier problems in [17,18].…”