Superrelaxation and the rate of convergence in minimizing quadratic functions subject to bound constraints

SUMMARYA Total FETI (TFETI)-based domain decomposition algorithm with preconditioning by a natural coarse grid of the rigid body motions is adapted to the solution of multibody contact problems of elasticity in 2D and 3D and proved to be scalable. The algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition parameter and the discretization parameter is bounded. The analysis is based on the classical results by Farhat, Mandel, and Roux on scalability of FETI with a natural coarse grid for linear problems and on our development of optimal quadratic programming algorithms for bound and equality constrained problems. The algorithm preserves parallel scalability of the classical FETI method. Both theoretical results and numerical experiments indicate a high efficiency of our algorithm. In addition, its performance is illustrated on a real-world problem of analysis of the ball bearing.

Section: Algorithm 52 Modified Proportioning With Reduced Gradient Pmentioning

confidence: 99%

Scalable TFETI algorithm for the solution of multibody contact problems of elasticity

Dostál

Kozubek

Vondrák

et al. 2009

“…Remark Both of ALG1 and DHS can be further accelerated by using overrelaxation techniques; the step size for these algorithms can be chosen slightly larger than

1 / λ_{\max} (\overline{F})

to achieve faster convergence. One may refer to Yamagishi–Yamada 42 and Dostál et al 43 for overrelaxation techniques for FISTA and MPRGP, respectively. In the above numerical experiments, for the sake of fair comparison, we did not apply overrelaxation to neither of these algorithms.…”

Section: Applicationsmentioning

confidence: 99%

A dual‐primal finite element tearing and interconnecting method for nonlinear variational inequalities utilizing linear local problems

Lee

Park

2021

We propose a novel dual‐primal finite element tearing and interconnecting method for nonlinear variational inequalities. The proposed method is based on a particular Fenchel–Rockafellar dual formulation of the target problem, which yields linear local problems despite the nonlinearity of the target problem. Since local problems are linear, each iteration of the proposed method can be done very efficiently compared with usual nonlinear domain decomposition methods. We prove that the proposed method is linearly convergent with the rate 1−r−1/2 while the convergence rate of relevant existing methods is 1−r−1, where r is proportional to the condition number of the dual operator. The spectrum of the dual operator is analyzed for the cases of two representative variational inequalities in structural mechanics: the obstacle problem and the contact problem. Numerical experiments are conducted in order to support our theoretical results.

“…Recent work has suggested that performance of this method may be improved by diagonal scaling and super relaxation , but these extensions have not been implemented in this work.…”

Section: Summary Of Algorithms Consideredmentioning

confidence: 99%

Using Krylov subspace and spectral methods for solving complementarity problems in many‐body contact dynamics simulation

Heyn

Anitescu

Tasora

et al. 2013

SUMMARYMany-body dynamics problems are expected to handle millions of unknowns when, for instance, investigating the three-dimensional flow of granular material. Unfortunately, the size of the problems tractable by existing numerical solution techniques is severely limited on convergence grounds. This is typically the case when the equations of motion embed a differential variational inequality (DVI) problem that captures contact and possibly frictional interactions between rigid and/or flexible bodies. As the size of the physical system increases, the speed and/or the quality of the numerical solution decrease. This paper describes three methods -the gradient projected minimum residual (GPMINRES) method, the preconditioned spectral projected gradient with fallback (P-SPG-FB) method, and the Kucera method -that demonstrate better scalability than the projected Jacobi and Gauss-Seidel methods commonly used to solve contact problems that draw on a DVI-based modeling approach.