Semiconductor Process Modeling

Numerical Linear Algebra App

2002

SUMMARYA preconditioning methodology for an iterative solution of discrete stress analysis problems based on a space decomposition and subspace correction framework is analysed in this paper. The principle idea of our approach is a decomposition of a global discrete system into the series of subproblems each of which correspond to the di erent Cartesian co-ordinates of the solution (displacement) vector. This enables us to treat the matrix subproblems in a segregated way. A host of well-established scalar solvers can be employed for the solution of subproblems. In this paper we constrain ourselves to an approximate solution using the scalar algebraic multigrid (AMG) solver, while the subspace correction is performed either in block diagonal (Jacobi) or block lower triangular (Gauss-Seidel) fashion. The preconditioning methodology is justiÿed theoretically for the case of the block-diagonal preconditioner using Korn's inequality for estimating the ratio between the extremal eigenvalues of a preconditioned matrix. The e ectiveness of the AMG-based preconditioner is tested on stress analysis 3D model problems that arise in microfabrication technology. The numerical results, which are in accordance with theoretical predictions, clearly demonstrate the superiority of a component decomposition AMG preconditioner over the standard ILU preconditioner, even for the problems with a relatively small number of degrees of freedom.

“…One of the principal sources of stress ÿelds in microfabrication technology is a mismatch between thermal expansion coe cients of di erent material layers [14], [36].…”

Section: Case Studymentioning

confidence: 99%

A component decomposition preconditioning for 3D stress analysis problems

Numerical Linear Algebra App

2002

“…However, the suggested methodology is not restricted in any way to this particular application. Microfabrication technology is concerned with the design and production of microelectronic and micromechanical components [9]. Having established superiority of the component based preconditioning over the standard ILU(0) preconditioning [10], we now examine the efficiency (in terms of wall clock time and memory requirements) of our methodology when various strategies are employed for the solution of local subproblems.…”

Section: Numerical Examplesmentioning

confidence: 99%

Efficiency Study of the “Black-Box” Component Decomposition Preconditioning for Discrete Stress Analysis Problems

Computational Science - ICCS 2004

2004

Abstract. Efficiency of the preconditioning methodology for an iterative solution of discrete stress analysis problems is studied in this article. The preconditioning strategy is based on space decomposition and subspace correction framework. The principle idea is to decompose a global discrete system into a sequence of scalar subproblems, corresponding to the different Cartesian coordinates of the displacement vector. The scalar subproblems can be treated by a host of direct and iterative techniques, however we restrict ourselves to a "black-box" application of the direct sparse solvers and the scalar algebraic multigrid (AMG) method. The subspace correction is performed in either block diagonal or block lower triangular fashion. The efficiency and potential limitations of the proposed preconditioning methodology are studied on stress analysis for 2D and 3D model problems from microfabrication technology.

“…Microfabrication technology employs sequences of processes that utilise a variety of structural elements by embedding, butting and overlaying material layers which have widely differing mechanical properties and intrinsic built-in stress distributions [10]. There is an increasing interest in simulating the stress evolution of microfabrication structures.…”

Section: Case Studymentioning

confidence: 99%

Component‐wise algebraic multigrid preconditioning for the iterative solution of stress analysis problems from microfabrication technology

Commun. Numer. Meth. Engng.

2001

SUMMARYA methodology for preconditioning discrete stress analysis systems using robust scalar algebraic multigrid (AMG) solvers is evaluated in the context of problems that arise in microfabrication technology. The principle idea is to apply an AMG solver in a segregated way to the series of scalar block matrix problems corresponding to different displacement vector components, thus yielding a block diagonal AMG preconditioner. We study the componentwise AMG preconditioning in the context of the space decomposition and subspace correction framework [22]. The subspace problems are solved approximately by the scalar AMG solver and the subspace correction is performed either in block diagonal (block Jacobi) or lower triangular (block Gauss-Seidel) fashion. In our test examples we use fully unstructured grids of different sizes. The numerical experiments show robust and efficient convergence of the Krylov iterative methods with component-wise AMG preconditioning for both 2D and 3D problems.