Effectiveness of GMRES-DR and OSP-ILUC for wave diffraction analysis of a very large floating structure (VLFS)

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Abstract

This paper presents the performance of iterative solvers and preconditioners for the non-Hermitian dense linear systems arising from the boundary value problem related to the diffraction wave field around a very large floating structure (VLFS). These systems can be solved iteratively using the GMRES with deflated restarting (GMRES-DR), which has Krylov subspaces with approximate eigenvectors as starting vectors. The number of iteration needed by GMRES or GMRES-DR can be significantly reduced using preconditioning techniques. Matrix-vector products are approximated by utilizing the fast multipole method (FMM), which need not directly calculate the dense matrix of the far field interactions. The combination of the operator splitting preconditioner (OSP) and the Crout version of the incomplete LU factorization (ILUC) does not require the dense matrix of the far field interactions. Numerical experiments from a hybrid-type VLFS, which is composed of pontoon-part and semi-submersible part, whose length is 3000 m are presented.

Introduction

The problem of the interaction between the global structural response of a very large floating structure (VLFS) and the associated diffraction wave field in an unbounded exterior domain is achieved here by matching a boundary element analysis of the exterior diffraction wave field with a finite element analysis of the elastic structure at the fluid-structure interface. We finally arrive at solving a large system of linear equations of the formAx=b,where the coefficient matrix A is non-Hermitian and dense (see Refs. [1], [2]).

The generalized minimal residual method (GMRES) [3] is a well-known iterative method for solving large non-Hermitian linear systems of equations. Since GMRES becomes increasingly expensive and requires more storage as the iteration proceeds, it generally uses restarting, which slows the convergence. The GMRES with deflated restarting (GMRES-DR) [4] is derived from restarting implicitly with an appropriate vector, which is a linear combination of the approximate eigenvectors obtained from the orthogonal projection method [5] onto Krylov subspaces [3], [4], [5]. The deflation of eigenvalues can greatly improve the convergence rate of restarted GMRES.

It is well known that the convergence rate of Krylov subspace methods for linear equations depends on the spectrum of A [6]. It is therefore natural to try to transform the original system (1) into one having the same solution but more favorable spectral properties. A preconditioner is a matrix that can be used to accomplish such a transformation. The operator splitting preconditioner (OSP) [7], [8] is an effective technique in solving dense linear systems arising from the boundary element method (BEM). OSP splits the coefficient matrix A into the sparse matrix Anear of the near field interactions and the dense matrix Afar of the far field interactions. Out of Anear, the preconditioner M is constructed using the Crout version of the incomplete LU factorization (ILUC) [9]. Matrix-vector products are approximated by utilizing the fast multipole method (FMM) [10], [11], which need not directly calculate Afar. The OSP-ILUC preconditioner does not require Afar. Therefore, it can be expected that the boundary element method using FMM will be further accelerated by OSP-ILUC.

In the first part of this paper, we apply GMRES-DR to the analysis of the boundary value problem related to the diffraction wave field around a very large floating structure (VLFS).

In the second part, we apply OSP-ILUC to our example. OSP has a parameter r, which is the minimum distance of far points. We show that OSP becomes ineffective if r is too large. We clarify the range of r where OSP is effective at the same time.

Section snippets

Integral equation formulation

Example that is investigated is shown in Fig. 1, which is a hybrid-type VLFS, which is composed of pontoon-part and semi-submersible part, floating in the open sea with constant depth h, whose length is L, whose width is B, and whose draft is d [1]. The coordinate system is defined such that the xy plane locates on the undisturbed free surface and the z-axis points upward. The longcrested harmonic wave with small amplitude is considered. The amplitude of the incident wave is defined by A, the

GMRES with deflated restarting

It is well known that the convergence rate of GMRES for linear equations depends on the distribution of eigenvalues of the coefficient matrix A. Therefore, deflating small eigenvalues can greatly improve the convergence rate. The GMRES with deflated restarting (GMRES-DR) [4] is derived from restarting implicitly with an appropriate vector, which is a linear combination of harmonic Ritz vectors [4].

Numerical experiments without preconditioning

Example that is investigated is the hybrid-type VLFS (See Fig. 1) floating in the open sea, whose main specifications are shown in Table 1. The wave diffraction analysis has been made for the oblique incident wave (β=π/3) whose wavelengths are λ=121.21 m or λ=88.77 m.

The eight-noded quadrilateral panel element is used. The pontoon-part is discretized by 10 m×10 m panels for the bottom surface and 10 m×1.5 m panels for the side surfaces. The number of elements is 8480 and the number of nodes is 25,921

Construction of the preconditioner

The number of iterations needed by GMRES may be significantly reduced using preconditioning techniques.

Let M be a preconditioner. We transform the original system (1) intoAM1y=b;x=M1y.

As is known, there are two somewhat self-conflicting requirements of the preconditioner. The first one is efficiency, that is, the extra work required to solve Mx=y should be easier than solving original system (1). Practically this means that M must be sparse or a product of sparse matrices. The second

Numerical experiments with preconditioning

Example that is investigated is the hybrid-type VLFS (See Fig. 1) floating in the open sea, whose main specifications are shown in Table 1. The wave diffraction analysis has been made for the oblique incident wave (β=π/3) whose wavelengths are λ=88.77 m or λ=55.03 m. Also, we have compared the spectrum of the coefficient matrix AM−1 transformed by OSP-ILUC with that of A without preconditioning.

The eight-noded quadrilateral panel element is used. The pontoon-part is discretized by 10 m×10 m panels

Conclusions

In the first part of this paper, we have applied GMRES-DR to the analysis of the boundary value problem related to the diffraction wave field around a very large floating structure (VLFS). We have succeeded in reducing the total computing times by 1/2 as compared with using restarted GMRES. GMRES-DR has the parameter l, which is the desired number of harmonic Ritz vectors. It has been found that the parameter l of GMRES-DR can be fixed on about l/m=1/10 regardless of incident wavelengths λ,

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