• International Journal of Aeronautical and Space Sciences
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• 제9권2호
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• pp.79-86
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• 2008

High performance direct-iterative hybrid linear solver for large scale finite element problem is developed. Direct solution method is robust but difficult to parallelize, whereas iterative solution method is opposite for direct method. Therefore, combining two solution methods is desired to get both high performance parallel efficiency and numerical robustness for large scale structural analysis problems. Hybrid method mentioned in this paper is based on FETI-DP (Finite Element Tearing and Interconnecting-Dual Primal method) which has good parallel scalability and efficiency. It is suitable for fourth and second order finite element elliptic problems including structural analysis problems. We are using the hybrid concept of theses two solution method categories, combining the multifrontal solver into FETI-DP based iterative solver. Hybrid solver is implemented for our general structural analysis code, IPSAP.

• Structural Engineering and Mechanics
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• 제83권3호
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• pp.353-361
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• 2022

Linear Elastic Fracture Mechanics (LEFM) has been developed by applying stress analysis to determine the stress intensity factor (SIF, K). The finite element method (FEM) is widely used as a standard tool for evaluating the SIF for various crack configurations. The prediction accuracy can be achieved by applying an adaptive Delaunay triangulation combined with a FEM. The solution can be solved using either direct or iterative solvers. This work adopts the element-by-element preconditioned conjugate gradient (EBE-PCG) iterative solver into an adaptive FEM to solve the solution to heal problem size constraints that exist when direct solution techniques are applied. It can avoid the formation of a global stiffness matrix of a finite element model. Several numerical experiments reveal that the present method is simple, fast, and efficient compared to conventional sparse direct solvers. The optimum convergence criterion for two-dimensional LEFM analysis is studied. In this paper, four sample problems of a two-edge cracked plate, a center cracked plate, a single-edge cracked plate, and a compact tension specimen is used to evaluate the accuracy of the prediction of the SIF values. Finally, the efficiency of the present iterative solver is summarized by comparing the computational time for all cases.

• 한국전산구조공학회논문집
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• 제32권2호
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• pp.117-124
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• 2019

대규모 유한요소 모델을 빠르게 해석하기는 위해서 병렬 희소 솔버를 필수적으로 적용해야 한다. 이 논문에서는 미세하게 변화하는 시스템 행렬을 대상으로 연속적으로 해를 구해야 하는 문제에서 효율적으로 적용가능한 반복-직접 희소 솔버 조합 기법을 소개한다. 반복-직접 희소 솔버 조합 기법은 병렬 희소 솔버 패키지인 PARDISO에 제안 및 구현된 기법으로 새롭게 행렬값이 갱신된 선형 시스템의 해를 구할 때 이전 선형 시스템에 적용된 직접 희소 솔버의 행렬 분해(factorization) 결과를 Krylov 반복 희소 솔버의 preconditioner로 활용하는 방법을 의미한다. PARDISO에서는 미리 설정된 반복 회수까지 해가 수렴하지 않으면 직접 희소 솔버로 해를 구하며, 이후 이어지는 갱신된 선형 시스템의 해를 구할 때는 최종적으로 사용된 직법 희소 솔버의 행렬 분해 결과를 preconditioner로 사용한다. 이 연구에서는 첫 번째 Krylov 반복 단계에서 소요되는 시간을 동적으로 계산하여 최대 반복 회수를 설정하는 기법을 제안하였으며, 주파수 영역 해석에 적용하여 그 효과를 검증하였다.

• Structural Engineering and Mechanics
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• 제45권5호
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• pp.613-629
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• 2013

In this article we will present a method to directly evaluate the critical point of a non-linear system by using the solution of a polynomial eigenvalue approximation as a starting point for an iterative non-linear system solver. This method will be used to evaluate out-of-plane buckling properties of truss structures for which the lateral displacement of the upper chord has been prevented. The aim is to assess for a number of example structures whether or not the linearized eigenvalue solution gives a relevant starting point for an iterative non-linear system solver in order to find the minimum positive critical load.

• Structural Engineering and Mechanics
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• 제30권4호
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• pp.467-480
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• 2008

An aggregation multigrid method (AMM) is a leading iterative solver in solid mechanics. Recently, AMM is applied for solving Schur Complement system in the FE analysis of shell structures. In this work, an extended application of AMM for solving Schur Complement system in the FE analysis of continuum elements is presented. Further, the performance of the proposed AMM in multiple load cases, which is a challenging problem for an iterative solver, is studied. The proposed method is developed by combining the substructuring and the multigrid methods. The substructuring method avoids factorizing the full-size matrix of an original system and the multigrid method gives near-optimal convergence. This method is demonstrated for the FE analysis of several elastostatic problems. The numerical results show better performance by the proposed method as compared to the preconditioned conjugate gradient method. The smaller computational cost for the iterative procedure of the proposed method gives a good alternative to a direct solver in large systems with multiple load cases.

• 대한기계학회논문집A
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• 제24권5호
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• pp.1215-1230
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• 2000

Parallel Approaches using Cray T3E which is NIPP (Massively Parallel Processors) machine are presented for the efficient computation of the finite element analysis of 3-D shape rolling processes. D omain decomposition method coupled with parallel linear equation solver is used. Domain decomposition is applied for obtaining element tangent stifffiess matrices and residual vectors. Direct and iterative parallel algorithms are used for solving the linear equations. Direct algorithm is_parallel version of direct banded matrix solver. For iterative algorithms, the well-known preconditioned conjugate gradient solver with Jacobi preconditioner is also employed. Moreover a new effective iterative scheme with block inverse matrix preconditioner, which is named by present authors, is presented and its results are compared with the one using Jacobi preconditioner. PVM and MPI are used for message passing and synchronization between processors. The performance and efficiency of each algorithm is discussed and comparisons are made among different algorithms.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제2권1호
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• pp.27-39
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• 1998

We describe a fast numerical method for non-separable elliptic equations in self-adjoin form on irregular adaptive domains. One of the most successful results in numerical PDE is developing rapid elliptic solvers for separable EPDEs, for example, Fourier transformation methods for Poisson problem on a square, however, it is known that there is no rapid elliptic solvers capable of solving a general nonseparable problems. It is the purpose of this paper to present an iterative solver for linear EPDEs in self-adjoint form. The scheme discussed in this paper solves a given non-separable equation using a sequence of solutions of Poisson equations, therefore, the most important key for such a method is having a good Poison solver. High performance is achieved by using a fast high-order adaptive Poisson solver which requires only about 500 floating point operations per gridpoint in order to obtain machine precision for both the computed solution and its partial derivatives. A few numerical examples have been presented.

• Tribology and Lubricants
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• 제28권4호
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• pp.160-166
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• 2012

This paper presents a Reynolds equation solver for hydrostatic gas bearings, implemented to run on graphics processing units (GPUs). The original analysis code for the central processing unit (CPU) was modified for the GPU by using the compute unified device architecture (CUDA). The red-black Gauss-Seidel (RBGS) algorithm was employed instead of the original Gauss-Seidel algorithm for the iterative pressure solver, because the latter has data dependency between neighboring nodes. The implemented GPU program was tested on the nVidia GTX580 system and compared to the original CPU program on the AMD Llano system. In the iterative pressure calculation, the implemented GPU program showed 20-100 times faster performance than the original CPU codes. Comparison of the wall-clock times including all of pre/post processing codes showed that the GPU codes still delivered 4-12 times faster performance than the CPU code for our target problem.

• 한국지구물리탐사학회:학술대회논문집
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• 한국지구물리탐사학회 2009년도 학술대회 초록집
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• pp.75-80
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• 2009

유한요소법을 이용한 3차원 모델링은 모형에 대한 기하학적 유연성에도 불구하고 대형 연립방정식에 대한 계산이 필요하여 실용성에는 많은 제약이 있었다. 본 연구에서는 최근 컴퓨터와 수치해석기술을 바탕으로 직접해법과 반복해법을 선택적으로 사용한 효율적인 유한요소 3차원 모델링을 구현하였다. 직접해법인 PARDISO는 멀티코어 계산환경에서 병렬 연산을 이용하여 계산속도를 현저히 감소시킬 수 있었고 특히 단딜 주파수 다중 송신의 계산에서 최대의 계산효율을 보였다. 소규모 메모리 환경에서는 BiCGSTAB(1)이 다른 반복해법들에 비해 빠르고 안정적으로 수치해를 계산하였다. 효율적인 3차원 모델링은 주어진 문제와 상황에 적합한 해법을 선택적으로 사용함으로써 가능하다. 모델링에는 다양한 형태의 인공송신원이 고려되어 있으며 향후 3차원 역산의 효율적인 엔진으로 활용할 수 있다.

• Coupled systems mechanics
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• 제7권6호
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• pp.755-774
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• 2018

The Ritz method is known as very successful strategy for discretizing continuous problems, but it has never been used for solving systems of algebraic equations. The Iterated Ritz Method (IRM) is a novel iterative solver based on the discretized Ritz procedure applied at each iteration step. With an appropriate choice of coordinate vectors, the method may be efficient in linear, nonlinear and optimization problems. Additionally, some iterative methods can be explained as special cases of this approach, which helps to understand advantages and limitations of these methods and gives motivation for their improvement in sense of IRM. In this paper, some ideas for generation of efficient coordinate vectors are presented. The algorithm was developed and tested independently and then implemented into the open source program FEAP. Method has been successfully applied to displacement based (even ill-conditioned) models of structural engineering practice. With this original approach, a new iterative solution strategy has been opened.