Solving an Elliptic PDE Eigenvalue Problem via Automated Multi-Level Substructuring and Hierarchical Matrices

We propose a new method for the solution of discretised elliptic PDE eigenvalue problems. The new method combines ideas of domain decomposition, as in the automated multi-level substructuring (short AMLS), with the concept of hierarchical matrices (short H-matrices) in order to obtain a solver that scales almost optimal in the size of the discrete space. Whereas the AMLS method is very effective for PDEs posed in two dimensions, it is getting very expensive in the three-dimensional case, due to the fact that the interface coupling in the domain decomposition requires dense matrix operations. We resolve this problem by use of data-sparse hierarchical matrices. In addition to the discretisation error our new approach involves a projection error due to AMLS and an arithmetic error due to H-matrix approximation. A suitable choice of parameters to balance these errors is investigated in examples. Mathematics Subject Classification (2000) 65F15, 65F30, 65F50, 65H17, 65N25, 65N5

Periodic steady state response of large scale mechanical models with local nonlinearities

AbstractLong term dynamics of a class of mechanical systems is investigated in a computationally efficient way. Due to geometric complexity, each structural component is first discretized by applying the finite element method. Frequently, this leads to models with a quite large number of degrees of freedom. In addition, the composite system may also possess nonlinear properties. The method applied overcomes these difficulties by imposing a multi-level substructuring procedure, based on the sparsity pattern of the stiffness matrix. This is necessary, since the number of the resulting equations of motion can be so high that the classical coordinate reduction methods become inefficient to apply. As a result, the original dimension of the complete system is substantially reduced. Subsequently, this allows the application of numerical methods which are efficient for predicting response of small scale systems. In particular, a systematic method is applied next, leading to direct determination of periodic steady state response of nonlinear models subjected to periodic excitation. An appropriate continuation scheme is also applied, leading to evaluation of complete branches of periodic solutions. In addition, the stability properties of the located motions are also determined. Finally, respresentative sets of numerical results are presented for an internal combustion car engine and a complete city bus model. Where possible, the accuracy and validity of the applied methodology is verified by comparison with results obtained for the original models. Moreover, emphasis is placed in comparing results obtained by employing the nonlinear or the corresponding linearized models

Distributed solution of Laplacian eigenvalue problems

The purpose of this article is to approximately compute the eigenvalues of the symmetric Dirichlet Laplacian within an interval $(0,\Lambda)$. A novel domain decomposition Ritz method, partition of unity condensed pole interpolation method, is proposed. This method can be used in distributed computing environments where communication is expensive, e.g., in clusters running on cloud computing services or networked workstations. The Ritz space is obtained from local subspaces consistent with a decomposition of the domain into subdomains. These local subspaces are constructed independently of each other, using data only related to the corresponding subdomain. Relative eigenvalue error is analysed. Numerical examples on a cluster of workstations validate the error analysis and the performance of the method.Comment: 28 page

개선된 다단계 부구조화 기법을 이용한 대형구조물의 재해석 방법

학위논문 (석사)-- 서울대학교 대학원 : 기계항공공학부(멀티스케일 기계설계전공), 2013. 8. 조맹효.유한요소법(FE)에서 대형 모델의 동적 해석 시 계산 효율 향상을 위해 부분구조합성법(Component Mode Synthesis, CMS)이 사용될 수 있다. 최근 새로운 부분구조합성법으로 개선된 다단계 부구조화 기법(Enhanced Multi-Level Substructuring Scheme, EMLS)이 제안되었다. 이 방법은 기존 고정경계 기반 부분구조합성법인 Craig-Bampton 방법보다 시스템 크기를 더 작게 축소할 수 있는 동시에 더 넓은 주파수 대역에서 정확한 해를 구할 수 있다. 본 연구에서는 개선된 다단계 부구조화 기법을 시스템 변형에 따른 반복적 해석 과정에 사용하였고 계산 효율을 극대화하기 위해 재해석 방법을 도입하였다. 이러한 개선된 다단계 부구조화 기법을 이용한 재해석 방법의 적절성과 계산 효율을 확인하기 위해 비행기 날개 구조 모델을 이용하여 크기 최적화 문제를 해결하였다.FEM has still challenge in dynamic analysis of large-scale model when it comes to computational costs. For it, Component Mode Synthesis (CMS) can be solution. New CMS based fixed interface normal mode presented. It is Enhanced Multi-Level Sub-structuring Scheme(EMLS). EMLS has higher accuracy than traditional method such as Craig-Bampton method through using dynamic constraint mode and sub-structuring from hierarchical. For the repeated calculation with EMLS, reanalysis method was adapted. It makes reduction basis from static analysis instead of dynamic analysis. To prove its robustness and effectiveness, sizing optimization for wing box model was solved.초록 1. 서론 2. 부구조화 기법 2.1. 부구조화 기법 2.2. 개선된 다단계 부구조화 기법 3. 재해석 방법 3.1. 재해석 방법 3.2. 재해석 방법의 개선된 다단계 부구조화 기법 적용 4. 수치 예제 5. 결론 참고문헌 부록 AbstractMaste

High-Performance Solvers for Dense Hermitian Eigenproblems

We introduce a new collection of solvers - subsequently called EleMRRR - for large-scale dense Hermitian eigenproblems. EleMRRR solves various types of problems: generalized, standard, and tridiagonal eigenproblems. Among these, the last is of particular importance as it is a solver on its own right, as well as the computational kernel for the first two; we present a fast and scalable tridiagonal solver based on the Algorithm of Multiple Relatively Robust Representations - referred to as PMRRR. Like the other EleMRRR solvers, PMRRR is part of the freely available Elemental library, and is designed to fully support both message-passing (MPI) and multithreading parallelism (SMP). As a result, the solvers can equally be used in pure MPI or in hybrid MPI-SMP fashion. We conducted a thorough performance study of EleMRRR and ScaLAPACK's solvers on two supercomputers. Such a study, performed with up to 8,192 cores, provides precise guidelines to assemble the fastest solver within the ScaLAPACK framework; it also indicates that EleMRRR outperforms even the fastest solvers built from ScaLAPACK's components