Stable and Reliable Computation of Eigenvectors of Large Profile Matrices

Independent eigenvector computation for a given set of eigenvalues of typical engineering eigenvalue problems still is a big challenge for established subspace solution methods. The inverse vector iteration as the standard solution method often is not capable of reliably computing the eigenvectors of a cluster of bad separated eigenvalues. The following contribution presents a stable and reliable solution method for independent and selective eigenvector computation of large symmetric profile matrices. The method is an extension of the well-known and well-understood QR-method for full matrices thus having all its good numerical properties. The effects of finite arithmetic precision of computer representations of eigenvalue/eigenvector solution methods are analysed and it is shown that the numerical behavior of the new method is superior to subspace solution methods

Matrix Iteration for Large Symmetric Eigenvalue Problems

Eigenvalue problems are common in engineering tasks. In particular the prediction of structural stability and dynamic behavior leads to large symmetric real matrices with profile structure, for which a set of successive eigenvalues and the corresponding eigenvectors must be determined. In this paper, a new method of solution for the eigenvalue problem for large real symmetric matrices with profile structure is presented. This method yields the eigenstates in the sequence of the absolute values of their eigenvalues. The profile structure is preserved during iteration, thus reducing the storage requirements and the computational effort. Deflation of the matrix in combination with spectral shifts and repeated preconditioning are used to accelerate the iteration. The method is capable of handling multiple eigenvalues and eigenvalues of equal magnitude but opposite sign. For large matrices, less than one decomposition of the matrix is required for each desired eigenvalue. The determination of the eigenvector corresponding to a given eigenvalue requires one decomposition of the matrix

A Parallel High-Order Fictitious Domain Approach for Biomechanical Applications

The focus of this contribution is on the parallelization of the Finite Cell Method (FCM) applied for biomechanical simulations of human femur bones. The FCM is a high-order fictitious domain method that combines the simplicity of Cartesian grids with the beneficial properties of hierarchical approximation bases of higher order for an increased accuracy and reliablility of the simulation model. A pre-computation scheme for the numerically expensive parts of the finite cell model is presented that shifts a significant part of the analysis update to a setup phase of the simulation, thus increasing the update rate of linear analyses with time-varying geometry properties to a range that even allows user interactive simulations of high quality. Paralellization of both parts, the pre-computation of the model stiffness and the update phase of the simulation is simplified due to a simple and undeformed cell structure of the computation domain. A shared memory parallelized implementation of the method is presented and its performance is tested for a biomedical application of clinical relevance to demonstrate the applicability of the presented method

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