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

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

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

Uncertainty quantification for personalized analyses of human proximal femurs

Computational models for the personalized analysis of human femurs contain uncertainties in bone material properties and loads, which affect the simulation results. To quantify the influence we developed a probabilistic framework based on polynomial chaos (PC) that propagates stochastic input variables through any computational model. We considered a stochastic E-ρ relationship and a stochastic hip contact force, representing realistic variability of experimental data. Their influence on the prediction of principal strains (ϵ1 and ϵ3) was quantified for one human proximal femur, including sensitivity and reliability analysis. Large variabilities in the principal strain predictions were found in the cortical shell of the femoral neck, with coefficients of variation of Math Eq. Between 60-80% of the variance in ϵ1 and ϵ3 are attributable to the uncertainty in the E-ρ relationship, while Math Eq are caused by the load magnitude and 5-30% by the load direction. Principal strain directions were unaffected by material and loading uncertainties. The antero-superior and medial inferior sides of the neck exhibited the largest probabilities for tensile and compression failure, however all were very small (Math Eq). In summary, uncertainty quantification with PC has been demonstrated to efficiently and accurately describe the influence of very different stochastic inputs, which increases the credibility and explanatory power of personalized analyses of human proximal femurs

The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries

We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase-field approximations of sharp domains. Leveraging the properties of the phase-field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase-field solutions of the Allen-Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase-field in the diffuse boundary region and a uniform mesh for the representation of the physics-based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty-type methods. In the context of imaging based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body

Die Bestimmung von Eigenzuständen mit dem Verfahren der Inversen Matrizeniteration

Die Inverse Matrizeniteration ist ein Verfahren zur Bestimmung der Eigenzustände reeller, symmetrischer Matrizen mit konvexer Profilstruktur. Das Verfahren zeichnet sich besonders durch den Erhalt der Profilstruktur und die Bestimmung der Eigenwerte in geordneter Reihenfolge aus. Die Iteration ermöglicht die gezielte Bestimmung mehrerer aufeinander folgender betragskleinster Eigenwerte, wie es im Bauwesen insbesondere für Stabilitäts- und Schwingungsanalysen erforderlich ist. Die Anwendung dieses Verfahrens hat gezeigt, daß ein Teil der gesuchten Eigenwerte mit wenigen Iterationszyklen bestimmt werden kann, während andere eine größere Anzahl von Iterationszyklen erfordern. Diese lokale Konvergenzverschlechterung kann durch eine Jacobi-Randkorrektur, wie sie im Beitrag näher erläutert wird, behoben werden

Status and First Operation of Gyrotron Teststand FULGOR at KIT

FULGOR, the new KIT gyrotron teststand for megawatt-class gyrotrons, will be presented. Results of initial experiments using a 1.5 MW 140 GHz short pulse pre-prototype gyrotron will be discussed

Multi-level hp-finite cell method for embedded interface problems with application in biomechanics

This work presents a numerical discretization technique for solving three-dimensional material interface problems involving complex geometry without conforming mesh generation. The finite cell method (FCM), which is a high-order fictitious domain approach, is used for the numerical approximation of the solution without a boundary-conforming mesh. Weak discontinuities at material interfaces are resolved by using separate FCM meshes for each material sub-domain, and weakly enforcing the interface conditions between the different meshes. Additionally, a recently developed hierarchical hp-refinement scheme is employed to locally refine the FCM meshes in order to resolve singularities and local solution features at the interfaces. Thereby, higher convergence rates are achievable for non-smooth problems. A series of numerical experiments with two- and three-dimensional benchmark problems is presented, showing that the proposed hp-refinement scheme in conjunction with the weak enforcement of the interface conditions leads to a significant improvement of the convergence rates, even in the presence of singularities. Finally, the proposed technique is applied to simulate a vertebra-implant model. The application showcases the method's potential as an accurate simulation tool for biomechanical problems involving complex geometry, and it demonstrates its flexibility in dealing with different types of geometric description