Application of hierarchical matrices for computing the Karhunen-Loève expansion

Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. The uncertainties in the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. To approximate random fields with as few random variables as possible, but still retaining the essential information, the Karhunen-Lo`eve expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of sparse hierarchical matrix techniques for this. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield an efficient and fast discretisation of the random fields presented

On general purpose software in structural reliability

Structural Reliability Development has come to a stage where it could be cast in easy-to-use software and hence accessible to the engineering practice (see e.g. (Schuëller 1997)). In this paper an overview is given over the current developments. Short descriptions of the structure and the capabilities of the various software packages will be presented. Based on the example of COSSAN TM (COSSAN 1996) the basic structure and scope of the software developments related to structural reliability will be elaborated on. Besides the accuracy of the algorithms, computational efficiency and ease-of-use are critical factors in the assessment of structural reliability software. One of the key issues is also the ability of the package to solve the corresponding deterministic problem; in this respect the possibility of interaction with external third-party FE codes proves to be a necessity (see e.g. (Schuëller 2004)

LEOPOLD-FRANZENS UNIVERSITY Chair of Engineering Mechanics

A novel procedure for the estimating the response sensitivity to input parameters of a complex FE model is presented. The method is specifically directed toward problems involving high-dimensional input parameter spaces, as they are encountered during uncertainty analysis with complex, refined FE-models. In these cases one is commonly faced with thousands of uncertain parameters and traditional techniques, e.g. finite difference, are unfeasible. In contrast, the presented method quickly filters out the most influential variables. This is achieved by generating a set of samples with direct MCS, which are closely scattered around the point at which the gradient is sought. From these samples, estimators of the sensitivities are synthesized and the most important ones are refined with a finite difference calculation. The numerical example, illustrating the procedure, involves a refined FE-model of a satellite structure provided by the European Space Agency (ESA). Results are shown which demonstrate the capability of the algorithm to deliver estimates with relatively few samples