Solving large-scale dynamic systems using band Lanczos method in Rockwell NASTRAN on CRAY X-MP

The improved cost effectiveness using better models, more accurate and faster algorithms and large scale computing offers more representative dynamic analyses. The band Lanczos eigen-solution method was implemented in Rockwell's version of 1984 COSMIC-released NASTRAN finite element structural analysis computer program to effectively solve for structural vibration modes including those of large complex systems exceeding 10,000 degrees of freedom. The Lanczos vectors were re-orthogonalized locally using the Lanczos Method and globally using the modified Gram-Schmidt method for sweeping rigid-body modes and previously generated modes and Lanczos vectors. The truncated band matrix was solved for vibration frequencies and mode shapes using Givens rotations. Numerical examples are included to demonstrate the cost effectiveness and accuracy of the method as implemented in ROCKWELL NASTRAN. The CRAY version is based on RPK's COSMIC/NASTRAN. The band Lanczos method was more reliable and accurate and converged faster than the single vector Lanczos Method. The band Lanczos method was comparable to the subspace iteration method which was a block version of the inverse power method. However, the subspace matrix tended to be fully populated in the case of subspace iteration and not as sparse as a band matrix

Eigenvalue and eigenmode synthesis in elastically coupled subsystems

A method to synthesize the modal characteristics of a system from the modal characteristics of its subsystems is proposed. The interest is focused on those systems with elastic links between the parts which is the main feature of the proposed method. An algebraic proof is provided for the case of arbitrary number of connections. The solution is a system of equations with a reduced number of degrees of freedom that correspond to the number of elastic links between the subsystems. In addition the method is also interpreted from a physical point of view (equilibrium of the interaction forces). An application to plates linked by means of springs shows how the global eigenfrequencies and eigenmodes are properly computed by means of the subsystems eigenfrequencies and eigenmodes.Peer ReviewedPostprint (author's final draft

Substructuring approach to the calculation of higher-order eigensensitivity

2012-2013 > Academic research: refereed > Publication in refereed journalAccepted ManuscriptPublishe

Recent developments in structural sensitivity analysis

Recent developments are reviewed in two major areas of structural sensitivity analysis: sensitivity of static and transient response; and sensitivity of vibration and buckling eigenproblems. Recent developments from the standpoint of computational cost, accuracy, and ease of implementation are presented. In the area of static response, current interest is focused on sensitivity to shape variation and sensitivity of nonlinear response. Two general approaches are used for computing sensitivities: differentiation of the continuum equations followed by discretization, and the reverse approach of discretization followed by differentiation. It is shown that the choice of methods has important accuracy and implementation implications. In the area of eigenproblem sensitivity, there is a great deal of interest and significant progress in sensitivity of problems with repeated eigenvalues. In addition to reviewing recent contributions in this area, the paper raises the issue of differentiability and continuity associated with the occurrence of repeated eigenvalues

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

Constraint interface preconditioning for topology optimization problems

The discretization of constrained nonlinear optimization problems arising in the field of topology optimization yields algebraic systems which are challenging to solve in practice, due to pathological ill-conditioning, strong nonlinearity and size. In this work we propose a methodology which brings together existing fast algorithms, namely, interior-point for the optimization problem and a novel substructuring domain decomposition method for the ensuing large-scale linear systems. The main contribution is the choice of interface preconditioner which allows for the acceleration of the domain decomposition method, leading to performance independent of problem size.Comment: To be published in SIAM J. Sci. Com

A high-accuracy optical linear algebra processor for finite element applications

Optical linear processors are computationally efficient computers for solving matrix-matrix and matrix-vector oriented problems. Optical system errors limit their dynamic range to 30-40 dB, which limits their accuray to 9-12 bits. Large problems, such as the finite element problem in structural mechanics (with tens or hundreds of thousands of variables) which can exploit the speed of optical processors, require the 32 bit accuracy obtainable from digital machines. To obtain this required 32 bit accuracy with an optical processor, the data can be digitally encoded, thereby reducing the dynamic range requirements of the optical system (i.e., decreasing the effect of optical errors on the data) while providing increased accuracy. This report describes a new digitally encoded optical linear algebra processor architecture for solving finite element and banded matrix-vector problems. A linear static plate bending case study is described which quantities the processor requirements. Multiplication by digital convolution is explained, and the digitally encoded optical processor architecture is advanced