Lanczos eigensolution method for high-performance computers

The theory, computational analysis, and applications are presented of a Lanczos algorithm on high performance computers. The computationally intensive steps of the algorithm are identified as: the matrix factorization, the forward/backward equation solution, and the matrix vector multiples. These computational steps are optimized to exploit the vector and parallel capabilities of high performance computers. The savings in computational time from applying optimization techniques such as: variable band and sparse data storage and access, loop unrolling, use of local memory, and compiler directives are presented. Two large scale structural analysis applications are described: the buckling of a composite blade stiffened panel with a cutout, and the vibration analysis of a high speed civil transport. The sequential computational time for the panel problem executed on a CONVEX computer of 181.6 seconds was decreased to 14.1 seconds with the optimized vector algorithm. The best computational time of 23 seconds for the transport problem with 17,000 degs of freedom was on the the Cray-YMP using an average of 3.63 processors

The use of Lanczos's method to solve the large generalized symmetric definite eigenvalue problem

The generalized eigenvalue problem, Kx = Lambda Mx, is of significant practical importance, especially in structural enginering where it arises as the vibration and buckling problem. A new algorithm, LANZ, based on Lanczos's method is developed. LANZ uses a technique called dynamic shifting to improve the efficiency and reliability of the Lanczos algorithm. A new algorithm for solving the tridiagonal matrices that arise when using Lanczos's method is described. A modification of Parlett and Scott's selective orthogonalization algorithm is proposed. Results from an implementation of LANZ on a Convex C-220 show it to be superior to a subspace iteration code

Sparse Equation-Eigen Solvers for Symmetric/Unsymmetric Positive-Negative-Indefinite Matrices with Finite Element and Linear Programming Applications

Vectorized sparse solvers for direct solutions of positive-negative-indefinite symmetric systems of linear equations and eigen-equations are developed. Sparse storage schemes, re-ordering, symbolic factorization and numerical factorization algorithms are discussed. Loop unrolling techniques are also incorporated in the coding to enhance the vector speed. In the indefinite solver, which employs various pivoting strategies, a simple rotation matrix is introduced to simplify the computer implementation. Efficient usage of the incore memory is accomplished by the proposed restart memory management schemes. A sparse version of the Interior Point Method, IPM, has also been implemented that incorporates the developed indefinite sparse solver for linear programming applications. Numerical performance of the developed software is conducted by performing the static analysis and eigen-analysis of several practical finite elements models, such as the EXXON Offshore Structure, the High Speed Civil Transport (HSCT) Aircraft, and the Space Shuttle Solid Rocket Booster (SRB). The results have been compared to benchmark results provided by the Computational Structural Branch at NASA Langley Research Center. Small to medium-scale linear programming examples have also been used to demonstrate the robustness of the proposed sparse IPM

A robust adaptive algebraic multigrid linear solver for structural mechanics

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size and ill-conditioned linear systems, especially when accurate results are sought for derived variables interpolated with lower order functions, like stress or deformation fields. Such task represents the most time-consuming kernel in commercial simulators; thus, it is of significant interest the development of robust and efficient linear solvers for such applications. In this context, direct solvers, which are based on LU factorization techniques, are often used due to their robustness and easy setup; however, they can reach only superlinear complexity, in the best case, thus, have limited applicability depending on the problem size. On the other hand, iterative solvers based on algebraic multigrid (AMG) preconditioners can reach up to linear complexity for sufficiently regular problems but do not always converge and require more knowledge from the user for an efficient setup. In this work, we present an adaptive AMG method specifically designed to improve its usability and efficiency in the solution of structural problems. We show numerical results for several practical applications with millions of unknowns and compare our method with two state-of-the-art linear solvers proving its efficiency and robustness.Comment: 50 pages, 16 figures, submitted to CMAM

Parallel-Vector Design Sensitivity Analysis in Structural Dynamics

In this study, the design sensitivity analysis is for the purpose of providing constraint derivative information for structural optimization under dynamic loads. Various existing formulations are reviewed, and the direct differentiation method is justified as the best one for design sensitivity analysis in structural dynamics. An alternative formulation for design sensitivity analysis with direct differentiation method is developed. The alternative formulation works efficiently with the reduced system of dynamic equations, and it eliminates the need for expensive and complicated eigenvector derivatives, which is required in the existing reduced system formulation. The relationship of the alternative formulation and the existing reduced system formulation is established originally, and it is proven analytically that the two approaches are identical, when the transformation is exact, i.e, when all the modes are included. The alternative approach is accurate, simple, and efficient. Eigenvectors are used as the base vectors in system reduction for both dynamic response analysis and the design sensitivity analysis. Lanczos algorithm is used for eigensystem solutions. A modified mode acceleration method is presented, thus, not only the displacements but also the velocities and accelerations are shown to be improved. The accuracy of the dynamic response is checked by comparing with the original full system solution, and the accuracy of the sensitivity information is verified by comparing with the sensitivity information obtained by finite difference method of the original full system. Numerical studies have verified that the alternative formulation proposed could yield excellent accuracy. Numerical studies also show that the modal acceleration method could very effectively reduce the computation cost for both dynamic response analysis and design sensitivity analysis. An efficient parallel-vector algorithm for design sensitivity analysis in large-scale structural dynamics is developed. Parallel computation can be achieved in both the global and local levels. The developed parallel-vector algorithm is then implemented in the Cray 2 and Cray Y-MP parallel computers using a parallel Fortran language called Force. The efficiency of the parallel-vector algorithm is illustrated by analyzing of large-scale structural systems and making comparison with the sequential version of the algorithm

Development of Flutter Constraints for High-fidelity Aerostructural Optimization

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143080/1/6.2017-4455.pd

Proceedings of the Fifth NASA/NSF/DOD Workshop on Aerospace Computational Control

The Fifth Annual Workshop on Aerospace Computational Control was one in a series of workshops sponsored by NASA, NSF, and the DOD. The purpose of these workshops is to address computational issues in the analysis, design, and testing of flexible multibody control systems for aerospace applications. The intention in holding these workshops is to bring together users, researchers, and developers of computational tools in aerospace systems (spacecraft, space robotics, aerospace transportation vehicles, etc.) for the purpose of exchanging ideas on the state of the art in computational tools and techniques