Sensitivity analysis for large-scale problems

Advances in Engineering Software, 1998

1990

This paper highlights the application to structural dynamics of the sensitivity analysis methods developed by numerical analysts and presents a historical development of first-and higherorder eigenvalue and eigenvector sensitivities. Different formulae for an eigenvalue sensitivity are presented and it is shown that all of these are implicitly the same. A condition number is presented to give the limited bound of application for the first-order eigenvalue and eigenvector sensitivities. An alternative structural modification method based on Rayleigh Quotient Iteration is presented. A lumped spring-mass system with 7 degrees-of-freedom (7DoF) is used to show the applicability of the Rayleigh quotient iteration method.

AIAA Journal, 2001

Combined approximations (CA) is an ef cient method for reanalysis of structures where binomial series terms are used as basis vectors in reduced basis approximations. In previous studies high-quality approximations have been achieved for large changes in the design, but the reasons for the high accuracy were not fully understood. In this work some typical cases, where exact and accurate solutions are achieved by the method, are presented and discussed. Exact solutions are obtained when a basis vector is a linear combination of the previous vectors. Such solutions are obtained also for low-rank modi cations to structures or scaling of the initial stiffness matrix. In general the CA method provides approximate solutions, but the results presented explain the high accuracy achieved with only a small number of basis vectors. Accurate solutions are achieved in many cases where the basis vectors come close to being linearly dependent. Such solutions are achieved also for changes in a small number of elements or when the angle between the two vectors representing the initial design and modi ed design is small. Numerical examples of various changes in cross sections of elements and in the layout of the structure show that accurate results are achieved even in cases where the series of basis vectors diverges.

Some innovative techniques applicable to sensitivity analysis of discretized structural systems are reviewed. These techniques include a finite-difference step-size selection algorithm, a method for derivatives of iterative solutions, a Green's function technique for derivatives of transient response, a simultaneous calculation of temperatures and their derivatives, derivatives with respect to shape, and derivatives of optimum designs with respect to problem parameters. Computerized implementations of sensitivity analysis and applications of sensitivity derivatives are also discussed. Finally, some of the critical needs in the structural sensitivity area are indicated along with Langley plans for dealing with some of these needs.

International Journal for Numerical Methods in Engineering, 2002

A preconditioned conjugate gradient (PCG) method that is most suitable for reanalysis of structures is developed. The method presented provides accurate results e ciently. It is easy to implement and can be used in a wide range of applications, including non-linear analysis and eigenvalue problems. It is shown that the PCG method presented and the combined approximations (CA) method developed recently provide theoretically identical results. Consequently, available results from one method can be applied to the other method. E ective solution procedures developed for the CA method can be used for the PCG method, and various criteria and error bounds developed for conjugate gradient methods can be used for the CA method. Numerical examples show that the condition number of the selected preconditioned matrix is much smaller than the condition number of the original matrix. This property explains the fast convergence and accurate results achieved by the method.

Mechanical Systems and Signal Processing, 2011

The sensitivity method is probably the most successful of the many approaches to the problem of updating finite element models of engineering structures based on vibration test data. It has been applied successfully to large-scale industrial problems and proprietary codes are available based on the techniques explained in simple terms in this article. A basic introduction to the most important procedures of computational model updating is provided, including tutorial examples to reinforce the reader's understanding and a large scale model updating example of a helicopter airframe.

8th Symposium on Multidisciplinary Analysis and Optimization, 2000

Combined Approximations (CA) is an efficient method for reanalysis of structures where binomial series terms are used as basis vectors in reduced basis approximations. In previous studies high quality approximations have been achieved for large changes in the design, but the reasons for the high accuracy were not fully understood. In this paper some typical cases, where exact and accurate solutions are achieved by the method are presented and discussed. Exact solutions are obtained in the general case where a basis vector is a linear combination of the previous vectors. Such solutions are obtained also in cases of low rank modifications to structures or scaling of the initial stiffness matrix. In general the CA method does not provide exact solutions, but the solutions presented in the paper explain the high accuracy achieved with only small number of basis vectors. Accurate solutions are achieved in many cases where the basis vectors come close to being linearly dependent. Such solutions are achieved also in cases of changes in a small number of elements or when the angle between the two vectors representing the initial design and modified design is small. Numerical examples of various changes in cross sections of elements and in the layout of the structure show that accurate results are achieved even in cases where the series of basis vectors diverges.

1994

Automatic differentiation tools (ADIFOR) is incorporated into a finite element based structural analysis program for shape and non-shape design sensitivity analysis of structural systems. The entire analysis and sensitivity procedures are parallelized and vectorized for high performance computation. Small-scale examples to verify the accuracy of the proposed program and a medium-scale example to demonstrate the parallel-vector performance on the multeiple Cray-C90-processors are included in the paper.

Structural and Multidisciplinary Optimization, 2010

The main aim of this work is the mathematical formulation, computational implementation and the application of the local version of the Response Function Method (RFM) to analyze structural design sensitivity in nonlinear structures and problems. This method is based on the Finite Element Method-based determination of the polynomial response function between design parameter and the structural state function like displacements or temperatures. One may use this numerical technique in its global version, where a single polynomial is determined for the entire computational domain or, in the case of nonlinear, transient analyses or the heterogeneous domains, in the local approach--where nodal response function are to be determined. The application of this methodology is illustrated with three examples--transient heat transfer in the homogeneous rod, the elastoplastic analysis of 2D truss as well as the eigenvibrations for a large scale 3D structure, where time, increment and eigenvalue dependent variations of the first and the second order sensitivities with respect to the physical and material parameters are computed. The first order gradients computed with the use of the RFM approach are contrasted with the finite difference computations.

Experimental validation of a new model updating method is presented for structural mass and stiffness estimation using vibration data. The method uses transfer functions for finite element model updating via a quasi-linear sensitivity equation of the structural response. Excitation frequencies are selected in the most sensitive ranges of transfer functions for robust updating of the structural parameters. In addition, noisy regions are omitted from measured transfer functions. A least squares algorithm with appropriate normalization is used for solving the overdetermined system of equations. The method is verified using experimental data from a one-bay, one-story aluminum frame. Fast and accurate prediction of stiffness and mass parameters using a subset of measured transfer functions in selected frequency ranges illustrated the success and robustness of the method in the presence of measurement and modeling errors.