Eigensolution reanalysis of modified structures using epsilon‐algorithm

Abstract

SUMMARYBased on the Neumann series expansion and epsilon-algorithm, a new eigensolution reanalysis method is developed. In the solution process, the basis vectors can be obtained using the matrix perturbation or the Neumann series expansion to construct the vector sequence, and then using the epsilon algorithm table to obtain the approximate eigenvectors. The approximate eigenvalues are computed from the Rayleigh quotients. The solution steps are straightforward and it is easy to implement with the general finite element analysis system. Two numerical examples, a 40-storey frame and a chassis structure, are given to demonstrate the application of the present method. By comparing with the exact solutions and the Kirsch method solutions, it is shown that the excellent results are obtained for very large changes in the design, and that the accuracy of the epsilon-algorithm is higher than that of the Kirsch method and the computation time is less than that of the Kirsch method.

An exact block-based reanalysis method for local modifications

“…Calculation of modified displacement r by the CA method involves the following steps. [2] m-rank changes (m small value) Invert matrix  The theorems of geometric variation [3,4] Geometric variation, 18.75% (144nodes) Linear, nonlinear Truss, beam, plate Extended SMW formula [5]  Nonlinear problem Truss Sub-structuring technique [6] Imposing boundary conditions Linear Frame Incremental Cholesky factorization [7]  Crack growth modeling A finite plate Medium First and second order convex approximation [10,11]  Optimization 2-bar, cantilever beam Two-point constraint approximation [12]   Test functions Response surface method [13]  Optimization  Epsilon-algorithm reanalysis method [14] Fixed parameters(Length, Thickness) Eigenvalue Frame/Chassis BFGS reanalysis method [15] Modified 216 nodes (Initial 542 nodes) Static Bracket Perturbation and Padé approximation [16] Fixed parameters(Material, Length) Static Beam Multi-sample compression algorithm [17] Thickness, Yong and Tangent modulus, Pressure…”

“…Using polynomial fitting and response surface method (RSM), Haftka and Unal used simple functions to replace the response functions [12,13]. After the year 2000, based on the Neumann series expansion and epsilon-algorithm, Chen and Wu developed a new eigenvalue reanalysis method [14]. Xu et al presented a method for the static reanalysis which uses super-linear convergence property of the Broyden-FletcherGoldfarb-Shanno (BFGS) quasi-Newton algorithm [15].…”

Efficient computation for dynamic responses of systems with time-varying characteristics

“…Up till now, there is few good reanalysis method for situations of very large modification. Epsilon-algorithm can give good results while the variations are fairly large [14,15], but they may deteriorate when K K K t+ t caused by time-varying characteristics becomes larger and larger.…”

“…(12)(13)(14), we can get lim j→∞ s s s j → (1, 1) T , which is the exact solution obtained from Ax = s s s 0 with A A A = I I I − G G G. Using the epsilon-algorithm, we obtain the following epsilon-algorithm table as shown in Fig. 3.…”

“…Recently, a mathematical algorithm named epsilonalgorithm [11][12][13] was introduced into the reanalysis problems for eigenproblem [14] and static displacement [15] in practical engineering applications. Using epsilon table to accelerate the convergence of Neumann series expansion, the accurate approximate results of eigenvalues and displacements can be obtained even the parameters have large modifications.…”