The theorems of geometric variation for finite element analysis

Abstract

SUMMARYThis paper illustrates the use of the theorems of geometric variation for the reanalysis of finite element structures. The formulations presented here form the basis for an efficient computer implementation.

The Theorems of Structural and Geometric Variation: A Review

“…The design studies and efficiency tests presented in Refs [l, 20,21,22, 231 indicate that the Theorems of Structural and Geometric Variation may be used efficiently for the reanalysis of structures. Potential applications include nonlinear analysis, interactive design studies, optimization and other structural sensitivity calculations.…”

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…”

“…In 1988, the theorems of geometric variation were developed for the reanalysis of finite element structures when variations in the co-ordinates of the nodes of the elements are considered by Topping and Kassim [3]. Continuously, the theorems of geometric variation were applied into the nonlinear reanalysis problems [4].…”

Exact and efficient isogeometric reanalysis of accurate shape and boundary modifications

“…Among all these three categories, we found several works that mentioned the details of the changes, such as the fixed DOFs with total DOFs, and highlight them in Table 1 [57]. For example, reanalysis methods for small change [46,47,58,59] represented change details that they handle. And for medium scales, the works [60,61,62,55] also mentioned how and to what extent the structures were modified, as well the updated triangular factorization algorithm [53] to handle large changes.…”

“…This method takes full use of the advantages of IGA in CAD/CAE integration and IFU in providing exact and efficient CAE reanalysis; and it also provides a powerful tool for future researchers in solving problems that are largescale or need iterations, as well as being extremely valuable in real world engineering problems. [47] m-rank changes (m small value) The theorems of geometric variation [58,59] Geometric variation, 18.75% (144nodes)…”