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This lecture, presented by Shu Shang, delves into the concepts of structural approximation and fast reanalysis in the context of linear static response analysis. It discusses how small design perturbations can lead to manageable changes in the response and stiffness matrix. The lecture explores efficient methods for approximating changes in response without recalculating the stiffness matrix entirely. Various approaches, including iterative scaling and the Sherman-Morrison formula, are highlighted, demonstrating their applications in vibration and buckling studies.
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Lecture 25Structural Approximation (Fast Reanalysis) EGM6365 Structural Optimization 03/12 Given by Shu Shang
Introduction • In static response,designperturbationfrom x0to x0+ Δxcan causechangeinresponsefromu0to u0+ Δuaswellas stiffness matrix from K0to K0+ ΔK • When the perturbation size Δx is small, we expect Δu and ΔK will also be small • Then, instead of rebuilding stiffness, K0+ ΔK, with a small perturbation, it is possible to approximate Δu with a reasonable accuracy. (Structural approximation) • Since K0is factorized already, we can use the factored matrix K0to approximate Δu. (Fast reanalysis)
Linear Static Response • Linear static response at the initial design x0 • At the perturbed design, • Subtracting the equilibrium equation of the initial design (1) • We approximate Δu into Δ u1by ignoring H.O.T. ΔKΔu (2) • Accurate when Δxis small • This process is fast because K0is already factorized
Improvement of Approximation • Subtracting (2) from (1) • ApproximateΔu2= Δu– Δu1and ignoring H.O.T. ΔK(Δu–Δu1) • If we repeat this process continuously Where the terms Δuiare obtained through the iterative process of solving
Approximation with Scaling • Kirsch and Taye introduced scaling and redistribution • Choose s to minimize ΔKsso that sK0is close to K0+ΔK • Calculate s to minimize the square sum of elements of ΔKs • Then, consider initial design to be sK0 instead of K0, and • Only consider the case where
Example: 1D Bar P • Design parameter: cross section area A • Initial design x0=1 • Perturbation Δx=0.25 • Stiffness • Tip displacement • Approximation L
One more iteration • Another approach
Eigenvalue Problem • Vibration or buckling response • At perturbed design • Subtract (3) from (4) and ignore H.O.T. • Pre-multiplying by u0T • Or pre-multiply (4) by and neglect some higher order terms
Example: Mass-spring system • Estimate the effect on the lowest frequency caused by doubling the left mass • Stiffness matrix: • Mass matrix: • The lowest eigenvalue and corresponding eigenvector are • Perturbation: • Exact result:
First approach • Another approach
Problem 1 • Estimate the effect on the lowest frequency caused by an 50% increase in the stiffness of the left spring P
Exact Reanalysis • Calculate Δuexactly (no approximation) • Still it needs to be fast • Let K be the original stiffness matrix, and we have its inverse • Here ΔKis rank one matrix and can be written as uvT • Sherman-Morrison formula • Popular for truss structures, since the change of one truss element leads to a rank-one modification of K
Mehmet A. Akgun,John H. Garcelonand Raphael T. Haftka, Fast exact linear and non-linear structural reanalysis and the Sherman-Morrison-Woodbury formulas, International journal for numerical methods in engineering • Review of the re-invention of the Sherman-Morrison method by different authors over the years from 1950 to 2000
Problem 2 • Solve problem 1 using Sherman-Morrison formula
