Lecture 25 Structural Approximation (Fast Reanalysis)

1. Lecture 25Structural Approximation (Fast Reanalysis) EGM6365 Structural Optimization 03/12 Given by Shu Shang

2. 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)

3. 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

4. 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

5. 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

6. Total change Δu predicted by this approach

7. Example: 1D Bar P • Design parameter: cross section area A • Initial design x0=1 • Perturbation Δx=0.25 • Stiffness • Tip displacement • Approximation L

8. One more iteration • Another approach

9. 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

10. 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:

11. First approach • Another approach

12. Problem 1 • Estimate the effect on the lowest frequency caused by an 50% increase in the stiffness of the left spring P

13. 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

14. 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

15. Problem 2 • Solve problem 1 using Sherman-Morrison formula