Efficient Re-Analysis Methodology for Probabilistic Vibration of Large-Scale Structures

1. Efficient Re-Analysis Methodology for Probabilistic Vibration of Large-Scale Structures Efstratios Nikolaidis, Zissimos Mourelatos April 14, 2008

2. Definition and Significance It is very expensive to estimate system reliability of dynamic systems and to optimize them • Vibratory response varies non-monotonically • Impractical to approximate displacement as a function of random variables by a metamodel

3. m k Failure occurs in many disjoint regions g<0: failure g>0: survival Perform reliability assessment by Monte Carlo simulation and RBDO by gradient-free methods (e.g., GA). This is too expensive for complex realistic structures

4. Solution • Deterministic analysis of vibratory response • Parametric Reduced Order Modeling • Modified Combined Approximations • Reduces cost of FEA by one to two orders of magnitude • Reliability assessment and optimization • Probabilistic reanalysis • Probabilistic sensitivity analysis • Perform many Monte-Carlo simulations at a cost of a single simulation

5. Outline • Objectives and Scope • Efficient Deterministic Re-analysis • Forced vibration problems by reduced-order modeling • Efficient reanalysis for free vibration • Parametric Reduced Order Modeling • Modified Combined Approximation Method • Kriging approximation • Probabilistic Re-analysis • Example: Vehicle Model • Conclusion

6. 1. Objectives and Scope • Present and demonstrate methodology that enables designer to; • Assess system reliability of a complex vehicle model (e.g., 50,000 to 10,000,000 DOF) by Monte Carlo simulation at low cost (e.g., 100,000 sec) • Minimize mass for given allowable failure probability

7. Scope • Linear eigenvalue analysis, steady-state harmonic response • Models with 50,000 to 10,000,000 DOF • System failure probability crisply defined: maximum vibratory response exceeds a level • Design variables are random; can control their average values

8. 2. Efficient Deterministic Re-analysis Problem: • Know solution for one design (K,M) • Estimate solution for modified design (K+ΔK, M+ΔM)

9. Modal Representation: Modal Basis: Issues: • Basis must be recalculated for each new design • Many modes must be retained (e.g. 200) • Calculation of “triple” product expensive Modal Model: 2.1 Solving forced vibration analysis by reduced basis modeling Reduced Stiffness and Mass Matrices

10. Practical Issues: • Basis must be recalculated for each new design • Many modes must be retained • Calculation of “triple” product can be expensive Kriging interpolation Solution Re-analysis methods: PROM and CA / MCA

11. p3 p2 p1 Reduced Basis Efficient re-analysis for free vibrationParametric Reduced Order Modeling (PROM) Idea: Approximate modes in basis spanned by modes of representative designs Design point Parameter Space

12. PROM (continued) • Replaces original eigen-problem with reduced size problem • But requires solution of np+1 eigen-problems for representative designs corresponding to corner points in design space

13. p3 p2 MCA Approximation Full Analysis p1 Parameter Space Modified Combined Approximation Method (MCA)Reduces cost of solving m eigen-problems • Exact mode shapes for only one design point • Approximate mode shapes for p design points using MCA • Cost of original PROM: (p+1) times full analysis • Cost of integrated method: 1 full analysis + np MCA approximations

14. Basis vectors MCA method Idea: Approximate modes of representative designs in subspace T • Recursive equation converges to modes of modified design. • High quality basis, only 1-3 basis vectors are usually needed. • Original eigen-problem (size nxn) reduces to eigen-problem of size (sxs, s=1 to 3) Approximate reduced mass and stiffness matrices of a new design by using Kriging

15. p3 p2 p1 Deterministic Re-Analysis Algorithm 1.Calculate exact mode shape by FEA 2.Calculate np approximate mode shapes by MCA 3.Form basis 4. Generate reduced matrices at a specific number of sample design points 5.Establish Kriging model for predicting reduced matrices Repeat steps 6-9 for each new design: 6. Obtain reduced matrices by Kriging interpolation 7. Perform eigen-analysis of reduced matrices 8. Obtain approximate mode shapes of new design 9. Find forced vibratory response using approximate modes

16. 3. Probabilistic Re-analysis • RBDO problem: Find average values of random design variables To minimize cost function So that psys ≤ pfall • All design variables are random • PRA analysis: estimate reliabilities of many designs at a cost of a single probabilistic analysis

17. 4. Example: RBDO of Truck • Model: • Pickup truck with 65,000 DOF • Excitation: • Unit harmonic force applied at engine mount points in X, Y and Z directions • Response: • Displacement at 5 selected points on the right door

18. Design variables

19. 583 hrs 28 hrs Example: Cost of Deterministic Re-Analysis Deterministic Reanalysis reduces cost to 1/20th of NASTRAN analysis

20. Re-analysis: Failure probability and its sensitivity to cabin thickness

21. RBDO • Find average thickness of chassis, cross link, cabin, bed and doors • To minimize mass • Failure probability pfall • Half width of 95% confidence interval  0.25 pfall • Plate thicknesses normal • Failure: max door displacement>0.225 mm • Repeat optimization for pfall : 0.005-0.015 • Conjugate gradient method for optimization

22. Optimum in space of design variables Baseline: mass=2027, PF=0.011 Feasible Region Mass decreases

24. 5. Conclusion • Presented efficient methodology for RBDO of large-scale structures considering their dynamic response • Deterministic re-analysis • Probabilistic re-analysis • Demonstrated methodology on realistic truck model • Use of methodology enables to perform RBDO at a cost of a single simulation.

25. x1 Feasible Region Increased Performance Optimum Failure subset x2 Solution: RBDO by Probabilistic Re-Analysis Iso-cost curves