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

4 Solution 1.Deterministic analysis of vibratory response –Parametric Reduced Order Modeling –Modified Combined Approximations –Reduces cost of FEA by one to two orders of magnitude 2.Reliability assessment and optimization –Probabilistic reanalysis –Probabilistic sensitivity analysis –Perform many Monte-Carlo simulations at a cost of a single simulation

5 Outline 1.Objectives and Scope 2.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 3.Probabilistic Re-analysis 4.Example: Vehicle Model 5.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 2.1 Solving forced vibration analysis by reduced basis modeling Modal Representation: Modal Basis: Modal Model:  Basis must be recalculated for each new design  Many modes must be retained (e.g. 200)  Calculation of “triple” product expensive Issues: Reduced Stiffness and Mass Matrices

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

11 Efficient re-analysis for free vibration Parametric Reduced Order Modeling (PROM) Parameter Space p1p1 p3p3 p2p2 Design point Reduced Basis Idea: Approximate modes in basis spanned by modes of representative designs

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 Modified Combined Approximation Method (MCA) Reduces cost of solving m eigen-problems p1p1 p3p3 p2p2 Parameter Space  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 Full Analysis MCA Approximation

14 Basis vectors 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) MCA method Approximate reduced mass and stiffness matrices of a new design by using Kriging

15 Deterministic Re-Analysis Algorithm p1p1 p3p3 p2p2 2. Calculate np approximate mode shapes by MCA 4. Generate reduced matrices at a specific number of sample design points 5. Establish Kriging model for predicting reduced matrices 3. Form basis 1. Calculate exact mode shape by FEA 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 Repeat steps 6-9 for each new design:

16 3. Probabilistic Re-analysis RBDO problem: Find average values of random design variables To minimize cost function So that p sys ≤ p f all 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 Example: Cost of Deterministic Re- Analysis 583 hrs 28 hrs Deterministic Reanalysis reduces cost to 1/20 th 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  p f all Half width of 95% confidence interval  0.25 p f all Plate thicknesses normal Failure: max door displacement>0.225 mm Repeat optimization for p f all : Conjugate gradient method for optimization

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

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24 5. Conclusion Presented efficient methodology for RBDO of large-scale structures considering their dynamic response 1.Deterministic re-analysis 2.Probabilistic re-analysis Demonstrated methodology on realistic truck model Use of methodology enables to perform RBDO at a cost of a single simulation.

25 Solution: RBDO by Probabilistic Re-Analysis Iso-cost curves Feasible Region Increased Performance x2x2 x1x1 Optimum Failure subset