An Investigation of Reliability-based Topology Optimization

1. An Investigation of Reliability-based Topology Optimization Chandan Mozumder Advisor: Dr. John E. Renaud 20th Aerospace and Mechanical Engineering Graduate Student Conference University of Notre Dame 19th October, 2006 20th Aerospace and Mechanical Engineering Graduate Student Conference

2. Synopsis • Introduction • Reliability-based Design Optimization (RBDO) Formulation • Reliability-based Topology Optimization (RBTO) • Formulation • Different approaches • RBTO with Hybrid Cellular Automata (HCA) method • Numerical experiments and results • Conclusion 20th Aerospace and Mechanical Engineering Graduate Student Conference

3. Why Reliability-based Design? • Mathematical modeling and simulation for design of systems • Optimization strategies to avoid burden of manual iterations, manipulating inputs and reviewing outputs • Models are only abstraction of realities • Deterministic optimization techniques do not consider impact of uncertainties • error in design decisions Introduction 20th Aerospace and Mechanical Engineering Graduate Student Conference

4. Reliability-based Approach • Deterministic Design: may lead to unsafe design • Factor of Safety Approach: lead to conservative design Introduction • Reliability-based Approach: design is insensitive to input and model uncertainties 20th Aerospace and Mechanical Engineering Graduate Student Conference

5. Reliability-based Design Optimization x = design variable p = fixed parameter P = failure probability min f(x, p, y(x, p)) subject to gR(V, η) ≥ 0 gjD(x, p, y(x, p)) ≥ 0 j = 1,…,Ndet xl ≤ x ≤ xu Reliability constraints RBDO • Reliability constraints can be formulated by Performance Measure Approach (PMA) or Reliability Index Approach (RIA) • PMA: grcare formulated as constraints on performance that satisfies a probability requirement • RIA: grcare formulated as constraints on probability of failure 20th Aerospace and Mechanical Engineering Graduate Student Conference

6. RBDO formulation • Probability of failure corresponding to a failure mode: • Approximation to the multi-dimensional integral using First Order Reliability Method (FORM), which computes the Most Probable Point (MPP) of failure RBDO • Rosenblatt Transformation: • random vector (V) to standard normal vector (U) • zero mean and unit variance • Limit state function:GiR(u,η) = 0 20th Aerospace and Mechanical Engineering Graduate Student Conference

7. u2 G = 0 G < 0 unsafe region βp safe region G > 0 u1 MPP of failure • Solve the following optimization problem in U-space min ||u|| subject to GR(u,η) = 0 • First order approximation to probability of failure Pf = Φ(-βp) where βp= ||u*|| RBDO 20th Aerospace and Mechanical Engineering Graduate Student Conference

8. Topology Optimization Topology Optimization • Optimization process systematically and iteratively eliminates and re-distributes material throughout a design domain to obtain an optimal structure • Homogenization approach by Bendsøe and Kikuchi [Bendsøe and Kikuchi’88] • Density approach or SIMP approach by Bendsøe [Bendsøe ’89] • Simpler to implement RBTO 20th Aerospace and Mechanical Engineering Graduate Student Conference

9. Reliability-based Topology Optimization • RBTO extends reliability notion to topology optimization • Reliability-based constraints with SIMP approach for continuum structure [Kharmanda et al. ’02, ’04] • improved reliability level of structure without increasing weight • RBTO using HCA for continuum structure [Patel et al. ’05] • increase in weight in resulting structure for increased reliability level • Reliability-based constraints using discrete frame elements [Mogami et al. ’06] RBTO 20th Aerospace and Mechanical Engineering Graduate Student Conference

10. RBTO approach by Kharmanda et al. • Initial sensitivity analysis to identify random variables which have significant effect on the objective function • Limit state function used is a linear combination of the random variables RBTO u1 = applied load u2, u3 = the number of elements used to discretize the design domain in 2D u4 = volume fraction no physical significance with respect to the failure probability of the structure [Kharmanda et al.’02, ’04] 20th Aerospace and Mechanical Engineering Graduate Student Conference

11. subject to G ≤ 0 Some observations … • Physical significance of limit state function? • Reliability analysis independent of boundary and loading condition? • Driving the random variables to satisfy the following equation irrespective of the problem definition: RBTO • Dependence on the initial point? 20th Aerospace and Mechanical Engineering Graduate Student Conference

12. Some observations … • Dependence on the initial point? RBTO 20th Aerospace and Mechanical Engineering Graduate Student Conference

13. X X 0 Empty N = 0 Von Neumann N = 4 Moore N = 8 Neighborhood: Boundary: Fixed Periodic Hybrid Cellular Automata (HCA) • Cellular Automata (CA) computing & control theory are used to distribute material within a discretized design domain • CAs are by definition, dynamical systems that are discrete in space and time and operate on a uniform, regular lattice. • CAs are characterized by local interactions. RBTO 20th Aerospace and Mechanical Engineering Graduate Student Conference

14. FEA S* S Material distribution rule Update HCA Algorithm RBTO [Tovar et al.’04] 20th Aerospace and Mechanical Engineering Graduate Student Conference

15. RBTO using HCA • Decoupled reliability and structural analysis • Strain energy density as target • PMA to search for MPP • Random variables: • modulus of the material E0 • the loads Pi on the structure • Limit-state function: • Failure mode with respect to maximum allowable displacement RBTO 20th Aerospace and Mechanical Engineering Graduate Student Conference

16. Start x0(0), P(0), E(0) Initial Density x0(t), P(t), E(t) Structural optimization (HCA) x(t+1) Reliability assessment P(t+1), E(t+1) Convergence test |uTu|<ε3 |*max(t+1)–*max (t)|<ε4 no yes End RBTO using HCA Results & Observations 20th Aerospace and Mechanical Engineering Graduate Student Conference

17. Some observations … • Gradient free method • No approximation of gradients • Less numerical instabilities • Limit state function is based on a physical failure mode 20th Aerospace and Mechanical Engineering Graduate Student Conference

18. Mitchell-type Structure Three-bar truss Numerical Experiments Numerical Experiments • Design domain discretized into 5000 elements • Maximum allowable displacement of 1cm for Mitchell-type and 2cm for three-bar truss • Standard deviation of 5% for the applied load(s) 20th Aerospace and Mechanical Engineering Graduate Student Conference

19. Results Results & Observations 20th Aerospace and Mechanical Engineering Graduate Student Conference

20. Numerical Verification • Monte-Carlo Simulation with 10,000 sample points Results & Observations 20th Aerospace and Mechanical Engineering Graduate Student Conference

21. Observations • Mass increases to obtain a six-sigma design as compared to deterministic design • Mitchell-type structure: 33.15% • Three-bar truss: 25.65% • Good correlation between expected and MC predicted reliability levels • Decoupled approach to reliability-based optimization with the HCA method for structural topology synthesis is an efficient approach to topology optimization of continuum structure with desired reliability level Results & Observations 20th Aerospace and Mechanical Engineering Graduate Student Conference

22. Future Studies … • Multiple failure criteria • Design of compliance mechanism considering geometric and material non-linearity Conclusions 20th Aerospace and Mechanical Engineering Graduate Student Conference

23. Thank You!!! 20th Aerospace and Mechanical Engineering Graduate Student Conference