Erdem Acar Sunil Kumar Richard J. Pippy Nam Ho Kim Raphael T. Haftka Approximate Probabilistic Optimization Using Exact-Capacity-Approximate-Response-Distribution (ECARD) Erdem Acar Sunil Kumar Richard J. Pippy Nam Ho Kim Raphael T. Haftka

Outline Introduction & Motivation Introduce characteristic stress and correction factor Details of Exact Capacity Approximate Response Distribution (ECARD) optimization method Demonstration on two Examples: Cantilever beam problem Ten bar truss problem Conclusion

Introduction: Design Optimization Deterministic Design governed by safety factor for loads, and knockdown factors for allowable stress and displacement. Suboptimal Risk allocation because of uniform safety factor Probabilistic Optimum risk allocation by probabilistic analysis Light weight components usually should have higher safety factors than heavy elements because, for them, weight for reducing risk is very small compared to heavier elements Computational expense involved in reliability assessment

Dealing with the Computational Cost Double loop optimization: Outer loop for design optimization, inner loop for reliability assessment by Lee and Kwak in 1987 Single loop methods: sequential deterministic optimizations by Du and Chen in 2004 known as Sequential Optimization and Reliability Assessment (SORA) method. ECARD Optimization Uses sequence of approximate inexpensive probabilistic optimizations It reduces computational cost by approximate treatment of expensive response distribution

Introduction to ECARD Model Limit State function can be expressed as F (response, capacity) = Capacity - Response CDF of capacity is usually easy to obtain from failure records : Required by Regulations ECARD uses Exact CDF of capacity It approximates the Response (e.g. stress ) Distribution (PDF) using Characteristic Response (* ) to estimate probability of failure for any given design Characteristic stress is an equivalent deterministic stress having the same failure probability for random capacity (e.g. failure stress)

Exact Capacity Approximate Response Distribution (ECARD) Model

Exact Capacity Approximate Response Distribution (ECARD) Model

Exact Capacity Approximate Response Distribution (ECARD) Model

Exact Capacity Approximate Response Distribution (ECARD) Model

Exact Capacity Approximate Response Distribution (ECARD) Model

Exact Capacity Approximate Response Distribution (ECARD) Model

Correction factor Correction factor, k, is defined as ratio of * &  It replaces derivatives of probability of failures in full probabilistic optimization and provides an approximate direction for optimizing objective function. Simplifying assumption: ‘k’ is constant

Linearity assumption between * &  If distribution shape does not change k can be approximated easily by shifting Nominal MCS values For lognormally distributed failure stress and normally distributed stress, the linearity assumption is quite accurate over the range -10%  10%.

Initial Steps of ECARD Method Calculate Characteristic stress,σp*, of the previous or given design using Calculate deterministic stresses σ0 for the initial design using the mean values of all input variables Calculate correction factor ‘k’ using finite differences. For instance:

ECARD approximate Optimization To calculate Pfapprox : ‘k’ is estimated before start of the ECARD optimization procedure As design changes in optimization procedure the changes in probability of failure are reflected by changes in Characteristic responses

Example 1: Cantilever Beam Problem Random variable Mean Coefficient of variation FX (lb) 500 20% FY (lb) 1,000 10% Young's Modulus, E (psi) 2.9107 5% Failure Stress,σf (psi) 40,000

Cantilever Beam Problem: Deterministic optimization where SFL(=1.5) is safety factor for loads, kc,1(=1) and kc,2 (=1) are knockdown factors for allowable stress and displacement. Width (in) Thickness Area (in2) 2.27 4.41 10.04 Optimum design :

Cantilever Beam Problem: Probabilistic optimization Ditlevsen’s First Order upper Bound Leads to 6% reduction in Area over Deterministic Optimum Design by reallocating risk between different failure modes Deterministic Design allocates Most of the risk to Displacement criteria but its cheaper to guard against Displacement constraint violation Width (in) Thickness Area (in2) PF(stress) PF(Displacement) PTotal Deterministic optimum 2.27 4.41 10.04 9.8 x 10-5 2.67x 10-3 2.7x 10-3 Probabilistic optimum 2.65 3.56 9.44 2.410-3 3.310-4 2.710-3

Cantilever Beam Problem: ECARD Optimization Only 5 Iterations of ECARD optimization needed Leads to 0.2% heavier Design than Probabilistic Optimum Design which was 6% lighter than deterministic Design by proper risk allocation. Probability of failure due to stress and displacement criteria have changed in opposite directions. Similar to full Probabilistic optimization. Width (in) Thickness Area (in2) PF(stress) PF(Displacement) PTotal # Response PDF Assessments Deterministic optimum 2.27 4.41 10.04 9.8x 10-5 2.67x 10-3 2.7x 10-3 Probabilistic 2.65 3.56 9.44 2.310-3 3.3110-4 2.710-3 455 ECARD 5th Iteration 2.50 3.80 9.50 1.7710-3 9.810-4 10

Cantilever beam Problem: Convergence Convergence of ECARD optimization technique to the full probabilistic optimum is not achieved exactly because of approximations in correction factor ‘k’.

Example 2: Ten-bar Truss Problem Aluminum Truss: Density = 0.1 lb/in³ Elasticity Modulus: E = 10,000 ksi Length: b = 360 in P1 = P2 = 100,000 lbs (includes a SF of 1.5)

Ten-bar Truss Problem: Deterministic Optimization where, W = Total Weight of Truss,  = Density, L = Length, A = Cross-sectional Area, N = Axial force in an element Constraints: Minimum Area = 0.1 in² Maximum Stress in all elements = 25 ksi , Except in Element 9,it is 75 ksi

Ten-bar Truss Problem: Deterministic Optimization Results Element Area (in2) Weight (lb) Stress (ksi) Pfd 1 7.9 284 25 2.1E-03 2 0.1 4 1.1E-02 3 8.1 292 -25 4.80E-04 3.9 140 2.19E-03 5 4.04E-04 6 1.07E-02 7 5.8 295 1.69E-03 8 5.5 281 1.89E-03 9 3.6 187 37.5 5.47E-13 10 Total --- 1498 -- 4.08E-02 Light weight elements account for 50% of total failure probability

Ten Bar Truss Problem: Probabilistic Optimization Results Errors in loads, cross sectional area, stress calculations and failure predictions leads to uncertainty Results of full probabilistic optimization using 10,000 samples of Separable MCS Element Deterministic Areas Probabilistic Pf 1 7.9 7.2 2.1E-03 5.9E-03 2 0.1 0.3 1.0E-02 3.1E-03 --- -- Totals: 1497.6 Ibs 1407.13 Ibs 4.10E-02

Ten Bar Truss Problem: ECARD Optimization Results Element Determ. Des. iter_01 iter_02 iter_03 iter_04 AREAS (in2) 1 7.9 7.45 7.48 2 0.1 ACTUAL PF 2.1E-03 5.5E-03 5.3E-03 5.2E-03 1.1E-02 3.1E-03 2.2E-03 Risk of failure of elements have changed in opposite direction Element Deterministic Areas Probabilistic Pf 1 7.9 7.2 2.1E-03 5.9E-03 2 0.1 0.3 1.0E-02 3.1E-03 Compare it with Full probabilistic optimization Computational costs Probabilistic Optimization ECARD Optimization # Expensive Response PDF Assessments 728 8 Cost Comparison

Conclusions A failure characteristic stress * is used to approximate changes in probability of failure with changes in design Using this, ECARD dispenses with most of the expensive structural response calculations. Cantilever beam: 455 to 10 expensive reliability assessments Ten bar truss: 728 to 8 expensive reliability assessments ECARD converges to near optima of allocated risk between failure modes much more efficiently than the deterministic optima

Any Questions or Comments? Thank you