Efficient Methodologies for Reliability Based Design Optimization Harish Agarwal Department of Aerospace and Mechanical Engineering University of Notre Dame, IN - 46556 Email :- Harish.Agarwal.6@nd.edu Presentation for Dr. Yoshimura and his research group

Overview Introduction. Deterministic design optimization. Motivation. Background on Reliability-based design optimization (RBDO). New Unilevel Method for RBDO. Example Problem. Summary.

Multidisciplinary Systems Design Linked simulation tools. Highly coupled. Complex information exchange. Computationally expensive. Highly nonlinear. Elastic Structures Suspension Aero-dynamics Crash-worthiness Occupant Dynamics Fuel Economy

Deterministic Design Optimization 90 70 Deterministic Optimum Reliable Optimum 50 Note that almost 75% of designs around the deterministic optimum fall in the failure domain and hence fail. Optimized deterministic designs are driven to the limit of the design constraints and can lead to catastrophic failure. A variety of different types of uncertainties are inherently present in engineering design. Deterministic design optimization does not account for the uncertainties. Therefore, there is a strong need for optimization under uncertainty.

Reliability Based Design Optimization (RBDO) In RBDO, the deterministic problem is reformulated and the failure driven constraints are replaced with reliability constraints. The reliability constraints can be formulated by the reliability index approach (RIA) or the performance measure approach (PMA). In RIA, are formulated as constraints on the probability of failure In PMA, are formulated as constraints on performances that satisfies a probability requirement The computation of and requires solution to optimization problems.

Reliability Analysis The probability of failure corresponding to a failure mode is given as It is almost impossible to compute the multi-dimensional integral. However, approximations to the probability of failure can be obtained using the First Order Reliability Method (FORM), which computes a Most Probable Point (MPP) of failure. (Failed) (Safe) Original Space Standard Space

First Order Reliability Method (FORM) The MPP is computed by solving the following optimization problem. The first order approximation to the probability of failure is The optimizer may fail to provide a solution to the equality constrained FORM problem (singularity). Limit state surface is far from the origin in U-space. The case never occurs at a given design setting (the design has a failure probability equal to zero or one). Padmanabhan and Batill [2002] addressed this problem by using a trust region algorithm for equality constrained problems. Gives solution to The performance measure approach (PMA) avoids the singularities through an inverse reliability analysis.

Performance Measure Approach (PMA) The reliability constraints are formulated in terms of the performance values that meets a given probability requirement The following inverse reliability analysis optimization problem is solved PMA formulation is robust compared to RIA. In current work, the PMA formulation has been used.

FORM for RIA and PMA Mean Value Design Point MPP Locus of MPP

Issues in Traditional RBDO Formulation It should be noted that irrespective of the formulation used to prescribe the reliability constraints (RIA or PMA), the traditional RBDO involves a nested optimization process. Each iteration of RBDO requires the evaluation of the reliability constraints which themselves require solution to optimization problems. Such a formulation is inherently computationally intensive for problems where the function evaluations are expensive (e.g., multidisciplinary systems). Moreover, the formulation becomes impractical as the number of hard constraints increases which is often the case in real-life design problems. To reduce the computational cost, a variety of RBDO techniques have been developed. Main Optimizer Objective Function and Soft Constraints Reliability Constraints Deterministic Analysis Engineering Simulation Model Inner Optimization Loops OR

Sequential Method for RBDO Inverse Reliability Assessment Deterministic Optimization Converge Yes Final Design Calculate Optimal Sensitivities of MPPs No To alleviate the computational cost associated with traditional RBDO, an improved sequential RBDO technique is developed in this investigation. The optimization and the reliability assessment are decoupled from each other. The methodology requires the gradient of the MPP with respect to the decision variables in order to update the MPP during the optimization phase.

Computation of optimal sensitivities of the MPP It should be noted that during the first iteration, the MPPs are set equal to the mean values of the random variables. This is equivalent to solving a deterministic optimization problem. In subsequent cycles, the MPP is updated based on reliability assessment at the previous optimal design setting. A linear post optimality analysis is performed to compute the post-optimal sensitivities of the MPP with respect to the design variables. This requires the Hessian of the limit state function at the MPP. Hessian update schemes can be employed (SR1, BFGS, etc.) If the Hessian is not available or is difficult to obtain, approximations to the limit state function can be employed to estimate the sensitivities. The methodology is tested for analytical problems and simple engineering design problems.

New Unilevel Method – Deriving KKT conditions The reliability constraints in PMA are formulated as The Lagrangian for IRA is The first order optimality conditions require that the gradient of the Lagrangian should be zero The following inverse reliability analysis (IRA) problem is solved The corresponding KKT conditions are

Unilevel Method for RBDO A unilevel formulation for RBDO is developed. The first order KKT conditions of the inverse reliability analysis optimization problem are imposed at the system level directly as equality constraints. Through algebraic manipulation, the first order KKT conditions for the inverse reliability optimization problem can be reduced to It should be noted that the optimization is performed at an augmented space that consist of the design variables and the MPPs.

Example Problem CA1 CA2 RBDO Inputs

The infeasible region in is shaded. -10 -8 -6 -4 -2 2 4 6 8 10 1 3 5 7 9 d 1000 850 700 550 400 250 100 50 Deterministic Optima Reliable Optima The figure shows the contours of the objective and the constraints at the mean values of the random variables. The infeasible region in is shaded. This problem has two optimal solutions which can be found by choosing different starting points.

Comparison of Different RBDO Methods Starting from the design [-5,3], an optimal solution of [-3.006,0.049] is obtained. For this starting point, the number of system analysis required in different RBDO methods is compared below. Starting from the design [5,3], an optimal solution of [2.9277,1.3426] is obtained. For this starting point, the number of system analysis required in different RBDO methods is compared below. Note that the traditional RBDO formulation that uses the RIA formulation to prescribe the probabilistic constraint fail to converge. It is also observed that the unilevel and the sequential RBDO methods developed in this research are computationally efficient compared to the traditional approaches.

Closure Traditional reliability based design optimization (RBDO) involves the solution to a nested optimization problem which is extremely computationally intensive. A sequential RBDO (work in progress) and a unilevel RBDO methodology are developed in this investigation. These methodologies avoid the numerical instability associated with the traditional RIA formulation for RBDO. Tests on an analytical problem show that these methodologies are significantly computationally efficient. Current efforts are focused towards testing these approaches for large-scale multidisciplinary problems.

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