1. Efficient Methodologies for Reliability Based Design Optimization
Harish Agarwal
Department of Aerospace and Mechanical Engineering
University of Notre Dame, IN -
Presentation for Dr. Yoshimura and his research group

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

3. 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

4. Deterministic Design Optimization90 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.

5. 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.

6. 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

7. 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.

8. 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.

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

10. 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

11. 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.

12. 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.

13. 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

14. 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.

15. Example Problem
CA1
CA2
RBDO
Inputs

16. 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.

17. 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.

18. 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.

19. Questions & Discussions