Optimization in Engineering Design: Enhancing Quality Through Iterative Analysis

1. Introduction • Optimization: Produce best quality of life with the available resources • Engineering design optimization: Find the best system that satisfies given requirements • Analysis versus design • Analysis: determine performance of given system • Design: Find system that satisfies given requirements. Design involves iterations in which many design alternatives are analyzed.

2. Objective function: measures performance of a design or a decision • Constraints: Requirements that a design must satisfy • Numerical optimization can be the only practical approach for most real-life problems

3. General optimization problem statement • Find design (decision) variables, X • To minimize objective function, F(X) • so that • g(X) no greater than zero (inequality constraints) • h(X)=0

4. Example: tubular column optimization • Design a column to minimize the mass so that the column does not fail under a given applied axial load • Three failure modes--three constraints: yielding, Euler buckling, local buckling • May not have unique optimum • At the optimum some constraints are active, i.e. applied stress is equal to failure stress

5. Active constraints g3=0 x2 Feasible region Weight increases g2=0 Optimum x1

6. A taxonomy of optimization problems Multiple objectives Dynamic Static One objective Deterministic Non deterministic

7. Taxonomy • Deterministic: know values of all input variables • Non deterministic: Only probability distribution of input variables known • Static: Solve one optimization problem • Dynamic: Solve sequence of optimization sub problems (e.g. chess) • Single objective • Multiple objectives

8. Iterative optimization procedure • Most real life optimization problems solved using iterations • Two steps is each iteration • Find search direction • One dimensional search -- find how far to go in a given direction

9. Necessary and sufficient conditions, unconstrained minimization Gradient =0 at X* Gradient =0 at X*, Hessian pos. def. at X* , X* local min Gradient =0 at X*, Hessian pos. def. everywhere , X* global min

10. Necessary condition for local optimum, constrained minimization. Example F=constant F=constant F F Feasible sector B Feasible sector A g2=0 g1 g2=0 -F g1=0 g1=0 -F g2 g1 g2 B is not a local minimum, the feasible sector and the usable sector intersect A is local minimum, there is no feasible and usable sector

11. Necessary condition for local optimum, constrained minimization (continued) • For the example, there exist two non negative numbers 1 and 2 such that: F+ 1g1+ 2g2=0 General case: There are non negative numbers j0, j=1,…,m F+ jgj+ k+mhk+m=0 where the first sum is for j=1,…,m and the second for k=1,…,l

12. Sufficient conditions global optimum K-T conditions satisfied Local optimum Global optimum Design space convex, K-T conditions satisfied

13. Sensitivity analysis • Allows one to find the sensitivity derivatives of the optimum solution and the optimum value of the objective function with respect the a problem parameter without solving the optimization problem many times. • Useful for finding important constraints and important design variables. • Very high sensitivity of objective function wrt design parameters; poor design

14. Equations for sensitivity analysisSensitivity derivatives of design variables • A: second order derivatives of objective function and active constraints (size nxn) • B: columns are gradients of constraints (size nxm) • c: second order derivatives of objective function and constraints wrt design variables and design parameter (size nx1) • d: derivatives of constraints wrt design parameter (size mx1) • X and : derivatives of design variables at optimum and Lagrange multipliers wrt parameter

15. Equations for sensitivity analysis (continued) • Chain rule for sensitivity derivatives of objective function • Sensitivity derivatives useful for predicting effect of small changes in problem parameters on solution