1. Design Optimization School of Engineering University of Bradford 1 Formulation of a design improvement problem as a formal mathematical optimization problem MATHEMATICAL OPTIMIZATION PROBLEM A formal mathematical optimization problem: to find components of the vector x of design variables: where F(x) is the objective function, g j (x) are the constraint functions, the last set of inequality conditions defines the side constraints.

2. Design Optimization School of Engineering University of Bradford 2 Specific Features of Shape Optimization MATHEMATICAL OPTIMIZATION PROBLEM CAD model generation is done once Optimization process modifies this CAD model and returns a valid CAD model that needs to be analysed The CAD model allows for the use of automatic tools (mesh generator, adaptive FE, etc) Example. Linking a FE mesh directly to optimization can violate the basic assumptions the model is based on:

3. Design Optimization School of Engineering University of Bradford 3 Formulation of a design improvement problem as a formal mathematical optimization problem Normalisation of constraints It is important to normalise the constraints and make them dimensionless. Example 1: stress constraint can be transformed to Example 2: buckling constraint can be transformed to MATHEMATICAL OPTIMIZATION PROBLEM

4. Design Optimization School of Engineering University of Bradford 4 Geometrical interpretation of a constrained maximization problem MATHEMATICAL OPTIMIZATION PROBLEM

5. Design Optimization School of Engineering University of Bradford 5 Geometrical interpretation of an optimization process MATHEMATICAL OPTIMIZATION PROBLEM

6. Design Optimization School of Engineering University of Bradford 6 Local and global minima Local and global one-dimensional optimization MATHEMATICAL OPTIMIZATION PROBLEM

7. Design Optimization School of Engineering University of Bradford 7 Optimality conditions Unconstrained optimization problem: MATHEMATICAL OPTIMIZATION PROBLEM Necessary conditions of a minimum:

8. Design Optimization School of Engineering University of Bradford 8 Optimality conditions Unconstrained optimization problem. Satisfaction of the necessary conditions does not guarantee the minimum. The second derivatives need to be checked MATHEMATICAL OPTIMIZATION PROBLEM

9. Design Optimization School of Engineering University of Bradford 9 Optimality conditions Optimization problem with equality constraints MATHEMATICAL OPTIMIZATION PROBLEM Necessary conditions of a minimum Lagrangian function:

10. Design Optimization School of Engineering University of Bradford 10 Kuhn-Tucker optimality conditions MATHEMATICAL OPTIMIZATION PROBLEM General optimization problem with inequality constraints Kuhn-Tucker optimality conditions Important: Kuhn-Tucker optimality conditions can only be used for active constraints.

11. Design Optimization School of Engineering University of Bradford 11 Kuhn-Tucker optimality conditions MATHEMATICAL OPTIMIZATION PROBLEM Another form of the Kuhn-Tucker optimality conditions: where  F(x) and  G j (x) are the gradient vectors of the objective function and the active constraints respectively. Here j are the Lagrangian multipliers. The greater j, the greater is the gain in relaxing the j-th constraint, or the greater penalty for tightening it up in terms of the objective function.

12. Design Optimization School of Engineering University of Bradford 12 Kuhn-Tucker optimality conditions MATHEMATICAL OPTIMIZATION PROBLEM G1(x)G1(x) G2(x)G2(x) -F (x)-F (x) F (x)F (x) F (x)=const G 2 (x)=1 G 1 (x)=1 Infeasible Feasible 2  G 2 (x) 1  G 1 (x) In this example 0 > 1 >1 and 2 >1

13. Design Optimization School of Engineering University of Bradford 13 Formulation of a multi-objective problem Pareto optimum set consists of the designs which cannot be improved with respect to all criteria at the same time MULTI-OBJECTIVE PROBLEMS A general multi-objective optimization problem

14. Design Optimization School of Engineering University of Bradford 14 A multi-objective problem MULTI-OBJECTIVE PROBLEMS Pareto optimum solutions correspond to the AB part of the contour

15. Design Optimization School of Engineering University of Bradford 15 Formulation of a multi-objective problem Basic approaches to the formulation of a combined criterion 1. Select the most important criterion and treat it as a single criterion. Impose constraints on the values of all remaining criteria. 2. Linear combination of all criteria. It is important to normalise the criteria using the best values of corresponding individual criteria. 3. The minimax criterion MULTI-OBJECTIVE PROBLEMS where F k * is the desirable value of the criterion F k.