There is always a need for improvement of products and processes. How?

1. MOTIVATION There is always a need for improvement of products and processes.How?

2. Use Design Optimization!

3. DESIGN PROCESS Traditional design by making and breaking: gradual improvement Example: four bridges over the Taff at Pontypridd built by William Edwards. Condition: to stand for 7 years. Cost: £500. First bridge (1746): three or four spans, collapsed two years and two months later, one of the piers was destroyed by a flood. Second bridge (about 1750): change of design, a single span of 42 m. When almost complete, the timber centering collapsed under the excessive weight of masonry. Third bridge (1754): lighter arch completed in September, collapsed in November, the light crown was forced upwards by heavy haunches. Fourth bridge (1755), still stands. Modifications: heavier crown, haunches lightened by large cylindrical openings.

4. DESIGN PROCESS Overview of the design process William Edwards’ fourth bridge in 1755

5. DESIGN PROCESS Overview of the design process William Edwards’ fourth bridge now

6. DESIGN PROCESS Overview of the design process Design optimization as systematic design improvement. Design optimization is a rational finding of a design that is the best of all possible designs for a chosen objective and a given set of geometrical and behavioural constraints. A minimalist’s (a realist’s?) view: design optimization is a systematic way of improvement of designs.

7. CLASSIFICATION Roman goddess Opes: the word optimus - the best - was derived from her name

8. MATHEMATICAL OPTIMIZATION PROBLEM Formulation of a design improvement problem as a formal 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, gj(x) are the constraint functions, the last set of inequality conditions defines the side constraints.

9. MATHEMATICAL OPTIMIZATION PROBLEM Formulation of a design improvement problem as a formal mathematical optimization problem Choice of design variables Design variables are selected to uniquely identify a design. They have to be be mutually independent. Typical examples: • Area of cross section of bars in a truss structure • Number of a specific steel section in a catalogue of UB sections • Coordinates of poles of B-splines defining the shape of an aerofoil • etc.

10. MATHEMATICAL OPTIMIZATION PROBLEM Specific Features of Shape Optimization • 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:

11. MATHEMATICAL OPTIMIZATION PROBLEM Formulation of a design improvement problem as a formal mathematical optimization problem Discrete and continuous problems Discrete problems: a design variable can only take specified positions from a given set. Specific case: integer variables. Example: a number of reinforcement bars. Continuous variables can take any real value on a given range. Example: coordinates of a nodal point. Mixed discrete-continuous problems are the hardest to solve.

12. MATHEMATICAL OPTIMIZATION PROBLEM Formulation of a design improvement problem as a formal mathematical optimization problem Example of a discrete problem Optimization of a steel structure where some of the members are described by 10 design variables. Each design variable represents a number of a UB section from a catalogue of 10 available sections. One full structural analysis of each design takes 1 sec. on a computer. Question: how much time would it take to check all the combinations of cross-sections in order to guarantee the optimum solution? Answer: 317 years

13. OPTIMIZATION TECHNIQUES Difficulty in solving a discrete problem

14. MATHEMATICAL OPTIMIZATION PROBLEM Formulation of a design improvement problem as a formal mathematical optimization problem Criteria of structural efficiency are described by the objective function F(x). Typical examples: • cost • weight • use of resources (fuel, etc.) • stress concentration • etc. Criteria of system’s efficiency

15. MATHEMATICAL OPTIMIZATION PROBLEM Formulation of a design improvement problem as a formal mathematical optimization problem Formulation of typical constraints on system’s behaviour Constraints can be imposed on: • equivalent stress • critical buckling load (local and global), can include postbuckling characteristics • frequency of vibrations (can be several) • cost • etc.

16. MATHEMATICAL OPTIMIZATION PROBLEM Formulation of a design improvement problem as a formal mathematical optimization problem Constrained and unconstrained problems Almost all realistic problems of design optimizatiion are constrained problems, i.e. there are some limitations on the performance characteristics of an engineering system. Still, it is important to learn how to solve efficiently an unconstrained problem because some of the optimization techniques treat a general constrained problem as an equivalent sequence of simpler unconstrained problems. Also, inverse problems can often be formulated as unconstrained problems.

17. MATHEMATICAL OPTIMIZATION PROBLEM 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

18. MATHEMATICAL OPTIMIZATION PROBLEM Geometrical interpretation of a constrained maximization problem

19. MATHEMATICAL OPTIMIZATION PROBLEM Geometrical interpretation of an optimization process

20. MATHEMATICAL OPTIMIZATION PROBLEM Classification of mathematical optimization problems - the optimization tree

21. EXAMPLES Sizing optimization Tractor-trailer combination Objective: to improve the ride characteristics Design variables: properties of the suspension system

22. EXAMPLES Sizing optimization Flight simulator Kinematic optimization of a Stewart platform manipulator for a flight simulator. The goal of optimization is to design a manipulator with maximum workspace whose characteristics are defined according to the manoeuvres of an aircraft.

23. EXAMPLES Sizing optimization Flight simulator (cont.) Kinematic optimization of a Stewart platform manipulator for a flight simulator.

24. EXAMPLES Sizing optimization Flight simulator Six design variables define the configuration of the platform.

25. EXAMPLES Sizing optimization problems Stirling engine Objective: to improve the thermodynamic efficiency. Constraint: power output. Design variables: parameters of the engine.

26. EXAMPLES Shape optimization Optimization of a spanner

27. EXAMPLES Shape optimization Optimization of a spanner A CAD model of a structure. Moves of the boundary are allowed at the indicated points

28. EXAMPLES Shape optimization Optimization of a spanner Initial and final designs. Courtesy of J. Rassmusen

29. EXAMPLES Shape optimization problems

30. EXAMPLES Shape optimization problems

31. EXAMPLES Shape optimization problems

32. EXAMPLES: SHAPE OPTIMIZATION Optimization of an aerofoil B-spline representation of the NACA 0012 aerofoil. The B-spline poles are numbered from 1 to 25. Design variables: x and y coordinates of 22 B-spline poles (N = 44). W.A. Wright, C.M.E. Holden, Sowerby Research Centre, British Aerospace (1998)

33. EXAMPLES: SHAPE OPTIMIZATION Problem definition (aerofoil, cont.) Problem formulation: • Objective function (to be minimized): drag coefficient at Mach 0.73 and Mach 0.76: F0 (x) = 2.0 Cdtotal (M=0.73) + 1.0 Cdtotal (M=0.76) • Constraints: on liftand other operational requirements (sufficient space for holding fuel, etc.) Techniques used: • Powell’s Direct Search (PDS) • Genetic Algorithm (GA) • MARS Carren M.E. HoldenSowerby Research Centre, British Aerospace, UK

34. EXAMPLES: SHAPE OPTIMIZATION Results (aerofoil, cont.) Results of MARS. Initial (dashed) and obtained (solid) configurations

35. EXAMPLES: SHAPE OPTIMIZATION Problem definition (optimization of a shell) A shell is described by a square reference plan. The mid-surface is described using square patches. At the keypoints the out-of-plane coordinate and its derivatives with respect to the in-plane coordinates have been specified.

36. EXAMPLES: SHAPE OPTIMIZATION Problem definition (optimization of a shell, cont.) The geometry is assumed to be symmetric with respect to the diagonals. The design variables are the out-of-plane coordinates of the keypoints and the corresponding derivatives (12 in total). The out-of-plane coordinates of the corners are fixed. Also, the thickness of the shell is taken as a design variable. The shell is supported at its corner nodes, for which all displacement components are prescribed. The shell is loaded by a uniform out-of-plane load. The optimization problem is formulated as minimization of the maximum displacement while the volume remains below the specified limit. Numerical studiy showed that this optimization problem has several local optima. Two designs corresponding to almost equally good optima are shown in the figures below.

37. EXAMPLES: SHAPE OPTIMIZATION Problem definition (optimization of a shell, cont.) First design, normalized constraint equals 0.993

38. EXAMPLES: SHAPE OPTIMIZATION Problem definition (optimization of a shell, cont.) Second design, normalized constraint equals 0.998

39. APPLICATIONS OF OPTIMIZATION TECHNIQUES Three-bay by four-bay by four-storey structure Discrete variables are numbers of sections from a catalogue

40. APPLICATIONS OF OPTIMIZATION TECHNIQUES Optimization of front wing of J3 Jaguar Racing Formula 1 car

41. APPLICATIONS OF OPTIMIZATION TECHNIQUES Optimization of front wing of J3 Jaguar Racing Formula 1 car

42. APPLICATIONS OF OPTIMIZATION TECHNIQUES Genetic Algorithm Front wing of J3 Jaguar Racing Formula 1 car

43. APPLICATIONS OF OPTIMIZATION TECHNIQUES Genetic Algorithm Schematic layup of the composite structure of the wing

44. APPLICATIONS OF OPTIMIZATION TECHNIQUES Optimization problem: minimize mass subject to displacement constraints (FIA and aerodynamics) Result of optimization by a genetic algorithm (GA): Obtained design weight: 4.95 Kg Baseline design weight: 5.2 Kg Improvement: 4.8%

45. EXAMPLES Material optimization problem (O. Sigmund, TU of Denmark) Design of a negative Poisson's ratio material (expands vertically when stretched horizontally) using topology optimization. Left: base cell. Centre: Periodic material composed of repeated base cells. Right: Test beam manufactured by Microelektronik Centret (Denmark)

46. EXAMPLES Material optimization problem (O. Sigmund, TU of Denmark) Design of a material with negative thermal expansion. It is composed of two materials with different thermal expansion coefficients 1 = 1 (blue) and 2 =10 (red) and voids. The effective thermal expansion coefficient is 0= - 4.17. Left: base cell. Centre: thermal displacement of microstructure subjected to heating. Right: periodic material composed of repeated base cells.