1. Design Optimization School of Engineering University of Bradford 1 There is always a need for improvement of products and processes. How? MOTIVATION

2. Design Optimization School of Engineering University of Bradford 2 Use Design Optimization!

3. Design Optimization School of Engineering University of Bradford 3 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. DESIGN PROCESS

4. Design Optimization School of Engineering University of Bradford 4 Overview of the design process DESIGN PROCESS William Edwards’ fourth bridge in 1755

5. Design Optimization School of Engineering University of Bradford 5 Overview of the design process DESIGN PROCESS William Edwards’ fourth bridge now

6. Design Optimization School of Engineering University of Bradford 6 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. DESIGN PROCESS

7. Design Optimization School of Engineering University of Bradford 7 CLASSIFICATION Roman goddess Opes: the word optimus - the best - was derived from her name

8. Design Optimization School of Engineering University of Bradford 8 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.

9. Design Optimization School of Engineering University of Bradford 9 Formulation of a design improvement problem as a formal mathematical optimization problem Choice of design variables MATHEMATICAL OPTIMIZATION PROBLEM 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. Design Optimization School of Engineering University of Bradford 10 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:

11. Design Optimization School of Engineering University of Bradford 11 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. MATHEMATICAL OPTIMIZATION PROBLEM

12. Design Optimization School of Engineering University of Bradford 12 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 MATHEMATICAL OPTIMIZATION PROBLEM

13. Design Optimization School of Engineering University of Bradford 13 Difficulty in solving a discrete problem OPTIMIZATION TECHNIQUES

14. Design Optimization School of Engineering University of Bradford 14 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. MATHEMATICAL OPTIMIZATION PROBLEM Criteria of system’s efficiency

15. Design Optimization School of Engineering University of Bradford 15 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. MATHEMATICAL OPTIMIZATION PROBLEM

16. Design Optimization School of Engineering University of Bradford 16 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. MATHEMATICAL OPTIMIZATION PROBLEM

17. Design Optimization School of Engineering University of Bradford 17 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

18. Design Optimization School of Engineering University of Bradford 18 Geometrical interpretation of a constrained maximization problem MATHEMATICAL OPTIMIZATION PROBLEM

19. Design Optimization School of Engineering University of Bradford 19 Geometrical interpretation of an optimization process MATHEMATICAL OPTIMIZATION PROBLEM

20. Design Optimization School of Engineering University of Bradford 20 Classification of mathematical optimization problems - the optimization tree MATHEMATICAL OPTIMIZATION PROBLEM

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

22. Design Optimization School of Engineering University of Bradford 22 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. EXAMPLES

23. Design Optimization School of Engineering University of Bradford 23 Sizing optimization Flight simulator (cont.) Kinematic optimization of a Stewart platform manipulator for a flight simulator. EXAMPLES

24. Design Optimization School of Engineering University of Bradford 24 Sizing optimization Flight simulator Six design variables define the configuration of the platform. EXAMPLES

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

26. Design Optimization School of Engineering University of Bradford 26 Shape optimization Optimization of a spanner EXAMPLES

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

28. Design Optimization School of Engineering University of Bradford 28 Shape optimization Optimization of a spanner Initial and final designs. Courtesy of J. Rassmusen EXAMPLES

29. Design Optimization School of Engineering University of Bradford 29 Shape optimization problems EXAMPLES

30. Design Optimization School of Engineering University of Bradford 30 Shape optimization problems EXAMPLES

31. Design Optimization School of Engineering University of Bradford 31 Shape optimization problems EXAMPLES

32. Design Optimization School of Engineering University of Bradford 32 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). EXAMPLES: SHAPE OPTIMIZATION W.A. Wright, C.M.E. Holden, Sowerby Research Centre, British Aerospace (1998)

33. Design Optimization School of Engineering University of Bradford 33 Problem definition (aerofoil, cont.) EXAMPLES: SHAPE OPTIMIZATION Problem formulation: Objective function (to be minimized): drag coefficient at Mach 0.73 and Mach 0.76: F 0 (x) = 2.0 C d total (M=0.73) + 1.0 C d total (M=0.76) Constraints: on lift and other operational requirements (sufficient space for holding fuel, etc.) Techniques used: –Powell’s Direct Search (PDS) –Genetic Algorithm (GA) –MARS Carren M.E. Holden Sowerby Research Centre, British Aerospace, UK

34. Design Optimization School of Engineering University of Bradford 34 Results (aerofoil, cont.) EXAMPLES: SHAPE OPTIMIZATION Results of MARS. Initial (dashed) and obtained (solid) configurations

35. Design Optimization School of Engineering University of Bradford 35 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. EXAMPLES: SHAPE OPTIMIZATION

36. Design Optimization School of Engineering University of Bradford 36 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. EXAMPLES: SHAPE OPTIMIZATION

37. Design Optimization School of Engineering University of Bradford 37 Problem definition (optimization of a shell, cont.) First design, normalized constraint equals 0.993 EXAMPLES: SHAPE OPTIMIZATION

38. Design Optimization School of Engineering University of Bradford 38 Problem definition (optimization of a shell, cont.) Second design, normalized constraint equals 0.998 EXAMPLES: SHAPE OPTIMIZATION

39. Design Optimization School of Engineering University of Bradford 39 APPLICATIONS OF OPTIMIZATION TECHNIQUES Three-bay by four-bay by four-storey structure Discrete variables are numbers of sections from a catalogue

40. Design Optimization School of Engineering University of Bradford 40 Optimization of front wing of J3 Jaguar Racing Formula 1 car APPLICATIONS OF OPTIMIZATION TECHNIQUES

41. Design Optimization School of Engineering University of Bradford 41 Optimization of front wing of J3 Jaguar Racing Formula 1 car APPLICATIONS OF OPTIMIZATION TECHNIQUES

42. Design Optimization School of Engineering University of Bradford 42 Genetic Algorithm APPLICATIONS OF OPTIMIZATION TECHNIQUES Front wing of J3 Jaguar Racing Formula 1 car

43. Design Optimization School of Engineering University of Bradford 43 Genetic Algorithm APPLICATIONS OF OPTIMIZATION TECHNIQUES Schematic layup of the composite structure of the wing

44. Design Optimization School of Engineering University of Bradford 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. Design Optimization School of Engineering University of Bradford 45 Material optimization problem (O. Sigmund, TU of Denmark) EXAMPLES 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. Design Optimization School of Engineering University of Bradford 46 Material optimization problem (O. Sigmund, TU of Denmark) EXAMPLES 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.