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

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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.

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.

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

MATHEMATICAL OPTIMIZATION PROBLEM

Formulation of a design improvement problem as a formal mathematical optimization problemA 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.

MATHEMATICAL OPTIMIZATION PROBLEM

Formulation of a design improvement problem as a formal mathematical optimization problemChoice 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.

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:

MATHEMATICAL OPTIMIZATION PROBLEM

Formulation of a design improvement problem as a formal mathematical optimization problemDiscrete 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

Formulation of a design improvement problem as a formal mathematical optimization problemExample 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

Formulation of a design improvement problem as a formal mathematical optimization problemCriteria 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

MATHEMATICAL OPTIMIZATION PROBLEM

Formulation of a design improvement problem as a formal mathematical optimization problemFormulation 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

Formulation of a design improvement problem as a formal mathematical optimization problemConstrained 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

Formulation of a design improvement problem as a formal mathematical optimization problemNormalisation 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

Sizing optimization

Tractor-trailer

combination

Objective: to

improve the ride

characteristics

Design variables:

properties of the

suspension system

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.

Sizing optimization

Flight simulator (cont.)

Kinematic optimization of a Stewart platform manipulator for a flight simulator.

Sizing optimization problems

Stirling engine

Objective: to improve the thermodynamic efficiency.

Constraint: power output.

Design variables: parameters of the engine.

Shape optimization

Optimization of a spanner

A CAD model of a structure. Moves of the boundary are

allowed at the indicated points

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)

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

Results (aerofoil, cont.)

Results of MARS. Initial (dashed) and obtained (solid) configurations

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.

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.

Problem definition (optimization of a shell, cont.)

First design, normalized constraint equals 0.993

Problem definition (optimization of a shell, cont.)

Second design, normalized constraint equals 0.998

APPLICATIONS OF OPTIMIZATION TECHNIQUES

Three-bay by four-bay by four-storey structure

Discrete variables are numbers of sections from

a catalogue

APPLICATIONS OF OPTIMIZATION TECHNIQUES

Genetic

Algorithm

Front wing of J3 Jaguar Racing Formula 1 car

APPLICATIONS OF OPTIMIZATION TECHNIQUES

Genetic

Algorithm

Schematic layup

of the composite structure of the wing

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%

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)

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.

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