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MOTIVATION. There is always a need for improvement of products and processes. How?. Use Design Optimization!. DESIGN PROCESS. Traditional design by making and breaking: gradual improvement Example: four bridges over the Taff at Pontypridd built by William
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Traditional design by making and breaking: gradual
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 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
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.
Choice of design variables
Design variables are selected to uniquely identify a design. They have to be be mutually independent.
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.
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
Criteria of structural efficiency are described by the objective function F(x). Typical examples:
Criteria of system’s efficiency
Formulation of typical constraints on system’s behaviour
Constraints can be imposed on:
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.
Normalisation of constraints
It is important to normalise the constraints and make them
Example 1: stress constraint
can be transformed to
Example 2: buckling constraint
can be transformed to
F0 (x) = 2.0 Cdtotal (M=0.73) + 1.0 Cdtotal (M=0.76)
Carren M.E. HoldenSowerby Research Centre, British Aerospace, UK
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.
First design, normalized constraint equals 0.993
Second design, normalized constraint equals 0.998
Three-bay by four-bay by four-storey structure
Discrete variables are numbers of sections from
Front wing of J3 Jaguar Racing Formula 1 car
of the composite structure of the wing
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
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)
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.