Design Optimization.ppt

Ken Youssefi Mechanical Engineering Dept. 1
Design Optimization
• Optimization is a component of design
process
• The design of systems can be formulated as
problems of optimization where a measure
of performance is to be optimized while
satisfying all the constraints.

Design Optimization
• Design variables – a set of parameters that describes
the system (dimensions, material, load, …)
• Design constraints – all systems are designed to
perform within a given set of constraints. The constraints
must be influenced by the design variables (max. or min.
values of design variables).
• Objective function – a criterion is needed to judge
whether or not a given design is better than another (cost,
profit, weight, deflection, stress, ….).

Optimum Design – Problem Formulation
The formulation of an optimization problem is
extremely important, care should always be
exercised in defining and developing expressions
for the constraints.
The optimum solution will only be as good as the
formulation.

Problem Formulation
Design of a two-bar structure
The problem is to design a two-member bracket to support a
force W without structural failure. Since the bracket will be
produced in large quantities, the design objective is to
minimize its mass while also satisfying certain fabrication and
space limitation.

Problem Formulation
In formulating the design problem, we need to define structural
failure more precisely. Member forces F1 and F2 can be used to
define failure condition.
Apply equilibrium conditions;
Σ Fy = 0
Σ Fx = 0

Problem Formulation
Design Variables
An important first step in the proper formulation of the problem
is to identify design variables for the system.
1. All design variables should be independent of each
other as far as possible.
2. All options of identifying design variables should
be investigated.
3. There is a minimum number of design variables
required to formulate a design problem properly.
4. Designate as many independent parameters as
possible as at the beginning. Later, some of the
design variables can be eliminated by assigning
numerical values.

Problem Formulation
Represent all the design variables for a problem in the vector x.
x3 = outer diameter of member 1
x4 = inner diameter of member 1
x1 = height h of the truss
x2 = span s of the truss

Problem Formulation
Objective Function
A criterion must be selected to compare various designs
1. It must be a scalar function whose numerical values
could be obtained once a design is specified.
2. It must be a function of design variables, f (x).
3. The objective function is minimized or maximized
(minimize cost, maximize profit, minimize weight,
maximize ride quality of a vehicle, minimize the cost
of manufacturing, ….)
4. Multi-objective functions; minimize the weight of a
structure and at the same time minimize the
deflection or stress at a certain point.

Problem Formulation
Objective Function
Mass is selected as the objective function
Mass = ( density )( area )( length )

Problem Formulation
Design Constraints
Feasible Design
A design meeting all the requirements is called a feasible
(acceptable) design. An infeasible design does not meet one
or more requirements
Implicit & Explicit Constraints
All restrictions placed on a design are collectively called
constraints. Some constraints are explicit (obvious, provided)
such as min. and max. values of design variables, some are
implicit (derived, deduced) which are usually more complex
such as deflection of a structure.

Problem Formulation
Linear and Nonlinear Constraints
Constraint functions having only first-order terms in design
variables are called linear constraints. More general
problems have nonlinear constraint functions as well.
A machine component must move precisely by a certain value
(equality). Stress must not exceed the allowable stress of the
material (inequality)
It is easier to find feasible designs for a
system having only inequality constraints.
Design problems may have equality as well as inequality
constraints. A feasible design must satisfy precisely all the
equality constraints.
Equality and Inequality Constraints

Problem Formulation
Constraints for the example problem are: member stress shall
not exceed the allowable stress, and various limitations on
design variables shall be met.
Constraint on stress
σ (applied) < σ (allowable)

Problem Formulation
Finally, the constraints on design variables are written as
Where xil and xiu are the minimum and maximum
values for the ith design variable. These constraints are
necessary to impose fabrication and physical space
limitations.

Problem Formulation
The problem can be summarized as follows
Find design variables x1, x2, x3, x4, x5, and x6 to minimize
the objective function,
subject to the constraints of the equations
x3 = outer diameter of
member 1
x4 = inner diameter of
member 1
x5 = outer diameter of
member 2
x6 = inner diameter of
member 2

Example – Design of a Beer Can
Design a beer can to hold at least the specified amount of beer
and meet other design requirement. The cans will be produced
in billions, so it is desirable to minimize the cost of
manufacturing. Since the cost can be related directly to the
surface area of the sheet metal used, it is reasonable to
minimize the sheet metal required to fabricate the can.

Fabrication, handling, aesthetic, shipping considerations and
customer needs impose the following restrictions on the size
of the can:
1. The diameter of the can should be no more
than 8 cm. Also, it should not be less than 3.5
cm.
2. The height of the can should be no more than
18 cm and no less than 8 cm.
3. The can is required to hold at least 400 ml of
fluid.

Design variables
D = diameter of the can (cm)
H = height of the can (cm)
Objective function
The design objective is to minimize the surface area
(Non-linear)

The constraints must be formulated in terms of design variables.
The first constraint is that the can must hold at least 400 ml of
fluid.
The problem has two independent design variable and five
explicit constraints. The objective function and first constraint
are nonlinear in design variable whereas the remaining
constraints are linear.
(Non-linear)
The other constraints on the size of the can are:
(linear)

Standard Design Optimization Model
The standard design optimization model is defined as
follows: Find an n-vector x = (x1, x2, …., xn) of design
variables to minimize an objective function
subject to the p equality constraints
and the m inequality constraints

Observations on the Standard Model
• The functions f(x), hj(x), and gi(x) must depend on
some or all of the design variables.
• The number of independent equality constraints
must be less than or at most equal to the number of
design variables.
• There is no restriction on the number of inequality
constraints.
• Some design problems may not have any
constraints (unconstrained optimization problems).
• Linear programming is needed If all the functions
f(x), hj(x), and gi(x) are linear in design variables x,
otherwise use nonlinear programming.

Helical Compression Spring
Free body diagram of axially loaded helical spring
D = coil diameter
d = wire diameter
Maximum shear stress

Shear stress correction factor Spring index
Maximum
shear stress

Deflection
The total strain energy, torsional and shear components
Total strain
energy

Castigliano’s theorem – deflection is the partial derivative of
the total strain energy.
Spring rate
Linear relationship

Helical spring end conditions
N (total) = Na (active number of coils) + Ni (inactive number of coils)

Example of unconstraint optimization problem
Design a compression spring of minimum weight, given the
following data:

Example – spring design
The weight of the compression spring is given by the
following equation:

The equation can be expressed in terms of the
spring index C = D/d
The maximum shear stress and the deflection in the
spring are given by the following equations

Substituting all of the equations into the weight equation, we
obtain the following expression in terms of spring index C:
The plot of the
objective function
W vs. C, the
spring index

Infeasible Problem
Conflicting requirements, inconsistent constraint equations
or too many constraints on the system will result in no
solution to the problem.
No region of design space
that satisfies all constraints.

Optimization using graphical method
A wall bracket is to be designed to support a load of W. The bracket
should not fail under the load.
W = 1.2 MN
h = 30 cm
s = 40 cm
1
2

Problem Formulation
Design variables:
Objective function:
Stress constraints:
Where forces on bar 1 and bar 2 are:

Graphical Solution

Numerical Methods for Non-linear
Optimization
Graphical and analytical methods are inappropriate for many
complicated engineering design problems.
1. The number of design variables and constraints can be
large.
2. The functions for the design problem can be highly
nonlinear.
3. In many engineering applications, objective and/or
constraint functions can be implicit in terms of design
variables.

Structural Optimization
Structural optimization is an automated synthesis of a mechanical
component based on structural properties.
For this optimization, a geometric modeling tool to represent the
shape, a structural analysis tool to solve the problem, and an
optimization algorithms to search for the optimum design are needed.
Structural Optimization
Geometric Modeling Structural Analysis Optimization
Finite element
modeling
Finite element
analysis
Nonlinear programming
algorithm

Structural Optimization Categories
• Size optimization
Deals with parameters that do not alter the location of nodal points in the
numerical problem, keeps a design’s shape and topology unchanged
while changing dimensions of the design.
• Shape optimization
Deals with parameters that describe the boundary position in the
numerical model. Design variables control the shape of the
design. The process requires re-meshing of the model and results in
freeform shapes
• Topology optimization
The goal of topology optimization is to determine where to place
or add material and where to remove material. The optimization
could determine where to place ribs to stiffen the structure.

Optimization Categories
Size and configuration optimization of a truss, design variables
are the cross sectional areas and nodal coordinates of the truss.
The truss could also be optimized for material.
The topology or connectivity of the truss is fixed.

Shape optimization of a torque arm. Parts of the
boundary are treated as design variables.

Topology optimization can be performed by using
genetic algorithm.
Optimum shapes of the cross section of a beam for
plastic (b), aluminum (c), and steel (d)

Design Optimization.ppt

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Design Optimization.ppt