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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.

Mechanical Engineering Dept.

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, ….).

Mechanical Engineering Dept.

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.

Mechanical Engineering Dept.

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.

Mechanical Engineering Dept.

Apply equilibrium conditions;ΣFx = 0, ΣFy = 0

Problem FormulationIn formulating the design problem, we need to define structural failure more precisely. Member forces F1 and F2 can be used to define failure condition.

Mechanical Engineering Dept.

Problem Formulation

Design Variables

An important first step in the proper formulation of the problem is to identify design variables for the system.

- All design variables should be independent of each other as far as possible.
- All options of identifying design variables should be investigated.
- There is a minimum number of design variables required to formulate a design problem properly.
- Designate as many independent parameters as possible as at the beginning. Later, some of the design variables can be eliminated by assigning numerical values.

Mechanical Engineering Dept.

x1 = height h of the truss

x2 = span s of the truss

x3 = outer diameter of member 1

x4 = inner diameter of member 1

x5 = outer diameter of member 2

x6 = inner diameter of member 2

Problem FormulationRepresent all the design variables for a problem in the vector x.

Mechanical Engineering Dept.

Problem Formulation

Objective Function

A criterion must be selected to compare various designs

- It must be a scalar function whose numerical values could be obtained once a design is specified.
- It must be a function of design variables, f (x).
- The objective function is minimized or maximized (minimize cost, maximize profit, minimize weight, maximize ride quality of a vehicle, minimize the cost of manufacturing, ….)
- Multi-objective functions; minimize the weight of a structure and at the same time minimize the deflection or stress at a certain point.

Mechanical Engineering Dept.

Problem Formulation

Objective Function

Mass is selected as the objective function

Mass = density x area x length

x1 = height h of the truss

x2 = span s of the truss

x3 = outer diameter of member 1

x4 = inner diameter of member 1

x5 = outer diameter of member 2

x6 = inner diameter of member 2

Mechanical Engineering Dept.

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 Constraints

All restrictions placed on a design are collectively called constraints. Some constraints are simple (explicit) such as min. and max. values of design variables, some are more complex and indirectly influenced by design variable (implicit constraints), deflection of a complex structure.

Mechanical Engineering Dept.

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.

Equality and Inequality Constraints

Design problems may have equality as well as inequality constraints. A feasible design must satisfy precisely all the equality constraints.

A machine must move precisely by a delta (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.

Mechanical Engineering Dept.

σ (applied) < σ (allowable)

Problem FormulationConstraints for the example problem are: member stress shall not exceed the allowable stress, and various limitations on design variables shall be met.

Mechanical Engineering Dept.

Problem Formulation

Finally, the constraints on design variables are written as

Where xiland xiu are the minimum and maximum values for the ith design variable. These constraints are necessary to impose fabrication and physical space limitations.

Mechanical Engineering Dept.

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

Mechanical Engineering Dept.

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.

Mechanical Engineering Dept.

Example – Design of a Beer Can

Fabrication, handling, aesthetic, and shipping considerations impose the following restrictions on the size of the can

- The diameter of the can should be no more than 8 cm. Also, it should not be less than 3.5 cm.
- The height of the can should be no more than 18 cm and no less than 8 cm.
- The can is required to hold at least 400 ml of fluid.

Mechanical Engineering Dept.

The design objective is to minimize the surface area

Example – Design of a Beer CanDesign variables

D = diameter of the can (cm)

H = height of the can (cm)

Mechanical Engineering Dept.

Example – Design of a Beer Can

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 other constraints on the size of the can are:

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.

Mechanical Engineering Dept.

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

Mechanical Engineering Dept.

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.

Mechanical Engineering Dept.

Example of unconstraint optimization problem

Design a compression spring of minimum weight, given the following data:

Mechanical Engineering Dept.

Example – spring design

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

Mechanical Engineering Dept.

Example – spring design

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

Mechanical Engineering Dept.

The plot of the objective function Wvs.C, the spring index

Example – spring designSubstituting all of the equations into the weight equation, we obtain the following expression in terms of spring index C:

Mechanical Engineering Dept.

No region of design space that satisfies all constraints.

Infeasible ProblemConflicting requirements, inconsistent constraint equations or too many constraints on the system will result in no solution to the problem.

Mechanical Engineering Dept.

W = 1.2 MN

A wall bracket is to be designed to support a load of W. The bracket should not fail under the load.

Optimization using graphical method1

h = 30 cm

2

s = 40 cm

Mechanical Engineering Dept.

Problem Formulation

Design variables:

Objective function:

Stress constraints:

Where forces on bar 1 and bar 2 are:

Mechanical Engineering Dept.

Graphical Solution

Mechanical Engineering Dept.

Numerical Methods for Non-linear Optimization

Graphical and analytical methods are inappropriate for many complicated engineering design problems.

- The number of design variables and constraints can be large.
- The functions for the design problem can be highly nonlinear.
- In many engineering applications, objective and/or constraint functions can be implicit in terms of design variables.

Mechanical Engineering Dept.

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

Mechanical Engineering Dept.

Optimization Categories

- Size optimization
keeps a design’s shape and topology unchanged while modifying specified dimensions of the design.

- Shape optimization
holds the topology constant while modifying the shape. Design variables control the shape of the design.

- Topology optimization
In order to obtain a globally optimal shape, topology must be also modified, allowing the creation of new boundaries (applies to structures

Mechanical Engineering Dept.

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.

Mechanical Engineering Dept.

Optimization Categories

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

Mechanical Engineering Dept.

Optimization Categories

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)

Mechanical Engineering Dept.

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