Chapter one introduction optimization

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Chapter one
Introduction
Introduction
What is optimization?
“ Optimization”
• The act of obtaining the best result under given circumstances
• In design, construction, and maintenance of any engineering system, engineers
have to take many technological and managerial decisions at several stages
• The ultimate goal of all such decisions is either to minimize the effort required
or to maximize the desired benefit
• Optimization can be defined as the process of finding the conditions that give
the maximum or minimum value of a function

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Introduction
Some typical applications Optimization from different engineering disciplines
Design of civil engineering structures such as frames, foundations, bridges, towers,
chimneys, and dams for minimum cost
Minimum-weight design of structures for earthquake, wind, and other types of
random loading
Design of water resources systems for maximum benefit
Optimum design of electrical networks
Shortest route taken by a salesperson visiting various cities during one tour
Optimal production planning, controlling, and scheduling
Allocation of resources or services among several activities to maximize the benefit
Introduction
What is Structural optimization?
“ Structure ”
Any assemblage of materials which is intended to sustain loads
“ Optimization”
• The act of obtaining the best result under given circumstances
Structural optimization is the subject of making an assemblage of materials
sustain loads in the best way

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The Design Process
• The design of many engineering systems is a complex process
• Assumptions must be made to develop realistic models that can be subjected to
mathematical analysis by the available methods, and the models must be verified
by experiments
• Many possibilities and factors must be considered during problem formulation
Economic considerations play an important role in designing cost-effective
systems
• To complete the design of an engineering system, designers from different fields
of engineering usually must cooperate
• The entire design project must be broken down into several sub problems, which
are then treated somewhat independently
• Design is an iterative process
The Design Process

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Conventional versus Optimum
design process
(a) conventional design method and (b) optimum design method
The Design Process
1. Function: What is the use of the product?
2. Conceptual design: What type of construction concept should
we use?
3. Optimization: Within the chosen concept, and within the
constraints on function, make the product as good as possible
4. Details: This step is usually controlled by market, social or
esthetic factors

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Types of Structural Optimization ProblemsTypes of Structural Optimization ProblemsTypes of Structural Optimization ProblemsTypes of Structural Optimization Problems
• Depending on the geometric feature, we divide structural optimization
problems into three classes:
1. Sizing optimization:
• Determine optimal size for a given design (connectivity)
Types of Structural Optimization ProblemsTypes of Structural Optimization ProblemsTypes of Structural Optimization ProblemsTypes of Structural Optimization Problems
2. Shape optimization:
• Given the domain, find the optimal shape of domain

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Types of Structural Optimization Problems
3. Topology optimization:
• Given a design domain in which to operate, find the optimal
Connectivity and boundaries of structure (design features of
structures)
Types of Structural Optimization Problems

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Optimization problem
Any optimization problem has three basic ingredients:
1.Optimization variables
• Also called design variables denoted as vector x
• Their values give us a design for the system-what we seek
• should be independent
• If related through an equality constraint, they are referred to as “state
variables”
State variable (y): For a given structure, i.e., for a given design x, y is a function
or vector that represents the response of the structure
For a mechanical structure, response means displacement, stress, strain or force
2.Objective function
• Denoted as f (x) , Function to be optimized
• If minimization, often referred to as “cost function”
• It has to be a function of the design variables,
• An “optimized design” has best value of the objective function
• Selection of the objective function dictates solution
Examples: Minimum weight, cost
Maximize stiffness, safety factor
• Multi-objective optimization: combination of objectives

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3.Constraints
• Expressed as equalities or inequalities denoted as (x)
• Restrictions of design and performance
Feasible design: All constraints are satisfied (required for optimality)
• Feasible region: design space of all feasible design
• Infeasible design: One or more of constraints is violated
• Types of constraints: Equality, in-equality, design variable bounds
• If no constraints (rare): “unconstrained problem”
Optimum Design Problem Formulation
Feasible Design
• Design meeting all requirements
• Feasible Region is the set of all
possible points (sets of values of the
choice variables) of an optimization
problem that satisfy the
problem's constraints, potentially
including inequalities, equalities,
and integer constraints

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• Design variables: variables with which the design problem is parameterized:
• Objective function( “cost function”): quantity that is to be minimized
(maximized) Usually denoted by:
• Constraint: condition that has to be satisfied
• Inequality constraint:
• Equality constraint:
( ) 0g x
( ) 0h x
( )f x
 1 2, , , nx x xx K
• General form of optimization problem:
 xxx
x
xh
xg
x
x




n
X
f
0)(
0)(:subject to
)(min

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Optimization ProblemsOptimization ProblemsOptimization ProblemsOptimization Problems
• Problems:
• Constrained vs. unconstrained
• Single level vs. multilevel
• Single objective vs. multi-objective
• Deterministic vs. stochastic
• Variables:
• Continuous vs. discrete (integer, ordered, non-ordered)
• Responses:
• Linear vs. nonlinear
• Convex vs. nonconvex
• Smooth vs. nonsmooth
• Proper definition and formulation of a problem take roughly 50 percent of
the total effort needed to solve it
• Optimum solution will be only as good as the formulation
Problem Formulation Process:
1: Project/problem description
2: Data and information collection
3: Definition of design variables
4: Optimization criterion
5: Formulation of constraints

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1.Project/problem description
• The statement describes the overall objectives of the project and
the requirements to be met
• By the project’s owner/sponsor
2. Data and information collection
• To develop a mathematical formulation for the problem, gather
information on
Material properties, performance requirements, resource limits,
cost of raw materials, and in addition analysis procedures and
analysis tools
3. Definition of Design Variables / Optimization variables
• Regarded as free because we should be able to assign any value to them
• The design variables should be independent of each other as far as possible
• The number of independent design variables gives the design degrees of
freedom for the problem
• A minimum number of design variables required to properly formulate a design
optimization problem must exist
• A numerical value should be given to each identified design variable to
determine if a trial design of the system is specified

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4: Optimization Criterion
• There can be many feasible designs for a system, and some are better than
others
• The criterion to select the better one should be a scalar function whose
numerical value can be obtained once a design is specified; that is, it must be a
function of the design variable vector x.
• Such a criterion is usually called an objective function for the optimum design
problem, and it needs to be maximized or minimized depending on problem
requirements
• Some objective functions are cost (to be minimized), profit (to be maximized)
,weight (to be minimized), energy expenditure (to be minimized),
5: Formulation of Constraints
• Design and fabricate with the given resources and must meet
performance requirements
For example, For structural members
• Should not fail under normal operating loads
• The vibration frequencies of a structure must be different from the
operating frequency of the machine it supports;
• Members must fit into the available space

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Linear and Nonlinear Constraints
• linear constraints ;which have only first-order terms in design variables
• Linear programming problems have only linear constraints and objective
functions
Equality and Inequality Constraints
Implicit Constraints
Example
MINIMUM-WEIGHT TUBULAR COLUMN DESIGN
STEP 1: PROJECT/PROBLEM DESCRIPTION
Straight columns are used as structural elements in civil, mechanical,
aerospace, agricultural, and automotive structures. Many such
applications can be observed in daily life. It is important to optimize the
design of a straight column since it may be mass-produced. The
objective of this project is to design a minimum-mass tubular column of
length l supporting a load P without buckling or overstressing. The
column is fixed at the base and free at the top, as shown in Figure. This
type of structure is called a cantilever column.

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Example
Example
STEP 2: DATAAND INFORMATION COLLECTION
• The buckling load (also called the critical load) for a cantilever column is
given as
I is the moment of inertia , E is the modulus of elasticity
• Material stress σ = P/A,
A is the cross-sectional area of the column.
σa =The material allowable stress under the axial load is, and
ρ = the material mass density is (mass per unit volume).

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Example
STEP 3: DEFINITION OF DESIGN VARIABLES
• For the first formulation, the following design variables are defined:
R = mean radius of the column
t = wall thickness
• Assuming that the column wall is thin (R >> t), the material cross
sectional area and moment of inertia are
STEP 4: OPTIMIZATION CRITERION
• The total mass of the column to be minimized is given as
Example
STEP 5: FORMULATION OF CONSTRAINTS
• Stress (P/A) should not exceed the material allowable stress σa to avoid material
failure ,Inequality σ≤σa
• The applied load P should not exceed the buckling load (i.e., P≤Pcr)
• Finally, the design variables R and t must be within the specified minimum and
maximum values:
Rmin ≤ R ≤ Rmax; tmin ≤ t ≤ tmax

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Example
Min (R,t)
Subjected to
Rmin ≤ R ≤ Rmax;
tmin ≤ t ≤ tmax
Weight Minimization of a Two-Bar Truss Subject to
Stress Constraints
• The bars have the same length L and Young’s modulus E. The force F >0,
and for the angle α we assume 0 ≤ α ≤ 90◦. We are to minimize the weight
under stress constraints. The design variables are the cross-sectional areas
A1 and A2. The objective function, i.e., the total weight of the truss,
becomes
f (A1,A2) = (A1 +A2)ρL
As design constraints the cross-sectional
areas must, for obvious physical reasons,
be nonnegative,
i.e., A1 ≥ 0, A2 ≥ 0

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• Equilibrium in the x- and y-directions gives
F cosα − σ1A1 = 0, F sinα −σ2A2 = 0,
• The state constraint involving stresses reads
|σi| ≤ σ0, i= 1, 2,
• where σ0 is a maximum allowed stress level, the same in both tension
and compression
−σ0A1 ≤ F cosα ≤ σ0A1,
−σ0A2 ≤ F sinα ≤ σ0A2.

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More Examples
An Introduction to Structural Optimization Peter W. Christensen · Anders

Chapter one introduction optimization

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