1.   Introduction

This chapter has the following sections:

1.1.        Function and parameter optimization

1.2.        Problem Formulation

1.2.1.Design variables


1.2.3.Objective function

1.3.        Graphical Optimization

1.4.        Exercises

1.5.        References


1.1.        Function and parameter optimization

Structural Optimization deals with the efficient design of structures.  Efficiency implies minimum cost or minimum weight while satisfying a variety of strength and stiffness requirements.  Until the early 1980s, structural optimization was mostly the domain of researchers in universities or government laboratories.  Then industry started making use of the techniques developed by researchers, and today these techniques are commonly applied in a large spectrum of industries, ranging from aerospace and civil to automotive engineering.

The origins of the optimization theory are formed as a study in Mathematics.  Early applications used the calculus of variations to seek functions that will minimize or maximize a functional.  A classical example of this approach was the so-called brachistochrone problem, of finding the functional form of a path that minimizes the time for a particle to roll from point A to point B in a vertical plane (Fox, 1987, p.24).  Early applications of structural optimization also used the calculus of variations and employed the differential equations that govern the behavior of the structure.  The structural response (e.g., displacements or stresses) was defined as functions over a continuum, and structural optimization sought functions describing the distribution of the optimal structural properties.  A typical application was the design of a beam for minimum weight, with the optimum defined in terms of the variation of moment of inertia over the length of the beam (e.g., Barnett, 1961)

In contrast, for most industrial applications, structural optimization uses analysis methods that discretize the structural domain, notably finite element methods.  The properties of the design are described by scalars associated with the form of discretization rather than by functions.  For example, a beam structure may have design variables that define the moment of inertia of each beam element that make up the structure.  The element is assumed to have a constant cross section along its length.  Figure 1.1.1 contrasts the function optimization and parameter optimization of the moment of inertia of the beam. 

The mathematical discipline that deals with parameter optimization is called mathematical programming.  The bulk of this courseware deals with mathematical programming techniques and their application to structural optimization problems defined by discretized models.  In particular, it is often implicitly assumed that the structure is modeled by finite elements.


Figure 1.1.1 Beam example: (a) Loading; (b) Function optimization for EI(x); (c) Parameter optimization for EI1 and EI2.

1.2.        Problem Formulation

We formulate a structural optimization problem as a mathematical optimization problem in a specific mathematical form.  The problem is to minimize or maximize a function f of n variables, xi, i=1, 2,…, n.  The variables are components of a vector denoted in boldface lowercase letters, 

                                                       .                                                (1.2.1)

A general mathematical optimization problem also includes a set of functions that are posed as constraints.  A more in-depth discussion of the physical significance of the constraints and the function to be minimized will be provided in the following sections.  It will suffice to present the mathematical form here as,


where ne is the number of constraint functions that are posed as strict equalities and ng is the number of  inequality constraints.  Properly casting a structural design optimization problem into this mathematical form is one of the crucial steps of a design optimization effort and will be described in the following.

1.2.1.     Design Variables  

The design of the structure is often a matter of sizing dimensions, such as thicknesses of plate and shell segments or cross-sectional areas of bars.  In some problems, the design involves selecting shapes, such as the optimal shape of a hole.  The dimensions and the shapes are usually defined in terms of a number of scalar parameters called design variables.  In this courseware, we denote the design variables by their mathematical representation, which are the components of a vector x, with the exception that the expanded form of the vector is shown with regular parentheses rather than curly ones and the transpose sign is dropped for convenience,


This vector represent a point in the design space  spanned by the design variables, that is x Î Rn.  In the sequel, we call this vector and the associated structure a design.


Figure 1.2.1  Cantilevered I-beam with two segments. (a) overall view; (b) cross-sectional shape

In the beam example of Figure 1.2.1, the moments of inertia  and  of the two segments can be two design variables.  After the optimization is concluded, the designer will need to design the details of the cross section, or find a commercially available cross section with a close match to the optimal moments of inertia.  This approach of having a single design variable per beam segment is common when the design is governed mostly by stiffness consideration.  When stresses in the beam cross-section are an important design consideration, then the details of the cross section may be included as design variables. 

If the beam is constructed by welding steel plates to form the I-section, as shown in Figure 1.2.1(b), it is natural to choose the plate thicknesses  as design variables, for a total of six design variables (three per beam segment).  The breadth  and height  of each cross section can be additional design variables (for a total of 10) or may be fixed as pre-assigned parameters.  It is possible to reduce the number of design variables by assuming both flange thicknesses to be equal, i.e.,  =  for each segment, reducing the total number back to eight.  The numerical solution of an optimization problem becomes more difficult when the number of design variables is increased.  On the other hand, if the optimum can be successfully found, the value of an objective function usually improves with additional design variables.


Another illustrative example, a nine-bar truss under two vertical loads, is shown in Figure 1.2.2. 



Figure 1.2.2: Nine bar truss


Again, several choices for design variables are possible.  Perhaps the most common choice consists of member areas  as design variables, with the dimensions H and L fixed.  The design variable vector is then


If the height and the span can also be changed, the design variable vector expands to


One approach to reduce the number of design variables in Eq. (1.2.5) is to form groups of bars, each having the same member area.  For example, we can set all the horizonal bars to have one design variable,; the two vertical bars to have another, and the three slanted bars to have the third,. Then the formulation with cross-sectional design variables, Eq. (1.2.4)  has only three components, and the formulation including the geometrical dimensions only five. 

            All the design variables described above assume values that are positive, because it does not make any sense for a physical dimension of a structure to be a negative number.  In some cases design variables that have negative values must be used.  [Rafi,  how do you feel about explaining how to represent (-)ve design variables for linear programming algorithms that work with only (+)ve variables in here? I think that this should wait until that chapter. Even then it should appear only in a web version that requires a click in order to see it.]

            It is important for the design variables to describe fully the design configuration.  However, it is also important that the will be independent. It is possible inadvertently to choose redundant variables.  For example, for the design of a tubular member, one may choose the inner and outer radii, ri and ro respectively, and the wall thickness, t, as design variables.  Clearly, one needs only two of the three variables to fully describe the cross-sectional geometry.  If it is convenient to use all three variables in the design formulation, the geometric relation between them, ro =  ri + t  must be maintained during the design process as a constraint.  However, more design variables and more constraints make the problem more difficult to solve. Therefore, whenever possible, it is better to avoid using redundant variables and the associated equality constraints.

Design variables may take continuous or discrete values.  Continuous design variables have a range of variation, and they can take any value in that range.  For example, the cross-sectional areas of the nine-bar truss in the example above would be mostly used as continuous design variables in some range of minimum and maximum cross-sectional areas.  Discrete design variables, on the other hand, usually take only a finite number of permissible values.  For example, if the bars for the truss must be made from commercially available cross sections, then only a finite number of cross-sectional areas are possible.  Material design variables that decide what material is used are typically assigned integer values corresponding to a list of materials.  For example, a value of one may correspond to aluminum, a value of two to steel, etc.

We often disregard the discrete nature of the design variables.  Once we obtain the optimum design, we adjust the values of the design variables to the nearest available discrete value.  This approach is popular because an optimization problem with discrete design variables is usually much more difficult to solve than a similar problem with continuous design variables.  Rounding off the design to the closest discrete solution may work well when the available values of the design variables are closely spaced.  Then changing the value of a design variable to the nearest discrete value does not change much the response of the structure.  In some cases, the discrete values of the design variables are spaced so far apart that we must solve the problem by using discrete variables.  For this we employ a branch of optimization theory called integer programming.  In this courseware, design variables are continuous unless otherwise stated.

Finite element models usually require elements that are small compared to the length scale of the variation in structural properties. Because design variables often define that length scale, we have to be careful not to have more design variables than the finite element model can support.  For example, the plate shown in Figure 1.2.3 was analyzed (Prasad and Haftka, 1979)  by a fixed  finite element mesh, with most design variables specifying the thickness of the individual elements.  While the  model was adequate for the initial design, which had a uniform thickness, it was not adequate for the final design shown in the figure.



Figure 1.2.3: Plate thickness optimization.


A similar problem may surface when the coordinates of nodes of the finite element model serve as design variables.  For example, the shape of the hole in the plate shown in Figure 1.2.4 was optimized (Braibant et al., 1983Ref_Barnett61) to reduce the stress concentration near the hole.  The coordinates of all the boundary nodes of the hole served as design variables.  Again, the finite element mesh depicted in the figure was adequate for the analysis of the initial square hole.  The final shape shown in Figure 1.2.4 clearly does not represent the optimal shape.  The optimization procedure took advantage of the large number of design variables to reduce the objective function by increasing the error in the computation rather than by reducing the true stress concentration.  In general, the distribution of design variables should be much coarser than the distribution of finite elements.  Only in skeletal structures, where finite elements often correspond to physical structural elements, a similar distribution for elements and design variables may be natural.



Figure 1.2.4: Hole shape optimization


1.2.2.     Constraints 

The design of a structure is normally subject to requirements or limitations.  Typical requirements specify that the structure must be safe, for example, that stresses do not exceed material limits.  Stress limits are also often specified by standards and design codes or by customers.  Manufacturing requirements may also constrain design freedom, for example, limiting the number of different sections that we can use.  In mathematical optimization, equality and inequality constraints introduce these requirements.  In this courseware, inequality constraints are expressed as


The constraint function gj(x) may in represent any economic, physical or technical quantity that depends on at least one design variable.  In structural optimization, typical constraints are functions of stresses, displacements, buckling loads, fatigue life, and natural vibration frequencies.  It is possible to reverse the sense of an inequality of the opposite type


to the standard form used in this courseware by multiplying it by .  Unfortunately, inequalities of both senses are widely used in the literature, and the choice affects the sign convention in many of the results obtained in this courseware.

When inequality constraints simply specify upper and lower limits on the design variables they are called side constraints and often treated separately from other constraints.  They are expressed as


where  is the given lower limit and  the given upper limit for design variable .

Equality constraints are not common in structural optimization.  They often appear in special formulations where the analysis and design are integrated into a single problem.  In such formulations, the structural response (e.g., displacement) components are used as additional design variables, and the equations of equilibrium are imposed as equality constraints.  In a more standard formulation, equality constraints appear when a response quantity, such as a frequency, must take a specified value.

Equality constraints are expressed here as,


The number of independent equality constraints must be naturally smaller than the number of design variables.  Otherwise, there is no design freedom and so no optimization problem to solve.  Each equality constraint in Eq.                                             (1.2.9) represents a hyper-surface on which the design should lie.  Equality constraints restrict design freedom more severely than inequality constraints and are more difficult to handle numerically.  It is desirable, whenever possible, to avoid equality constraints by reformulating the problem.

A design that satisfies all the constraints is called a feasible design or feasible solution.  The part of the design space where all constraints are satisfied is called feasible region or feasible set.  In the following,  is used for the feasible set, which is defined as follows


A feasible point x Î W can satisfy an inequality constraint as an equality or with a margin.  In the former case, the constraint gj(x) = 0 is called active.  In the latter case gj(x) < 0 , and the constraint is said to be passive or inactive.  If a design x violates at least one constraint, gj(x) > 0 , then this design is called infeasible.  In most numerical optimization software, the boundary between the feasible domain and infeasible domain is not so sharp.  That is, a constraint is considered satisfied even if it is violated by a small amount.  Similarly, a constraint is considered active even if it is satisfied by a small margin.  These tolerances cater to small errors in evaluating constraints, and often improve the performance of the optimization solution procedure.


1.2.3.     Objective function

In order to find the best feasible design, we must have a criterion, such as cost, weight, stiffness, or strength that ranks designs.  This criterion as a function of design variables is called objective function.  If the objective function is better when large (e.g., stiffness), then the optimization problem is a maximization problem.  If the objective function is better when small (e.g., weight) than it is a minimization problem.  The objective is denoted here by f(x), with f : Rn ® R.  Other terms, such as cost function, merit function or performance index have also been used for objective function.  In this courseware, all problems are formulated as minimization but this does not exclude the consideration of maximization problems.  Since

                                                  ,                                     (1.2.11)

any maximization problem may be easily converted into a minimization problem.

For many practical problems, however, the choice of just one objective function to rank designs is difficult.  Instead, several competing criteria should be considered simultaneously.  In structural optimization, cost, weight, maximum stresses, displacements, vibration frequencies, buckling or limit loads, and the probability of failure are typical criteria.  It is possible to have several objective functions using multicriterion optimization.  This topic is treated in detail in Chapter xx, and in the rest of the courseware, a single objective function is assumed.  Material volume or structural weight is the most common objective function found in the literature of structural optimization.  It is a simple and often linear function of design variables. 


Example 1.2.1

As an introductory example, consider a two-bar truss shown in Figure 1.2.5 (a).  The structure is loaded by vertical force F = 20 kN at the free node.  Minimize the material volume, V, of the truss when the allowable stresses in the bars are s u = 200 N/mm2 in tension and s l = 200 N/mm2 in compression.  In addition, the vertical displacement of the loaded node is limited to be below Du = 2.5 mm.  Cross-sectional areas of both bars are used as design variables.  The truss is made of steel with a modulus of elasticity E = 2.0´105 N/mm2, and the length L = 1000 mm.  Ignore possible buckling of the compression bar. 


 In this statically determinate case, explicit expressions for the stresses and the vertical displacement are easily obtained.  Denoting the member area of bar 1 by x1 and of bar 2 by x2, the following optimization problem is obtained,





Here V is the objective function, which represents the material volume of the truss.  It is a linear function of the design variables and is minimized subject to displacement constraint (1.2.13) as well as stress constraints (1.2.14) and (1.2.15).  In this simple case, an explicit problem formulation without equality constraints can be obtained.  Constraint functions defined in the standard form                                   (1.2.6) are




where x = (x1, x2) is the design variable vector.  With only two design variables, we can solve the problem graphically.  This is shown in Figure 1.2.5 (b) where the feasible set W has been depicted by drawing curves gj = 0, j = 1, 2, 3 in the design plane R2 and checking that all constraints (1.2.12-14) are satisfied simultaneously.  The feasible set W is obtained as an intersection of those three regions, which correspond to cases g1 £ 0, g2 £ 0 and g3 £ 0.  It is the unshaded region in the top right part of the figure.


Figure 1.2.5: Two bar truss.

In addition, two lines of constant objective function, called objective function contours, which are straight lines according to                                           (1.2.12) are depicted in Figure 1.2.5 (b).  Along these lines, the material volume has a constant value.  The lower contour, which just touches the feasible set, corresponds to the optimum value Vmin = 360,000 mm2.  The optimal solution is achieved at point A where x1* = 120 mm2 and x2* = 170 mm2.  In this problem, only displacement constraint


is active, i.e., g1(x*) = 0. 

1.3.        Graphical Optimization

The mathematical formulation presented in the previous section can best be solved using numerical methods, which will be described in some details in the following chapters.  However, when the design problem can be formulated in terms of only two design variables, graphical methods can be used to solve the problem as demonstrated by Example 1.2.1.  Graphical methods are not the most efficient methods of solution, because they require a large number of evaluations of the objective functions and constraints, but they help designers gain a valuable understanding of the nature of the design space.  Moreover, because visualizing the design space is a powerful tool for understanding the trade-offs associated with a design problem, graphical methods are often used even when the number of design variables exceeds two.  In that case, we look at special forms or reformulations of the design problem. For example, all of the design variables may be fixed except two that are allowed to vary.

There is a variety of media for performing graphical solutions of an optimization problem, the most traditional being paper and manual drawing tools.  However, most modern engineering software, such as Mathematica and Matlab, provide functions that allow convenient graphical representation of an optimization problem. 

The first step in solving an optimization problem graphically is to visualize the dependence of the objective function on the two design variables.  The standard approach that mapmakers have used for generations is often used for this purpose.  Contour lines as in Figure 1.2.5(b) representing lines with equal value of the function are drawn.  In weather maps, these are usually isotherms (lines of equal temperature) or isobars (lines of equal atmospheric pressures). For graphical optimization, these lines are called function contours.  Function contours can be created in many different ways.  We can take the expression for the function, set it equal to a constant, and solve for one of the variables in terms of the other one.  This will give us an equation for one of the curves.  Repeating the process of other values of the constant completes the contour lines.  Alternatively, we can cover the region of interest with a dense grid of points, calculate the function value at each one of the points, and find by interpolation where the function has a desired value to draw the line associated with that value.  Again, the process is repeated for other values of the function to complete the contour lines.

Once the contour lines are drawn, regions where the objective function is high are denoted in different shadings or colors than regions where the objective function is low.  In a weather map, for example, high temperatures are denoted in red and low temperatures in blue.  The software packages mentioned earlier can draw function contours automatically and include numerous options to change the appearance of the contour map.

If there are no constraints other than side constraints, the plane region defined by the ranges of the design variables used in the graphical representation represents all possible combinations of the two design variables.  This region is typically referred to as the design space.  For constrained problems, parts of the design space are unacceptable.  Therefore, the next step in graphical solution is to draw a boundary line for each of the constraint functions (both inequality or equality constraint) superimposed over the objective function contour plot.  Unlike the objective function, which we do not know its value at optimum, the value of the constraint function along the boundary line is zero for a problem specified in the standard form of Eqs.                                   (1.2.6) or                                          (1.2.9).  The difference between the equality and inequality constraints is in the interpretation of the design space.  Constraints divide the design space into feasible and infeasible parts.  A section of the design space that violates a constraint function is termed infeasible region.  Because the equality constraint requires the value of the constraint function to be identical to zero, any other contour line, hi ≠ 0, which is a line on either side of the hi = 0 line is in the infeasible region.  Hence, the only feasible portion of the two-dimensional space is the line segment corresponding to an equality constraint line.  In the case of inequality constraints, only one side of the gi = 0 line, which correspond to the constraint function values greater than zero (the positive values) is unacceptable.  To graphically indicate the region of the design space that is infeasible, the convention is to draw hatch lines on the side of the constraint that is infeasible. 

Once the feasible and infeasible portions of the design space are identified, the final step is to identify the optimal solution by inspection.  Naturally, if we are trying to minimize the objective function, we will move in the feasible domain to a point that has the smallest objective function value.  This can be accomplished by moving to the contour line with the lowest value of the objective function in the feasible domain.  If that lowest contour line is entirely in the feasible domain, the minimum lies inside the closed domain formed by that contour line and is away from the constraint boundaries.  In this case, none of the inequality constraints dictates the location of the optimal solution; hence, they are referred to as being inactive constraints.  This situation is rarely encountered in structural optimization because most problems have important constraints that are active at the optimum.

If the lowest contour line intersects the boundary of one or more inequality constraints, then some or all of these constraints dictate the location of the optimal solution and are called active constraints.  The active constraints are identically satisfied at the optimum, gi = 0. 

If the determination of the optimal solution via inspection is difficult because of graphical resolution, then there are two possible remedies.  The first one is to zoom in on the region around the place of expected optimum by redefining the range of the two design variables and drawing the objective function contours with finer increments.  This can be done repeatedly until a desired level of resolution in the design variables is achieved. 

The second remedy involves mathematical manipulation of the objective function and the active constraint equations.  If there are two active constraints, then it may be possible to solve the two equations and determine the exact value of the optimal design variables.  If there are more then two active constraints, then any two of the constraints may be picked up for solution, and the solution should satisfy the other active constraints. 

In the case of a single active constraint, a solution involving a two-step process is possible when the objective function and constraints are differentiable.  First, since the constraint is satisfied exactly, the active constraint equation may be solved for one of the design variables in terms of the second one.  Substituting the solution of the constraint equation into the objective function reduces it to a single variable.  Finding the optimum value of a function with a single variable can then be accomplished by equating its derivative with respect to the variable to zero. 

There is one more possibility in case of a single active constraint.  If, upon substitution of the solution of the constraint equation for one of the variables in terms of the other one reduces the objective function to a constant value (rather then reducing it to a function of the other variable), then the objective function contours are parallel to the constraint boundaries.  This implies that all the points along the constraint boundary are optimal solutions.

Finally, if the optimum is unconstrained and the objective function is differentiable, then its exact location may be found by setting the derivatives of the objective function to zero.

We will use the following example of the design of a soft-drink-can to discuss some of the basic concepts associated with graphical optimization described above.


Example 1.3.2

A soft-drink company has a new drink that they plan to market, and through preliminary research it was found that the cost C to produce and distribute a cylindrical can for the drink is approximately



where Vo is the volume in fluid ounces, and S is the surface area of the can in square inches.  They also determined that they can sell the drink in cans ranging from 5 to 15 ounces, and then the price P in cents that can be charged for a can is estimated as




Based on their past experience they will consider only a can with diameter D between 1.5 and 3.5 inches, and their market research has shown that soft-drink cans have to have an aspect ratio of at least 2.0 to be easy to drink from.  That is, the height H of the can has to be at least twice the diameter.


The objective of the company is to design the can so as to maximize their profit from the sales of the soft drink.  Two measures of profit are considered.  One is the profit per can, and the other is the profit per ounce of the drink.  The first measure is more useful if we assume that consumers will buy a fixed number of cans, and the other if we expect consumers to buy a fixed volume of drink.




We first consider the three elements that go into formulating this design example as a mathematical optimization problem (see Section SecOptForm).  First, the design variables that are needed to specify the shape of the can.  In this case, the diameter D and the height H are natural design variables.  However, we could use the radius of the can instead of the diameter, and conceivably even use the volume and surface area as design variables and calculate from them the diameter and aspect ratio of the can.


Next we consider the two objective functions, the profit per can and the profit per ounce, as functions of the design variables. We start by expressing the volume and the surface area in terms of the diameter and the height of the can




where the 1.8 factor is used to convert from cubic inches to fluid ounces.  The first objective function, profit per can pc, may be then written as




We can get an explicit expression for pc in terms of the design variables by substituting P and C into this equation to obtain




However, it is not necessary to obtain such explicit expressions. Instead the collection of equations for substituting P and C constitutes a procedure for calculating pc as a function of D and H that we can denote as pc (D, H).  In fact, if we write a computer program to evaluate pc, using the individual expressions is more convenient, more error free, and simpler to understand for somebody reading the program than using the implicit expression given above.


The second objective function, the profit per ounce, can be obtained by dividing pc by the volume.




We can again substitute the expression for pc and the expression for Vo to obtain an explicit expression for pc (D,H).   However, this is not needed.


Finally, let us consider the expressions for the constraints in this problem.  The first set consists side constraints that are the lower bound and an upper bound for the diameter




Next is the aspect-ratio constraint,




Finally we have the constraints on the volume,




By substituting the expression for the volume, we can rewrite this expression as




Both the side constraints and the aspect ratio constraint are inequality constraints.  We can now formulate the soft-drink-can example in standard format using normalized constraints.  Using the profit per can as objective function, and changing the sign to transform the problem to minimization, we get



The aspect ratio constraint and the volume bound constraints are normalized as


To help in converting the formulation to the standard format, we will write the entire problem with D replaced by x1, H replaced by x2, and the objective function denoted as f.  Then we get






The first step in solving an optimization problem graphically is to visualize the dependence of the objective function on the 2 design variables. Regions where the objective function is high are denoted in different shadings or different colors as compared to regions where the objective function is low.


The following figure Figure 1.3.1 shows a map of the dependence of the objective function (minus of the profit per can) on the diameter D and the height H of the can. The range used for the diameter is based on the side constraints specified.  The range of the height in the figure is based on the condtraint that required the height not ro exceed twice the diameter.  The region where the profit is low (high objective function) is light, while the region where the profit is high (low objective function) is dark. The distance between the successive contour lines correspond to about 1.5 cents in profit.  A quick look at the figure reveals that the profit is high when both the diameter and the height are large and is low when both are small.  This, of course is not surprising, as we would expect to make more profit when we sell a large can compared to a small can.


Figure 1.3.1:  Profit-per-can contours for the soft-drink-can example.


Next figure Figure 1.3.2 shows the contours of the profit per ounce for the same example. Again, light regions correspond to low profit and dark regions to high profit.  This time the most profitable soft-drink-can would appear to be around 2.5 inch diameter and 3 inches of height.  In fact, the figure hints that higher profits could possibly be realized if we could go to heights below 3 inches.  This is an indication of the need to depict the constraints on the map in order to find the most profitable can that satisfies the constraints, or the most profitable feasible can.


Figure 1.3.2: Profit-per-ounce contours for the soft-drink-can example.


Figure 1.3.3 shows the constraints boundaries superimposed on the profit contours.  The constraint that the diameter must range between 1.5 and 3.5 inches is accomodated by the vertical boundaries of the map.  The minimum volume constraint (5 ounces) and the maximum volume constraint (15 ounces) are two curves that are almost parallel to the cost contours.  Finally, the aspect ratio constraint, requiring that the height be at least twice the diameter is the straight line connecting the two corners of the map.  The black line for each constraint depicts the boundary and was drawn that way. For example, the boundary of the aspect-ratio constraint, g1, is the line H = 2D drawn in the figure.  Similarly, the minimum volume constraint, g2, gives rise to the boundary 11.46  = H D2, and the maximum volume constraint, g3, gives rise to the boundary 34.38 = H D2.  The convention is to draw hatch lines on the side of the constraint that is infeasible.  In the figure, thick red lines indicate the infeasible side of the constraint boundaries.


Figure 1.3.3: Profit-per-can contours and constraints for soft-drink-can example.


The feasible domain in Figure 1.1.1 is the pentagon with two slightly curved boundaries in the upper left corner of the design space.  Actually, the top line H = 7 in. is not a boundary of the feasible domain, as there is no constraint on a maximum value for H.  However, the region for large values of H is light, denoting low profit, so that there is no reason to extend the figure in that direction.


The other four boundaries correspond to four constraints.  Designs to the left of the D = 1.5 in. line have diameters which are too small.  Designs in the bottom left corner of the design space do not have enough (5 ounces) volume.  However, these two constraints are not important, because they bound off regions of design space where profit is low anyhow.  Designs in the bottom right half of the design space are too squat, they do not satisfy the aspect ratio constraint, and designs in the top right corner are too big (more than 15 ounces).


The last two constraints certainly limit the profit; they prevent access to the most profitable (darkest regions) of the design space. It is clear from the figure that the optimum design is found at the intersection of these two constraints, that is for H = 2D and Vo = 15 ounces.  We can find this intersection easily by substituting H = 2D into the expression for the volume, and setting it equal to 15 ounces to get



Solving this equation we get D = 2.581 in., which corresponds to a height H = 5.162 in. By using equations for the price-per-can, the cost of a can, and profit-per-can, we find that the solution corresponds to selling a can for 33 cents, a cost of 17.23 cents, and a profit of 15.77 cents per can.  


The design obtained above maximizes the profit per can of the soft drink.  However, this design does not correspond to the largest possible profit per ounce.   The constrained design space for the profit per ounce of the soft drink is shown in the following figure, Figure 1.3.4.


Figure 1.3.4: Constrained design space for the profit-per-ounce of soft drink.


As opposed to the previous solution, where two constraints were active, the darker region of the design space, where the profit is the largest is, in the middle lower section of the design space an is limited by only the aspect ratio constraint.  In fact, the contour line bounding the darkest section of the design space almost touches the aspect ration constraint, indicating the possible location of the solution.   Hence, in this case only the aspect ratio constraint is active, H = 2D.  Substituting this into the equation for profit-per-ounce, we get




Now, we have a cost function with one variable.  By taking the derivative of this function with respect to the can diameter and equating zero, we can find the solution to be D = 2.036 in., which corresponds to a height H = 4.072 in. 

1.4.         Exercises

1. A simply supported beam, 40 feet long, need to carry a uniformly distributed load of w = 1500 lb/ft.  You would like to design a minimum weight beam with rectangular cross section (with b denoting the width and h the height of the cross section).  Consider two failure modes, one in shear, and one bending.  The maximum shear stress in the beam is


where A = b h is the cross-sectional area, and L=40 ft is the length.  The maximum bending stress is


You can choose to make the beam from steel, aluminum, or wood.  The allowable shear stresses are 55,000 psi, 32,000 psi, and 1,000 psi, respectively.  The allowable bending stresses are 100,000 psi, 60,000 psi, and 5,000 psi, respectively.  The densities of the three materials are 0.28 lb/in3, 0.1 lb/in3, and 0.016 lb/in3.  Additionally, we will run into problems of stability if the aspect ratio of the cross section is too high.  Consequently, we do not permit the ratio between b and h to be larger than 5 or smaller than 0.2.

(a) Formulate the problem of designing the lightest weight steel beam in standard format, and using normalized constraints.

(b) Formulate the problem of designing the lightest weight beam using any of the three materials as a single optimization problem.

Hint: You can define three zero/one design variables that indicate whether or not you use any of the three candidate material, xs = 1 if you use steel and xs = 0 if you do not.  Similarly, xa = 1 if you use aluminum, and xa = 0 if you do not, and for xw.  Then the density of the material you use may be written as

r = 0.28 xs + 0.1 xa + 0.016 xw ,

with similar expressions for the allowable shear stress and bending stresses.  You will need a constraint that will ensure that you use only a single material (that is that only one of the three variable can be equal to one), and this can be written as a simple equality constraint in terms of xs, xa, and xw.


2. Solve graphically the beam design problem of Exercise 1.  For each material, draw a map of weight contours with the boundaries of the feasible domain superimposed, and each constraint clearly labeled.



1.5.        References


Barnett, R.L., “Minimum Weight Design of Beams for Deflection,"  J. EM Division, ASCE,
Vol. EM1, 1961, pp. 75-95.


Braibant, V., Fleury, C., and Beckers, P., “Shape Optimal Design: An Approach Matching
C.A.D. and Optimization Concepts," Report SA-109, Aerospace Laboratory of the University of Liege, Belgium, 1983.


Fox, C., An Introduction to the Calculus of Variations, Dover Publications, 1987.


Prasad, B, and Haftka, R. T., “Optimal Structural Design with Plate Finite Elements," ASCE J. Structural Division, 105, pp. 2367-2382, 1979.