In this chapter, we describe the interactive design optimization process. The role of designer interaction and algorithms for interaction are described, especially for advanced users who would prefer to interact with the optimization process. Desired interactive capa- bilities and decision-making facilities are discussed and simple examples are used to demon- strate their use in the design process. These discussions essentially lay out the specifications for an interactive design optimization software. 13.1 Role of Interaction in Design Optimization 13.1.1 What Is Interactive Design Optimization? In Chapter 1 we described the engineering design process. The differences between the conventional and the optimum design process were explained. The optimum design process requires sophisticated computational algorithms. However, most algorithms have some uncertainties in their computational steps. Therefore, it is sometimes prudent to interactively monitor their progress and guide the optimum design process. Interactive design optimiza- tion algorithms are based on utilizing the designer’s input during the iterative process. They are in some sense open-ended algorithms in which the designer can specify what needs to be done depending on the current design conditions. They must be implemented into inter- active software that can be interrupted during the iterative process and that can report the status of the design to the user. Relevant data and conditions must be displayed at the designer’s command at a graphics workstation. Various options should be available to the designer to facilitate decision making and change design data. It should be possible to restart or terminate the process. With such facilities, designers have complete control over the design optimization process. They can guide it to obtain better designs and ultimately the best design. It is clear that for interactive design optimization, proper algorithms must be implemented into highly flexible and user-friendly software. It must be possible for the designer to inter- act with the algorithm and change the course of its calculations. We describe later in Section 13.2 algorithms that are suitable for designer interaction. Figure 13-1 is a conceptual flow diagram for the interactive design optimization process. It is a modification of Fig. 1-2 in which an interactive block has been added. The designer interacts with the design process through this block. We shall discuss the desired interactive capabilities and their use later in this chapter. 13.1.2 Role of Computers in Interactive Design Optimization As we have discussed earlier, the conventional trial-and-error design philosophy is chang- ing with the emergence of fast computers and computational algorithms. The new design methodology is characterized by the phrase model and analyze. Once the design problem is properly formulated, numerical methods can be used to optimize the system. The methods are iterative and generate a sequence of design points before converging to the optimum solu- tion. They are best suited for computer implementation to exploit the speed of computers for performing repetitive calculations. It is extremely important to select only robust optimization algorithms for practical appli- cations. Otherwise, failure of the design process will undoubtedly result in the waste of com- puter resources and, more importantly, the loss of the designer’s time and morale. An optimization algorithm involves a limiting process, because some parameters go to zero or infinity as the optimum design is approached. The representation of such limiting processes is difficult in computer implementation as it may lead to underflow or overflow. In other words, the limiting processes can never be satisfied exactly on a computer and quantities such as zero and infinity must be redefined as very small and large numbers, respectively, on the computer. These quantities are relative and machine-dependent. 434 INTRODUCTION TO OPTIMUM DESIGN Often, the proof of convergence or rate of convergence of an iterative optimization algo- rithm is based on exact arithmetic and under restrictive conditions. Thus, the theoretical behavior of an algorithm may no longer be valid in practice because of inexact arithmetic causing round-off and truncation errors in computer representation of numbers. This discus- sion highlights the fact that proper coding and interactive monitoring of theoretically con- vergent algorithms are equally important. 13.1.3 Why Interactive Design Optimization? The design process can be quite complex. Often the problem cannot be stated in a precise form for complete analysis and there are uncertainties in the design data. The solution to the problem need not exist. On many occasions, the formulation of the problem must be devel- oped as part of the design process. Therefore, it is neither desirable nor useful to optimize an inexact problem to the end in a batch environment. It would be a complete waste of valu- able resources to find out at the end that wrong data were used or a constraint was inadver- tently omitted. It is desirable to have an interactive algorithm and software capable of designer interaction. Such a capability can be extremely useful in a practical design envi- ronment because not only can better designs be obtained, but more insights into the problem behavior can be gained. The problem formulation can be refined, and inadequate and absurd designs can be avoided. We shall describe some interactive algorithms and other suitable capabilities to demonstrate the usefulness of designer interaction in the design process. Interactive Design Optimization 435 Identify: (1) Design variables (2) Cost function to be minimized (3) Constraints that must be satisfied Collect data to describe the system Estimate initial design Analyze the system Check the constraints Stop No Yes Does the design satisfy convergence criteria? Change the design using an optimization method Interactive session FIGURE 13-1 Interactive optimum design process. 13.2 Interactive Design Optimization Algorithms It is clear from the preceding discussion that for a useful interactive capability, proper algo- rithms must be implemented into well-designed software. Some optimization algorithms are not suitable for designer interaction. For example, the constrained steepest descent method of Section 10.5 and the quasi-Newton method of Section 11.4 are not suitable for the inter- active environment. Their steps are in a sense closed-ended allowing little opportunity for the designer to change course from the iterative design process. However, it turns out that the QP subproblem and the basic concepts discussed there can be utilized to devise algo- rithms suitable for the interactive environment. We shall describe these algorithms and illus- trate them with examples. Depending on the design condition at the current iteration, the designer may want to ask any of the following four questions: 1. If the current design is feasible but not optimum, can the cost function be reduced by g percent? 2. If the starting design is infeasible, can a feasible design be obtained at any cost? 3. If the current design is infeasible, can a feasible design be obtained without increasing the cost? 4. If the current design is infeasible, can a feasible design be obtained with only d percent penalty on the cost? We shall describe algorithms to answer these questions. It will be seen that the algorithms are conceptually quite simple and easy to implement. As a matter of fact, they are modifica- tions of the constrained steepest descent (CSD) and quasi-Newton methods of Sections 10.5 and 11.4. It should also be clear that if interactive software with commands to execute the foregoing steps is available, the designer can actually use the commands to guide the process to successively better designs and ultimately an optimum design. 13.2.1 Cost Reduction Algorithm A subproblem for the cost reduction algorithm can be defined with or without the approxi- mate Hessian H. Without Hessian updating, the problem is defined in Eqs. (10.25) and (10.26) and, with Hessian updating, it is defined in Eqs. (11.48) to (11.50). Although Hessian up- dating can be used, we shall define the cost reduction subproblem without it to keep the discussion and the presentation simple. Since the cost reduction problem is solved from a feasible or almost feasible point, the right side vector e in Eq. (10.26) is zero. Thus, the cost reduction QP subproblem is defined as minimize (13.1) subject to (13.2) (13.3) The columns of matrices N and A contain gradients of equality and inequality constraints, respectively, and c is the gradient of the cost function. Equation (13.2) gives the dot product of d with all the columns of N as zero. Therefore, d is orthogonal to the gradients of all the equality constraints. Since gradients in the matrix N are normal to the corresponding con- straint surfaces, the search direction d lies in a plane tangent to the equality constraints. The right side vector b for the inequality constraints in Eq. (13.3) contains zero elements corre- sponding to the active constraints and positive elements corresponding to the inactive con- Ad b T £ Nd 0 T = f TT =+cd dd05. 436 INTRODUCTION TO OPTIMUM DESIGN straints. If an active constraint remains satisfied at the equality (i.e., a (i) ·d = 0), the direction d is in a plane tangent to that constraint. Otherwise, it must point into the feasible region for the constraint. The QP subproblem defined in Eqs. (13.1) to (13.3) can incorporate the potential con- straint strategy as explained in Section 11.1. The subproblem can be solved for the cost reduc- tion direction by any of the available subroutines cited in Section 11.2. In the example problems, however, we shall solve the QP subproblem using KKT conditions. We shall call this procedure of reducing cost from a feasible point the cost reduction (CR) algorithm. After the direction has been determined, the step size can be calculated by a line search on the proper descent function. Or, we can require a certain reduction in the cost function and determine the step size that way. For example, we can require a fractional reduction g in the cost function (for a 5 percent reduction, g = 0.05), and calculate a step size based on it. Let a be the step size along d. Then the first-order change in the cost using a linear Taylor’s expansion is given as a|c·d|. Equating this to the required reduction in cost |g f |, the step size is calculated as (13.4) Note that c·d should not be zero in Eq. (13.4) to give a reasonable step size. The cost reduc- tion step is illustrated in Example 13.1. a g = ◊ f cd Interactive Design Optimization 437 EXAMPLE 13.1 Cost Reduction Step Consider the design optimization problem minimize subject to From the feasible point (4, 4), calculate the cost reduction direction and the new design point requiring a cost reduction of 10 percent. Solution. The constraints can be written in the standard form as The optimum solution for the problem is calculated using the KKT conditions as xu x*,; * ,,,; * .= () = ()() =-6 3 17 16 0 0 55 5f gx 42 0=- £ gx 31 0=- £ gxx 2 1 12 12 2100=+ () -£. gxx 1 1 3 12 10 0=- () -£. xx 12 0, £ xx 12 212+£ xx 12 3-£ f xxx x xxx () =- + - - 1 2 12 2 2 12 345106. 438 INTRODUCTION TO OPTIMUM DESIGN At the given point (4, 4), Therefore, constraint g 2 is active, and all the others are inactive. The cost function is much larger than the optimum value. The constraints for the problem are plotted in Fig. 13-2. The feasible region is identified as 0ABC. Several cost function contours are shown there. The optimum solution is at the point B (6, 3). The given point (4, 4) is identified as D on the line B–C in Fig. 13-2. The gradients of cost and constraint functions at the point D (4, 4) are calculated as These gradients are shown at point D in Fig. 13-2. Each constraint gradient points to the direction in which the constraint function value increases. Using these quantities, the QP subproblem of Eqs. (13.1) to (13.3) is defined as minimize subiect to The solution for the QP subproblem using KKT conditions or the Simplex method of Section 10.4 is At the solution, only the first constrain is active having a positive Lagrange multiplier. The direction d is shown in Fig. 13-2. Since the second constraint is inactive, a (2) ·d must be negative according to Eq. (13.3) and it is (-0.625). Therefore, direction d du=- - () = ( ) 05 35 435 0 0 0., . ; ., , , 1 3 1 3 1 12 1 6 1 2 10 01 1 0 4 4 - - - È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ È Î Í ˘ ˚ ˙ £ È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ d d fdddd=- + () ++ () 14 18 0 5 12 1 2 2 2 . a 4 01 () =- () , a 3 10 () =- () , a 2 1 12 1 6 () = () , a 1 1 3 1 3 () =- () , =- () 14 18, c = +- () 2 3 10 3 9 6 12 12 xx xx, g 4 40 0=- < () . inactive g 3 40 0=- < () . inactive g 2 0= () active g 1 10 0=- < () . inactive f 44 24, () =- Interactive Design Optimization 439 points toward the feasible region with respect to the second constraint, which can be observed in Fig. 13-2. The step size is calculated from Eq. (13.4) based on a 10 percent reduction (g = 0.1) of the cost function as Thus, the new design point is given as which is quite close to point D along direction d. The cost function at this point is calculated as which is approximately 10 percent less than the one at the current point (4, 4). It may be checked that all constraints are inactive at the new point. Direction d points into the feasible region at point D as can be seen in Fig. 13-2. Any small move along d results in a feasible design. If the step size is taken as 1 (which would be obtained if the inaccurate line search of Section 11.3 was performed), then the new point is given as (3.5, 0.5), which is marked as E in Fig. 13-2. At point E, constraint g 1 is active and the cost function has a value of -29.875, which is smaller than the previous value of -26.304. If we perform an exact line search, then a is com- puted as 0.5586 and the new point is given as (3.7207, 2.0449)—identified as point E¢ in Fig. 13-2. The cost function at this point is -39.641, which is still better than the one with step size as unity. f x 1 26 304 () () =- . x 1 4 4 03 7 05 35 3 979 3 850 () = È Î Í ˘ ˚ ˙ + - - È Î Í ˘ ˚ ˙ = È Î Í ˘ ˚ ˙ . . . . . a = - () - () ◊- - () = 01 24 14 18 0 5 3 5 03 7 . ,.,. . 7 7 8 9 10 11 12 13 14 15 30 6 6 Cost contours 5 5 4 4 3 3 2 2 1 1 0 0 –1 –2 –3 x 2 x 1 0 –30 –52 –56 –60 –70 C B a (4) a (3) a (2) c d a (1) A g 2 = 0 g 1 = 0 x 1 + 2x 2 = 12 x 1 – x 2 = 3 D E E¢ FIGURE 13-2 Feasible region for Example 13.1. Cost reduction step from point D. Example 13.2 illustrates the cost reduction step with potential constraints. 440 INTRODUCTION TO OPTIMUM DESIGN EXAMPLE 13.2 Cost Reduction Step with Potential Constraints For Example 13.1, calculate the cost reduction step by considering the potential inequality constraints only. Solution. In some algorithms, only the potential inequality constraints at the current point are considered while defining the direction finding subproblem, as discussed previously in Section 11.1. The direction determined with this subproblem can be dif- ferent from that obtained by including all constraints in the subproblem. For the present problem, only the second constraint is active (g 2 = 0) at the point (4, 4). The QP subproblem with this active constraint is defined as minimize subject to Solving the problem by KKT optimality conditions, we get Since the Lagrange multiplier for the constraint is zero, it is not active, so d =-c is the solution to the subproblem. This search direction points into the feasible region along the negative cost function gradient direction, as seen in Fig. 13-2. An appro- priate step size can be calculated along the direction. If we require the constraint to remain active (i.e., d 1 /12 + d 2 /6 = 0), then the solu- tion to the subproblem is given as This direction is tangent to the constraint, i.e., along the line D–B in Fig. 13-2. d =- () =-18 4 9 2 52 8., . ; .u d =- () =14 18 0,;u 1 12 1 1 6 2 0dd+£ fdddd=- + () ++ () 14 18 0 5 12 1 2 2 2 . 13.2.2 Constraint Correction Algorithm If constraint violations are very large at a design point, it may be useful to find out if a fea- sible design can be obtained. Several algorithms can be used to correct constraint violations. We shall describe a procedure that is a minor variation of the constrained steepest descent method of Section 10.5. A QP subproblem that gives constraint correction can be obtained from Eqs. (10.25) and (10.26) by neglecting the term related to the cost function. In other words, we do not put any restriction on the changes in the cost function, and define the QP subproblem as minimize (13.5) subject to (13.6) (13.7) Ad b T £ Nd e T = f T = 05. dd A solution to the subproblem gives a direction with the shortest distance to the constraint boundary (linear approximation) from an infeasible point. Equation (13.5) essentially says: find a direction d having the shortest path to the linearized feasible region from the current point. Equations (13.6) and (13.7) impose the requirement of constraint corrections. Note that the potential set strategy as described in Section 11.1 can also be used here. After the direc- tion has been found, a step size can be determined to make sure that the constraint violations are improved. We shall call this procedure the constraint correction (CC) algorithm. Note that constraint correction usually results in an increase in cost. However, there can be some unusual cases where constraint correction is also accompanied by a reduction in the cost function. The constraint correction step is illustrated in Example 13.3. Interactive Design Optimization 441 EXAMPLE 13.3 Constraint Correction Step For Example 13.1, calculate the constraint correction step from the infeasible point (9, 3). Solution. The feasible region for the problem and the starting point (F) are shown in Fig. 13-3. The constraint and cost gradients are also shown there. At the point F (9, 3), the following data are calculated: g 3 90=- < () inactive g 2 025 0=> () . violation g 1 10=> () violation f 9 3 67 5,. () =- FIGURE 13-3 Feasible region for Example 13.3. Constraint correction and constant cost steps from point F; constant cost step from point I. 7 7 9 10 12 13 14 15 30 6 6 Cost contours 5 5 4 4 3 3 2 2 1 1 0 0 –1 –2 –3 x 2 x 1 0 –30 –52 –56 –60 C J a (3) c a (1) a (4) a (4) a (3) a (2) a (1) a (2) G L I A F H K d d c g 2 = 0 g 1 = 0 x 1 + 2x 2 = 12 x 1 – x 2 = 3 B –70 8 11 13.2.3 Algorithm for Constraint Correction at Constant Cost In some instances, the constraint violations are not very large. It is useful to know whether a feasible design can be obtained without any increase in the cost. This shall be called a constant cost subproblem, which can be defined by adding another constraint to the QP subproblem given in Eqs. (13.5) to (13.7). The additional constraint simply requires the current linearized cost to either remain constant or decrease; that is, the linearized change in cost (c·d) be nonpositive, which is expressed as (13.8) The constraint imposes the condition that the direction d be either orthogonal to the gradi- ent of the cost function (i.e., c·d = 0), or make an angle between 90 and 270° with it (i.e., c·d < 0). We shall see this in the example problem discussed later. cd◊£0 442 INTRODUCTION TO OPTIMUM DESIGN and gradients of the constraints are the same as in Example 13.1. Cost and constraint gradients are shown at point F in Fig. 13-3. Thus, the constraint correction QP sub- problem of Eqs. (13.5) to (13.7) is defined as minimize subject to Using the KKT necessary conditions, the solution for the QP subproblem is given as Note that the shortest path from Point F to the feasible region is along the line F–B, and the QP subproblem actually gives this solution. The new design point is given as which is point B in Fig. 13-3. At the new point, constraints g 1 and g 2 are active, and g 3 and g 4 are inactive. Thus, a single step corrects both violations precisely. This is due to the linearity of all the constraints in the present example. In general several itera- tions may be needed to correct the constraint violations. Note that the new point actu- ally represents the optimum solution. x 1 9 3 3 0 6 3 () = È Î Í ˘ ˚ ˙ + - È Î Í ˘ ˚ ˙ = È Î Í ˘ ˚ ˙ du=- () = () 30 61200,; ,,, 1 3 1 3 1 12 1 6 1 2 10 01 1 025 9 3 - - - È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ È Î Í ˘ ˚ ˙ £ - - È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ d d . fdd=+ () 05 1 2 2 2 . c =- - () 16, g 4 30=- < () inactive If Inequality (13.8) is active (i.e., the dot product is zero, so d is orthogonal to c), then there is no change in the linearized cost function value. However, there may be some change in the original cost function due to nonlinearities. If the constraint is inactive, then there is actually some reduction in the linearized cost function along with correction of the con- straints. This is a desirable situation. Thus, we observe that a constant cost problem is also a QP subproblem defined in Eqs. (13.5) to (13.8). It seeks a shortest path to the feasible region that either reduces the linearized cost function or keeps it unchanged. We shall call this procedure the correction at constant cost (CCC) algorithm that is illustrated in Examples 13.4 and 13.5. Note that the constant cost QP subproblem can be infeasible if the current cost function contour does not intersect the feasible region. This can happen in practice, so a QP sub- problem should be solved properly. If it turns out to be infeasible, then the constraint of Eq. (13.8) must be relaxed, and the linearized cost function must be allowed to increase to obtain a feasible point. This will be discussed in the next subsection. Interactive Design Optimization 443 EXAMPLE 13.4 Constraint Correction at Constant Cost For Example 13.3, calculate the constant cost step from the infeasible point (9, 3). Solution. To obtain the constant cost step from point F in Fig. 13-3, we impose an additional constraint of Eq. (13.8) as (c·d) £ 0 on the QP subproblem given in Example 13.3. Substituting for c, the constraint is given as (13.9) which imposes the condition that the linearized cost function either remain constant at -67.5 or decrease further. From the graphical representation for the problem in Fig. 13-3, we observe that the cost function value of -67.5 at the given point (9, 3) is below the optimum cost of -55.5. Therefore, the current cost function value represents a lower bound on the optimum cost function value. However, the linearized cost func- tion line, shown as G–H in Fig. 13-3, intersects the feasible region. Thus, the QP subproblem of Example 13.3 with the preceding additional constraint has feasible solutions. The inequality of Eq. (13.8) imposes the condition that direction d be either on the line G–H (if the constraint is active) or above it (if the constraint is inactive). In case it is inactive, the angle between c and d will be between 90 and 270°. If it is below the line G–H, it violates Inequality (13.8). Note that the shortest path from F to the feasible region is along the line F–B. But this path is below the line G–H and thus not feasible for the preceding QP subproblem. Solving the problem using KKT conditions, we obtain the solution for the preced- ing QP subproblem as Thus the new point is given as At the new point (G in Fig. 13-3), all the constraints are inactive except the second one (g 2 ). The constant cost condition of Eq. (13.8) is also active, which implies that x = () =-45375 346., . .with f du=- () = () 4 5 0 75 0 83 25 0 0 2 4375.,. ; , . ,,,. £dd 12 60 [...]... Spring Design Problem in Batch Environment Iter Max vio Conv parm Cost d D N 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 9.6 179 1E-01 2.48814E+00 6.89 874 E-01 1.60301E-01 1.23963E-02 1. 973 57E-05 9.25486E-06 2. 271 39E-04 5.14338E-03 8 .79 064E-02 9. 070 17E-02 7. 2 070 5E-02 6 .74 501E-02 2.8 179 2E-02 1. 578 25E-02 5.85935E-03 1.49687E-04 0.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.42246E-01 8.92216E-03 6. 477 93E-03... 6. 477 93E-03 3.21448E-02 7. 68889E-02 8.80280E-02 8. 870 76E-02 6.66881E-02 7. 90647E-02 6.86892E-02 4.50482E-02 1.94256E-02 4.93063E-03 2.69244E-05 9 .76 924E-08 2.08000E-01 1.30122E-02 1.22613E-02 1.2 079 8E-02 1 .72 814E-02 1 .76 475 E-02 1 .76 248E-02 1 .75 088E-02 1.69469E-02 1.44839E-02 1.31958E-02 1.26517E-02 1.22889E-02 1.24815E-02 1.25465E-02 1.26254E-02 1.2 677 2E-02 1.2 678 7E-02 2.0000E-01 5.0000E-02 5 .74 91E-02 6.2522E-02... 2.0000E-01 5.0000E-02 5 .74 91E-02 6.2522E-02 6.8435E-02 6. 877 0E-02 6. 873 2E-02 6.8542E-02 6 .76 35E-02 6.3848E-02 6.0328E-02 5 .75 19E-02 5.4 977 E-02 5.3497E-02 5.2424E-02 5. 179 0E-02 5.1698E-02 5.1699E-02 1.3000E+00 1.3000E+00 9. 274 3E-01 7. 7256E-01 9.1481E-01 9.2 373 E-01 9.2208E-01 9.1385E-01 8 .74 86E-01 7. 170 6E-01 5.9653E-01 5.1028E-01 4.3814E-01 4.0092E-01 3 .74 13E-01 3.5896E-01 3.5692E-01 3.5695E-01 2.0000E+00... 2.0460E+00 2. 078 2E-00 2.2346E+00 2.9549E+00 4. 078 1E+00 5.4942E+00 7. 279 8E+00 8. 878 1E+00 1.0202E+01 1.1113E+01 1.1289E+01 1.1289E+01 Constraint activity No Active Value Lagr mult 1 2 3 4 Yes Yes No No -4.66382E-09 -2.46286E-09 -4.0 479 2E+00 -7. 275 68E-01 1. 077 17E-02 2.44046E-02 0.00000E+00 0.00000E+00 Design variable activity No Active Design Lower Upper Lagr mult 1 2 3 Lower Lower No 5.16987E-02 3.56950E-01... 6.89 874 E-01 1.60301E-01 3 .70 554E-01 5.06054E-01 0.00000E+00 3.53358E-02 4.24950E-04 1.08957E-04 0.00000E+00 l.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1.00000E+00 1. 676 23E-02 1. 676 23E-02 1. 676 23E-02 1. 676 23E-02 5.49055E-05 2.08000E-01 1.30122E-02 1.22613E-02 1.2 079 8E-02 1.03315E-02 7. 96802E-03 1. 473 52E-02 1.19085E-02 1. 272 98E-02 1. 272 90E-02 1. 273 00E-02 2.0000E-01 5.0000E-02 5 .74 91E-02... close to the true optimum However, the design point is somewhat different It turns out that there are several near optimum designs in the neighborhood of the true optimum for this example problem 456 INTRODUCTION TO OPTIMUM DESIGN TABLE 13-3 Interactive Solution Process for the Spring Design Problem Iter Algor Max vio Conv parm Cost d D N 1 2 3 4 5 6 7 8 9 10 11 CC CC CC CC CC CC CR CCC CCC CCS CR 9.6 179 1E-01... optimization can also be handled IDESIGN has several facilities that permit the engineer to interact with and control the optimization process The designer can backtrack to any previous design or manually input 450 INTRODUCTION TO OPTIMUM DESIGN a new trial design Design information can be displayed in a variety of ways or represented in graphs The system has been designed to accommodate both experienced... It is also useful to monitor the optimum process through the interactive session Histories of the cost function, constraint functions, design variables, maximum constraint violation, and convergence parameter should be monitored When these histories are graphically dis- 448 INTRODUCTION TO OPTIMUM DESIGN played, they can be of great help in certain cases of decision making If the design process is... much Design variable 2 Interactive Design Optimization 4 57 11 10 Design variable value 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Iteration no FIGURE 13-5 History of design variables for the spring design problem •, d; ᭺, D; ᭝, N 3.0 Maximum violation 2.5 2.0 1.5 1.0 0.5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Iteration no FIGURE 13-6 History of the maximum constraint violation for the spring design. .. the spring design problem Using this graph, we can locate feasible designs For example, designs after iteration seven are feasible Designs at all previous iterations had some violation of constraints 458 INTRODUCTION TO OPTIMUM DESIGN 0.30 Cost function 0.25 0.20 0.15 0.10 0.05 0.00 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 12 13 Iteration no FIGURE 13 -7 History of the cost function for the spring design problem . = () =-45 375 346., . .with f du=- () = () 4 5 0 75 0 83 25 0 0 2 4 375 .,. ; , . ,,,. £dd 12 60 444 INTRODUCTION TO OPTIMUM DESIGN the direction d is orthogonal to the cost gradient vector c. As. with potential constraints. 440 INTRODUCTION TO OPTIMUM DESIGN EXAMPLE 13.2 Cost Reduction Step with Potential Constraints For Example 13.1, calculate the cost reduction step by considering the potential inequality. with commands to execute the foregoing steps is available, the designer can actually use the commands to guide the process to successively better designs and ultimately an optimum design. 13.2.1