Materials Science and Engineering Handbook Part 5 docx

design, shown in Fig. 2, meets the requirements if it is interpreted colloquially. If the requirement is "Load2 shall be ON if and only if switch1 is ON," then the design indicated is wrong because load1 can also be ON if switch2 and switch3 and switch4 are ON. This is a trivial example of an error due to a "sneak path." Today most requirements are not stated with the precision that distinguishes "if then" from "if and only if then." Because the requirement is often ambiguous, the circuit paths that implement the "if then" form can be highlighted by a heuristic "sneak" error detection process or tool as a possible error to be examined more closely by the designer. A circuit in Fig. 2 is an example of a common sneak pattern, which occurs when two initially independent circuits of switches and loads are drawn near each other on the page, sharing a common power source at the top and a common power return at the bottom, and then a design change introduces a switched path between these previously independent circuits. This is called an "H" pattern because of its resemblance to a letter "H." Errors like these are easy to make in large electromechanical systems with incremental requirements and numerous switches and relays. Fig. 2 An automotive example of a "sneak path" Design errors of this kind can be detected by formal methods where both the requirements and the design are described in languages with semantics that are very well defined. An algorithmic process can then be used to prove whether or not the design requirement is met by the design. This algorithmic process can be automated. There is some progress in this area, but it has been difficult to express requirements in a language that is both precise enough to be mechanically compared to the design and clear enough to discuss with the customer. Likewise, the design language must be precise while not interfering with the creativeness inherent in most good design. References cited in this section 5. L.W. Nagel, SPICE2: "A Computer Program to Simulate Semiconductor Circuits," Electronic Research Laboratory Report ERL-M520, cited in Semiconductor Device Modeling with SPICE, P. Antognetti and G. Massobrio, Ed., McGraw-Hill, 1988 6. V.P. Nelson et al., Digital Logic Circuit Analysis and Design, Prentice Hall, 1974 Computer-Aided Electrical/Electronic Design Shaun S. Devlin, Ford Motor Company Physical Phase Printed Circuit Board Layout The determination of the placement of components and the routing of conductors between them is one of the most important computer-aided capabilities available to the engineer. The algorithms to make the process more automatic have been studied for many years. The process has a blend of electrical and mechanical/geometric aspects, and there are several logical steps. Choice of Board Material, Outline, and Number of Layers. Most printed circuit boards are made of phenolic. However, unusual requirements of temperature range, thermal dissipation, strength, or dielectric constant may indicate other materials. Although the temperature range may be known a priori, the details of the other considerations may require that the material be chosen before the corresponding analysis is made. For example, the magnitude and location of heat-generating components on a board is known only after placement and simulation. Likewise, the contribution of the conductor to the spreading of heat is determined by the layout. Hence, it may be necessary to run a thermal analysis of the board with all components on it and in an enclosure for several choices of board material. Differential thermal expansion has caused reliability problems, particularly with surface-mounted components. The board size and outline are determined by the size of the circuit to be mounted but are often constrained by the size of the enclosure and the connector arrangement. If the circuit is too large to fit on one board in the chosen enclosure and the partitioning of circuit elements is not obvious (i.e., dictated by a bus architecture), then computer aids are sometime used to allocate the circuit to multiple boards, minimizing the interconnect. The cost of a board rises steeply with the number of layers. Hence, it is desirable to try to lay out the board with a low number of layers and then increase the number of layers if it is not feasible to route. Feasibility is conditional on how much manual routing is acceptable in the local engineering process. A layout tool that can minimize the need to try several alternatives is very desirable. Placement of components must account for conductor lengths, thermal dissipation, insertion feasibility, and many other factors. Each of these considerations may be the subject of analysis, and while an optimal set of locations can be calculated for one criterion, no tools take into account multiple considerations. A layout system must first have a library of the "footprints" (a projection of the part outline, including pins on the board, and preferably with recommended pad geometry and drill hole patterns) for each part in the proposed design. A comprehensive layout system provides predesigned drill hole patterns and corresponding electrical pads for each component in the design library. A more comprehensive library will also include the constraints on intercomponent spacing imposed by the insertion process, whether automatic or human. The library system must certainly allow the addition of new components. It is desirable but much more difficult to be able to add new aspects or attributes of the components when those design attributes become important. For example, if the current library system provides for storing only part shapes that are parameterized by three or four numbers (e.g., cylinders, parallelepipeds, and pyramids), it may be difficult to extend it to be able to describe more complex shapes such as transformers, heat sinks, and other shapes requiring many more parameters, if they can be described parametrically at all. Similarly, a library of component models of geometry for manufacturing and electrical/thermal simulation may be difficult to extend for use with a vibration analysis of a populated board. Hence these possibly future needs should be considered when acquiring a new computer-aided electrical design system. Routing of Conductors. The conductors must be routed between the appropriate pins on the placed components according to the intent expressed in the netlist (the set of sets of interconnected components and connector pins) but subject to the constraints of the size of the board, minimum widths and spacings, and the number of layers available. Routing of the conductive strips is both the most tedious hand process and the one most investigated theoretically (Ref 7). The netlist provides the connectivity desired. The router attempts to provide that connectivity subject to the constraints of the number of layers and the kind of vias (interlayer connections) that the manufacturing process allows. Interlayer connections should be minimized because they tend to be expensive. Modern routing tools can fully route average boards but may require manual assistance with particularly large designs (large number of component pins per unit area). All tools allow manual routing of difficult or special cases. Systems Interconnected with Wire and Cable. Electrical/electronic equipment often requires wiring to interconnect circuit boards and other electrical equipment (sensors, switches, motors, etc.). If the equipment in which the electronics is housed is mass produced, the wiring is prebuilt and installed with the electronic boards or modules in the larger system. Examples include a complete computer workstation, a telephone switch, or an automobile. If the product is manufactured in low volume, the wiring is usually installed wire by wire and connectors are attached during the installation process. These two applications have many design similarities but the manufacturing techniques are very different. Large electrical/electronic systems often consist of printed circuit boards interconnected by more or less organized wiring within a cabinet, vehicle, building, or larger complex. The process of using the netlist as the expression of the desired connectivity is common. The process of determining the routing of the wiring in three-dimensional space is subject to many nonquantifiable constraints related to installation and serviceability. The physical problem of routing individual wires or bundles of wires in three-dimensional space is very similar to the problem of routing piping. In fact, several systems simplify the problem to routing conduits (pipe to protect wiring) or channels (imaginary conduit). Initially only the centerline of the channel path is determined, and the diameter is determined after the number of wires and their diameter are fixed. The capability to use the lengths, spacings, and curvatures defined in the routing process in a calculation of the resulting "parasitic" remittances, capacitances, and inductances is not available in any commercial product in a manner that can be used easily in simulation. In principle, there is an interaction between the routing and the diameter (if there is a voltage drop constraint), but it rarely causes an experimental router to iterate. Simulation. The metal foil interconnect between a set of pins is not a perfect conductor. Its geometry and the dielectric properties of the board material can lead to (usually) undesired parasitic remittances, and to interconductor capacitance and inductance. For most circuits to operate as intended, these must be below some acceptably low level, determined by the current and frequency levels and the desired function of the circuit. These levels can normally be reached by following the spacing rules that the router uses. Parasitic extraction is the process of calculating the values of these stray parameters from the geometry of the conductors and the parameters of the material and calculating the equivalent lumped circuit elements (resistance, capacitance, inductance) and inserting them in the original netlist. This new actual netlist with the calculated parasitic parameters can now be resimulated to ensure that the circuit meets its requirements. At very high frequencies (>110 MHz), the parasitics are treated not as lumped circuit elements but as transmission lines, and the tailoring of their properties is an essential part of the design process. Test Design A stimulus file and the results of a simulation with that stimulus should be usable as an expected results file. The physical design can be verified by subjecting it to a stimulus file in real time and comparing the measured electrical outputs with those predicted by simulation. The detailed physical design files of the unit being tested can be used to help design the mechanical fixtures and electrical probes for the test system. Reference cited in this section 7. N. Sherwani, S. Bhingardi, and A. Punyan, Routing in the Third Dimension, IEEE Press, 1995 Computer-Aided Electrical/Electronic Design Shaun S. Devlin, Ford Motor Company Standards It should be clear from this discussion that it is unlikely that a single computer-aided tool (or family of tools) will provide all the electrical/electronic design, manufacturing engineering, and test engineering functions required in a complex enterprise. Standards of representation of the product design and behavior are aimed at providing vendor-neutral file formats or other mechanisms of transferring the required information from one computer system or another. Modern standards of data representation should have a careful definition of the semantics of their terms so that a developer of an application who writes a product description has a clear understanding of the meaning of each term and so that the developer of a reader will have the same understanding. Many standards use the language Express for that purpose (Ref 8). The STEP family of standards developed by ISO TC184-SC4, "Industrial Data," is the most ambitious. It is an evolving series of standards, of which ISO 10303-210, "Electronic Design and Assembly" (AP210) (primarily printed circuit boards and assemblies), and ISO 10303-212 (AP212), "ElectroTechnical Plants," may be of interest to electrical engineers and computer-aided tool developers. The languages VHDL and VERILOG for the description of the behavior and structure of digital systems are now U.S. standards, and VHDL is now an International Electrotechnical Commission (IEC) standard. The EIA/EDIF series of standards (Ref 9) have evolved from a pure netlist standard to include schematics and printed circuit boards with multichip modules. There is vigorous ongoing work in this area, and it is becoming more important as design and supplier relationships become global and organizations can no longer depend on a single supplier. The process of review required in establishing a national or international standard helps clarify any ambiguities in the meaning of a standard and helps ensure that it will allow description of product aspects that are larger than the scope of any one tool. References cited in this section 8. "Express I Language Reference," ISO 10303-11: 1994, ISO 10303- 11:1994, International Organization for Standardization 9. "Electronic Design Interchange Format," EDIF 200 (EIA 548-1988), EDIF 300 (EIA 618- 1994), and EDIF 400 (EIA 682-1996), Electronic Industries Association, Arlington, VA Computer-Aided Electrical/Electronic Design Shaun S. Devlin, Ford Motor Company References 1. D.E. Whitney, Why Mechanical Design Cannot Be Like VLSI Design, Res. Eng. Des., Vol 8, 1996, p 125- 139 2. W K. Chen, Chap 2, Applied Graph Theory, North Holland Publishing Co., 1971 3. VHDL Language Reference Manual, IEEE 1076-1993, Institute of Electrical and Electronic Engineers 4. VERILOG Hardware Description Language Reference Manual, IEEE 1364- 1995, Institute of Electrical and Electronic Engineers 5. L.W. Nagel, SPICE2: "A Computer Program to Simulate Semiconductor Circuits," Electronic Research Laboratory Report ERL-M520, cited in Semiconductor Device Modeling with SPICE, P. Antognetti and G . Massobrio, Ed., McGraw-Hill, 1988 6. V.P. Nelson et al., Digital Logic Circuit Analysis and Design, Prentice Hall, 1974 7. N. Sherwani, S. Bhingardi, and A. Punyan, Routing in the Third Dimension, IEEE Press, 1995 8. "Express I Language Reference," ISO 10303-11: 1994, ISO 10303- 11:1994, International Organization for Standardization 9. "Electronic Design Interchange Format," EDIF 200 (EIA 548-1988), EDIF 300 (EIA 618- 1994), and EDIF 400 (EIA 682-1996), Electronic Industries Association, Arlington, VA Design Optimization Douglas E. Smith, Ford Motor Company Introduction ENGINEERING DESIGN involves the reallocation of materials and energy to improve the quality of life. This occurs in all fields of engineering, including civil, mechanical, electrical, and so forth, and often involves trade-offs based on the requirements of each application. The idea of design optimization suggests that for a given set of possible designs and design criteria, there exists a design that is the best or optimal. Optimization is a part of everyone's life, either consciously or subconsciously. It is our nature to optimize. Investors want the largest return with the least investment or risk. Marathon runners adjust their pace to achieve the best overall time. This article discusses tools that provide a method for systematic optimization of engineering designs. The primary focus here is on the practical application of optimization technology in a computer-aided engineering (CAE) environment. The role of the CAE simulation tool is very important in CAE-based design optimization. Computer-aided-engineering- based design optimization does in fact turn CAE analysis tools into CAE design tools by replacing traditional trial-and- error design approachs with a systematic design-search methodology. Thus, CAE computations that quantify the performance of a particular design are enhanced with information on how to modify the design to better achieve important performance criteria. It is impossible to cover in detail the broad field of optimal design in this short article. The goal here, therefore, is to acquaint the reader with CAE-based design optimization and to provide direction on where to find additional information on the topic. Although CAE-based optimal design is applicable to a wide array of engineering design problems, much of its development has focused on structural optimization. This fact reflects the greater emphasis devoted to structural optimization in this article. Background in numerical optimization is discussed, and emphasis is placed on identifying specific challenges that are encountered when computing optimal designs with traditional CAE analysis tools. Trends in optimal design for CAE applications are also considered through a discussion of emerging technologies in this area. The interested reader is encouraged to consult the cited and Selected References at the end of the article for more information. Other approaches to engineering design that also seek the best design solution can be found elsewhere (see, for example, Taguchi methods in the article "Robust Design" in this Volume). Design Optimization Douglas E. Smith, Ford Motor Company Numerical Optimization Methods A key component of CAE-based design optimization is the numerical optimization algorithm. These algorithms solve optimization problems with mathematical programming techniques independent of the physical application. Better designs are computed based on the design definition and the performance measures that evaluate the goodness of a design. This section focuses on numerical optimization algorithms and is intended to provide some background on how these algorithms make decisions when searching for the optimal design. The interested reader is encouraged to find more information in Ref 1, 2, 3, 4, 5. The Nonlinear Constrained Optimization Problem To formulate the design-optimization problem, the notion of having design parameters (often referred to as design variables) and performance measures is first considered. Design parameters define the process or structure of interest and thus provide a means for changing it to improve its performance. Performance measures that are defined as functions of the design parameters quantify the effectiveness of a given design and enter the optimization problem through the objective function (sometimes referred to as the cost function) and the constraints. The goal when solving an optimization problem is to determine the design parameters that give more desirable objective function and constraint values. The most general single objective optimization problem is one that minimizes or maximizes an objective function F defined over the N design parameters b i , i = 1, 2, . . ., N while satisfying both equality and inequality constraints. In mathematical terms, find: b = {b 1 , b 2 , . . ., b N } T To minimize or maximize F(b) such that g j (b) 0 j = 1, 2, . . ., n g h k (b) = 0 k = 1, 2, . . ., n h b i i = 1, 2, . . ., N (Eq 1) where superscript T indicates the transpose. Constraint functions g and h are defined that mark the boundaries between which designs are allowed and which are infeasible. These constraints are divided into two groups, n g inequality constraints g j , j = 1, 2, . . ., n g , and n h equality constraints h k , k = 1, 2, . . ., n h . The design parameters are assembled in a vector b, and initial values are chosen for each component. Side constraints define the upper and lower bounds for each design variable b i as and , respectively. The set of all possible designs that can be generated by adjusting the design variables between their respective upper and lower limits is called the design space. A simple structural optimization example is given in Ref 1, where the cross-sectional areas of the truss shown in Fig. 1 are adjusted to obtain a design with minimum mass while satisfying constraints on the maximum allowable tension and compression stresses in each member. In this example, the cross-sectional areas A 1 , A 2 , and A 3 are the design variables and the mass of the structure is the objective function. Inequality constraints are formed to represent the limits on the maximum stress, and side constraints bound the range of cross-sectional areas to be considered. Truss member stresses are computed from a deformation analysis of the structure under the given loads. Fig. 1 Three-bar truss example. Source: Ref 1 Aspects of Numerical Optimization Many aspects of the optimization problem play a major role in algorithm selection and performance, and ultimately in the success or failure of the optimization process. Below are some considerations which should be addressed when formulating an optimization problem. The Nature of the Design Parameters. Design parameters define the structure or process being optimized and thus provide a means to alter or change it. They are often classified as continuous or discrete. Continuous design parameters are permitted to take any value over a predetermined range. A hole or fillet radius, for example, or the location of a joint in a truss structure can take any value over its permissible range. Discrete or integer variables are restricted to a finite set of values. Discrete design variables are required when design components are limited to sizes that are available, such as sheet metal gage thickness or tubing diameter. Material modulus is also a discrete design variable because there is only a finite set of materials and thus moduli that exist. Commonly used optimization algorithms accept only continuous design variables. Therefore, the discrete variables are often treated as continuous and the discrete value that is closest to the optimal design is selected. This approach works relatively well when the distance between discrete values is small. Otherwise, integer programming, which limits the design variables to the predefined finite set, may be required when the distance between permissible values is large, such as when the variable defines the choice of material (Ref 1). The Nature of the Performance Measures. Performance measures provide a quantitative measure of the goodness of a design. For each set of design parameters, there is a corresponding set of values, one for each of the performance measures. Performance measures may enter the optimization problem as an objective function or as a constraint. As the objective function, the performance measure is either minimized or maximized. For example, one may wish to minimize the mass of a structure. When the performance measure forms a constraint, it is expressed as a limit on a critical value. In structural optimization, for example, mass is commonly minimized while it is specified that the maximum stress remain below the yield value of the structure's material. The nature of the performance measures is problem dependent. Objective and constraint functions can be linear or nonlinear functions of the design parameters and can be smooth or even discontinuous functions. General nonlinear mathematical programming algorithms are used when the order and/or smoothness of the performance measures is unknown (Ref 3). Alternatively, special algorithms have been developed when the nature of the objective and constraint functions is known. For example, linear programming methods, though applied to nonlinear problems, are very efficient when solving optimization problems where the objective and constraint functions are linear. Other specialized methods exist to efficiently solve least-squares problems that often arise when analytical models are adjusted to match experimental data. Local and Global Optimum. The minimization problem in Eq 1 may exhibit a single global minimum and possibly many local minima. The goal of solving Eq 1 is to find the global minimum; however, in practice, in the absence of certain convexity conditions (Ref 3), one can only ensure that the solution is a local minimum. Descent search methods (discussed later in this article) and necessary conditions for an optimum are restricted to values of nearby points, which results in a focus on local or relative minima. In practice, starting with various initial designs alleviates the concern that the global minima is missed; however, local minima are rarely a hindrance to the success of solving practical optimization problems. Constrained versus Unconstrained Optimization. Optimization problems are classified as constrained or unconstrained. When the range of feasible designs is not restricted, the optimization problem is defined as unconstrained. Alternatively, when limitations are placed on the range of feasible designs through simple bounds on the design variables or with functions of the design variables, the problem is constrained. This distinction plays a significant role in optimization algorithm selection. Most engineering optimization problems are constrained. Indeed, to minimize the mass of the truss structure in Fig. 1 without constraints on the stress or displacement is of little use. The unconstrained solution is meaningless because all of the areas would simply be reduced to their respective lower bounds. Deformations and stresses in such a design would be well outside their useful ranges, revealing that an ill-defined optimization problem was chosen. Constraints can significantly affect the computed optimal design because they often force the objective function to assume a higher value. Such constraints are considered to be active because if they were removed, the objective function would decrease in value. It is useful, therefore, to know how a particular constraint influences the value of the optimal objective function value. For smooth, continuous objective functions and constraints, this is accomplished with Lagrange multipliers, which measure the sensitivity of the optimal design to changes in the constraints. Large Lagrange multipliers suggest that even slight changes in the associated constraint limit would result in a significant reduction in the objective function, whereas Lagrange multipliers near zero indicate that the constraint has very little effect on the optimal design. Multiple- and Single-Objective Optimizations. Many numerical methods have been developed to solve the unconstrained minimization problem given in Eq 1 for a single-objective function F. Often in design, however, there are multiple objectives F i that may need to be considered. For example, a design may be desired that minimizes both stress and weight. Solution techniques for multiple-objective problems, however, have not been developed to the same level as those for single-objective formulations. Two common methods are used to convert a multiple-objective problem into one that can be solved using single-objective algorithms. The first method defines a new objective function as the weighted sum of each of the individual objectives F i . Weighting coefficients are selected to reflect the relative importance of each F i , and care must be taken when using this method because the individual objective with the largest sensitivity always dominates the optimization. The second method chooses the most important F i as the objective function and defines limits on those remaining, which are then included as constraints in the optimization problem. In the latter, many single-objective optimization problems are typically solved, each with different constraint limits, to understand the behavior of the optimal design. To a lesser extent, Edgeworth-Pareto optimization has been used to solve multiple-objective problems when it is difficult to determine the relative importance of the performance measures (Ref 6, 7). Additionally, compromise programming avoids the sensitivity issues of the weighted objective method by minimizing the difference between each individual objective and its respective target value in a least-squares sense (Ref 8). Optimization Algorithms When F is an algebraic function of the design variables, classical methods from elementary calculus can be used to compute the optimal design. For example, when Eq 1 is unconstrained, the design b*, which satisfies F(b*) = 0, and certain criteria on higher-order derivatives comprise the minimum. However, when CAE tools are used to compute the performance of a design, the convenience of having a simple algebraic function is lost because the F is not an explicit function of the design b. Instead, the performance measures are implicitly dependent on b through a CAE solution. In this case, classical methods may not be applied and iterative schemes that search the design space for the optimal design parameter values must be adopted. Searching for the Minimum. Most CAE optimal design implementations are based on computationally expensive numerical simulations to evaluate the performance measures and use descent methods to move through the design space. Commonly used descent methods are based on the same underlying structure when systematically adjusting the design variables while searching for a minimum (Ref 3). For unconstrained minimizations, an initial starting design is specified. A search direction is then determined based on some fixed rule, followed by a one-dimensional line search, which minimizes the function along that direction in the design space. This new minimum serves as a starting point for another iteration, and the process is terminated when the objective function cannot be further reduced. The primary difference between descent algorithms is the rule used to define the search direction and the line search minimization technique. Additional distinctions are made between constrained algorithms based on the manner in which they handle the constraints. Descent methods iteratively update designs as: b I + 1 = b I + I s I (Eq 2) where I is the iteration number and s I and I are the corresponding search direction and step length, respectively. The only requirement is that a positive movement along s I , that is, I > 0, reduces the value of the objective function. Once the search direction s I is selected, I is computed from a one-dimensional search that minimizes F(b I + I s I ). The method of steepest descent is one of the simplest unconstrained descent algorithms that provides a satisfactory result. This method is rarely used in practical problems because of its poor performance, but it is discussed here to demonstrate the basics of descent algorithms. Furthermore, more advanced descent methods have been motivated by a desire to improve the steepest descent method. The search direction of Eq 2 for the method of steepest descent is the negative of the objective function gradient, that is: s I = - F(b I ) (Eq 3) Note that in this case s I represents the direction of largest decrease in the objective function F. For each iteration, the objective function F and its gradient F = -s I are evaluated. Multiple-function evaluations are then performed during the one-dimensional line search. More advanced algorithms use higher-order information to compute search directions. Quasi-Newton methods, for example, are popular because they approximate the matrix of second-order sensitivities (the Hessian matrix) with gradient information, thus avoiding its direct computation. As an example, Fig. 2 shows the iterative solution path for the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton algorithm on Rosenbrock's function (Ref 4). Fig. 2 Unconstrained optimization procedure using BFGS search directions. Shown is the two- dimensional Rosenbrock function F(b) = 100(b 2 - ) 2 + (1 - b 1 ) 2 , which has a unique minimum at (1,1). i, initial design; *, optimal design. Source: Ref 4 In addition to general-purpose optimization algorithms, efficient techniques of limited scope have also been developed for specific applications. The fully stressed design technique (Ref 1), for example, minimizes the mass of truss structures subject to stress constraints alone. New designs are updated based on optimality criteria, which works well in this case for lightly redundant single-material structures. The limitations of many specific optimization methods render them useless for general applications and thus receive little attention today. Convergence Criteria. Because numerical optimization is iterative, it is important to know when to stop, that is, when the optimization process has converged to the optimal design. Specifying the maximum number of allowable optimization iterations guarantees that the optimization process terminates; however, it does not ensure convergence is achieved. One convergence criterion is to monitor absolute and relative changes of the objective and constraint functions and the design parameters (Ref 2). Convergence can then be indicated when changes in the performance measures and/or design parameters between successive optimization iterations are within a predefined tolerance. For example, one can choose to terminate an optimization when a new design results in a reduction of mass that is within 1% of the mass for the initial design. Another important convergence criterion is provided by the Kuhn-Tucker necessary conditions for optimality (Ref 1, 2, 3). For unconstrained problems, this criterion simply requires that at the optimal design b*, the objective function gradient F(b*) is less than a small specified constant. The Kuhn-Tucker conditions generalize for constrained optimization problems where a linear combination of the objective function gradient and the constraint gradients are used to indicate convergence (Ref 2, 3). Analysis Solutions and Optimization Solutions. The solution of an optimization problem differs significantly from that of a typical CAE simulation. Computer-aided-engineering simulations compute the response or state of a product or process, for example, displacement or temperature; whereas the goal of an optimization solution is to define the product or process itself. Additionally, when analyzing a structure, for example, the displacement solution is almost always guaranteed and under certain conditions, it is unique. On the other hand, the existence and uniqueness of an optimal design is not ensured. Quite possibly, a design may not exist that will merely satisfy the constraints, let alone, be optimal. Furthermore, numerical methods used to solve optimization problems are often sensitive to the initial guess, and solution methods are algorithm dependent. The CAE engineer attempting to optimize his or her design should not be discouraged if the first try is not as successful as expected. Algorithm Selection. Optimization algorithms are classified by the derivative information that they require to compute s I in Eq 2, for example, zero-, first-, and second-order methods. Common unconstrained algorithms include the random search, Powell's conjugate direction, and sequential simplex methods (zero-order); steepest descent, Fletcher-Reeves' conjugate direction, variable metric, Davidon-Fletcher-Powell (DFP), and BFGS methods (first-order); and Newton's method (second-order) (Ref 1, 2, 3, 4, 5). Constrained first-order methods include reduced gradient, feasible direction, and sequential linear and quadratic programming methods (Ref 1, 2, 3, 4, 5). In CAE-based design optimization, efficient algorithms are desired because each iteration requires one or more computationally expensive numerical simulations. Higher-order algorithms are generally more efficient, that is, they require fewer iterations; however, higher-order derivatives may be impractical to evaluate. First-order methods are typically used in CAE-based design optimization because they require far fewer function evaluations than zero-order methods and avoid the Hessian evaluations required for second-order methods. Reference 4 provides further guidance for algorithm selection when solving unconstrained and linearly and nonlinearly constrained optimization problems. References cited in this section 1. R.T. Haftka and Z. Gürdal, Elements of Structural Optimization, 3rd ed., Kluwer Academic Publishers, 1992 2. G.N. Vanderplaats, Numerical Optimization Techniques for Engineering Design: with Applications, McGraw-Hill, 1984 3. D.G. 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The languages VHDL and VERILOG for the description of the behavior and structure of digital systems are now U.S. standards, and VHDL is now