Introduction to Design Optimization

Transcription

1 Introduction to Design Optimization James T. Allison University of Michigan, Optimal Design Laboratory June 23, Introduction Technology classes generally do a good job of helping students learn analysis techniques, i.e. methods for predicting the behavior of some artifact. In fact, sophisticated tools such as finite element analysis have been introduced in technology classes. Certainly predictive capabilities are valuable, but for what exactly? Predictive tools allow us to model in the virtual world things that exist in the physical world. Without modeling, we would be forced to rely on full-scale prototypes, which may: be to expensive to build more than one require substantial time for each realization risk human safety during testing Suppose you are an engineer working for a company that produces some widget, and you modeled this widget and predicted its performance. Maybe it is expected to be safe, or perhaps it is predicted to fail. In either case, what do you do next? What questions should you ask? Safe prediction: Can we improve performance or reduce cost without risking failure? Failure prediction: How can we change the widget in order to prevent failure? 1

2 What common thing does each question seek? In either case, a change in the characteristics, or design, of the product is sought. A new design needs to be proposed that we hope performs better than the last design. What are some ways to generate new and improved designs? The first two suggestions employ a trial-and-error approach: Use intuition or experience-based knowledge to propose a new design Change one aspect of the product at a time and see what happens Each approach will require iteration, and even if this iteration is performed in the less expensive and more safe virtual world, we may not arrive at the results we want as quickly as we would like. We may end up testing a large portion of the set of possible designs, and at the end still not know if there exists some other design that might still be better. In addition, many modern products, such as computers or automobiles, are so complex that no single person is capable of possessing the intuition required to progress toward more ideal designs. A more scientific approach is needed. Optimization, a branch of mathematics, provides the necessary tools to move efficiently toward a design that is superior to all other alternatives. This course provides an introductory overview of design optimization, and motivates its use in secondary education curricula. 2 Example 1: Optimal Design of a Trebuchet We begin with the first of two computational examples, the design of a trebuchet, in order to introduce some basic concepts and to further motivate the study of design optimization. A trebuchet, which launches a projectile using only force from a weight, is frequently the object of student design projects. A design objective might be to maximize the distance a projectile is launched, subject to some kind of limitations on the design such as size, weight, or building materials. An important aspect of trebuchet design is the launch angle θ of the projectile (Figure 1). Projectile motion is a classical topic usually studied in high-school physics, and we can use the basic equations of projectile motion to predict how far an object will be launched for a given launch angle and launch speed. Neglecting air resistance, the horizontal component of the projectile velocity is constant, so v x = v 0 cos θ, and the horizontal position as a function 2

3 θ Figure 1: Schematic of a simple trebuchet and projectile launch angle θ of time is x = v 0 cos θt. In the vertical direction the only force acting on the projectile is gravity, so the horizontal velocity varies with time, and is described by the equation v y = v 0 sin θ gt, where g is the gravitational acceleration. The vertical position is given by y = v 0 sin θt 1 2 gt2. The maximum horizontal distance traveled is found by setting the vertical position to zero, solving for t, and substituting the expression for t into the horizontal position equation. The result is given in Equation (1). x max = 2 g v2 0 sin θ cos θ (1) Since we are considering how to maximize launch distance by selecting the ideal value of the launch angle, we can formalize the design optimization statement as shown in Equation (2). max θ 2 g v2 0 sin θ cos θ (2) Using mathematical optimization techniques based on calculus, it is possible to find the solution to this optimization problem directly. It can be shown that the solution to Equation (2) is found by satisfying the equation sin θ = cos θ. The only angle between 0 and 90 that satisfies this relation 3

4 is 45. Without knowledge of the optimization techniques that led to this answer, how might a designer approach this problem? One might use some software like MS Excel TM to rapidly calculate the launch distance for a given angle, or even to graph the relationship between θ and x max and visually find the optimum. Figure 2 illustrates how MS Excel TM can be used for these investigations. This works fine for such a simple example, but what happens if each function evaluation requires minutes, hours, or even days (as may be the case with finite element analysis or computational fluid dynamics)? Or, what if more than two items are being varied at a time in the design and it is impossible to visualize where the optimum is? A more intelligent approach is required in these cases. An expansive array of optimization algorithms have been developed to solve equations such as the one shown in Equation 2. MS Excel TM has an add-in package called Solver that employs some of these algorithms to solve such problems. By defining the which values are to varied, what should be optimized, and what constraints might exist, these algorithms can automatically find the best design, even if the number of design variables is large. This course will explore in a little more depth what design optimization is, how to formulate optimization problems, and some additional extensions for optimization. Figure 3 provides a definition and framework for the rest of the course. 4

5 Example 1: Trebuchet Launch Angle Optimization g 9.81! 45 v 0 15 x max Trajectory Visualization x y 0 0 Projectile Trajectory Horizontal Position (m) xmax Launch Angle-Launch Distance Relationship! x max E-15 Vertical Position (m) 5 Figure 2: Using MS Excel TM to determine the optimal launch angle.

6 Optimization: Effective and efficient decision making based on quantitative metrics 'Making the right decision' 'What decisions should be made' 'Quantifiably predict the outcome of a decision' Problem Formulation 'Making decisions quickly' Governing Equations Empirical Models Optimality Conditions Algorithms and Convergence Design variables vs. parameters Finding the best design without testing them all Numerical Approximations Figure 3: Definition of optimization 6

7 3 Making Effective Design Decisions In design optimization we seek the design that is superior to all others. The way we judge this superiority depends on how the problem is formulated and how the quantitative metrics for comparison are obtained. This section discussed the first item, while the latter pertains to modeling, and is discussed next. Optimization problems may be described in a standard form called the negative null form: min x f(x) (3) subject to g(x) 0 h(x) = 0 The objective is always posed as a minimization problem (a maximization problem can be transformed to minimization by negating the objective function), and constraints are grouped into inequality and equality constraints. The vector x is comprised of all design variables, i.e., values that are to be changed during the optimization process. Any input values that are held fixed during the optimization process are called design parameters. The objective and constraint functions are all dependent on x. The way we choose to formulate the problem will affect the resulting solution. For example, if we are designing a car, we might want to minimize both 0 60 time and fuel consumption. We could use fuel economy as the objective and minimum acceleration as a constraint, or acceleration as the objective and minimum fuel economy as a constraint. The mathematical techniques will help us find the answer, but we are the ones that define the problem. The answer to the design optimization problem is only as good as the problem definition and the models used in the solution. Now that we know formally what the problem is, how do we define when something is optimal? Refer back to the θ vs. x max plot in Figure 2. What do you observe about the optimal point? The slope of the function at that point is zero, or in other words a line tangent to the curve at the optimal point is horizontal. This is a necessary condition for optimality. This idea can be extended to higher dimensions using calculus, and these optimality conditions are the basis for many optimization algorithms. Other more complicated conditions can be investigated to determine whether or not we can guarantee that a design is optimal. 7

8 Many examples exist in nature of making ideal or optimal design decisions. A general principle of natural systems is to seek the lowest energy state. Chemical reactions proceed only if they are energetically favorable. Soap bubbles are round because of the energy associated with surface area, and a spherical bubble has the smallest surface area for a given volume of contained air. In addition, a single large bubble has less surface area than many smaller bubbles with the same collective volume, explaining why soap bubbles in you sink tend to coalesce. Crystals form based on similar principles. If minerals or metals are allowed to cool slowly, the crystals tend to be larger and the total interfacial area between crystals will be smaller. Survival of the fittest phenomena in nature in effect optimizes organisms or entire ecosystems, and in fact a class of algorithms is based on these evolutionary processes. 4 Making Design Decisions Quickly Now that we have defined more carefully what an optimal design is, we can discuss how to find this design more quickly. Earlier we described how the set of possible designs can be described as a curved function (Figure 2). This idea extends to higher dimensions. If we have two design variables instead of just one, the objective function will be a surface. Suppose we are in a fishing boat on a pond and we want to find the lowest point of the pond. We cannot see the bottom, so the only information we have is the depth at individual locations. We could use trial and error, but that could take a very long time. Another approach would be to create a grid and measure each point on the grid, but then we will never be sure if another point exists that might be lower. What is a better approach to finding the lowest point? We ideal want to find the lowest point, without having to test all points. Suppose at each point we take a few measurements nearby the original measurement point. This will give us some idea of the slope of the bottom surface of the pond, and what is the downhill, or steepest descent, direction. We then head that direction, taking measurements along the way, until we find the lowest point in that direction. Then we take some extra measurements there to determine the downhill direction, and repeat the process until we discover there is no more downhill direction. At that point we have met the necessary conditions for optimality, and stop the process. This steepest descent method works well for many optimization prob- 8

9 lems. It converges quickly in many cases, and is simple to implement. Other more sophisticated algorithms that can converge even more quickly exist that are still based on information about slope and downhill direction. The existence and rate of convergence are important properties of an optimization algorithm. Existence of convergence means that the algorithm will terminate (ideally at the optimal point), and rate refers to how many function evaluations, or measurements, must be made before convergence occurs. So far the discussion has focused on design problems that are continuous, i.e., they are able to take on any value within a range. Some variables are discrete, and other classes of algorithms have been developed to deal with discrete optimization problems. Discrete variables can either be integer (countable, can be ordered), or categorical (cannot be ordered). In designing a table continuous design variables could be the height, width, or length of the table. An integer variable could be the number of table legs, and a categorical variable could be the type of wood used. Figure 4 illustrates the family of optimization algorithm categories based on variable type. Algorithms may also be classified by the properties of the objective and constraint functions: linear vs. nonlinear, noisy vs. smooth, or convex vs. non-convex. Optimization Continuous Discrete Mixed Categorical Integer Figure 4: Taxonomy of optimization algorithm classifications based on variable type 9

10 5 What Decisions are to be Made? We previously introduced the design vector x, which is a set of numbers represents a design. Describing a design completely by a set of numbers may be a new idea for some individuals. Initial design stages where the basic configuration of a design is generated is sometimes called conceptual design. Once this conceptual design is determined, it can be described completely by a set of numeric parameters and relations between these parameters. This is sometimes referred to as parametric design. An example of parametric design is a CAD drawing that has been created using relations between important dimensions such that if a change is made to any of these dimensions (or parameters), the rest of the design is automatically updated. It is desirable to minimize the number of parameters used to describe a design. When we consider how to formulate the design optimization problem, an important decision is what parameters to select as design variables (values that are changed in the optimization process), and what parameters should be left fixed as design parameters. It is easier to solve a problem with fewer design variables, but limiting the number of design variables may cause us to overlook certain designs that may be more desirable. 6 Modeling: Quantification of Designs A critical link exists between the success of a design optimization endeavor and the models (or analysis) used in solving the optimization problem. The accuracy of design optimization solution is only as accurate as the models used to represent the artifact. This motivates the usage of high-fidelity models. However, these models are computationally expensive. A designer may not have enough time to wait for such an optimization to finish. This tradeoff is difficult to make. One approach is to perform preliminary optimization studies using simple models in order to narrow down the number of design configurations and design variables, and then move on to high-fidelity modeling for final optimization. 7 Example 2: Air Flow Sensor Design The second example considers the design of an air flow sensor. A springloaded panel of length l and width w deflects as air flows past it. The amount 10

11 of angular deflection θ is proportional to the air velocity v, the frontal area of the panel A f = wl cos θ, and the torsional spring has stiffness k. In addition, the drag force on the panel is limited to F max because of structural limitations, and the area of panel must equal a specified value A. This design problem ties together several topics discussed so far: optimization in more than one variable, optimization with equality and inequality constraints, modeling, and choice of design variables. v F 1/2 cos k cos Figure 5: Illustration of a simplified airflow sensor The objective is to calibrate the sensor such that a specific amount of deflection θ T is achieved for a given air velocity. The formal optimization problem is given in negative null form in Equation (4). We have chosen the length l and the width w of the panel as design variables. The spring constant k, maximum force F max, air velocity v, target deflection θ T, and the prescribed panel area A are all fixed parameters in this example. min l,w subject to F F max 0 (θ θ T ) 2 (4) lw A = 0 To predict the behavior of this design we must consider both aerodynamic analysis and structural analysis. In the aerodynamic analysis we must 11

12 predict the drag force given the air velocity and frontal area of the panel. Using a simple analysis technique, Equation (5) provides us with the means to quantify this relationship. C is a constant that is related to the drag coefficient of the panel and the density of air. F = CA f v 2 = Clw cos θv 2 (5) In the structural analysis we need to determine the deflection as a function of drag force. Assuming the spring is linear, Equation (6) approximates this relationship. Note that we cannot solve explicitly for θ, so a numerical method must be employed in the structural analysis. kθ = 1 F l cos θ (6) 2 The structural and aerodynamic analyses are interdependent, i.e., the output of the structural analysis is the input to the aerodynamic analysis, and vice-versa. This can be overcome using a recursive solution process such as fixed point iteration. A guess is made for the value of one of the inputs to start the process. Let s say we choose to guess a value for the deflection in order to execute the aerodynamic analysis. This produces an approximation for the drag force, which is then used as an input to the structural analysis to generate an updated value for the deflection. This process is repeated until the coupling variables, θ and F, converge to stable values. Figure 6 illustrates the information flow between the two different disciplinary analyses. l Structural Analysis F l,w Aerodynamic Analysis Figure 6: Information flow between structural and aerodynamic analyses 12

13 Table 1: Parameter values used for the air flow sensor problem parameter value units θ T 0.25 radians F max 7.0 newtons A 0.01 meters k 0.50 newtons/radian C 1.0 kilograms/meter 3 v 40 meters/second In order to visualize the space of feasible designs and the corresponding performance, the objective and constraint functions were computed over a range of width and length values, and plotted in Figures 7 and 8. The parameter values used are shown in Table 1. Figure 7 allows us to visualize the nature of the objective function. Obviously small values of l do a good job of minimizing the objective, but we need to be mindful of constraints. Figure 8 illustrates the constraints. We must stay on the solid line, which represents the equality constraint, and on the upper right side of the dashed line, which represents the inequality constraint. Observing the downhill directions, we see that the optimal design exists at the intersection of these two constraint lines. The graphical approach to solving the air flow sensor problem is instructive, but can be time consuming and uses up unnecessary resources. We can identify the optimum design with far fewer analysis evaluations, and in much less time using one of the many available software tools. 8 Summary of Basic Topics The primary value of optimization in a product design context is that is enables us to find to best possible design in a relatively small amount of time. However, there are limitations to optimization techniques. The optimal design is only optimal with respect to how the problem was formulated, to what values were chosen as design variables, and to the accuracy of the modeling used. A practitioner needs to ask the following questions: Does the objective function accurately reflect what is really wanted? Is the modeling accurate enough. 13

14 Airflow Sensor Design Space w l Figure 7: Three-dimensional view of the air flow sensor objective function Contour Plot of Design Space w l Figure 8: Contour lines of the objective function and constraints. Equality constraint is the solid line, and the inequality constraint is the dashed line. 14

15 Are there constraints or interactions in the real physical design that were overlooked in the formulation of the optimization problem? Is there uncertainty or variation in: Manufacturing processes or supplies? Product usage? Environment the product exists in? Modeling or analysis? 9 Additional Topics This section outlines additional topics for potential discussion, which may be covered as time permits. 1. Organizations and Economies as optimization processes 2. Complex system optimization (a) Analytical Target Cascading: coordinating between system-level and component-level thinking (b) Multidisciplinary Design Optimization: integration of many different disciplinary analysis into an overall system optimization (c) Software for large-scale optimization: Optimus, isight 3. Additional optimization applications (a) Fitting models to experimental results (b) Design of experiments (c) Operations Research: military operations planning, airport problem, diet problem, project management (d) Manufacturing: optimization of CNC tool paths, tolerance allocation 4. Multi-objective optimization 5. Product family/product platform design 15

16 6. Local vs. global optima 7. Generalized reduced gradient method (algorithm used in MS Excel TM ) 16