Rocket Nozzle Design - CiteSeerX

Rigorous Global Search Working Note 5 GlobSol Case Study: Rocket Nozzle Design (MacNeal-Schwendler)

Frank Fritz, George F. Corliss, Andrew Johnson, Donald Prohaska, and Jonathan Harty DRAFT 2.2 29 Jan. 1998

Contents

1 Introduction 2 What { Rocket Engine Nozzle Design

2.1 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Involute Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Who { MacNeal-Schwendler Why { Signi cance How { Mathematical Model New Directions

6.1 Stochastic Method . . . . . . . . . 6.1.1 Bayesian Approach . . . . . 6.1.2 Kriging . . . . . . . . . . . 6.1.3 Modi ed Kriging Algorithm 6.1.4 Correlation . . . . . . . . . 6.2 First-Principles FEA Method . . . 6.2.1 Description of Problem . . 6.2.2 Solving the Problem . . . . 6.2.3 Bene ts and Drawbacks . .

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7 Set-up: How to Apply GlobSol

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7.1 Using GlobSol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7.1.1 Make le . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7.1.2 Codelist program: bar.f90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Department of Mathematics, Statistics, and Computer Science, Marquette University, P.O. Box 1881, Milwaukee, WI 53201{1881 y 49 Maracay, San Clemente, CA 92672 

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7.1.3 OVERLOAD.CFG 7.1.4 bar.DT1 . . . . . . 7.1.5 OPTTBND.CFG . 7.1.6 bar.OT1 . . . . . . 7.2 Results . . . . . . . . . . .

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8 Interpretation and pictures 9 Conclusion

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27 27 Abstract

This paper provides a description of a research optimization problem conducted at Marquette University. The problem was developed from one suggested by the MacNeal-Schwendler Corporation. The design process has traditionally consisted of applying theory to a speci c engineering problem, developing a detail design, manufacturing a working prototype, and testing the operation of the prototype. Depending on the prototype performance, what followed was usually a time-consuming reiteration of the detail design, prototype construction and testing process, until the design produced the results speci ed by the customer. With the advent of the desk-top computer, numerical analysis methods like Finite Element Analysis (FEA) conveniently allowed designers to eectively model their ideas on a computer, exploring and improving a design before any prototype was constructed. Although Finite Element Analysis provides a signi cant advantage in reducing the time and cost required to develop a design over traditional design methodologies, Finite Element Analysis can only provide the designer with speci c information about the model as constructed by the designer. Finite Element Analysis programs currently lack the capability to intelligently modify or optimize a parameter of the original model to automatically achieve a desired design goal. The optimization problem suggested by MacNeal-Schwendler is a Finite Element Analysis model study of the internal stress level of a rocket nozzle exhaust cone. The FEA program called CPATCHES was internally modi ed to produce a sensitivity derivative at every node constructed within the FEA model. These sensitivity derivatives signi ed the amount of change in displacement each model node experienced with respect to the amount of change in a particular design variable. The sensitivity derivative information was then feed into a special optimization program called MICRODOT. From Hook's law and the model node displacements, the stresses within the nozzle were determined by the MICRODOT optimizer. If the stress levels were not minimized, MICRODOT modi ed the model design variables, and restarted the Finite Element program, producing a new set of sensitivity derivatives. This process continued until MICRODOT could not determine any further improvement to the stress levels within the design. This optimization methodology produced a design in which the margin of safety for the stresses was maximized. The research project conducted at Marquette University attempted to employed the same CPATCHES Finite Element Analysis of the rocket nozzle, but was to use a Global Optimization program based on the interval arithmetic methodology suggested by R. Baker Kearfott, instead of the MICRODOT program. The bene ts of Global Optimization using interval arithmetic as compared to current techniques is that a solution is guaranteed to exist within the bounds speci ed to the problem, and the belief that the solution will require substantially less processing time. During the initial research phase of the project, a thorough examination of Hart's work reviled that the objective function passed to the MICRODOT optimizer was not a continuous, dierentiable function, and could not be directly applied to Kearfott's interval optimizer. This restriction required the research project to change direction and seek an equivalent interval solution in one of two speci c approaches. The rst approach involves using a linear regression function to perform an interpolation between previously acquired FEA point values to determine where a global minimum might

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exist. From the interpolated point data, the GlobSol optimizer will be employed to nd the local minima within a small segment of the problem. This process will be repeated as necessary until the entire FEA point eld has been analyzed, providing the global minimumto the problem. The second approach will involve the development of a rst-principles FEA program that computes the FEA dierential equation in interval form directly from the fundamental stress equations (Hook's Law). The FEA dierential equation is feed into GlobSol as the objective function, with the element matrix parameters entered as constraints.

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Rocket Nozzle Design (MacNeal-Schwendler) 1 Introduction

Let f be a continuous, dierentiable objective function X  Rn ! R. Consider the global optimization problem, min f (x); x2X

possibly with linear or nonlinear equality or inequality constraints g(x). We seek validated, tight bounds for the set of minimizers X  and/or the optimum value f  . Linear systems Ax = b and nonlinear systems G(x) = 0 for G : Rn ! Rn, are also solved by components in our tool box. This report is a kind of \memory dump" of everything we have done on this project. The wise reader will select speci c portions of interest. The primary audience of this report are scientists and engineers in (possibly distantly) related areas with optimization problems. They want to know whether the GlobSol technology is something they should explore. Secondary audience is MacNeal Schwendler. They want to be assured that we understood their problem and solved the right problem. They want to know what we did and how they can apply our approach to their related problems. Another secondary audience are researchers in the optimization and more general scienti c computation community. They are skeptical of validated techniques. They want evidence that we can do what we claim on tough problems. Another secondary audience are researchers in the interval community. They believe and seek evidence to support their beliefs. Our goal is to solve some very challenging problems so that our clients and others will be genuinely impressed if we are able to solve them. This case study describes one challenging problem we received from MacNeal-Schwendler and how we solved it. The research project uses interval arithmetic techniques and software for the veri ed solution of nonlinear systems of equations, and for rigorous, deterministic unconstrained and constrained global optimization.

2 What { Rocket Engine Nozzle Design A rocket engine exhaust nozzle's primary function is to direct the outward ow of high-pressure exhaust gas produced by the engine. In this manner, the thrust of the engine is focused, thus achieving the jet propulsion necessary to drive the vehicle forward. By slightly altering the exit direction of the exhaust gases relative to the space craft, the forces generated by the exhaust gases can be vectored to steer the rocket on a desired path, providing in- ight directional control of the space vehicle. While in operation, a rocket engine nozzle cone experiences a tremendous amount of internal material stress due to the massive thrust pressure and temperature created by the ignition of the rocket fuel within the engine. This material stress called hoop-stress, tends to act on the nozzle cone in a circumferential direction, trying to blow the nozzle apart. This type of material stress is exactly the same stress that causes a toy balloon to rupture from excessive internal air pressure. The task of the rocket nozzle designer is to design a nozzle cone that is strong enough to withstand the extreme forces and temperatures generated within the nozzle during engine operation, while keeping the nozzle weight to a minimum. Design problems like the rocket engine nozzle stress pose incredible challenges to aerospace and materials engineers. To reliably solve these dicult problems in very complex shapes, engineers use several analysis tools to model accurately the behavior of their designs on a computer before a prototype is actually constructed. One of the most powerful numerical analysis tools used by designers is the Finite Element Method.

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Figure 1: NASA photo of shuttle engines and exhaust nozzles

2.1 Finite Element Analysis

The Finite Element Method is implemented in a computer program referred to as Finite Element Analysis (FEA). In a Finite Element Analysis program, the designer constructs a model of the design within the FEA program. This design is modeled as a skeleton type of structure, consisting of a series of nodes, each interconnected to its surrounding nodes by a mesh of elements. Once the designer has de ned the pro le of the model to the FEA program with nodes, elements, and material properties of the elements, the designer can specify the internal pressure or temperature that the nozzle experiences in operation. With this information de ned to the FEA program, the numerical analysis can begin. The task of the FEA program is to calculate the amount and location of internal stress within the rocket nozzle cone while in operation. Then the designer can compare these FEA calculated stress values with the measured maximum allowable stress properties for the material from which the nozzle is manufactured. If the computed stress levels are above the maximum safe stress levels, the wall of the nozzle may rupture, resulting in a catastrophic failure. Therefore, the designer must alter the shape of the nozzle or change some design parameter to reduce the nozzle's internal stress to a level below the maximum allowable stress level of the material. Using Finite Element Analysis, the designer must solve this stress design problem without violating some other nozzle design speci cation such shape. Finite Element modeling has limitations: 1. A stress solution is provided only for the exact model parameters de ned to the FEA program.

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2. Design optimization techniques are not available to the designer from within the FEA software. The designer must evaluate the model results and through engineering experience, estimate where design changes must occur to maximize the strength or achieve a desired speci cation of the nozzle without sacri cing safety. The Finite Element Analysis program by itself cannot intelligently modify the model to optimize some desired feature. The designer can never truly know if the model is indeed at its optimum con guration, or if further improvements can be obtained by changing some other combination of design parameters.

2.2 Involute Structures

We want to optimize the design of material geometry for involute structures of rocket engine nozzle exit cones. The objective is to maximize the safety factor of the nozzle design in terms of the maximum stress seen by the nozzle during operation, to the stress level that the nozzle can withstand before material failure, with design constraints of stress (maximum stress criterion), and manufacturing (orientation of the carbon ply pattern). MacNeal-Schwendler suggested we consider the optimization problem posed by Jonathan Hart in his doctoral dissertation [2]. Hart used a nite element program (CPATCHES) with internal code modi cations to provide sensitivity derivatives at each design node. These sensitivity derivatives provided displacement, strain, and stress of an element node with respect to each design variable X (dU=dX ). From these sensitivity derivatives, Hart generated the constraint derivatives. Hart then passed these sensitivity and constraint derivatives to an optimizer called MICRODOT. MICRODOT evaluated the derivatives by iterating the design with the product of ply thickness and ply count being incremented by a prede ned step size. The FEA model was then re-run with the new design conditions. Eventually, the optimizer determined the optimum ply thickness, ply count, and ber orientation angle resulting in the rede nition of the ply pattern geometry used to construct the nozzle exit cone. This new ply pattern geometry provided the greatest margin of safety by minimizing the level of internal stresses seen by the nozzle. Typically, a rocket motor nozzle exit cone is built up of many cloth plys made of carbon- ber material. Carbon-carbon is used because it is one of the few structural materials that does not vaporize at 5000 degrees Fahrenheit. These cloth plys are sandwiched together in an involute spiral to layer-build the shape of the nozzle cone around a mold. One edge of each cloth ply is positioned at the inside diameter of the nozzle, and the opposite edge of the ply curves outward in an involute spiral to form the outside diameter of the nozzle. The involute structures modeled in the test problem were cloth fabric-reinforced laminated composites formed into rocket engine nozzle exit cones. Involute construction is commonly used to manufacture high-temperature bodies of revolution such as rocket motor exhaust nozzles, since the involute plys oer better control of the ber orientation and compact to a uniform ply thickness when cured. After all the cloth plys have been arranged and layered around the mold, the entire structure is placed in a furnace to cure the plys together to form a single, solid nozzle exit cone structure. The design of the involute surface needed to produce a ply pattern that has a minimum of ply fabric distortion is extremely dicult to create. An involute curve is a compound surface, which is extremely dicult to visualize from the manufacturing standpoint. Therefore, designers rely on an inexact involute surface, which readily lends itself to the construction of rocket motor exit nozzles. The inexact involute surface is developed into a cloth ply pattern from a series of design parameters. When parameters are given to a ply-generating computer program, the ply patterns displayed in gure 3 are produced. The design variables used to create the ply con guration

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Figure 2: Top view of an involute curve constructed about a cylinder are:

t N 

The thickness of a ply (inches). The number of plys used to layer-build the nozzle cone. The helix angle of the ply pattern (i.e. the angle the carbon ber makes with the axis of revolution (degrees)).

Start-line geometry. The start-line is an involute surface meridian line. The construction of this line is the fundamental method for establishing a workable construction technique to build an inexact involute surface. The start-line is geometrically constructed using the variables: t ply thickness (inches) N ply count  helix angle (degrees) From a series of equations, the start-line is de ned in an R ? Z (axisymmetric) coordinate system, and directly transferred into the geometric modeler of the CPATCHES FEA software module. A computer program determines the ply pattern con guration from the input variables N , t, and . It is the development of the ply pattern geometry through the optimization of the design variables N , t, and , with respect to stress, that was the focus of the work previously done by Hart.

3 Who { MacNeal-Schwendler The MacNeal-Schwendler Corporation is the world's largest provider of mechanical computeraided engineering (MCAE) strategies, software, and services. MacNeal-Schwendler is the largest single provider of nite element analysis (FEA) products.

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Figure 3: Typical ply pattern MacNeal-Schwendler's solutions are used by scientists, engineers, designers and analysts in industry, research laboratories, and universities. The majority of MacNeal-Schwendler's business centers on aerospace and automotive markets, as well as many other growth industries. For over 30 years, companies worldwide depend on MacNeal-Schwendler software and services to help bring their products to market faster, reduce costs, and increase product quality and performance. MacNeal-Schwendler provides MCAE solutions that allow designers greater freedom to innovate design concepts, explore \what-if" scenarios, optimize complex solutions, and exploit materials as a design variable. In their dedication to provide their customers with the widest selection of computer software and hardware, MacNeal-Schwendler has developed relations with computer hardware suppliers, as well as other CAD and MCAE software companies. These relationships have developed into the MacNeal-Schwendler Global Partners Program. This program facilitates increased interoperability between MacNeal-Schwendler software and third-party complementary CAD and MCAE software. Hardware partnerships are focused both on performance improvements of MacNeal-Schwendler software on existing hardware and ensuring that MacNeal-Schwendler's software products will perform well on new hardware platforms.

4 Why { Signi cance MacNeal-Schwendler software solutions are regularly used in the most challenging areas of industry. MacNeal-Schwendler FEA software provides the necessary solutions to the most demanding problems encountered from aerospace to automotive applications. Customers of MacNealSchwendler not only expect MacNeal-Schwendler software to be robust and accurate, but project managers depend on MacNeal-Schwendler software to produce solutions to complicated problems with a minimum of computer processor time, resulting in development projects that run on schedule. As the demands and complexity placed on existing nite element modeling increases, it becomes more and more dicult for traditional FEA packages to provide the rapid prototyping environment desired by designers. The primary bottleneck to achieving rapid design

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Figure 4: Start line geometery used to model nozzle solutions usually isn't due to the modeling package, but rather in the human analysis of the model output. Designers typically must rely on their own judgment and experience to acquire the necessary model changes from information received from a previous FEA session. Without the assistance of global optimization, these engineering modeling decisions can be a lengthy, costly experience. To demonstrate the capabilities of the global optimization software, a sample design problem was selected by MacNeal-Schwendler to be used as a research tool. This research project is being conducted to provide a prototypical general design methodology for the solution to complex optimization problems. One of the fundamental bene ts of this emerging methodology in comparison to existing technologies is a decrease in the computing time required to obtain an exact solution. Bene ts of this nature are extremely important to MacNeal-Schwendler's customers, who rely on MacNeal-Schwendler products to solve complex design and optimization problems with a minimum of processor time.

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5 How { Mathematical Model

Let f be a continuous, dierentiable objective function X  Rn ! R. We will consider a global optimization problem, min f (x); x2X possibly with linear or nonlinear equality or inequality constraints g(x). We seek validated, tight bounds for the set of minimizers X  and/or the optimum value f  . Linear systems Ax = b and nonlinear systems G(x) = 0 for G : Rn ! Rn, are also solved by components in our tool box. The optimization problem solved by Hart incorporated three constraint functions:  Response or stress constraint.  Manufacturing constraints.  Side constraints. The independent variables of the involute design problem are t ply thickness, N ply count, and  helix angle. The start-line geometry is derived from them. The dependent variables of the involute design problem are the node displacements (U ), the node strain (), and the node stress (), from which the margin of safety can be calculated. We wanted to solve the optimization problem by using the previous research of Jody Hart [2]. Design variables of ply thickness, ply count, and helix angle were de ned to generate the start-line geometry. With the start-line geometry de ned, the Euler angles and ply pattern were developed. The geometry of the ply pattern was entered into the CPATCHES FEA program. The CPATCHES program had been modi ed by Hart to produce the sensitivity derivatives for each node of the FEA model. The sensitivity derivatives were used to compute the constraint derivatives. The sensitivity and constraint derivatives were feed into the MICRODOT optimizer. From the derivative information, MICRODOT performed an iterative calculation that repetitively modi ed the design variables and then restarted the FEA process. This process continued until a design was found that was farthest from the constraint surfaces, thereby minimizing the risk of the design (maximized the margin of safety). Our initial intention was to solve the problem that Hart solved, by utilizing the same CPATCHES code which produces the sensitivity derivatives at each node, but wanted to replace the MICRODOT iterative optimizer with the single-pass interval arithmetic optimizer GlobSol [3]. Every node created by the designer in a nite element model must be identi ed to the computation algorithm. Whether node generation within a FEA program is automatic or manual, nodes are typically identi ed by their absolute position relative to the origin of the model coordinate system. In a three dimensional model, each node will be assigned a reference identi er consisting of the node's X , Y , and Z position within the model, (i.e. node(i, j , k), where i, j , and k correspond to the X , Y , and Z axis position). For a body of perfect revolution, like a rocket nozzle, only two reference positions are required { the node's distance from the center-line axis of rotation (R), and the node's relative position along the Z axis. Models constructed in this manner are said to be built in the axisymmetric, or the R ? Z coordinate system. A node in the R ? Z coordinate system is identi ed by the notation { node(i, j ), where i is the radius distance, and j is the position of the node along the Z axis, from the origin of the Z axis. When stress () and strain () are directly proportional { that is, when the stresses imposed on the material are below the material yield point, the linear relationship between stress and strain can be represented by Hook's law [6], which is:  =   E; where (1)

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Figure 5: FEA model of rocket nozzle  is the normal stress in pounds per square inch  is the strain in inches per inch E is the modulus of elasticity, or Young's modulus in pounds per square inch { a constant for a given material. From Hook's law, the equation describing the relationship between force and displacement of a node(i, j ) within a Finite Element mesh can be described as: Fi;j = Ki;j  Ui;j ; (2) where F is the resultant force at node(i, j ) in pounds K is the material stiness matrix at node(i, j ) in pounds per inch U is the displacement of the node(i, j ) in inches With the displacement of each node in the Finite Element mesh calculated, the strain energy () existing at each node(i, j ) can be determined from the equation: x (3) (xx )i;j = @U @X y (4) (yy )i;j = @U @Y ;

where

(xx)i;j is the strain in the x plane at node(i, j ) in inches per inch. (yy )i;j is the strain in the y plane at node(i, j ) in inches per inch.

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With the strain energy calculated at each node(i, j ), the stress () at each node(i, j ) can be calculated using Hook's law for two dimensional normal stress as follows [6]: (5) (xx )i;j = 1 ?Eu2
((xx )i;j + u(yy )i;j ) (yy )i;j = 1 ?Eu2
((yy )i;j + u(xx)i;j ); (6)

where

(xx)i;j is the stress in the x plane at node(i, j ) in pounds per square inch. (yy )i;j is the stress in the y plane at node(i, j ) in pounds per square inch. E is the modulus of elasticity, or Young's modulus in pounds per square inch { a constant for a given material. u is Poisson's ratio { dimensionless. The sensitivity derivative developed by Hart are: @U = K ?1  @F ? @K U  ; (7) @X @X @X where K is the stiness matrix, @F is the element load vector derivative, @X @K is the element stiness vector derivative, and @X U is the displacement eld. From this equation, Hart developed the constraint derivatives which, along with the sensitivity derivative, were feed into the MICRODOT optimizer.

6 New Directions The work of Hart focused on an optimization algorithm incorporating the MICRODOT program. MICRODOT took a sensitivity derivative from the FEA program in the form of (dU=dX ), and calculated the new vector of the model variable in question for the next FEA run. While this technique works very well for an ad hoc optimizer like MICRODOT, the sensitivity derivative information of (dU=dX ) is inadequate for an interval optimizer like Kearfott's GlobSol. The power of an interval optimizer can only be achieved when a continuous, dierentiable objective function is available. The output function of Hart's code was not in this form, but instead was represented as a simple oating point value (dU=dX ). The main obstacle encountered is that we need to pass interval arguments to GlobSol instead of oating point values, and we need to get back bounds rather than oating point values to code the GlobSol objective function. To complicate the development of the research project still further, we learned that Hart had made signi cant modi cations to the FORTRAN source code in an attempt to port his project to the UNIX and DOS operating systems. Hart no longer had the original source code that he had used to develop the FEA optimizer for his doctoral dissertation. The Marquette research development team had been given a 400,000 line program from MacNeal-Schwendler that had not run in its present condition on any operating system and appeared to require a signi cant eort from Hart just to get the software back to the original VMS environment. Hart had no idea at the time he began his code modi cations that one day MacNeal-Schwendler would request Hart to forward a copy of his FORTRAN code to the Marquette research team for an intensive optimization project. Faced with what initially appeared to be several insurmountable problems, the Marquette research team began to explore other alternative approaches to solving the Finite Element optimization problem. We considered two separate development paths, stochastic methods and an interval nite element analysis, that promised an interval solution. We outline them brie y

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here, and then describe them more fully. At the end of December, we decided to pursue only the interval nite element analysis in detail. The rst approach was to take advantage of the inherent point value computed by the existing (non-running) FEA program and apply a stochastic point evaluation regression function. This technique acquires point samples of the objective function from the FEA program over a speci c range of the model input variables. From these points, a regression function makes a rst-order prediction (surrogate model) of the objective function through interpolation. The surrogate model of the objective is optimized to identify values of the independent variables at which small values of the true objective are likely to occur. The objective is sampled at these new points, and the process is repeated until the global minimum is located from the entire problem space.

Figure 6: Stochastic Approach The second approach was to develop a rst-principles FEA program that computed the FEA dierential equation in interval form directly from the fundamental stress equations (Hook's Law). GlobSol is asked to minimize the maximum stress, subject to equality constraints expressing the nite element equations. In parallel with the two primary directions being pursued, several additional research activities are being investigated.

6.1 Stochastic Method

One of the several approaches that we took on the rocket nozzle problem was that of a statistical method to make inferences about the model at points that were not tried using the FEA program. A black box approach was decided upon. This method simply means that new parameter values are inserted into an algorithm, and a value is returned that is a statistical prediction of the objective function evaluated at the same parameter values. Two dierent statistical techniques were explored. These methods are the Bayesian Approach and Kriging. Both methods have strong advantages and disadvantages.

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6.1.1 Bayesian Approach

The Bayesian approach [1] expresses prior knowledge of the quantity to be estimated as a probability distribution. This and the likelihood data are then combined using Bayes' Theorem to get a posterior distribution of the unknown quantity. The most important property is that this technique depends on the a priori distribution de ned as P . This is both the main strength and the main weakness of this method. If the researcher has a knowledge of the random process involved, then this is an excellent method. Without such advanced knowledge, there is much uncertainty on how to de ne P . In the rocket nozzle problem, the information we have available are a nite number of the values of the parameters, and the corresponding maximumstress. From this limited information, there is no good way to de ne P . The need for a prior de nition of P does not lend itself well to the black box approach [5].

6.1.2 Kriging

The second method studied was the technique known as Kriging [5]. The word Kriging is synonymous with optimal prediction. This method makes inferences on the unobserved values of the random process from the data at known spatial locations (the key word here is spatial). Kriging assumes that the observed values, in our case the maximum stress, are the a random process. The algorithm uses the known values to predict values at unobserved locations. Weights are assigned to the known value in such a manner that the points close to the point being estimated have more weight. Therefore, severe outliers do not have an adverse eect on the predictor as in other statistical methods. Kriging assigns weights to the known values by the use of the variogram de ned by 2 (h) = var (Z (s + h) ? Z (s)); where Z (
) is the random process. Here, the quantity h is an actual step size. Consider the model Z (s) =  +  (s); where Z (s) is the value of the random process at the spatial location s, which is equal to the mean of the process plus the spatial variation. This leads to the following simple Kriging predictor n X P (Z ) = i Z (si ); i=1 P n where i=1 i = 1. Here the i are Lagrange multipliers, and the constraint that the sum = 1 guarantees unbiasedness [5]. There are several advantages of this method. Kriging is directly related to splines [5]. No prior knowledge of the probability distribution of the process is required as in the Bayesian method. This technique also minimizes the mean squared error (MSE), and is minimumvariance unbiased estimator. The algorithm uses vector and matrix algebra, hence is well adapted to computation. We decided that the Kriging method was more suitable for the black box approach we are attempting to develop.

6.1.3 Modi ed Kriging Algorithm

In the rocket nozzle problem, we are not dealing with variables in a spatial context. Hence, a modi ed Kriging technique [5], is employed, which uses a dierent form correlation which is de ned in a later section. Let s = fs1 ; :::; skg be the independent variables and y(s) = fy(s1 ); :::; y(sn)g be the corresponding values of the objective function. If we assume that y(s) is a random process and

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de ne Y (s) as a random variable, where Ys = fY (s1 ); :::; Y (sn )g. We can then consider the linear predictor y(x) = cT (x)y(s) of y(x) at an untried x. If we replace y(s) by the corresponding random quantity Ys = [Y (s1 ); :::; Y (sn )], treat y(s) as random, and compute the mean squared error averaged over the random process, the best linear predictor is obtained by choosing the n  1 vector c(x) to minimize MSE [y (x)] = E [cT (x)Ys ? Y (x)]2 ; subject to the unbiased constraints

E [cT (x)Ys ] = E [Y (x)]: (8) The model treats the deterministic response y(x) as a realization of a random function (stochastic process), Y (x) that includes a regression model Y (x) =

k X j =1

j fj (x) + Z (x):

The random process Z (:) has a mean of 0 and covariance V (w; x) = 2 R(w; x) between Z (w) and Z (x). Here, 2 is the process variation, and R(w; x) is a one-dimensional distance correlation. The model consists of two components, a generalized least-squares predictor, and an interpolation of the residuals as if there were no regression model. The following de nitions are required for the modi ed Kriging algorithm. De ne f (x) = [f1(x); :::; fk(x)]T . These are the k functions in the regression used to model the interaction eects if a block or random square design are used. In our case, they are all equal to 1. Next, de ne the matrix F as the matrix of the values of the independent variables

F = [f (s1 )T ; :::; f (sn)T ]T : Each row i is a set of parameter values, and Ys is the value of the maximum stress observed for these values. R is an n  n matrix de ned as follows. R = fR(si ; sj )g, where 1  i  n, and 1  j  n, a matrix of stochastic process correlations between each set of independent variables. De ne r(x) as a vector of correlations between evaluated independent variables and the vector x, a new set of independent variables for which we wish to predict the response. r(x) = [R(s1 ; x); :::;R(sn; x)]. With these de nitions, the MSE is MSE = 2 [1 + c(x)T Rc(x) ? 2c(x)T r(x)]; where an estimate of 2 is n1 (ys ? F )T R?1 (ys ? F ). Here, = (F T R?1 F )?1F T R?1Ys . To determine c(x) subject to the constraints of Equation (8), we must solve the following linear system. Here, the (x) are the Lagrange multipliers for the constrained minimization of the MSE.      0 FT (x) = f (x) F R c(x) r(x) Our predictor can now be written as y = + r(x)T R?1(Ys ? F ): The two terms on the right are uncorrelated. Therefore, this model can be thought to have two stages, a general linear regression and then an interpolation of the residuals as if there were no regression model.

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6.1.4 Correlation

The model uses weighted known values to estimate the values at unobserved locations/parameter values. This is accomplished using a one-dimensional distance correlation

R(w; x) = kj=1e?(wj ?xj ) : The parameter values are adjustable 0    2 and 1    2. In our case, we set  =  = 2 for robustness. With these parameter values, the quantity (wj ? xj ) is large if 1  abs(wj ? xj ), and the corresponding value for R(w; x) is small. This ensures that very little weight is given for points that \not near to each other". Note that 0  R(w; x)  1. After several new points have been estimated, these parameters may be re ned using maximum likelihood estimates [5]. The only term of the predictor y = + r(x)T R?1 (Ys ? F ) that contains a variable is r(x). All the other terms are data. Our plan is to use GlobSol to minimize the predictor y(x). Once we have these values, the FEA program will be run to generate a new value of the maximum stress. These values form a new row in the matrix F , and there is a corresponding new value in the vector Y s. The predictor is minimized again, and the process is repeated. In this manner, we hope to improve the results and determine a set of parameter values that improve on the current results. We will not be a able to guarantee that this is the optimum solution, but it should be an improvement. GlobSol will also be used to obtain better estimates of p and  in the calculation of the correlation matrix R. Here we must minimize (det R) n1 2 , which is a function of the correlation parameters and the data. With these new parameter values, the algorithm will be run again, with improved accuracy.

6.2 First-Principles FEA Method

We have just described stochastic methods we explored. Now we turn to the second possible approach, the derivation of an \all-together" nite element method, derived from rst principles, and suitable for use in an interval algorithm. In order to better understand the \all-together" FEA, we rst apply it to a VERY simple problem.

6.2.1 Description of Problem

Consider an axially loaded elastic bar. What cross-sectional area would result in the smallest stress at any point in the bar? ! Wall: ! #### ! #### ! ####BBBBBBBBBBBBBBBBBBBBB --> Right_End_Force (F) ! #### ! #### ! | | ! x = 0 L ! ! A = A(x) = Area (in^2) ! E = E(x) = Young's modulus (lb/in^2) ! u(x) = displacement due to the force (in) ! strain(x) = epsilon = delta u / delta x (in/in) ! stress(x) = sigma = E * strain(x) (lb/in^2) ! applied load (F) = Right_End_Force (lb) ! bar length (L) (in)

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We partition the bar into elements of equal length. The bar is xed on the right side. There is a force on the left side that is compressing the bar, and there is an axial load that acts on the entire length of the bar. The equation that describes this system [4] is d AE du  + b = 0; dx dx where A = cross-sectional area of the bar (in2 ), E = Young's or Elastic modulus (lb/in2) (an engineering constant that depends on the material), b = the axial loading (lb), and u = displacement (in). A nite element formulation of the problem looks like

Ku = F; where

n = number of nodes, x = n-dimensional vector representing the nodes u = n-dimensional vector representing the displacements (in), K = n  n elastic matrix (lb/in), and F = n-dimensional force vector (lb). The nite element formulation gives the displacement at each node. Once that information is determined, it is a relatively easy task to determine the strain on each element by taking the dierence of the displacement of the two ends of the element and dividing by the length of the element. Then we take the maximum of these strains to get the maximum strain. Given the strain, we can calculate the stress. To state the optimization problem more precisely now, we wish to minimize the maximum strain given by the nite element formulation given above. As a prototype for the design optimization, we view the area of the bar as a design variable, and minimize the maximum stress. Of course, we know that the stress goes down as the area goes up, so the \optimal" design will have the largest area we allow.

6.2.2 Solving the Problem

The problem as stated above poses a couple of major problems. 1. Evaluating the objective function as it is presented above involves solving a system of equations. 2. The objective function as stated above is not dierentiable due to the min-max nature of the problem. This poses a number of problems for the optimization software that we are using. These two problems are both overcome by restating the problem with more constraints. Instead of making the solution of the system of equations part of the objective function, we pass it to the optimizer as a system of equality constraints. This means we give the optimizer control over the components of the displacement vector. The idea is to solve the optimization and the system of equations all at once. In some sense the system of equations is never really solved, we simply pass the system in as a set of constraints. However, the optimizer is given the task now of nding points that meet the constraints. This means the optimizer must nd points that solve the system of equations. This does have a downside of adding dimensions to the problem, but it allows us to use the optimizer to solve the system of equations at the same time that it searches for an optimum point. The min-max problem was also dealt with by restating the problem with more constraints. we added another variable, the maximum strain. The maximum strain is constrained to be

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greater than the strain of any given element. Again, this adds more dimensions to the problem because we need to add a slack variable so that each inequality constraint could be speci ed as an equality constraint. After making the above changes, we can now restate the problem as below: See the Maple worksheet in Appendix A. Independents: u(1 : n), slack(1 : n ? 1), Area, Max strain() Objective: Max stress() = E * Max strain() Constraint: FEA equations: Sum of forces = 0 at each node u1 = 0 Bar is xed at the wall Node 1 ?AEu1 + AEu2 =
x1  Right End Force AEui?1 ? 2AEui + AEUi+1 = 0 Nodes i = 2 : n ? 1 AEun?1 ? AEun = ?
xn?1  Right End Force Node n In principle, we solve these (over-determined) equations for the displacement u(1 : n). Then we compute ? ui ; for each element i = 1 : n ? 1 straini := xui+1 ? x i+1

i

Constraint: Min/Max: Max strain - straini = slacki  0.

6.2.3 Bene ts and Drawbacks

There are two major bene ts to approaching a nite element optimization problem this way: 1. The optimization software will provide automatic optimization of the problem in one run. Traditionally, a designer will try out a number of dierent designs and run them through a nite element program to see what happens. This can be a very time-consuming process and there is often no guarantee that the designer will nd the optimal design. Using our new approach, the designer will let the computer nd the optimum design. This should result in better designs being produced more quickly. 2. The optimization software gives guaranteed bounds on the answer. This means, assuming the program is coded correctly, the answer is guaranteed to lie within the bounds put out by the optimizer. In the case of the rocket nozzle design, where there can be many local optima, this method should be able to say which point is really the optimum point. There are also some drawbacks: 1. The method requires that the system of equations that are at the heart of the nite element method be stated in such a way so that the coecients are expressed in terms of the design variables. Since, in general, developing these coecients can involve integration, this is not an easy task. In addition, this will mean that converting an existing nite element program to use this method may require rewriting a major portion of the program. 2. A low dimensional optimization problem becomes a high-dimensional problem. (In my example problem, a one-dimensional optimization problem became an eleven-dimensional problem.) In a model where there may be two or three dimensions to the displacement, this problem becomes two or three times worse. This high-dimensionality will require extensive CPU-time. However, since running a nite element program a number of times is also a process that requires a large amount of computation, it may be that our new approach still turns out to be reasonably fast.

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7 Set-up: How to Apply GlobSol GlobSol, a rigorous global search tool for continuous problems written in Fortran 90 by Kearfott [3], is used as the optimizer for the simple FEA problem Independents: u(1 : n), slack(1 : n ? 1), Area, Max strain() Objective: Max stress() = E * Max strain() Constraint: FEA equations: Sum of forces = 0 at each node u1 = 0 Bar is xed at the wall Node 1 ?AEu1 + AEu2 =
x1  Right End Force AEui?1 ? 2AEui + AEUi+1 = 0 Nodes i = 2 : n ? 1 AEun?1 ? AEun = ?
xn?1  Right End Force Node n

7.1 Using GlobSol

The steps described here are compressed from [7], where a more detailed step-by-step use of GlobSol is described.

7.1.1 Make le

Here is the Unix Make le we use to control the running of this example: # File: .../Examples/Constrained/Bar2/Makefile 19 Jan 1998 # # Author: George F. Corliss, Marquette University MSCS, 15 May 1997 # # Target: PROB = bar DATA = # Version (DT1, DT2, etc): VERS = 1 # INTOPT directories and files: INTOPT_DIR = /home/globsol/GLOB0.7 INTLIB_LIB = $(INTOPT_DIR)/intlib.alt/intlib.a OVERLOAD_LIB = $(INTOPT_DIR)/overload/overload.a INTOPT_BIN = $(INTOPT_DIR)/f90intbi # Fortran 90 compiler commands: # (Set to the appropriate commands for the compiler / OS) -# The following two commands are for Sun f90, with debugging and profiling #Compile = f90 -g -M$(INTOPT_DIR)/overload/ -c -pg #Link = f90 -g -M$(INTOPT_DIR)/overload/ -pg # The following two commands are for Sun f90, with optimization #Compile = f90 -O -M$(INTOPT_DIR)/overload/ -c -pg #Link = f90 -O -M$(INTOPT_DIR)/overload/ -pg # # The following two commands are for NAG f90, with debugging and profiling #Compile = f90 -g -I$(INTOPT_DIR)/overload/ -c #Link = f90 -g -I$(INTOPT_DIR)/overload/ # The following two commands are for NAG f90, with optimization Compile = f90 -O -I$(INTOPT_DIR)/overload/ -c Link = f90 -O -I$(INTOPT_DIR)/overload/

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# Operating system commands: OBJ = o Copy = cp Delete = /bin/rm -f Nice = nice

# Here is where the work gets done: DO_DIR: $(PROB).OT$(VERS)

############################################################ # Step 1. Create a code list for the objective function and # constraints. See also p. 85. # You MUST supply $(PROB).f90 # You may edit OVERLOAD.CFG ############################################################ $(PROB).CDL : $(PROB) $(DATA) OVERLOAD.CFG ./$(PROB) $(PROB): $(PROB).$(OBJ) $(INTLIB_LIB) $(Link) $(PROB).$(OBJ) $(OVERLOAD_LIB) $(INTLIB_LIB) -o $(PROB) $(PROB).o: $(PROB).f90 $(OVERLOAD_LIB) $(Compile) $(PROB).f90

############################################################ # Step 2. Symbolically differentiate the code list to # produce a gradient code list. See also p. 91. ############################################################ $(PROB)G.CDL: $(PROB).CDL $(PROB).NAM $(PROB)G.NAM \ $(INTOPT_DIR)/overload/makegrad \ $(INTOPT_DIR)/overload/optimize_codelist $(INTOPT_DIR)/overload/makegrad < $(PROB).NAM $(INTOPT_DIR)/overload/optimize_codelist < $(PROB)G.NAM $(Copy) $(PROB)GO.CDL $(PROB)G.CDL # The following two *.NAM files are used to avoid interactive # response to prompts from makegrad, optimize_codelist, and opttest: $(PROB).NAM : echo $(PROB) > $(PROB).NAM echo $(VERS) >> $(PROB).NAM $(PROB)G.NAM : echo $(PROB)G > $(PROB)G.NAM

############################################################ # Step 3. Create the box data file. See p. 201. # You MUST supply the file $(PROB).DT1

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############################################################

############################################################ # Step 4. Run RUN_GLOBAL_OPTIMIZATION ############################################################ $(PROB).OT$(VERS): $(PROB)G.CDL $(PROB).DT$(VERS) $(PROB).NAM \ OVERLOAD.CFG INTNEWT.CFG OPTTBND.CFG \ $(INTOPT_BIN)/opttest $(Nice) $(INTOPT_BIN)/opttest < $(PROB).NAM echo echo view $(PROB).OT$(VERS) to see the results

############################################################ # Clean up generated files ############################################################ clean: $(Delete) $(Delete) $(Delete) $(Delete) $(Delete) $(Delete) $(Delete) $(Delete) $(Delete) $(Delete)

$(PROB).$(OBJ) $(PROB) $(PROB).CDL $(PROB).NAM $(PROB)G.NAM $(PROB)G.CDL $(PROB)GO.CDL $(PROB).OT$(VERS) OPTTEST.TBL gmon.out

7.1.2 Codelist program: bar.f90

We need to write the Fortran 90 program that de nes the optimization problem. ! File: .../Examples/Constrained/Bar2/bar.f90 PROGRAM FEA_of_Axially_Loaded_Bar USE CODELIST_CREATION ! ! ! ! ! ! ! ! ! ! ! ! ! !

Purpose: Minimize the maximum stress by altering the design AREA Of course, we minimize the stress by taking AREA as large as possible. Our goal is to get SOME FEA running with GlobSol Version: All-together, 4 FEA nodes Authors: Andy Johnson, Frank Fritz, and George Corliss, Marquette University Modifications: 23 Dec 1997 George Corliss, Fortran code 22 Dec 1997 George Corliss, Maple worksheet 22 Dec 1997 Frank Fritz, Maple worksheet 10 Dec 1997 Andy Johnson, Maple worksheet

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! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

With help from http://www.nd.edu/~batill/www.ame446/ Element Formulation - Axial Rod I'll take the force in tension. Positive is to the right. Wall: #### #### ####BBBBBBBBBBBBBBBBBBBBB --> Right_End_Force (F) #### #### | | x = 0 L L = Bar length (in) F = Load (lbs) A = A(x) = Area (in^2) E = E(x) = Young's or Elastic modulus (lbs / in^2) u(x) = displacement due to the force (in) strain(x) = epsilon = delta u / delta x (in / in) stress(x) = sigma = E * strain(x) (lbs / in^2) Details of the analysis of a finite element are in the Maple worksheet bar1.mws Known solution: u(x) = F x / (A E) (in) strain(x) = u / x (in / in) stress(x) = epsilon * E, independent of x (lbs / in^2) Maximum stress = epsilon * E (lbs / in^2) Minimize the maximum stress by taking large Area Strategy: Formulate a Finite Element Model Let FEA equations be equality constraints Let Max strain - strain(x) = slack(x) >= 0 Problem: Vary AREA (up to a maximum limit) minimize Max stress subject to solving the FEA problem As posed to GlobSol: Finite element nodes: Independents: u(1:n), Objective: Max stress Constraints: FEA equations: Sum Min/Max: Max

x(1:n) slack(1:n-1), Area, Max strain = E * Max strain of forces = 0 at each node strain - strain(x) = slack(x) >= 0

IMPLICIT NONE

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INTEGER, PARAMETER :: NUMBER_OF_NODES = 4 INTEGER, PARAMETER :: NUMBER_OF_ELEMENTS = NUMBER_OF_NODES - 1 INTEGER, PARAMETER :: AREA = NUMBER_OF_NODES + NUMBER_OF_ELEMENTS + 1 INTEGER, PARAMETER :: MAX_STRAIN = NUMBER_OF_NODES + NUMBER_OF_ELEMENTS + 2 DOUBLE PRECISION, PARAMETER :: LENGTH_OF_BAR = 6.0D0 ! Rod length DOUBLE PRECISION, PARAMETER :: RIGHT_END_FORCE = 6.6D4 ! Applied load DOUBLE PRECISION, PARAMETER :: E = 3.0D7 ! Elastic modulus INTEGER :: I DOUBLE PRECISION, DIMENSION(NUMBER_OF_NODES) :: X DOUBLE PRECISION, DIMENSION(NUMBER_OF_ELEMENTS) :: DELTA_X TYPE(CDLVAR), DIMENSION(MAX_STRAIN) :: Y ! u(i) = Y(i), i = 1, NUMBER_OF_NODES ! slack(i) = Y(NUMBER_OF_NODES+i), i = 1, NUMBER_OF_ELEMENTS ! Area = Y(NUMBER_OF_NODES + NUMBER_OF_ELEMENTS + 1) ! Max_strain = Y(NUMBER_OF_NODES + NUMBER_OF_ELEMENTS + 2) TYPE(CDLLHS), DIMENSION(1) :: OBJECTIVE ! Maximum stress TYPE(CDLEQ), DIMENSION(NUMBER_OF_NODES + NUMBER_OF_ELEMENTS) :: CONSTRAINT ! MUST come before any use of CDL variables OUTPUT_FILE_NAME = 'bar.CDL' CALL INITIALIZE_CODELIST (Y) DO I = 1, NUMBER_OF_NODES X(I) = LENGTH_OF_BAR * (I - 1) / NUMBER_OF_ELEMENTS END DO DO I = 1, NUMBER_OF_ELEMENTS DELTA_X(I) = X(i+1) - X(i) END DO ! Minimize maximum stress = Young's modulus * maximum strain OBJECTIVE(1) = E * Y(MAX_STRAIN) ! Subject to boundary condition u[1] = 0 ! enforced by bound constraints ! Subject to sum of forces at the Wall or left end ! Contributions: External + element 1 CONSTRAINT(1) = - RIGHT_END_FORCE & + Y(AREA) * E * (Y(2) - Y(1)) / DELTA_X(1) ! Subject to sum of forces at each interior node ! Contributions: F(i+1) from Element i-1 + F(i) from Element i ! Sum_of_Forces := ! + A * E * u[i-1] / delta_X[i-1] ! - A * E * u[i] / delta_X[i-1] ! - A * E * u[i] / delta_X[i] ! + A * E * u[i+1] / delta_X[i]; DO I = 2, NUMBER_OF_NODES - 1 CONSTRAINT(I) = Y(AREA) * E & * ( (Y(I-1) - Y(I)) / DELTA_X(I-1) & + (Y(I+1) - Y(I)) / DELTA_X(I) ) END DO

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! Subject to sum of forces at the free or right end ! Contributions: F(i+1) from Element i-1 + External ! Sum_of_Forces := Right_End_Force ! + A * E * u[i-1] / delta_X[i-1] ! - A * E * u[i] / delta_X[i-1]; ! This equation is redundant, but it MUST also hold I = NUMBER_OF_NODES CONSTRAINT(I) = Y(AREA) * E * (Y(I-1) - Y(I)) / DELTA_X(I-1) & + RIGHT_END_FORCE ! Subject to Maximum strain >= strain(i) ! or Maximum strain = strain(i) + slack(i), slack(i) >= 0 ! ! The strain at each ELEMENT, not at each NODE is ! strain := (u[i+1] - u[i]) / delta_X[i], ! i = 1 .. Number_of_Elements DO I = 1, NUMBER_OF_ELEMENTS CONSTRAINT (NUMBER_OF_NODES+I) = Y(MAX_STRAIN) & - (Y(I+1) - Y(I)) / DELTA_X(I) - Y(NUMBER_OF_NODES+I) END DO CALL FINISH_CODELIST CALL SUN_DEPENDENT__CLEAR_UNDERFLOW END PROGRAM FEA_of_Axially_Loaded_Bar

7.1.3 OVERLOAD.CFG NEQMAX, NROWMAX, NCONSTMX, BRANCHMX, BINARY_CODELIST, NSTRINGMX 1000 100000 2000 1000 T 100

7.1.4 bar.DT1

We need a box data le de ning the search region. We take a symmetric region around the known correct answer: 1.0D-15 ! .../Examples/Optimization/Bar2/bar.DT1 0 0 ! Boundary condition u(1) = 0 0.002 0.005 ! e (element 1) in/in 4 nodes 0.004 0.01 ! e (element 2) 0.008 0.02 ! e (element 3) 0 0.001 ! Slack variable for element 1 0 0.001 ! Slack variable for element 2 0 0.001 ! Slack variable for element 3 0.5 1.0 ! Area. Only "REAL" variable 0 0.02 ! Maximum calculated strain T T ! Boundary condition F F F F F F ! 4 nodes T F ! Slack variable T F T F ! 3 elements T T F F

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0 0.0044 0.0088 0.0132 0 0 0 1.0 0.0022 T

! These are the correct answers

7.1.5 OPTTBND.CFG

The le OPTTBND.CFG contains con guration parameters for the optimization step. prnt ctrl: OPT._TEST, RED._INT._NWT, REDGS, PR_LANCE, FIND_APP_OPT, CERT._FEAS., PR_CONSTR., PRBDY, PRNTCMPL, INFLATE, MAXITR, newton? midpt? quad, full, epsx USE_GRAD_TEST USE_SUBSIT GOU PRT_LENGTH PRINT_SUBSIT LEAST_SQUARES_FUNCTIONS 0 0 0 0 0 0 0 0 0 0 10000 T T T F 1d-6 T F 7 1 0 F

7.1.6 bar.OT1 Output from RUN_GLOBAL_OPTIMIZATION on DATA WAS TAKEN FROM DATA FILE: bar.DT1 Initial box: 0.000000000000000000D+00 0.200000000000000004D-02 0.400000000000000008D-02 0.800000000000000017D-02 0.000000000000000000D+00 0.000000000000000000D+00 0.000000000000000000D+00 0.500000000000000000D+00 0.000000000000000000D+00

BOUND_CONSTRAINT: T T F F F F

F F

T F

01/29/1998

at

0.000000000000000000D+00 0.500000000000000010D-02 0.100000000000000002D-01 0.200000000000000004D-01 0.100000000000000002D-02 0.100000000000000002D-02 0.100000000000000002D-02 0.100000000000000000D+01 0.200000000000000004D-01

T F

T F

T T

F F

--------------------------------------CONFIGURATION VALUES: EPS_DOMAIN: 0.1000D-14 MAXITR: DO_INTERVAL_NEWTON: T QUADRATIC: T VERY_GOOD_INITIAL_GUESS: T USE_SUBSIT: T OUTPUT UNIT: 7 PRINT_LENGTH: 2 Default point optimizer was used.

5000 FULL_SPACE: F

LIST OF SMALL BOXES:

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21:23:31.

Box no.: 1 Box coordinates: 0.000000000000000000D+00 0.439968377223398308D-02 0.879968377223398247D-02 0.131996837722339836D-01 0.000000000000000000D+00 0.000000000000000000D+00 0.000000000000000000D+00 0.100000000000000000D+01 0.219968377223398294D-02

0.000000000000000000D+00 0.440031622776601745D-02 0.880031622776601512D-02 0.132003162277660162D-01 0.000000000000000000D+00 0.000000000000000000D+00 0.000000000000000000D+00 0.100000000000000000D+01 0.220031622776601645D-02

PHI: 0.659905131670194532D+05 0.660094868329805176D+05 B%LIUI(1,*): F F F F F F F F F B%LIUI(2,*): F F F F F F F F F B%SIDE(*): T F F F T T T T F B%PEEL(*): F F F F F F F F F Level: 1313165645 Box contains the following approximate root: 0.000000000000000000D+00 0.440000000000000027D-02 0.879999999999999880D-02 0.131999999999999999D-01 0.000000000000000000D+00 0.000000000000000000D+00 0.000000000000000000D+00 0.100000000000000000D+01 0.219999999999999970D-02 OBJECTIVE ENCLOSURE AT APPROXIMATE ROOT: 0.659999999999999418D+05 0.660000000000000291D+05 Unknown = T Contains_root = F Changed coordinates: F F F F F F F F F ------------------------------------------------THERE WERE NO BOXES CORRESPONDING TO VERIFIED FEASIBLE POINTS. ALGORITHM COMPLETED WITH LESS THAN THE MAXIMUM NUMBER, 5000 OF BOXES. No. dense interval residual evaluations -- gradient code list: Number of orig. system C-LP preconditioner rows: 22 Total number of forward_substitutions: 3348 Number of Gauss--Seidel steps on the dense system: 22 Number of gradient evaluations from a gradient code list: 108 Total number of dense slope matrix evaluations: 181 Total number second-order interval evaluations of the original function: 161 Total number dense interval constraint evaluations: 1204 Total number dense interval constraint gradient component evaluations: 889 Total number dense point constraint gradient component evaluations: 5824

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243

Total number dense interval reduced gradient evaluations: 17 Total number of calls to FRITZ_JOHN_RESIDUALS: 6 Number of times a box was rejected in the interval Newton method due to an empty intersection: 4 Number of times the interval Newton method made a coordinate interval smaller: 11 Total time spent doing linear algebra (preconditioners and solution processes): 8.6956977844238281E-02 Number of times the approximate solver was called: 213 Number Fritz-John matrix evaluations: 5 BEST_ESTIMATE: 0.179769313486231571+309 Total number of boxes processed in loop: 5 Overall CPU time: 0.533692776039242744D+01 CPU time in PEEL_BOUNDARY: 0.483129392936825752D+01 CPU time in REDUCED_INTERVAL_NEWTON: 0.149774551391601562D+00

7.2 Results

The results of this simple FEA problem show that GlobSol successfully solved the optimization problem requiring 1.75 seconds of CPU time running on the Marquette University STUDSYS computer.

8 Interpretation and pictures 9 Conclusion Tie results back to Section 4, Why is this important?

References [1] Noel A. C. Cressie. Statistics for Spatial Data. J. Wiley and Sons, New York, 1993. [2] Jonathan Kmita Hart. Material Geometry Optimization of Involute Structures Using Sensitivity Derivatives With Respect to Global Design Variables. PhD thesis, University of California, Los Angeles, 1988. [3] R. Baker Kearfott. Rigorous Global Search: Continuous Problems. Kluwer Academic Publishers, Dordrecht, Netherlands, 1996. [4] Niels Ottosen and Hans Petersson. Introduction to the Finite Element Method. Prentice Hall, Hertfordshire, England, 1992. [5] Jerome Sacks, William J. Welch, Toby J. Mitchell, and Henry P. Wynn. Design and analysis of computer experiments. Statistical Science, 4(4.409{435):409{427, 1989. [6] M. F. Spotts. Design of Machine Elements. Prentice-Hall, Inc., Englewood Clis, N.J., 1978. [7] Paul J. Thalacker, Kristie Julien, Peter G. Toumano, Joseph P. Daniels, George F. Corliss, and R. Baker Kearfott. Globsol case study: Portfolio management (Banc One). Technical Report No. ???, Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, Wisc., 1998. Global Solutions Working Note 6.

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