Multidisciplinary Design Optimization – Some Formal Methods, Framework Requirements, and Application to Vehicle Design

Srinivas Kodiyalam

SGI HPC Applications & Market Development

Server & Supercomputing Business 1600 Amphitheatre Parkway, MS 405

Mountain View, California 94043-1351. e-mail: skodiyal@sgi.com

& Jaroslaw Sobieszczanski-Sobieski

Manager, Computational AeroSciences & Multidisciplinary Research Coordinator

NASA Langley Research Center Mail Stop: 139

Hampton, Virginia 23681. Email: j.sobieski@larc.nasa.gov

1. Abstract: A vehicle is an engineering system whose successful design requires harmonization of a number of objectives and constraints that, in principle, can be modeled as a constrained optimization in the space of design variables. However, dimensionality of such optimization and the complexity and expense of the underlying analysis suggest a decomposition approach to enable concurrent execution of smaller and more manageable tasks. In order to preserve the couplings that naturally occur among the elements of the whole problem, such optimization by various types of decomposition must include a degree of coordination at the system level. Multidisciplinary Design Optimization (MDO) is a body of methods and techniques for performing the above optimization so as to balance the design considerations at the system and detail levels. The paper is an overview of a few MDO methods selected for their applicability to vehicle systems.

Keywords: Optimization, MDO Methods, MDO Requirements, High Performance Computing (HPC).

Reference to this paper should be made as follows: Kodiyalam, S. and Sobieszczanski-Sobieski, J. (2001) ‘Multidisciplinary design optimization – some formal methods, framework requirements, and application to vehicle design’, Int. J. Vehicle Design (Special Issue), pp. 3–22.

2. Introduction: It is an indisputable fact of physics that in an engineering system such as a road vehicle there are interactions among the physical phenomena and the vehicle hardware parts. These interactions make the vehicle a synergistic whole that is greater than the sum of its parts. Taking advantage of that synergy is the mark of a good design but the web of interactions is difficult to untangle. That difficulty combined with the need to partition the work into subtasks executed simultaneously to compress the project time gave rise to the practice of dividing the detailed design work into specialty areas, each area centered on a physical phenomenon, e.g., stress and strain, or on a hardware subsystem, e.g., the car suspension. The above practice has achieved its purpose of developing a broad work front and compressing the project calendar time but on the downside it impeded trade-offs across

the subtasks boundaries making the design of the vehicle fall somewhat short of optimal. To give a limited example of a trade-off, suppose that the system-level objective is to minimize the vehicle cost while maintaining a certain minimum of the acceleration capability. Obviously, acceleration benefits from both the lighter structure and more power in the engine, but the cost of reducing the structure weight or adding power may not be the same. Thus, the interplay of weight, power, and costs form a three-way trade-off that can be formalized as a mathematical optimization at the system level. Underlying that optimization are subsystem optimizations in the areas of propulsion and structure that engage large number of detailed design variables and considerable specialized expertise. The Multidisciplinary Design Optimization (MDO) has evolved as a new discipline (Sobieszczanski-Sobieski, 1995) that provides a body of methods and techniques to assist engineers in moving engineering system design closer to optimum in a situation illustrated by the above example. Parallel to the development of the above methodology, a number of software packages have been created to facilitate integration of codes, data, and user interface. These packages are often referred to as frameworks; one may consult (Salas and Townsend, 1998) for a few examples. Experience accumulated from these framework implementations and applications has now reached the point where it may be translated into requirements to guide the future framework developments. Accordingly, the focus of this paper is on Multidisciplinary Design Optimization (MDO), including formal methods, framework requirements, and application to vehicle design. 3. System Optimization Problem and Formal MDO Methods: A general system, consisting of three Black Boxes (BB) representing three disciplines such as Aerodynamics, Structures and Performance, is shown in Figure 1.0. The Black Boxes are coupled by exchanging information Y while the variables X and Z, local and system-level, govern their design. Formal MDO methods are intended for synthesis of such generic, multidisciplinary engineering systems, such as aircraft or automotive vehicle, whose design is governed by multiple disciplines [Sobieszczanski-Sobieski and Haftka, 1997, Balling and Sobieszczanski-Sobieski, 1996]. Several MDO methods exists, including, All-in-One (A-i-O) method, Individual Discipline Feasible (IDF) method [Cramer et al., 1994], Concurrent SubSpace Optimization (CSSO) using the Global Sensitivity Equations (GSE) [Sobieszczanski-Sobieski, 1988, Renaud and Gabriele, 1993], Collaborative Optimization (CO) method [Braun and Kroo, 1997], and Bi-Level Integrated System Synthesis (BLISS) method [Sobieszczanski-Sobieski, Agte and Sandusky, 1998]. Variants of several of these methods using polynomial or neural networks based response surfaces have also been investigated [Sobieski and Kroo, 1998, Renaud and Gabriele, 1994, Kodiyalam and Sobieszczanski-Sobieski, 1999]. The key concept in several of these MDO methods is a decomposition of the design task into subtasks performed independently in each of the modules, and a system-level or coordination task giving rise to a two-level optimization. In general, decomposition was

motivated by the obvious need to distribute work over many people and computers to compress the task calendar time. Equally important benefit from the decomposition is granting autonomy to the groups of engineers responsible for each particular subtask in choosing their methods and tools for the subtask execution. As an additional advantage, the concurrent execution of the subtasks fits well the technology of massively concurrent processing that is now becoming available. The general system optimization problem is stated in the following form: Given a set of design variables, X, (1) Find: ∆X Minimize: Φ (X,Y(X)) Satisfy: G(X,Y(X)) Bounds on X. In the problem defined by (1), Y(X) represents the behavior (state) variables, Φ represents the design objective function and G represents the design constraints. A brief description of some of the formal MDO methods used to solve the system optimization problem is provided in the following sub-sections. 3.1 All-in-One Method

The All-in-One method (also referred to as Multidisciplinary Feasibility (MDF) in Cramer et al., 1994) is the most common way of approaching the solution of MDO problems. In this method, the vector of design variables XD is provided to the coupled system of analysis disciplines and a complete multidisciplinary analysis (MDA) is performed via a fixed-point iteration with that value of XD to obtain the system (MDA) output variable U(XD) that is then used in evaluating the objective F(XD, U(XD)) and the constraints g(XD, U(XD)). The optimization problem is:

Minimize: F(XD, U(XD)) Subject to: g(XD, U(XD)) < 0 and, bounds on design variable, XD. If a gradient-based method is used to solve the above problem, then a complete MDA is necessary not just at each iteration, but at every point where the derivatives are to be evaluated. Thus, attaining multidisciplinary compatibility can be prohibitively expensive in realistic application.

Figure 2 shows the data flow in a A-i-O analysis and optimization. In this figure, µij is some spline coefficients obtained using a “fit” Fij of the output of discipline j. Fij may be either an interpolation or an approximation fit. The mapping Eij is an evaluation of the spline representation from discipline j into a form suitable for use by discipline i (for example, calculating structural loads from aerodynamic pressures).

3.2. Individual Discipline Feasible (IDF) Method

The IDF formulation provides a way to avoid a complete MDA at optimization. IDF maintains individual discipline feasibility, while allowing the optimizer to drive the individual disciplines to multidisciplinary feasibility and optimality by controlling the interdisciplinary coupling variables. In IDF, the specific analysis variables that represent communication, or coupling, between analysis disciplines are treated as optimization variables and are in fact indistinguishable from design variables from the point of view of a single analysis discipline solver. The IDF formulation is:

Minimize: F(XD, U(X)) with respect to X = (XD, Xµ) Subject to: g(XD, U(X)) < 0

C(X) = Xµ − 0=µ and bounds on optimization variable, X. XD is the set of design variables and Xµ is the set of interdisciplinary coupling variables. C is referred to as the interdisciplinary constraint. For implementation purposes, we use Jj = Cj

2 < 0.0001, j = 1, number of disciplines. It is important to note that an evaluation of U(X) involves executing all the single discipline analysis codes independently with simultaneously available multidisciplinary data X. Therefore, the analysis computations can be performed concurrently.

Figure 3 shows the data flow in an IDF analysis and optimization. The notations in Figure 3 are similar to those in Figure 2.

3.3. Collaborative Optimization (CO) The CO formulation is a two-level hierarchical scheme for MDO, with the top level being the system optimizer that optimizes on the multidisciplinary variables (or, system level targets, z) to satisfy the interdisciplinary compatibility constraints (J*) while minimizing the system objective (F). The objective of each subsystem optimizer is to minimize in a least squares sense the discrepancy between the subset of subspace design variables (xi) and subspace analysis computed responses (yj) that are common to more than one subspace analysis block and the system level values of these variables, z, while satisfying the subspace constraints (gj). The system level design variables, z, are considered to be fixed within a subspace problem. A distinction is made between the disciplinary design variables xsj, only of importance to subspace analysis j, and the interdisciplinary design variables xj, which are common to more than one subspace analysis block. For implementation purposes, the interdisciplinary compatibility constraints (J's) were formulated as inequality constraints (J < 0.0001) as against strict equality constraints (J = 0.0). J is defined as: Jj = | Xj – Zj

s |**2 + | Yj – Zjc |**2

where, Z = {Zs, Zc}; Zs represents the system design variable and Zc represents the system coupling variable. The collaborative optimization formulation is intended for cases when the number of disciplinary variables xsj is much larger than the number of interdisciplinary variables xj..

In other words, this formulation is intended for solving design problems with loosely coupled analyses of individually large dimension. Figure 4 shows the data flow in a CO analysis and optimization. The variables used in Figure 4 are defined in the CO method description provided under Section 2.3. 3.4. Concurrent Sub Space Optimization (CSSO) CSSO is a non-hierarchic system optimization algorithm that optimizes decomposed subspaces concurrently, followed by a coordination procedure for directing system problem convergence and resolving subspace conflicts. This corresponds to common design practice where individual design teams optimize their local component designs and compromises are made at the integrated product team or system level. In CSSO, each subspace optimization problem is a system level problem formulated with respect to a subset of the total system design vector. Within the subspace optimization, the non local states that are required to evaluate the objective and constraint functions are approximated using the Global Sensitivity Equations (GSE). Please refer to Sobieszczanski-Sobieski, 1988, for a detailed description of the GSE. The CSSO method provides for multidisciplinary analysis feasibility at each cycle but deals with all the design variables simultaneously at the system/coordination problem level. 3.5. Bi-Level Integrated System Synthesis (BLISS) The recently introduced BLISS method uses a gradient-guided path to reach the improved system design, alternating between the set of modular design subspaces (disciplinary problems) and the system level design space. BLISS is an A-i-O like method in that a complete system analysis performed to maintain multidisciplinary feasibility at the beginning of each cycle of the path. With BLISS, the general system optimization problem is decomposed into a set of local optimizations dealing with a large number of detailed local design variables (X) and a system level optimization dealing with a relatively small number of global variables (Z) in comparison with the other MDO methods. In optimization it is useful to distinguish between X and Z because: • The X variables are associated with individual components and, therefore, they

tend to be clustered. Also, the constraints they govern directly, e.g., the stringer buckling in built-up, thin-walled structures typical of aerospace vehicles, tend to be highly nonlinear. The total number of the X variables in a typical airframe is in thousands but their number in an individual substructure is likely to be quite small.

• The number of Z variables is much smaller than the total number of X variables. • Nonlinearity of the overall behavior constraints, such as displacements, with respect

to X and Z tends to be weaker than that of the local strength and buckling constraints.

With BLISS, the solution of the system level problem is obtained using either (i) the optimum sensitivity derivatives of the behavior/state (Y) variables with respect to system level design variables (Z) and the Lagrange multipliers of the constraints obtained at the solution of the disciplinary optimizations, or (ii) a response surface constructed using either the system analysis solutions or the subsystem optimum solutions. A flow chart of the BLISS method is shown in Figure 5. More details of these different MDO methods can be obtained from the above mentioned references. 4.0 MDO Framework Requirements: Several requirements exist for a framework to provide an easy-to-use and robust MDO environment. Sobieszczanski-Sobieski (1999) lists the key attributes for a MDO environment as follows: Computer Speed, Computer Agility, Task Decomposition, Sensitivity Analysis, Human Interface and Data Transmission. The framework requirements for MDO application development have been outlined in the work of Salas and Townsend (1998) as follows: Architectural Design, Problem Formulation, Problem Execution, and Access to Information. This paper proposes several key requirements for process integration and problem solving capabilities in a MDO framework. These include: • Provide for quick and easy linking of analysis tools. The set of analysis tools to be

linked could involve such tools as COTS software (CAD, CAE, CAM), legacy (in house) codes, spreadsheets, databases, and tools to capture user’s knowledge.

In relation to CAD tools, it would be highly desirable to support both a CAD-centric approach with the CAD tool as the master as well as an MDO-centric approach with the CAD tool as a slave to the MDO framework. In relation to CAE tools, the framework should provide for efficient transfer, storage and access of data, including analysis responses as well as behavior sensitivities.

• Provide effective support for geographically distributed modeling and optimization, through CORBA client-server compliancy of the software tools and models, facilitating both tight and loose collaboration, ranging from OEMs, customers, suppliers and consultants. The internet provides a means to link designers, engineers, tools and data in a geographically distributed setting. This then implies that gathering information through internet is essential for collaborative design. The MDO framework should facilitate integration with the world wide web and simplify information/content identification and manipulation through using markup languages such as XML that is more robust than parsing the HTML files.

• Access to efficient parametric study capability such as design of experiments (DOE) based procedures, including full factorial designs, fractional factorial designs (orthogonal arrays), central composite designs and latin hypercube designs.

As opposed to varying design variables (factors) independently or randomly varying numerous factors, formal Design of Experiments (DOE) offers a systematic approach to study the effects of multiple variables/factors on product/process performance by providing a structured set of analyses in a design matrix. More specifically, the use of DOE provides an efficient and effective method for determining the most significant factors and interactions in a given design problem.

• Access to a full range of optimization search strategies ranging from gradient based numerical optimization, simulated annealing and genetic algorithms and most importantly, an optimization advisor that can appropriately recommend the optimization algorithm or a combination of algorithms (hybrid optimization plan) to be used for solution of the user problem.

The key advantage of an optimization advisor is that it empowers less experienced users with the benefits of optimization techniques. The advisor should guide the user toward the choice of appropriate optimization search techniques to solve the user’s given problem. Unlike the black box approach, the optimization advisor should be based upon a set of intelligent problem formulation heuristics, such as the design space information, analysis tools information, and user knowledge about the optimization problem domain. Design space information could consist of details regarding number of design variables, number of design constraints, type of parameters (real, integer, discrete, or mixed), magnitude of variable variance, the existence of equality constraints, the existence of discontinuous feasible spaces and the nonlinearity of optimization constraint functions. Analysis tools related information such as simulation code type (linear or nonlinear), simulation code execution time (low or high) and code precision could also be critical in the choice of optimization algorithms. In addition, user’s knowledge of the design problem could be beneficial in defining design rules and effective utilization of such rules in the optimization process.

• Access to a full range of model approximation techniques such as polynomial,

Kriging, or neural networks based response surfaces, sensitivity based Taylor series linearization, and variable complexity models.

Creating a simple mathematical model to approximate the behavior of the expensive computational tool and using it during optimization for gradient calculations and finding the next optimum search direction can reduce the number of "expensive and exact" analyses by a factor of 10. In addition, these models can improve convergence for “noisy” computational tools by smoothing the response function. Such mathematical models can be based on representing the function of interest as a low degree polynomial with coefficients found by the regression analysis. This approach

is the basis of the Response Surface Methodology (RSM), widely used in many MDO applications. Another approach is to use gradients of the response function for constructing a Taylor Series Approximation. Several variations of TSA include linear, reciprocal, and hybrid approximations as well as two-point approximations which use function value and function gradients at the previous design point to account for the function curvature. The basis of another type of approximation, Variable Complexity Model (VCM), is using two computational tools modeling the same physical phenomenon with different degrees of fidelity. A VCM approximation is basically a formal way of keeping the scaling factors between the results of the two analysis tools and applying them to the outputs of the less expensive code when an "approximate" analysis is required.

• Provide the ability to perform trade-off studies between different design responses.

Multi-objective optimization procedure that utilizes compromise programming (CP) approaches and response surfaces should be available within the framework to assist exploration of efficient (Pareto) solutions and better decision making under multiple objectives.

• Provide support to easy description and set up of MDO problems using formal,

decomposition based MDO methods, such as, Global Sensitivity Equations (GSE) based Optimization, Collaborative Optimization (CO), and Bi-Level Integrated system Synthesis (BLISS).

The methods themselves are discussed briefly in the prior section. However, apart from the method implementation and its performance, it is extremely important that the framework provide an easy way of describing and setting up the optimization problem, through GUIs and templates, for use with these formal methods.

• Provide the ability to account for uncertainties in design using probabilistic

constraints and robust design formulations. It is essential to provide a mechanism to quantify a variety of design variability (uncertainty), such as the inaccuracies associated with simulation models, variations of design parameters, and uncertainties in the desired level of product performance. Probabilistic-based optimization models, such as the robust optimization and reliability-based optimization, according to different quality requirements should be available for modeling such stochastic characteristics and design reliable products that are insensitive or tolerant to design variations.

• Framework should provide support for parallel computing, including parallel

invocations of simulation codes as well as subsystem optimizations and intelligent load balancing.

• Provide effective support of visualization of design data both at runtime and post-processing stages. The design data here include design variables, objectives and constraints as well as response surfaces, Pareto curves and surfaces, etc..

• Provide effective support for database management for through SQL (Structured

Query Language) interface for data storage/access/manipulation both at the local (subsystem) and global (system) levels.

• The framework should be easy to use in terms of user interface for MDO, extensible

for user addition of optimization solvers, scalable for large scale problem solving and provide for robust performance.

Among the several commercials tools that specialize in Process Integration and Exploration, the Java and C++ based AML (Automated Modeling Language) from TechnoSoft, Inc. (www.technosoft.com) and ModelCenter v2.0 from Phoenix Integration, Inc. (www.phoenix-int.com) focus on Product Modeling and Ease of Tools and Process Integration across distributed and heterogeneous computing environments. The relative strength of Tcl/Tk based iSIGHT v5.0 from Engineous Software, Inc. (www.engineous.com) and OptiStruct from Altair Computing, Inc. (www.altair.com) is in the different problem solving capabilities. iSIGHT’s problem solving capabilities include Approximation Concepts, Design of Experiments and an Advisor based recommendation of Optimization search strategies while OptiStruct provides for sizing, shape and topology optimization capabilities. A combination of these different tools is required to effectively meet the requirements of a MDO framework. 5.0 Applications: 5.1 Electronic Packaging Design: The electronic packaging is a multidisciplinary problem with coupling between electrical and thermal subsystems. Component resistance is influenced by operating temperatures; the temperatures depend on resistance. The objective of the problem is to maximize the watt density for the electronic package subject to constraints. The constraints require the operating temperatures for the resistors to be below a threshold temperature and the current through the two resistors to be equal. More details of the problem can be obtained in Renaud, 1993. For the A-i-O approach, the optimization problem is given as follows: Maximize: Y1 (Watt Density) Subject to: h1 = Y4 – Y5 = 0.0 (branch current equality) g1 = Y11 – 85.0 < 0 (component 1 reliability) g2 = Y12 – 85.0 < 0 (component 2 reliability) The A-i-O problem has 8 design variables that are the following: 0.05 < heat sink width (x1) < 0.15

0.05 < heat sink length (x2) < 0.15 0.01 < fin length (x3) < 0.10 0.005 < fin width (x4) < 0.05 10.0 < resistance #1 (x5) < 1000.0 0.004 < temperature coefficient (x6) < 0.009 10.0 < resistance #2 (x7) < 1000.0 0.004 < temperature coefficient (x8) < 0.009 The Collaborative Optimization problem formulation for the Electronic Packaging problem is as follows: The system level optimization problem is stated as: Maximize: Z1 Subject to: J1 < 0.0001, J2 < 0.0001, Bounds on design variables. The system level CO problem has 5 design variables that are coupling parameters: Z1, Z2, Z3, Z11, Z12. The system level sensitivities are computed analytically. The thermal subsystem optimization task is given by: Minimize: J1 (where, J1 = (Y11-Z11)2 + (Y12-Z12)2 + (Y2-Z2)2 + (Y3-Z3)2 + (Y1-Z1)2 ) Subject to: h1 = 0.0, g1 = Y11 – 85.0 < 0, g2 = Y12 – 85.0 < 0, and Bounds on design variables. The thermal task includes 6 design variables: Xi; i = 1,4 & Y2, Y3. The electrical subsystem optimization task is given by: Minimize: J2 (where, J2 = (Y11-Z11)2 + (Y12-Z12)2 + (Y2-Z2)2 + (Y3-Z3)2 ) Subject to: g1 = Y11 – 85.0 < 0, g2 = Y12 – 85.0 < 0, and Bounds on design variables. The thermal task includes 6 design variables: Xi; i = 5,8 & Y11, Y12. The BLISS method formulation for the Electronic Packaging problem is as follows: The system level objective function is to maximize Watt Density (Y1). The system level optimization task has a total of 4 design variables (Z2, Z3, Z11, Z12) that are the coupling parameters between the 2 disciplines and physically represent the resistances and component temperatures. The BLISS system optimization problem is stated as: Find the set of system variables, Z, Maximize: Y1 (Watt Density) Subject to: Bounds on Z The 2 subsystem optimization problems are stated as follows. The thermal subsystem optimization task is given as:

Maximize: φ = D(Y1,X1). ∆X1 + (Y11-Z11)2 + (Y12 – Z12)2 Subject to: g Y1 11 850 0= − ≤. g Y2 12 850 0= − ≤. The thermal task has 4 design variables: X1

i; i = 1,4 The Electrical subsystem optimization task is given as: Maximize: φ = D(Y1,X2). ∆X2 + (Y2-Z2)2 + (Y3 – Z3)2 Subject to: h1 = Y4 – Y5 = 0.0 The Electrical task has 4 design variables: X2

i; i = 5,8 The Electronic Packaging problem was solved using the A-i-O, IDF, CO, and BLISS approaches and the results are provided in Table 1. For A-i-O, we report the total number of multidisciplinary analyses (MDA), including those necessary to compute the finite-difference derivatives. We also account the average number of fixed-point iterations taken to achieve each MDA. Thus, the average number of function evaluations for each run of A-i-O is equal to the number of MDA times the average number of fixed-point iterations per MDA times the number of disciplines. For CO, we report the sum of the number of function evaluations in each subsystem, including those required for finite-difference evaluations, and the number of iterations taken by the system-level optimization problem. For IDF, we report the total number of function evaluations, including those taken for finite-difference computation, times the number of disciplines. For BLISS, we report the total number of system analysis as well as the total number of all the subsystem/disciplinary analyses.

4.2 Automotive Vehicle NVH and Crashworthiness Design: To be competitive on the today’s market, cars have to be as light as possible while meeting the Noise, Vibration, and Harshness (NVH) requirements and conforming to Government-mandated crash survival regulations. Noise, Vibration, and Harshness (NVH) is one of the most important attributes for car product development. A vehicle with a good NVH often results in a much higher customer’s satisfaction. In car product development process, different NVH models are used for different purposes so that the quality of the NVH is high and the cost is at minimum. A car body called Body-In-Prime (BIP) is used for this study. The BIP is a trimmed body without all the closures (door, hood, deck lid) and other sub-systems (steering column, fuel tank, and seats) and trim items (carpeting, battery, etc.). A trimmed body structure may be thought of as a vehicle without the suspension and power train sub-systems. The BIP can also be thought as the "Body-In-White" with glass. The BIP plays an important role in determining the dynamic characteristics of the vehicle.

The BIP normal modes, static bending and static torsion analyses were conducted using the MSC/NASTRAN. The full scale NVH finite element model is shown in Figure 7. The total number of shell elements is close to 68,000. The total number of nodes is about 69,000. The normal modes were calculated under the free-free condition. The static bending analysis was conduced with front (yz and z) and rear (xz and xyz) shock towers constrained while for the static torsion rear shock tower supports (xz and xyz) and a mid point of the lower radiator support (z) were constrained. The bending stiffness calculated using a load applied at the front rocker locations was 3,676 N/mm while the torsion stiffness calculated using a torque applied at the front shock tower locations was 9092 N-m/Deg. The free-free normal mode analysis show that the overall torsion at 26.7 Hz and overall bending at 38.9 Hz. The total number of design variables for the NVH model is 19, including 10 for backlite glasses and sheet metal thickness, 9 for the stiffness of connection between the backlite glass and structures. The thickness design variables contain floor panels, jacking/towing on quarter panel, backlite glass, shotgun and radiator support. The torsion frequency for the BIP free-free normal mode is to increase by 10% to 29.32 Hz. The upper bounds for static torsion and static bending displacements are chosen as 3.3 mm and 1.1 mm, i.e., 10% improvement from the initial design. Vehicle roof crush is a federally mandated requirement intended to enhance passenger protection during a rollover event. The test procedure is clearly defined in the Federal Motor Vehicle Safety Standards (FMVSS 216). The finite element roof crush model for this study is converted from a NVH model. The Crash finite element model is shown in Figure 8 and the summary of its characteristics is shown below: NUMBER OF NODAL POINTS. . . . . . . . . . . . 128826 NUMBER OF BOUNDARY CONDITIONS . . . . . . . . 6085 ANALYSIS TYPE: 0=3D, 1=AXISYM, 2=PLANE STRAIN . 0 NUMBER OF 2D SOLID ELEMENTS . . . . . . . . . 0 NUMBER OF 3D SOLID ELEMENTS . . . . . . . . . 0 NUMBER OF 3D SHELL ELEMENTS (4 -NODES) . . . . 124868 NUMBER OF 3D TRUSS ELEMENTS . . . . . . . . . 0 NUMBER OF PROPERTY SETS . . . . . . . . . . . 286 NUMBER OF 3D BEAM ELEMENTS . . . . . . . . . 2 NUMBER OF 3D SPRING ELEMENTS. . . . . . . . . 2484 NUMBER OF 3D SHELL ELEMENTS (3 -NODES). . . . 0 The explicit finite element dynamic software RADIOSS was used for crush simulation. Some unnecessary parts in the NVH model are deleted and some missing parts are added in the roof crush model, e.g., very detailed side doors were added and the glasses are refined. The total number of elements for roof crush is about 120,000. A 72 inches by 30 inches square ram is added to perform the roof crush as specified by the FMVSS 216. The longitudinal axis of the ram (see Figure 8) is at a forward angle (side view) of 5 degrees below the horizontal, and is parallel to the vertical plane through the vehicle’s longitudinal centerline. The lateral axis is at a lateral outboard angle, in the front view projection, of 25 degrees below the horizontal. The lower surface is tangent to the surface

of the vehicle and initial contact point is on the longitudinal centerline of the lower surface of the ram and 10 inches from the forward most point of the centerline. In the RADIOSS simulation, the ram normal speed was set to 7.5 MPH. As described in the FMVSS 216, the force generated by vehicle resistance must be greater than 5000 lb (22,240 N) or 1.5 times the vehicle weight, which ever is less, through 5 inches of ram displacement. In this study, the roof crush resistant force was set to be 5,400 lb. The door thickness and material yield stress parameters are chosen as the design variables. The common set of design variables (Z) for the NVH and roof crush problems are windshield, A-pillar, B-pillar, C-pillar and roof thickness. The multidisciplinary NVH-Crashworthiness Optimization problem can be specifically stated as: Given the set of system (Z) and local (X) design variables, Find: ∆X and ∆Z Minimize: Weight of the Car Body Satisfy: Static torsion displacement < 3.3 mm Static bending displacement < 1.2 mm Frequency (Mode3) 26.65 < ω3 < 29.32 Hz Crash Force at Interface 2 (Normal) > 24kN Bounds on the design variables, X and Z In this optimization task, the NVH discipline has 29 local design variables while the crash discipline has 20 local design variables. A subset of these design variables (Z=10) are common to both the NVH and crash disciplines. As mentioned earlier, the design variables are primarily sizing (thickness) variables, spring stiffness, and, material yield stress parameters. The solution procedure is a piecewise approximation based optimization method, referred to as OMDAA (Optimization by a Mix of Dissimilar Analysis and Approximations) [Sobieszczanski-Sobieski et. al., 2000]. It involves using multiple approximation models, sensitivity based approximation model for NVH responses and polynomial response surface approximation model for Crash responses. The OMDAA solution procedure is executed on a SGI Origin 2000 High Performance Computing (HPC) server, with 256 processors, located at the NASA Ames Research Center. The Origin 2000 is a cache-coherent non-uniform access multiprocessor (ccNUMA) architecture where the memory is physically distributed among the nodes but is globally addressable to all the processors through the interconnection network. The Origin 2000 platform provided an effective computing platform for the OMDAA process involving compute intensive analyses and methods. The effectiveness is in terms of robustness of the operating system, dynamic load balancing across the multiple hundred threads, as well as handling of large I/O-intensive MSC.Nastran and RADIOSS jobs.

The results in terms of the design variable and behavior function values are documented in Table 2. As seen from Table 2, the initial design is an infeasible design with NVH discipline Static Torsion constraint violations of over 10%. The final design is a feasible design with a weight reduction of 15 kg relative to the initial design. Table 2 also shows that the optimization procedure was very judicious in choosing which variables should increase or decrease. The result is a mix of changes in both directions, for instance, variable #33, A pillar yield stress parameter, has been increased by 67 %, while variable #28, C pillar dimension, has been reduced by 33 %. Convergence of optimization based on approximations, such as OMDAA, depends on the accuracy of the predictions the approximations make in regard to the behavior data requested by the optimizer. The accuracy can be assessed by comparing the behavior predicted by the analysis performed at the outset of a new cycle with the predictions generated by the approximations in the previous cycle. Table 3 displays such comparisons by showing the data obtained by analysis, labeled Actual, and those based on the polynomial crash response surface model, labeled Approximate. The errors are relatively quite small attesting to the effectiveness of the approximations used in this application. To incorporate optimization such as the one reported herein in the actual design process would require close scrutiny of the results after each OMDAA cycle and a free exercise of human judgment and intervention in the process. That intervention might include adding and removing design variables, constraints, and changes to the mathematical model as the non-linear crash process unfolds. The changes to the design itself may also be called for as suggested by physical insight in the process gained by examination of the intermediate results. For example, changes of the cross-section from one type to another say from a closed tube to an open channel, may be in order dependent whether the critical constraints are those of strength or stiffness. It is by such a symbiosis of engineer's judgment with computational algorithm that superior designs can best be achieved. 5.0 Summary: An overview of selected Multidisciplinary Design Optimization methods have been presented to show how the formal methodology may help in solving the designer’s dilemma. The dilemma is that to optimize a vehicle as a system one needs to consider a few, system-level variables to evaluate a limited number of the vehicle principal characteristics, but to be correct that system-level analysis must also engage a very large number of locally important detailed design variables and subsystem analyses. To make the problem tractable, the conventional practice separates the system-level analysis and optimization from the detailed, component level design by substituting at the system-level the statistical data and simplified analyses and estimates for the data that should be provided by the detailed analyses. The latter are simply too time-consuming to be available on-line to support directly the system level optimization. The MDO methods solve the above dilemma by decomposing the system opimization into sub-optimizations, each pertaining to a physical phenomenon or a hardware part, while preserving the mathematical linkages (couplings) between the sub-optimizations

and the system-level optimization. Preservation of the couplings enables the system-level optimization and its underlying analysis to communicate with the detailed sub-optimizations their analyses, eliminating the need for substituting simplifications and estimates for hard data. Each of the few selected MDO methods reviewed in the paper approaches the above decomposition problem in a different manner that gives rise to a different execution algorithm unique to the method. However, they have a common denominator of creating opportunity for concurrent execution of numerical subtasks to compress the project time. That opportunity is being assisted by availability of frameworks – software packages that integrate codes, data, and user interfaces. Discussion of requirements for such frameworks has been included to offer assistance in future developments. 6.0 References: 1. R. J. Balling and J. Sobieszczanski-Sobieski, “Optimization of Coupled Systems: A

Critical Overview of Approaches,” AIAA Journal, Vol. 34, No. 1, pp. 6-17, Jan. 1996.

2. R. D. Braun and I. Kroo, “Development and Application of the Collaborative Optimization Architecture in a Multidisciplinary Design Environment,” Multidisciplinary Design Optimization, State of the Art, Edited by: N. Alexandrov and M. Y. Hussaini, SIAM, 1997.

3. E. J. Cramer, J. E. Dennis, P. D. Frank, R. M. Lewis and G. R. Shubin, “Problem Formulation for Multidisciplinary Design Optimization,” SIAM Journal on Optimization, 4(4), pp. 754-776, November 1994.

4. S. Kodiyalam, and Sobieski, J.S., “Bi-Level Integrated System Synthesis with Response Surfaces,” AIAA Journal, Vol. 38, No. 8, August 2000.

5. S. Kodiyalam, Evaluation of Methods for Multidisciplinary Design Optimization (MDO), Phase 1, NASA Contractor Report, NASA/CR-1998-208716, September 1998.

6. J. E. Renaud and G. A. Gabriele, “Improved Coordination in Non-Hierarchic System Optimization,” AIAA Journal, Vol. 31, Number 12, pp. 2367-2373, 1993.

7. J. E. Renaud and G. A. Gabriele, “Approximation In Non-Hierarchic System Optimization,” AIAA Journal, Vol. 32, Number 1, pp. 198-205, 1994.

8. A. O. Salas and J. C. Townsend, “Framework Requirements for MDO Application Development”, AIAA Paper No. AIAA-98-4740, 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, MO, September 2-4, 1998.

9. I. P. Sobieski and I. Kroo, Collaborative Optimization using Response Surface Estimation, 36th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 1998. AIAA Paper No. AIAA-98-0915.

10. J. Sobieszczanski-Sobieski, “Multidisciplinary Design Optimization: an Emerging, New Engineering Discipline”; Advances in Structural Optimization; J. Herskovits (Ed.); Kluwer Academic Publishers, 1995; pp.483-496.

11. J. Sobieszczanski-Sobieski, “Optimization by Decomposition: A Step from Hierarchic to Non-hierarchic Systems,” Proceedings, 2nd NASA/USAF Symposium

on Recent Advances in Multidisciplinary Analysis and Optimization, Hampton, Virginia, 1988. NASA CP-3031. Also, NASA TM-101494, 1988.

12. J. Sobieszczanski-Sobieski and R. T. Haftka, “Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments,” Structural Optimization, pp. 1-23, Vol. 14, No. 1, August 1997.

13. J. Sobieszczanski-Sobieski, J. Agte and R. Sandusky, Jr., “Bi-Level Integrated System Synthesis (BLISS),” Proceedings, 7th AIAA/USAF/NASA/iSSMO Symposium on Multidisciplinary Analysis and Optimization, AIAA, St. Louis, Missouri, September 1998. AIAA Paper No. 98-4916. To appear in AIAA Journal.

14. J. Sobieszczanski-Sobieski, J. Agte and R. Sandusky, Jr., “Bi-Level Integrated System Synthesis (BLISS),” NASA/TM-1998-208715, NASA Langley Research Center, Hampton, Virginia, August 1998.

15. J. Sobieszczanski-Sobieski, “Multidisciplinary design optimization MDO methods their synergy with computer technology in the design process,” The Aeronautical Journal, 1999.

16. J. Sobieszczanski-Sobieski, S. Kodiyalam, and R. J. Yang, “Optimization of Car Body for Noise, Vibration and Harshness (NVH) and Crash”, Proceedings of the AIAA/ASME/ASCE/AHS/ASC 41st Structures, Structural Dynamics and Materials Conference, AIAA, Atlanta, April 2000. AIAA Paper Number: AIAA-2000-1521.

Figure 1.0 A general system

Figure 2: A-i-O Model

Optimizer

Analysis 1

U1, U2

U1

U2

( )µ21 21 1= F U

( )Y E21 21 21= µ

( )µ12 12 2= F U

( )Y E12 12 12= µ

XD

Analysis 2

Figure 3: IDF Model

Figure 4: CO Model

Optimizer

Analysis 1

U1, U2,

µ21

( )Y E X21 21 21= µ

( )Y E X12 12 12= µ

X X XD, ,µ µ12 21

Analysis 2µ12

µ µ12 21,

U2

U1

System OptimizerMin F(z)s.t. Jj

*(z) = 0, j = 1, N

Subspace Optimizer 1Min J1(x1) = |x1-z1

s|2

+ |y1-zc1|2

s.t. g1(x1,xs1) < 0

Subspace Analysis 1

x1, xs1 y1, g1

z1 J1*

Subspace Optimizer 2

s.t. g2(x2,xs2) < 0

SubspaceAnalysis 2

x2, xs2 y2, g2

z2 J2*

Subspace Optimizer NMin JN(xN) = |xN-zs

N|2

+ |yN-zcN|2

s.t. gN(xN,xsN) < 0

SubspaceAnalysis N

xN, xsN yN, gN

zNJN*

------

System Analysis

Min J2(x2) = |x2-z2s|2

+ |y2-z2c|2

Figure5: BLISS Model

Figure 6: Data Flow in Electronic Packaging Problem

System Analysis

Convergence

BBSABB i

BBSABB j

System SensitivityAnalysis

BBOPTBB i

BBOPTBB j

Optimum Sensitivity Analysis or RSM

System Optimization

Update Variables

Initialize X & Z X = Xo + ∆Xopt

Z = Zo + ∆Zopt

∆Xopt

∆Zopt

Final Design

BLISScycle

Y

D(Y,X), D(Y,Z)

ELECTRICAL

THERMAL

x5, x6, x7,x8

Y2, Y3

Y11, Y12

Y1

x1, x2, x3, x4

Figure 7: NVH Model of Automotive Vehicle

Figure 8: Roof Crush Finite Element Model of Automotive Vehicle

Case Initial Design Objective

Initial Design Max Constraint Violation

Final Design Objective

Final Design Max Constraint Violation

Work

A-i-O 6836.3 -2.89560D-01 639720.0 +1.21880D-03 264

IDF 6836.3 -0.289 653670.0 +0.0001 8976

CO 6836.3 -0.289 657162.9 +0.00023 19872 (123 system iterations)

BLISS 6836.3 -2.89560D-01(3) 639720.0 +1.22D-03 207 (5 system analyses)

Table 1: Electronic Packing Application – Results

Response Name Response Values % Error between Actual &

Approximate Values Actual Approximate Weight (kg) 1522.73 1522.69 0.0 Mode 3 Frequency (hz) 29.32 29.32 0.0 Static Torsion (mm) 3.29 3.30 0.3 Static Bending (mm) -0.935 -0.895 4.3 Crash Normal Force (kN) 29.43 30.57 3.9 Internal Energy 2400.97 2617.9 9.0

Table 3: NVH and Crash Approximation Model Errors

Number Attribute Name Initial Design Cycle 1 Cycle 2 NVH Design Variables 1 Rear floor panel 0.76 1.0 1.0 2 Rear floor cross

member 1.4 2.0 2.0

3 Front floor pan 0.76 0.5 0.5 4 Front floor inner 1.07 1.0923 1.1926 5 Jacking/towing 0.8 1.5 1.5 6 Quarter panel 0.8 0.8876 0.8876 7 Backlite glass 3.8 2.6 2.6 8 Rear tire cover 0.75 1.0 1.0 9 Shotgun 1.22 1.3681 1.2643 10 Radiator support 0.76 0.5 0.5 11 Top edge

(x-component) 1073.3 1070.11 1193.41

12 Top edge (y-component)

366.9 477.95 478.0

13 Top edge (z-component)

2733.6 2733.37 2734.24

14 Bottom edge (x-component)

1424.5 1417.75 1438.51

15 Bottom edge (y-component)

487.0 484.41 629.97

16 Bottom edge (z-component)

3628.3 3627.95 3632.52

17 Side edges (x-component)

1521.0 1518.30 1914.19

18 Side edges (y-component)

520.0 513.96 675.0

19 Side edges (z-component)

3874.0 3873.39 3886.59

Common Design Variables 20 Windshield 3.8 2.6 2.6 21 Roof Panel 0.7 0.4 0.4 22 Roof rail 0.8 0.6 0.6 23 Roof Cross Member

Front 0.8 0.6 0.6

24 Roof Cross Member Rear

0.7 0.5 0.5

25 A Pillar 0.8 1.0998 1.0971 26 B Pillar 1 0.8 0.7944 0.7788 27 B Pillar 2 0.8 0.5 0.5 28 B Pillar 3 1.35 0.9 0.9 29 C Pillar 0.8 0.5 0.5 Crash Design Variables 30 Front door thickness 0.7 0.4 0.4 31 Front door inner

thickness 0.7 0.4 0.4

32 Rear door thickness 1.0 0.7 0.7 33 A-Pillar 1 0.207 0.345 0.192 34 A-Pillar 2 0.207 0.345 0.345 35 A-Pillar 3 0.207 0.345 0.192

36 B-Pillar 1 0.207 0.345 0.345 37 B-Pillar 2 0.345 0.192 0.345 38 Front door inner 1 0.207 0.192 0.192 39 Front door inner 2 0.207 0.345 0.345 MDO Objectives and Constraints related Responses: 1 NVH Weight (kg) 282.44 282.70 282.53 2 Crash Weight (kg) 1255.65 1240.3 1240.2 3 Mode 3 (Hz) 26.65 29.32 29.32 4 Static Torsion-

Z displacement: (mm) 3.67 (Violated) 3.29 3.29

5 Static Torsion- Z displacement: (mm)

-3.68 (Violated)

-3.31 -3.31

6 Static Bending- Z displacement: Maximum (mm)

-0.97 -0.97 -0.935

7 Crash: NF – Normal reaction at I/F 2 (kN)

34.69 28.82 29.43

8 Internal Energy 3015.79 2331.7 2400.97

Table 2: Design Variables, Objectives and Constraint: Initial and Optimized Values