1 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Multidisciplinary System Design Optimization (MSDO) Multidisciplinary Design and Analysis Problem Formulation Lecture 2 9 February 2004 Karen Willcox 2 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Today’s Topics ? MDO definition MDO disciplines Optimization problem elements Optimization problem formulation MDO in the design process MDO challenges 3 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox MDO Definition What is MDO ? A methodology for the design of complex engineering systems and subsystems that coherently exploits the synergism of mutually interacting phenomena Optimal design of complex engineering systems which requires analysis that accounts for interactions amongst the disciplines (= parts of the system) “How to decide what to change, and to what extent to change it, when everything influences everything else.” Ref: AIAA MDO website http://endo.sandia.gov/AIAA_MDOTC/main.html 4 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Design Disciplines Spacecraft: Astrodynamics Thermodynamics Communications Payload & Sensor Structures Optics Guidance & Control Automobiles: Engines Body/chassis Aerodynamics Electronics Hydraulics Industrial design others Aircraft: Aerodynamics Propulsion Structures Controls Avionics/Software Manufacturing others Fairly mature, but advances in theory, methodology, computation and application foster substantial payoffs 5 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Multidisciplinary Aspects of Design Emphasis is on the multidisciplinary nature of the complex engineering systems design process. Aero- space vehicles are a particular class of such systems. Structures Aerodynamics Control Emphasis in recent years has been on advances that can be achieved due to the inter- action of two or more disciplines. 6 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox System Level Optimization Why system-level, multidisciplinary optimization ? Disciplinary specialists tend to strive towards improvement of objectives and satisfaction of constraints in terms of the variables of their own discipline In doing so they generate side effects - often unknowingly- that other disciplines have to absorb, usually to the detriment of the overall system performance Example: High wing aspect ratio aircraft designs 7 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Concurrent Engineering Disciplines Must also include the broader set of concurrent engineering (CE) disciplines. Manufacturing: Supportability: Cost: Model manufacturing tools and processes as a function of part geometry, materials, and assemblies Model parts reliability and failure rates, estimated down-time due to repairs etc... Estimate development, manufacturing and operations costs. Often cost-estimation relationships (CER’s) Prerequisite: Development of realistic, reliable and easy to use mathematical models for these disciplines - difficult 8 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Supporting Disciplines Multidisciplinary design optimization of aerospace vehicles cannot take place without substantial contributions from supporting disciplines: Human Interface Aspects of Design Intelligent and Knowledge-Based Systems Computing Aspects of Design Information Integration and Management. 9 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Human Interface Aspects of Design It is wrong to think of MDO as “automated” or “push- button” design: The human strengths (creativity, intuition, decision- making) and computer strengths (memory, speed, objectivity) should complement each other The human will always be the Meta-designer Challenges of defining an effective interface – continuous vs. discrete thinking Challenges of visualization in multidimensional space, e.g. search path from initial design to final design Human element is a key component in any successful system design methodology 10 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Quantitative vs. Qualitative Human mind is the driving force in the design process, but mathematics and computers are indispensable tools AIAA Technical Committee on Multidisciplinary Design Optimization (MDO). White Paper on Current State of the Art. January 15, 1991. 11 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Quantitative vs. Qualitative Qualitative effort stream Quantitative disciplinary models high or low aspect ratio wing? aero=high, structures=low ? Qualitative effort stream Quantitative multidisciplinary model high or low aspect ratio wing for min weight? overall best aspect ratio MDO is a way of formalizing the quantitative tool to apply the best trade-offs. The question provides a metric; the answer accounts for both disciplinary and interaction information. 12 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Optimization Aspects of Design Optimization methods have been combined with design synthesis and parametric analysis for ca. 40 years Traditionally used graphical methods to find maximum or minimum of a multivariate function (“carpet plot”), but…. Graphics break down above 3-4 dimensions Objective J(x) D es i gn v ar i a b l e x 2 D e s ig n v a r ia b le x 1 Where is max J(x) ? Caution: local extrema ! “peaks” Where is min J(x) ? 13 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Combinatorial Explosion For n > 3 a combinatorial “explosion” takes place and the design space cannot be computed and plotted in polynomial time Numerical optimization offers an alternative to the graphical approach and “brute force” evaluation Any design can be defined by a vector in multidimensional space, where each design variable represents a different dimension During past two decades much progress has been made in numerical optimization 14 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Design Variables 1 2 3 aspect ratio [-] transmit power [W] # of apertures [-] orbital altitude [km] control gain [V/V] i n x x x x x aoao ???? ?? ?? ???? ?? ?? ???? ?? ???? x # # Design vector x contains n variables that form the design space During design space exploration or optimization we change the entries of x in some rational fashion to achieve a desired effect i x can be ….. Integer: i x ?i x ?] {0,1} i x ? {true, false} i x ? Real: Binary: Boolean: Design variables are “controlled” by the designers 15 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Objectives The objective can be a vector J of z system responses or characteristics we are trying to maximize or minimize 1 2 3 cost [$] range [km] weight [kg] data rate [bps] ROI [%] i z J J J J J aoao ???? ?? ?? ???? ?? ?? ???? ?? ???? J # # Often the objective is a scalar function, but for real systems often we attempt multi-objective optimization: xJ(x)6 Some objectives can be conflicting. 16 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Parameters Parameters p are quantities that affect the objective J, but are considered fixed, i.e. they cannot be changed by the designers. Sometimes parameters p can be turned into design variables x i to enlarge the design space. Sometimes parameters p are former design variables that were fixed at some value because they were found not to affect any of the objectives J i or because their optimal level was predetermined. 17 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Constraints Constraints act as boundaries of the design space x and typically occur due to finiteness of resources or technological limitations of some design variables. Often, but not always, optimal designs lie at the intersection of several active constraints 1 2 ,, 01,2, 01,2, 1, 2, , d dd ! ! ! j k iLB i iUB gjm hkm xxxi n x x Inequality constraints: Equality constraints: Bounds: Objectives are what we are trying to achieve Constraints are what we cannot violate Design variables are what we can change 18 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Constraints versus Objectives It can be difficult to choose whether a condition is a constraint or an objective. For example: should we try to minimize cost, or should we set a constraint stating that cost should not exceed a given level. The two approaches can lead to different designs. Sometimes, the initial formulation will need to be revised in order to fully understand the design space. In some formulations, all constraints are treated as objectives (physical programming). 19 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Example Problem Statement Minimize the take-off weight of the aircraft by changing wing geometric parameters while satisfying the given range and payload requirements at the given cruise speed. objective function constraints parameter design variables 20 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Formal Notation Quantitative side of the design problem may be formulated as a problem of Nonlinear Programming (NLP) ,, 1,..., ) min , s.t. , 0 ,=0 ( d dd iLB i iUB inxxx Jxp g(x p) h(x p) This is the problem formulation that we will discuss this semester. >@ 1 2 1 1 1 1 where () () () () ao ?? ao ?? ao ?? " "" " " T z T in T m T m JJ xxx gg hh Jx x x gx x hx x 21 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Group Exercise... Identify five complex engineering systems: 1. 2. 3. 4. 5. Consider the preliminary design phase. Identify: -important disciplines -potential objective functions -potential design variables -constraints and bounds -system parameters 22 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox What MDO really does MDO mathematically traces a path in the design space from some initial design x o towards improved designs (with respect to the objective J). It does this by operating on a large number of variables and functions simultaneously - a feat beyond the power of the human mind. The path is not biased by intuition or experience. This path instead of being invisible inside a “black box” becomes more visible by various MDO techniques such as sensitivity analysis and visualization Optimization does not remove the designer from the loop, but it helps conduct trade studies 23 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox MSDO Framework Discipline A Discipline B Discipline C I n p u t O u t p u t Simulation Model Tradespace Exploration (DOE) Optimization Algorithms Multiobjective Optimization Numerical Techniques (direct and penalty methods) Heuristic Techniques (SA,GA) 1 2 n x x x ao ?? ?? ?? ?? ?? ?? # Design Vector Coupling 1 2 ao ?? ?? ?? ?? ?? # z J J J Approximation Methods Coupling Sensitivity Analysis Isoperformance Objective Vector Output Evaluation 24 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Simulation versus Optimization There are two distinct components of the MSDO process: The optimization algorithm decides how to move through the design space. The simulation model evaluates designs chosen by the optimizer. Both objective functions and constraints must be evaluated. Sometimes, disciplinary simulation models can be used in an optimization framework, but often they are not appropriate. There are several different approaches to couple the optimizer and the simulation models (Lecture 5). 25 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Typical Process in MDO (1) Define overall system requirements (2) Define design vector x, objective J and constraints (3) System decomposition into modules (4) Modeling of physics via governing equations at the module level - module execution in isolation (5) Model integration into an overall system simulation (6) Benchmarking of model with respect to a known system from past experience, if available (7) Design space exploration (DoE) to find sensitive and important design variables x i (8) Formal optimization to find min J(x) (9) Post-optimality analysis to explore sensitivity and tradeoffs: sensitivity analysis, approximation methods, isoperformance, include uncertainty 26 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox In Practice... (i) Step through (1)-(8) (ii) The optimizer will use an error in the problem setup to determine a mathematically valid but physically unreasonable solution OR The optimizer will be unable to find a feasible solution (satisfies all constraints) (iii) Add, remove or modify constraints and/or design variables (iv) Iterate until an appropriate model is obtained Although MDO is an automated formalization of the design process, it is a highly interactive procedure... 27 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox MDO in the Design Process baseline design optimized design WingMOD CFDconfigurator outer mold line performance propulsion weights aerodynamics engine deck weights configuration drawing MDO is only one part of the design process couples with other design tools invaluable but not always complete economics 28 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox MDO Uses The ‘MD’ portion of ‘MDO’ is important on its own Often MDO is used not to find the truly optimal design, but rather to find an improved design, or even a feasible design ... Range of design objectives Feasible Improved Optimal Pareto from Giesing, 1998 29 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox MDO Challenges Fidelity/expense of disciplinary models Fidelity is often sacrificed to obtain models with short computation times. Complexity Design variables, constraints and model interfaces must be managed carefully. Communication The user interface is often very unfriendly and it can be difficult to change problem parameters. Flexibility It is easy for an MDO tool to become very specialized and only valid for one particular problem. How do we prevent MDO codes from becoming complex, highly specialized tools which are used by a single person (often the developer!) for a single problem? 30 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Fidelity vs. Expense high fidelity (e.g. CFD,FEM) Level of MSDO Fidelity Level i n c r e a s i n gd i f f ic u l t y can we do better? can the results be believed? how to implement? intermediate fidelity (e.g. vortex lattice, beam theory) empirical models trade studies limited optimization/iteration full MDO from Giesing, 1998 31 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Breadth vs. Depth System Breadth Disciplinary Depth intermediate fidelity (e.g. vortex lattice, beam theory) high fidelity (e.g. CFD,FEM) i n c r e a s i n gd i f f ic u l t y is design practical? can the results be believed? how to implement? empirical relations focus on a subsystem all critical constraints complete system 32 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox MDO Pros/Cons Advantages reduction in design time systematic, logical design procedure handles wide variety of design variables & constraints not biased by intuition or experience Disadvantages computational time grows rapidly with number of dv’s numerical problems increase with number of dv’s limited to range of applicability of analysis programs will take advantage of analysis errors to provide mathematical design improvements difficult to deal with discontinuous functions 33 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Data Management Need some kind of database to store design variables, constraints, objectives ... e.g. GenIE database ISight Would like to keep interface general and user friendly -don’t “hard-code” problem specific details Can be a serious problem for large systems 34 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Lecture summary MDO is not a stand-alone, automated design process MDO is a valuable tool that requires substantial human interaction and complements other design tools Elements of an MDO framework MDO Challenges Guidelines of how decomposition and integration of modules can be done is the subject of Lecture 4 35 ? Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox References Kroo, I.: “MDO applications in preliminary design: status and directions,” AIAA Paper 97-1408, 1997. Kroo, I. and Manning, V.: “Collaborative optimization: status and directions,” AIAA Paper 2000-4721, 2000. Sobieski, I. and Kroo, I.: “Aircraft design using collaborative optimization,” AIAA Paper 96-0715, 1996. Balling, R. and Wilkinson, C.: “Execution of multidisciplinary design optimization approaches on common test problems,” AIAA Paper 96-4033, 1996. Giesing, J. and Barthelemy, J.: “A summary of industry MDO applications and needs”, AIAA White Paper, 1998. AIAA MDO Technical Committee: “Current state-of-the-art in multidisciplinary design optimization”, 1991.