Multidisciplinary design optimization
Encyclopedia
Multi-disciplinary design optimization (MDO) is a field of engineering
that uses optimization
methods to solve design
problems incorporating a number of disciplines. As defined by Prof. Carlo Poloni, MDO is "the art of finding the best compromise". It is also known as multidisciplinary optimization and multidisciplinary system design optimization (MSDO).
MDO allows designers to incorporate all relevant disciplines simultaneously. The optimum of the simultaneous problem is superior to the design found by optimizing each discipline sequentially, since it can exploit the interactions between the disciplines. However, including all disciplines simultaneously significantly increases the complexity
of the problem.
These techniques have been used in a number of fields, including automobile
design, naval architecture
, electronics
, computer
s, and electricity distribution
. However, the largest number of applications have been in the field of aerospace engineering
, such as aircraft
and spacecraft
design. For example, the proposed Boeing
blended wing body
(BWB) aircraft concept has used MDO extensively in the conceptual and preliminary design stages. The disciplines considered in the BWB design are aerodynamics
, structural analysis
, propulsion
, control theory
, and economics
.
, or minimum structural weight.
Between 1970 and 1990, two major developments in the aircraft industry changed the approach of aircraft design engineers to their design problems. The first was computer-aided design, which allowed designers to quickly modify and analyse their designs. The second was changes in the procurement policy of most airline
s and military organizations, particularly the military of the United States
, from a performance-centred approach to one that emphasized lifecycle
cost issues. This led to an increased concentration on economic factors and the attributes known as the "ilities
" including manufacturability, reliability, maintainability
, etc.
Since 1990, the techniques have expanded to other industries. Globalization has resulted in more distributed, decentralized design teams. The high-performance personal computer
has largely replaced the centralized supercomputer
and the Internet
and local area network
s have facilitated sharing of design information. Disciplinary design software in many disciplines (such as NASTRAN
, a finite element analysis program for structural design) have become very mature. In addition, many optimization algorithms, in particular the population-based algorithms, have advanced significantly.
, dating back to Isaac Newton
, Leonhard Euler
, Daniel Bernoulli
, and Joseph Louis Lagrange
, who used them to solve problems such as the shape of the catenary
curve, numerical optimization reached prominence in the digital age. Its systematic application to structural design dates to its advocacy by Schmit in 1960. The success of structural optimization in the 1970s motivated the emergence of multidisciplinary design optimization (MDO) in the 1980s. Jaroslaw Sobieski championed decomposition methods specifically designed for MDO applications. The following synopsis focuses on optimization methods for MDO. First, the popular gradient-based methods used by the early structural optimization and MDO community are reviewed. Then those methods developed in the last dozen years are summarized.
The gradient methods unique to the MDO community derive from the combination of optimality criteria with math programming, first recognized in the seminal work of Fleury and Schmit who constructed a framework of approximation concepts for structural optimization. They recognized that optimality criteria were so successful for stress and displacement constraints, because that approach amounted to solving the dual problem for Lagrange multipliers
using linear Taylor series approximations in the reciprocal design space. In combination with other techniques to improve efficiency, such as constraint deletion, regionalization, and design variable linking, they succeeded in uniting the work of both schools. This approximation concepts based approach forms the basis of the optimization modules in modern structural design software ASTROS, MSC.Nastran, Genesis, I-DEAS, iSight.
Approximations for structural optimization were initiated by the reciprocal approximation Schmit and Miura for stress and displacement response functions. Other intermediate variables were employed for plates. Combining linear and reciprocal variables, Starnes and Haftka developed a conservative approximation to improve buckling approximations. Fadel chose an appropriate intermediate design variable for each function based on a gradient matching condition for the previous point. Vanderplaats initiated a second generation of high quality approximations when he developed the force approximation as an intermediate response approximation to improve the approximation of stress constraints. Canfield developed a Rayleigh Quotient approximation to improve the accuracy of eigenvalue approximations. Barthelemy and Haftka published a comprehensive review of approximations in 1993.
methods in several broad areas in the last dozen years. These include decomposition method
s, approximation
methods, evolutionary algorithm
s, memetic algorithm
s, response surface methodology
, reliability-based optimization, and multiobjective optimization
approaches.
The exploration of decomposition methods has continued in the last dozen years with the development and comparison of a number of approaches, classified variously as hierarchic and non-hierarchic, or collaborative and non-collaborative.
Approximation methods spanned a diverse set of approaches, including the development of approximations for surrogate models, variable fidelity models, and trust region management strategies. The development of multipoint approximations blurred the distinction with response surface methods. Kriging
methods became popular.
Response surface methodology
, developed extensively by the operations research community, received much attention in the MDO community in the last dozen years. A driving force for their use has been the development of massively parallel systems for high performance computing, which are naturally suited to distributing the function evaluations from multiple disciplines that are required for the construction of response surfaces. Distributed processing is particularly suited to the design process of complex systems in which analysis of different disciplines may be accomplished naturally on different computing platforms and even by different teams.
Evolutionary methods led the way in the exploration of non-gradient methods for MDO applications. They also have benefited from the availability of massively parallel high performance computers, since they inherently require many more function evaluations than gradient-based methods. Their primary benefit lies in their ability to handle discrete design variables and the potential to find globally optimal solutions.
Reliability-based optimization (RBO) is a growing area of interest in MDO. Like response surface methods and evolutionary algorithms, RBO benefits from parallel computation, because the numeric integration to calculate the probability of failure requires many function evaluations. One of the first approaches employed approximation concepts to integrate the probability of failure. The classical first-order reliability method (FORM) and second-order reliability method (SORM) are still popular. Grandhi used appropriate normalized variables about the most probable point of failure, found by a two-point adaptive nonlinear approximation to improve the accuracy and efficiency. Southwest Research Institute
has figured prominently in the development of RBO, implementing state-of-the-art reliability methods in commercial software. RBO has reached sufficient maturity to appear in commercial structural analysis programs like MSC's Nastran
.
Utility-based probability maximization (Bordley and Pollock, Operations Research, Sept, 2009, pg.1262) was developed in response to some logical concerns (e.g., Blau's Dilemma) with reliability-based design optimization. This approach focuses on maximizing the joint probability of both the objective function exceeding some value and of all the constraints being satisfied. When there is no objective function, utility-based probability maximization reduces to a probability-maximization problem. When there are no uncertainties in the constraints, it reduces to a constrained utility-maximization problem. (This second equivalence arises because the utility of a function can always be written as the probability of that function exceeding some random variable.) Because it changes the constrained optimization problem associated with reliability-based optimization into an unconstrained optimization problem, it often leads to computationally more tractable problem formulations.
). Design problems with continuous variables are normally solved more easily.
Design variables are often bounded, that is, they often have maximum and minimum values. Depending on the solution method, these bounds can be treated as constraints or separately.
generated by a wing must be equal to the weight of the aircraft. In addition to physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.
.
of aircraft prices, theoretical models, such as from computational fluid dynamics
, or reduced-order models of either of these. In choosing the models the designer must trade off fidelity with analysis time.
The multidisciplinary nature of most design problems complicates model choice and implementation. Often several iterations are necessary between the disciplines in order to find the values of the objectives and constraints. As an example, the aerodynamic loads on a wing affect the structural deformation of the wing. The structural deformation in turn changes the shape of the wing and the aerodynamic loads. Therefore, in analysing a wing, the aerodynamic and structural analyses must be run a number of times in turn until the loads and deformation converge.
where
is an objective, 
is a vector of design variables, 
is a vector of inequality constraints, 
is a vector of equality constraints, and 
and 
are vectors of lower and upper bounds on the design variables. Maximization problems can be converted to minimization problems by multiplying the objective by -1. Constraints can be reversed in a similar manner. Equality constraints can be replaced by two inequality constraints.
-based algorithms, population-based algorithms, or others. Very simple problems can sometimes be expressed linearly; in that case the techniques of linear programming
are applicable.
Most of these techniques require large numbers of evaluations of the objectives and the constraints. The disciplinary models are often very complex and can take significant amounts of time for a single evaluation. The solution can therefore be extremely time-consuming. Many of the optimization techniques are adaptable to parallel computing
. Much current research is focused on methods of decreasing the required time.
Also, no existing solution method is guaranteed to find the global optimum
of a general problem (see No free lunch in search and optimization). Gradient-based methods find local optima with high reliability but are normally unable to escape a local optimum. Stochastic methods, like simulated annealing and genetic algorithms, will find a good solution with high probability, but very little can be said about the mathematical properties of the solution. It is not guaranteed to even be a local optimum. These methods often find a different design each time they are run.
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
that uses optimization
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
methods to solve design
Design
Design as a noun informally refers to a plan or convention for the construction of an object or a system while “to design” refers to making this plan...
problems incorporating a number of disciplines. As defined by Prof. Carlo Poloni, MDO is "the art of finding the best compromise". It is also known as multidisciplinary optimization and multidisciplinary system design optimization (MSDO).
MDO allows designers to incorporate all relevant disciplines simultaneously. The optimum of the simultaneous problem is superior to the design found by optimizing each discipline sequentially, since it can exploit the interactions between the disciplines. However, including all disciplines simultaneously significantly increases the complexity
Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
of the problem.
These techniques have been used in a number of fields, including automobile
Automobile
An automobile, autocar, motor car or car is a wheeled motor vehicle used for transporting passengers, which also carries its own engine or motor...
design, naval architecture
Naval architecture
Naval architecture is an engineering discipline dealing with the design, construction, maintenance and operation of marine vessels and structures. Naval architecture involves basic and applied research, design, development, design evaluation and calculations during all stages of the life of a...
, electronics
Electronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...
, computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...
s, and electricity distribution
Electricity distribution
File:Electricity grid simple- North America.svg|thumb|380px|right|Simplified diagram of AC electricity distribution from generation stations to consumers...
. However, the largest number of applications have been in the field of aerospace engineering
Aerospace engineering
Aerospace engineering is the primary branch of engineering concerned with the design, construction and science of aircraft and spacecraft. It is divided into two major and overlapping branches: aeronautical engineering and astronautical engineering...
, such as aircraft
Aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air, or, in general, the atmosphere of a planet. An aircraft counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines.Although...
and spacecraft
Spacecraft
A spacecraft or spaceship is a craft or machine designed for spaceflight. Spacecraft are used for a variety of purposes, including communications, earth observation, meteorology, navigation, planetary exploration and transportation of humans and cargo....
design. For example, the proposed Boeing
Boeing
The Boeing Company is an American multinational aerospace and defense corporation, founded in 1916 by William E. Boeing in Seattle, Washington. Boeing has expanded over the years, merging with McDonnell Douglas in 1997. Boeing Corporate headquarters has been in Chicago, Illinois since 2001...
blended wing body
Blended wing body
Blended Wing Body aircraft have a flattened and airfoil shaped body, which produces most of the lift, the wings contributing the balance. The body form is composed of distinct and separate wing structures, though the wings are smoothly blended into the body...
(BWB) aircraft concept has used MDO extensively in the conceptual and preliminary design stages. The disciplines considered in the BWB design are aerodynamics
Aerodynamics
Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with...
, structural analysis
Structural analysis
Structural analysis is the determination of the effects of loads on physical structures and their components. Structures subject to this type of analysis include all that must withstand loads, such as buildings, bridges, vehicles, machinery, furniture, attire, soil strata, prostheses and...
, propulsion
Air propulsion
Air propulsion is the generation of thrust during flight by an aircraft or a creature such as a bird, bat or insect.-Aircraft:An aircraft propulsion system must serve two purposes. First, the thrust from the propulsion system must balance the drag of the airplane when the airplane is cruising...
, control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
, and economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
.
History
Traditionally engineering has normally been performed by teams, each with expertise in a specific discipline, such as aerodynamics or structures. Each team would use its members' experience and judgement to develop a workable design, usually sequentially. For example, the aerodynamics experts would outline the shape of the body, and the structural experts would be expected to fit their design within the shape specified. The goals of the teams were generally performance-related, such as maximum speed, minimum dragDrag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...
, or minimum structural weight.
Between 1970 and 1990, two major developments in the aircraft industry changed the approach of aircraft design engineers to their design problems. The first was computer-aided design, which allowed designers to quickly modify and analyse their designs. The second was changes in the procurement policy of most airline
Airline
An airline provides air transport services for traveling passengers and freight. Airlines lease or own their aircraft with which to supply these services and may form partnerships or alliances with other airlines for mutual benefit...
s and military organizations, particularly the military of the United States
Military of the United States
The United States Armed Forces are the military forces of the United States. They consist of the Army, Navy, Marine Corps, Air Force, and Coast Guard.The United States has a strong tradition of civilian control of the military...
, from a performance-centred approach to one that emphasized lifecycle
Product lifecycle management
In industry, product lifecycle management is the process of managing the entire lifecycle of a product from its conception, through design and manufacture, to service and disposal...
cost issues. This led to an increased concentration on economic factors and the attributes known as the "ilities
Ilities
Within systems engineering, quality attributes are non-functional requirements used to evaluate the performance of a system. These are sometimes named "ilities" after the suffix many of the words share...
" including manufacturability, reliability, maintainability
Maintainability
In engineering, maintainability is the ease with which a product can be maintained in order to:* isolate defects or their cause* correct defects or their cause* meet new requirements* make future maintenance easier, or* cope with a changed environment...
, etc.
Since 1990, the techniques have expanded to other industries. Globalization has resulted in more distributed, decentralized design teams. The high-performance personal computer
Personal computer
A personal computer is any general-purpose computer whose size, capabilities, and original sales price make it useful for individuals, and which is intended to be operated directly by an end-user with no intervening computer operator...
has largely replaced the centralized supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...
and the Internet
Internet
The Internet is a global system of interconnected computer networks that use the standard Internet protocol suite to serve billions of users worldwide...
and local area network
Local area network
A local area network is a computer network that interconnects computers in a limited area such as a home, school, computer laboratory, or office building...
s have facilitated sharing of design information. Disciplinary design software in many disciplines (such as NASTRAN
Nastran
NASTRAN is a finite element analysis program that was originally developed for NASA in the late 1960s under United States government funding for the Aerospace industry. The MacNeal-Schwendler Corporation was one of the principal and original developers of the public domain NASTRAN code...
, a finite element analysis program for structural design) have become very mature. In addition, many optimization algorithms, in particular the population-based algorithms, have advanced significantly.
Origins in structural optimization
Whereas optimization methods are nearly as old as calculusCalculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, dating back to Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
, Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
, Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...
, and Joseph Louis Lagrange
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
, who used them to solve problems such as the shape of the catenary
Catenary
In physics and geometry, the catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight. The curve is the graph of the hyperbolic cosine function, and has a U-like shape, superficially similar in appearance to a parabola...
curve, numerical optimization reached prominence in the digital age. Its systematic application to structural design dates to its advocacy by Schmit in 1960. The success of structural optimization in the 1970s motivated the emergence of multidisciplinary design optimization (MDO) in the 1980s. Jaroslaw Sobieski championed decomposition methods specifically designed for MDO applications. The following synopsis focuses on optimization methods for MDO. First, the popular gradient-based methods used by the early structural optimization and MDO community are reviewed. Then those methods developed in the last dozen years are summarized.
Gradient-based methods
There were two schools of structural optimization practitioners using gradient-based methods during the 1960s and 1970s: optimality criteria and mathematical programming. The optimality criteria school derived recursive formulas based on the Karush–Kuhn–Tucker (KKT) necessary conditions for an optimal design. The KKT conditions were applied to classes of structural problems such as minimum weight design with constraints on stresses, displacements, buckling, or frequencies [Rozvany, Berke, Venkayya, Khot, et al.] to derive resizing expressions particular to each class. The mathematical programming school employed classical gradient-based methods to structural optimization problems. The method of usable feasible directions, Rosen’s gradient projection (generalized reduce gradient) method, sequential unconstrained minimization techniques, sequential linear programming and eventually sequential quadratic programming methods were common choices. Schittkowski et al. reviewed the methods current by the early 1990s.The gradient methods unique to the MDO community derive from the combination of optimality criteria with math programming, first recognized in the seminal work of Fleury and Schmit who constructed a framework of approximation concepts for structural optimization. They recognized that optimality criteria were so successful for stress and displacement constraints, because that approach amounted to solving the dual problem for Lagrange multipliers
Lagrange multipliers
In mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...
using linear Taylor series approximations in the reciprocal design space. In combination with other techniques to improve efficiency, such as constraint deletion, regionalization, and design variable linking, they succeeded in uniting the work of both schools. This approximation concepts based approach forms the basis of the optimization modules in modern structural design software ASTROS, MSC.Nastran, Genesis, I-DEAS, iSight.
Approximations for structural optimization were initiated by the reciprocal approximation Schmit and Miura for stress and displacement response functions. Other intermediate variables were employed for plates. Combining linear and reciprocal variables, Starnes and Haftka developed a conservative approximation to improve buckling approximations. Fadel chose an appropriate intermediate design variable for each function based on a gradient matching condition for the previous point. Vanderplaats initiated a second generation of high quality approximations when he developed the force approximation as an intermediate response approximation to improve the approximation of stress constraints. Canfield developed a Rayleigh Quotient approximation to improve the accuracy of eigenvalue approximations. Barthelemy and Haftka published a comprehensive review of approximations in 1993.
Recent MDO methods
MDO practitioners have investigated optimizationOptimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
methods in several broad areas in the last dozen years. These include decomposition method
Decomposition method
In constraint satisfaction, a decomposition method translates a constraint satisfaction problem into another constraint satisfaction problem that is binary and acyclic. Decomposition methods work by grouping variables into sets, and solving a subproblem for each set...
s, approximation
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...
methods, evolutionary algorithm
Evolutionary algorithm
In artificial intelligence, an evolutionary algorithm is a subset of evolutionary computation, a generic population-based metaheuristic optimization algorithm. An EA uses some mechanisms inspired by biological evolution: reproduction, mutation, recombination, and selection...
s, memetic algorithm
Memetic algorithm
Memetic algorithms represent one of the recent growing areas of research in evolutionary computation. The term MA is now widely used as a synergy of evolutionary or any population-based approach with separate individual learning or local improvement procedures for problem search...
s, response surface methodology
Response surface methodology
In statistics, response surface methodology explores the relationships between several explanatory variables and one or more response variables. The method was introduced by G. E. P. Box and K. B. Wilson in 1951. The main idea of RSM is to use a sequence of designed experiments to obtain an...
, reliability-based optimization, and multiobjective optimization
Multiobjective optimization
Multi-objective optimization , also known as multi-criteria or multi-attribute optimization, is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints....
approaches.
The exploration of decomposition methods has continued in the last dozen years with the development and comparison of a number of approaches, classified variously as hierarchic and non-hierarchic, or collaborative and non-collaborative.
Approximation methods spanned a diverse set of approaches, including the development of approximations for surrogate models, variable fidelity models, and trust region management strategies. The development of multipoint approximations blurred the distinction with response surface methods. Kriging
Kriging
Kriging is a group of geostatistical techniques to interpolate the value of a random field at an unobserved location from observations of its value at nearby locations....
methods became popular.
Response surface methodology
Response surface methodology
In statistics, response surface methodology explores the relationships between several explanatory variables and one or more response variables. The method was introduced by G. E. P. Box and K. B. Wilson in 1951. The main idea of RSM is to use a sequence of designed experiments to obtain an...
, developed extensively by the operations research community, received much attention in the MDO community in the last dozen years. A driving force for their use has been the development of massively parallel systems for high performance computing, which are naturally suited to distributing the function evaluations from multiple disciplines that are required for the construction of response surfaces. Distributed processing is particularly suited to the design process of complex systems in which analysis of different disciplines may be accomplished naturally on different computing platforms and even by different teams.
Evolutionary methods led the way in the exploration of non-gradient methods for MDO applications. They also have benefited from the availability of massively parallel high performance computers, since they inherently require many more function evaluations than gradient-based methods. Their primary benefit lies in their ability to handle discrete design variables and the potential to find globally optimal solutions.
Reliability-based optimization (RBO) is a growing area of interest in MDO. Like response surface methods and evolutionary algorithms, RBO benefits from parallel computation, because the numeric integration to calculate the probability of failure requires many function evaluations. One of the first approaches employed approximation concepts to integrate the probability of failure. The classical first-order reliability method (FORM) and second-order reliability method (SORM) are still popular. Grandhi used appropriate normalized variables about the most probable point of failure, found by a two-point adaptive nonlinear approximation to improve the accuracy and efficiency. Southwest Research Institute
Southwest Research Institute
Southwest Research Institute , headquartered in San Antonio, Texas, is one of the oldest and largest independent, nonprofit, applied research and development organizations in the United States...
has figured prominently in the development of RBO, implementing state-of-the-art reliability methods in commercial software. RBO has reached sufficient maturity to appear in commercial structural analysis programs like MSC's Nastran
Nastran
NASTRAN is a finite element analysis program that was originally developed for NASA in the late 1960s under United States government funding for the Aerospace industry. The MacNeal-Schwendler Corporation was one of the principal and original developers of the public domain NASTRAN code...
.
Utility-based probability maximization (Bordley and Pollock, Operations Research, Sept, 2009, pg.1262) was developed in response to some logical concerns (e.g., Blau's Dilemma) with reliability-based design optimization. This approach focuses on maximizing the joint probability of both the objective function exceeding some value and of all the constraints being satisfied. When there is no objective function, utility-based probability maximization reduces to a probability-maximization problem. When there are no uncertainties in the constraints, it reduces to a constrained utility-maximization problem. (This second equivalence arises because the utility of a function can always be written as the probability of that function exceeding some random variable.) Because it changes the constrained optimization problem associated with reliability-based optimization into an unconstrained optimization problem, it often leads to computationally more tractable problem formulations.
Problem formulation
Problem formulation is normally the most difficult part of the process. It is the selection of design variables, constraints, objectives, and models of the disciplines. A further consideration is the strength and breadth of the interdisciplinary coupling in the problem.Design variables
A design variable is a specification that is controllable from the point of view of the designer. For instance, the thickness of a structural member can be considered a design variable. Another might be the choice of material. Design variables can be continuous (such as a wing span), discrete (such as the number of ribs in a wing), or boolean (such as whether to build a monoplane or a biplaneBiplane
A biplane is a fixed-wing aircraft with two superimposed main wings. The Wright brothers' Wright Flyer used a biplane design, as did most aircraft in the early years of aviation. While a biplane wing structure has a structural advantage, it produces more drag than a similar monoplane wing...
). Design problems with continuous variables are normally solved more easily.
Design variables are often bounded, that is, they often have maximum and minimum values. Depending on the solution method, these bounds can be treated as constraints or separately.
Constraints
A constraint is a condition that must be satisfied in order for the design to be feasible. An example of a constraint in aircraft design is that the liftLift (force)
A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...
generated by a wing must be equal to the weight of the aircraft. In addition to physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models. Constraints can be used explicitly by the solution algorithm or can be incorporated into the objective using Lagrange multipliers.
Objectives
An objective is a numerical value that is to be maximized or minimized. For example, a designer may wish to maximize profit or minimize weight. Many solution methods work only with single objectives. When using these methods, the designer normally weights the various objectives and sums them to form a single objective. Other methods allow multiobjective optimization, such as the calculation of a Pareto frontPareto efficiency
Pareto efficiency, or Pareto optimality, is a concept in economics with applications in engineering and social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution.Given an initial allocation of...
.
Models
The designer must also choose models to relate the constraints and the objectives to the design variables. These models are dependent on the discipline involved. They may be empirical models, such as a regression analysisRegression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
of aircraft prices, theoretical models, such as from computational fluid dynamics
Computational fluid dynamics
Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...
, or reduced-order models of either of these. In choosing the models the designer must trade off fidelity with analysis time.
The multidisciplinary nature of most design problems complicates model choice and implementation. Often several iterations are necessary between the disciplines in order to find the values of the objectives and constraints. As an example, the aerodynamic loads on a wing affect the structural deformation of the wing. The structural deformation in turn changes the shape of the wing and the aerodynamic loads. Therefore, in analysing a wing, the aerodynamic and structural analyses must be run a number of times in turn until the loads and deformation converge.
Standard form
Once the design variables, constraints, objectives, and the relationships between them have been chosen, the problem can be expressed in the following form:- find
that minimizes 
subject to 
, 
and 
where
is an objective, is a vector of design variables, is a vector of inequality constraints, is a vector of equality constraints, and and are vectors of lower and upper bounds on the design variables. Maximization problems can be converted to minimization problems by multiplying the objective by -1. Constraints can be reversed in a similar manner. Equality constraints can be replaced by two inequality constraints.Problem solution
The problem is normally solved using appropriate techniques from the field of optimization. These include gradientGradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
-based algorithms, population-based algorithms, or others. Very simple problems can sometimes be expressed linearly; in that case the techniques of linear programming
Linear programming
Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...
are applicable.
Gradient-based methods
- Adjoint equationAdjoint equationAn adjoint equation is a linear differential equation, usually derived from its primal equation using integration by parts. Gradient values with respect to a particular quantity of interest can be efficiently calculated by solving the adjoint equation. Methods based on solution of adjoint...
- Newton's methodNewton's methodIn numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
- Steepest descent
- Conjugate gradient
- Sequential quadratic programmingSequential quadratic programmingSequential quadratic programming is an iterative method for nonlinear optimization. SQP methods are used on problems for which the objective function and the constraints are twice continuously differentiable....
Population-based methods
- Genetic algorithmGenetic algorithmA genetic algorithm is a search heuristic that mimics the process of natural evolution. This heuristic is routinely used to generate useful solutions to optimization and search problems...
- Memetic algorithmMemetic algorithmMemetic algorithms represent one of the recent growing areas of research in evolutionary computation. The term MA is now widely used as a synergy of evolutionary or any population-based approach with separate individual learning or local improvement procedures for problem search...
- Particle swarm optimizationParticle swarm optimizationIn computer science, particle swarm optimization is a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality...
- Harmony searchHarmony searchIn computer science and operations research, harmony search is a phenomenon-mimicking algorithm inspired by the improvisation process of musicians...
Other methods
- Random search
- Grid search
- Simulated annealingSimulated annealingSimulated annealing is a generic probabilistic metaheuristic for the global optimization problem of locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete...
- Direct searchBrute-force searchIn computer science, brute-force search or exhaustive search, also known as generate and test, is a trivial but very general problem-solving technique that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problem's...
- IOSOIOSOIOSO is a multiobjective, multidimensional nonlinear optimization technology.-IOSO approach:IOSO Technology is based on the response surface methodology approach....
(Indirect Optimization based on Self-Organization)
Most of these techniques require large numbers of evaluations of the objectives and the constraints. The disciplinary models are often very complex and can take significant amounts of time for a single evaluation. The solution can therefore be extremely time-consuming. Many of the optimization techniques are adaptable to parallel computing
Parallel computing
Parallel computing is a form of computation in which many calculations are carried out simultaneously, operating on the principle that large problems can often be divided into smaller ones, which are then solved concurrently . There are several different forms of parallel computing: bit-level,...
. Much current research is focused on methods of decreasing the required time.
Also, no existing solution method is guaranteed to find the global optimum
Global optimization
Global optimization is a branch of applied mathematics and numerical analysis that deals with the optimization of a function or a set of functions to some criteria.- General :The most common form is the minimization of one real-valued function...
of a general problem (see No free lunch in search and optimization). Gradient-based methods find local optima with high reliability but are normally unable to escape a local optimum. Stochastic methods, like simulated annealing and genetic algorithms, will find a good solution with high probability, but very little can be said about the mathematical properties of the solution. It is not guaranteed to even be a local optimum. These methods often find a different design each time they are run.
Commercial MDO Tools
- BOSS quattro (SAMTECH)
- FEMtools Optimization (Dynamic Design Solutions)
- HEEDS MDO (Red Cedar TechnologyRed Cedar TechnologyRed Cedar Technology is a software development and engineering services company. Red Cedar Technology was founded by Michigan State University professors Ron Averill and Erik Goodman in 1999. The headquarters is located in East Lansing, MI, near MSU's campus. Red Cedar Technology develops and...
) - HyperWorks HyperStudy (Altair EngineeringAltair EngineeringAltair Engineering is a product design and development, engineering software and cloud computing software company. Altair was founded by Jim Scapa, George Christ, and Mark Kistner in 1985. Over its history, it has had various locations near Detroit, Michigan, USA...
) - IOSOIOSOIOSO is a multiobjective, multidimensional nonlinear optimization technology.-IOSO approach:IOSO Technology is based on the response surface methodology approach....
(Sigma Technology) - Isight (Dassault Systèmes Simulia)
- LS-OPT (Livermore Software Technology Corporation)
- modeFRONTIERModeFRONTIERmodeFRONTIER is a multi-objective optimization and design environment, written to couple CAD/computer aided engineering tools, finite element structural analysis and computational fluid dynamics software. It is developed by ESTECO Srl and provides an environment for product engineers and designers...
(Esteco) - ModelCenter (PhoenixIntegration)
- NexusNexus (Process integration and optimization)Nexus is software for process integration. It is designed to solve multi-disciplinary and multi-objective optimization problems via a flowchart representation validated on the fly. Nexus is developed by iChrome Ltd., a British engineering and software company that specializes mainly in mathematical...
- Optimus (Noesis Solutions)
- OptiYOptiYOptiY is a design environment providing modern optimization strategies and state of the art probabilistic algorithms for uncertainty, reliability, robustness, sensitivity analysis, data-mining and meta-modeling.-History:...
(OptiY e.K.) - VisualDOC (Vanderplaats Research and Development)
- SmartDO (FEA-Opt Technology)