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.
Traditionally engineering has normally been performed by teams, each with expertise in a specific discipline, such as aerodynamics or structures.
This led to an increased concentration on economic factors and the attributes known as the "ilities" including manufacturability, reliability, maintainability, etc.
The high-performance personal computer has largely replaced the centralized supercomputer and the Internet and local area networks have facilitated sharing of design information.
Whereas optimization methods are nearly as old as calculus, 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.
Jaroslaw Sobieski championed decomposition methods specifically designed for MDO applications.
First, the popular gradient-based methods used by the early structural optimization and MDO community are reviewed.
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.
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.
Fadel chose an appropriate intermediate design variable for each function based on a gradient matching condition for the previous point.
At present, many researchers are striving to arrive at a consensus regarding the best modes and methods for complex problems like impact damage, dynamic failure, and real-time analyses.
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.
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.
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.
Professor Ramana 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 Altair's Optistruct and MSC's Nastran.
Utility-based probability maximization was developed in response to some logical concerns (e.g., Blau's Dilemma) with reliability-based design optimization.
[4] This approach focuses on maximizing the joint probability of both the objective function exceeding some value and of all the constraints being satisfied.
In the marketing field there is a huge literature about optimal design for multiattribute products and services, based on experimental analysis to estimate models of consumers' utility functions.
In addition to physical laws, constraints can reflect resource limitations, user requirements, or bounds on the validity of the analysis models.
Once the design variables, constraints, objectives, and the relationships between them have been chosen, the problem can be expressed in the following form: where
Also, no existing solution method is guaranteed to find the global optimum of a general problem (see No free lunch in search and optimization).