Multidisciplinary design optimization
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Multidisciplinary design optimization (MDO) is a field of engineering that uses optimization methods to solve design problems incorporating a number of disciplines. 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, computers, 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.
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History
Traditional engineering design 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 drag, 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 airlines and the military, 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": 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 networks 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.
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 numeric value 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. 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 biplane). 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 lift 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 front.
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 analysis 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.
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:
minimize <math>J(\mathbf{x})<math>
subject to <math>\mathbf{g}(\mathbf{x})\leq\mathbf{0} <math>
<math>\mathbf{x}_{lb}\leq \mathbf{x} \leq \mathbf{x}_{ub} <math>
where <math>J<math> is an objective, <math>\mathbf{x}<math> is a vector of design variables, <math>\mathbf{g}<math> is a vector of constraints, and <math>\mathbf{x}_{lb}<math> and <math>\mathbf{x}_{ub}<math> 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 gradient-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.
Gradient-based methods
Population-based methods
Other methods
- Random search
- Grid search
- Simulated annealing
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 solution method currently exists that is guaranteed to find a global extremum of a problem. 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.