Multidisciplinary optimization

The multidisciplinary Optimization (OMD or English MDO) is a field of Ingénierie which uses methods of optimization in order to solve problems of design implementing several disciplines.

The MDO makes it possible to the originators to incorporate the effects of each discipline at the same time. The total optimum thus found is better than the configuration found by optimizing each discipline independently of the others, because one takes into account the interactions between the disciplines. However that involves a overcost on the level of the time calculation and complexity of the problem.

These techniques are used in several scopes of application, of which the design Automobile, naval, electronic and data-processing. However the greatest application remains in the field of the aeronautical design. For example, the project of Hang-glider of Boeing ( Blended wing body ) with largely used techniques MDO within the framework of the design preliminary draft. The disciplines considered in this project are the Aérodynamique, the mechanics of the structures, the propulsion and the economy.

History

Traditionally, the aeronautical design is carried out by teams, having each expert and of specific knowledge in a precise field, like aerodynamics or strucure. Each team endeavors to obtain a point of acceptable design while working sequentially. For example aerodynamics wishes to obtain a certain field of pressure on the wing and the strcuture will seek a profile of wing which will reach that point. The goal to reach are generally to minimize the Traînée or the weight.

Between 1970 and 1990, two factors changed the approach of the design. The Computer-aided design (CAD), which makes it possible to modify and analyze quickly the points of design. The second is the policy change of management in the majority of the airline companies and the military organizations, with performances concerning the costs generated during the life cycle of the vehicles. This involved creation again economic factors the such Faisabilité, the possibility of machining and of maintaining the configurations obtained, etc

Since the Years 1990, these techniques migrated towards other industries. The Mondialisation led to increasingly distributed and decentralized designs. The powers of the microcomputers replace the centralized supercomputeurs, the communication and the division of information are facilitated by Internet and the lans. The tools for analysis high fidelity, of type finite elements, start to be very powerful. Moreover, there are large projections in the field of the algorithms of Optimization.

Formulations MDO

This part is most difficult in these processes. One seeks a compromise between a completely coupled system, i.e. where interdisciplinary physics is well defined for each point of the optimization, but which generates a great cost on the level of the computing time and a system where one release the variable of Couplage and where interdisciplinary feasibility is only required with convergence.

The methods of resolution of a multidisciplinary problem of optimization can be divided into two categories: mononiveaux methods (or individual-level) and multilevel methods. The mononiveaux methods call upon a single optimizer.

Among the mononiveaux methods, one finds methods AAO (All At Ounce), MDF (Multi Discipline Feasible), IDF (Individual Discipline Feasible).
Among the multilevel methods: CO (Collaborative Optimization), CSSO (Competitor Sub Space Optimization), BLISS (Bi Level Integrated System Synthesis).

Variable of design

A variable of design is a numerical value that the designor can modify. For example the thickness of the box or the arrow can be regarded as variable of design. These variables can be continuous, or discrete (number of word on the plane).

One can also separate the local variables, which intervene only in one discipline, of the shared variables, which have an influence on several disciplines.

Constraint

A Contrainte is a condition which must be satisfied to obtain an acceptable point.

An example in the aeronautical design: the Portance of the plane must be equal to the weight. In addition to translating physical phenomena these constraints can also represent limitations of resource, requirements of the schedules of conditions, or limits of validity of the model of analysis.

Objectives

An objective is a numerical value which must be optimized. For example, the originator can choose to maximize the profit, the range operating, or to minimize the total weight. Other problems treat Multiobjectif, often contradictory. The solution will then be chosen by expert by finding a compromise using face of pareto.

Models

The originators must also choose models in order to represent the behavior of the constraints and objectives according to the variables of design. One can have empirical models, as a regressive analysis on the cost of the plane, or many more pushed ideal models, as the finite element or the CFD, or even of the models reduced in any kind. The choice of these models results from a compromise meanwhile of execution and présision of the analysis.

The multidisciplinary character of the problem strongly complicates the choice of the models and their implementation. Very often of many iterations are necessary before finding the good values of the functions costs and the constraints. Indeed, the field of pressure will modifira the shape of the wing and the latter will affect this same field. It is thus necessary sequentially to evaluate these values, before being able to converge towards the good configuration of physics.

Standart formulation

Once one chose the variables of design, the constraints, the objectives and the relations between each discipline, the general problem can be expressed in the following way:

to find $\ mathbf {X}$ which minimizes $J (\ mathbf {X})$ under constraint $\ mathbf {G} (\ mathbf {X}) \ Leq \ mathbf {0}$ and $\ mathbf {X} _ {lb} \ Leq \ mathbf {X} \ Leq \ mathbf {X} _ {ub}$

or $J$ is an objective, $\ mathbf {X}$ is a vector of variable of design, $\ mathbf {G}$ is the vector of the constraints, and $\ mathbf {X} _ {lb}$ and $\ mathbf {X} _ {ub}$ is the terminals of the variables of design.

Resolution of the problem

The problem is then solved by applying the suitable techniques of optimization. That can be algorithms containing gradient or many genetic algorithms.

The majority of these methods require a great number of evaluation of the objective and constraints. Each model can be rather expensive in computing times. It is then necessary to use technique of parallelization of calculation and surfaces of answer.

It should be specified finally that there does not exist miracle solution guaranteeing to find the optimum total in a reasonable time. The methods containing gradient are found easily wedged in a local optimum. Stochastic methods, like the genetic algorithms one of strong chance to obtain a correct solution, requiring a number of evaluation of functions exponentially proportional to the number of variable of design and not guaranteeing good properties of optimality. A hybrid method could stage these two difficulties.

Multidisciplinary optimization is sails very about it in research in aeronautics, data processing and mathematics applied.

Random links: