DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

THE IMMEDIATE DUALITY AS THE MOST SIMPLE SENSOR
FOR SOLVING SMOOTH MULTIDISCIPLINARY ELLIPTIC
PROBLEMS (DOMAIN VARIATION)
V. KAMINSKY
The paper deals with numerical methods for multidisciplinary optimization (MDO)
problems. Different traditional approaches for MDO problems on the base of Lagrange’s
multipliers (LM) have troubles with numerical calculation of the LM values in view of
nonlinear equations systems on each step of the recursion. The approach proposed in the
paper makes it possible to get these values almost “free of charge” on each step of the
recursion by more full use of the results of the problem linearization.
Copyright © 2006 V. Kaminsky. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
This paper has represented new approach in solving MDO problems which contain (1)
some boundary value problem (BVP) (as condition of the connection), (2) a convex object
functional, and (3) traditional constraints on all parameters (design variables -(DV))
and variables. Such problems are represented widely in design optimization [1, 2, 9, 14–
16], wherein the DV parameters can be present in differential operators, in the right part
of differential equations, in boundary conditions, and in description of the domain. A
kernel of the methods for MDO problems is usually the sensitivity analysis with respect
to the DV for succeeding recursion [2, 11, 16] (the so-called material derivatives). The
Main distinction between traditional and proposed approaches lies in the fact that we use
the linearization result more fully, applying for this purpose the pair of linear interrelated
problems: the primary (P-) problem and dual (D-) problem. Then the proposed technique
of improvement of the quality criteria becomes simpler and more precise specifically
in two situations: (a) the sensitivity matrix may be obtained semianalytically [10],
and (b) the description of DV variation is sufficiently simple (domain variation in BVP or
rightpartofdifferential equation variation in BVP, see please [1]) for elliptic operators.
2. Problem formulation
In what follows, the names of all the assumed conditions have been designed by the capital
character Λ □ □ with two indexes: (upper) for a number of the condition and (lower) for
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 467–475