A gradient-based transformation method in multidisciplinary design optimization

Abstract

Multidisciplinary design optimization (MDO) has become essential for solving the complex engineering design problems. The most common approach is to “divide and conquer” the MDO problem, that is, to decompose the complex problem into several sub-problems and to collect the local solutions to give a new design point for the original problem. In 1990s, researchers have developed some decomposition strategies to find or synthesize the optimal model of the optimization structure in order to evenly distribute the computational workloads to multiple processors. Several MDO methods, such as Collaborative Optimization (CO) and Analytical Target Cascading (ATC), were then developed to solve the decomposed sub-problems and coordinate the coupling variables among them to find the optimal solution. However, both the synthesis of the decomposition structure and the coordination of the coupling variables require additional function evaluations, in terms of evaluating the functional dependency between each sub-problem and determining the proper weighting coefficients between each coupling functions respectively. In this paper, a new divide-and-conquer strategy, Gradient-based Transformation Method (GTM), is proposed to overcome the challenges in structure synthesis and variable coordination. The proposed method first decomposes the MDO problem into several sub-systems and distributes one constraint from the original problem to each sub-system without evaluating the dependency between each sub-system. Each sub-system is then transformed to the single-variate coordinate along the gradient direction of the constraint. The total function evaluations equal the number of constraints times the number of variables plus one in every iteration. Due to the monotonicity characteristics of the transformed sub-problems, they are efficiently solved by Monotonicity Analyses without any additional function evaluations. Two coordination principles are proposed to determine the significances of the responses based on the feasibility and activity conditions of every sub-problem and to find the new design point at the average point of the most significant responses. The coordination principles are capable of finding the optimal solution in the convex feasible space bounded by the linearized sub-system constraints without additional function evaluations. The optimization processes continue until the convergence criterion is satisfied. The numerical examples show that the proposed methodology is capable of effectively and efficiently finding the optimal solutions of MDO problems.

Appendix: Mathematical tools for vectors, tensors, and dot products (Dill 2006)

In this paper, a column vector ${\bf A} =[A_{1}, A_{2}, \ldots , A_{N}]^{T}$ can be represented as the sum of N vectors along the base vectors, as in

$$ \label{eq49} {{\bf A}}=\sum\limits_{i=1}^N {A_i {{\bf e}}_i } $$

(49)

where e _i is the ith normal basis. To avoid having to repeatedly exhibit the summation sign, the (49) is further simplified using the Einstein summation convention:

$$ \label{eq50} {{\bf A}}=\sum\limits_{i=1}^N {A_i {{\bf e}}_i \equiv A_i {{\bf e}}_i } $$

(50)

where the repetition of any index means to take the sum of all terms obtained by letting that index take on its complete range of values. Following the summation convention, a N×N tensor C is represented by

$$ \label{eq51} {{\bf C}}=C_{ij} {{\bf e}}_i {{\bf e}}_j $$

(51)

where i and j both sum from 1 to N.

The dot product of two normal bases is given by

$$ \label{eq52} {{\bf e}}_i \cdot {{\bf e}}_j ={{\bf e}}_i^T {{\bf e}}_j =\delta_{ij} =\left\{ {{\begin{array}{@{}l@{\quad}l} 1 & {\mbox{for}\;i=j} \\[0.5em] 0 & {\mbox{otherwise}} \\ \end{array} }} \right. $$

(52)

where δ _ij is the Kronecker delta. Therefore, the dot product of A and another vector B = B _j e _j is given as

$$ \label{eq53} {{\bf A}}\cdot {{\bf B}}={{\bf A}}^T{{\bf B}}=A_i B_i $$

(53)

The tensor product of the vectors A and B, different from dot product in (53), produces a tensor, as in

$$ \label{eq54} {{\bf AB}}=A_i {{\bf e}}_i B_j {{\bf e}}_j =A_i B_j {{\bf e}}_i {{\bf e}}_j $$

(54)

The tensor C operating on the vector A produces a vector, shown as follows:

$$ \label{eq55} {{\bf C}}\cdot {{\bf A}}=C_{ij} {{\bf e}}_i {{\bf e}}_j \cdot A_k {{\bf e}}_k =C_{ij} A_k \delta_{jk} {{\bf e}}_i =C_{ik} A_k {{\bf e}}_i $$

(55)

The product of the tensors C and D = D _ij e _i e _j produces a tensor, as in

$$\begin{array}{lll} \label{eq56} {{\bf C}}\cdot {{\bf D}}&=&C_{ij} {{\bf e}}_i {{\bf e}}_j \cdot D_{kl} {{\bf e}}_k {{\bf e}}_l =C_{ij} D_{kl} \delta_{jk} {{\bf e}}_i {{\bf e}}_l \\ &=&C_{ik} D_{kl} {{\bf e}}_i {{\bf e}}_l \end{array} $$

(56)