gemseo / problems / scalable / parametric

gemseo.problems.scalable.parametric.core

• design_space - source
• models - source
• problem - source
• variables - source

Scalable module from Tedford and Martins (2010)

The modules located in this directory offer a set of classes relative to the scalable problem introduced in the paper:

Tedford NP, Martins JRRA (2010), Benchmarking multidisciplinary design optimization algorithms, Optimization and Engineering, 11(1):159-183.

Overview

This scalable problem aims to minimize an objective function quadratically depending on shared design parameters and coupling variables, under inequality constraints linearly depending on these coupling variables.

System discipline

A system discipline computes the constraints and the objective in function of the shared design parameters and coupling variables.

The discipline takes the global design parameters \(z\) and the coupling variables \(y_1,y_2,\ldots,y_N\) as inputs and returns the objective function value \(f(x,y(x,y))\) to minimize as well as the inequality constraints ones \(c_1(y_1),c_2(y_2),\ldots,c_N(y_N)\) which are expressed as:

\[f(z,y) = |z|_2^2 + \sum_{i=1}^N |y_i|_2^2\]

and:

\[c_i(y_i) = 1- C_i^{-T}Iy_i\]

Strongly coupled disciplines

The coupling variables are the outputs of strongly coupled disciplines.

Each strongly coupled discipline computes a set of coupling variables linearly depending on local design parameters, shared design parameters, coupling variables from other strongly coupled disciplines, and belonging to the unit hypercube.

The i-th discipline takes local design parameters \(x_i\) and shared design parameters \(z\) in input as well as coupling variables \(\left(y_i\right)_{1\leq j \leq N\atop j\neq i}\) from \(N-1\) elementary disciplines, and returns the coupling variables:

\[y_i =\frac{\tilde{y}_i+C_{z,i}.1+C_{x_i}.1}{\sum_{j=1 \atop j \neq i}^NC_{y_j,i}.1+C_{z,i}.1+C_{x_i}.1} \in [0,1]^{n_{y_i}}\]

where:

\[\tilde{y}_i = - C_{z,i}.z - C_{x_i}.x_i + \sum_{j=1 \atop j \neq i}^N C_{y_j,i}.y_j\]

Scalability

This problem is said “scalable” because several sizing features can be chosen by the user:

• the number of local design parameters for each discipline,

• the number of shared design parameters,

• the number of coupling variables for each discipline,

• the number of disciplines.

A given sizing configuration is called “scaling strategy” and this scalable module is particularly useful to compare different MDO formulations with respect to the scaling strategy.