Distributed Design Approach

Competitive engineering systems require tight integration of multiple disciplines (mechanics, electrics, software) ^{[20, 21, 22]}, which increases development complexity. A coherent early-phase conceptual design is therefore crucial to shorten timelines and avoid costly later corrections ^{[23, 24, 25]}. Multidisciplinary Design Optimization (MDO) uses algorithms to find feasible and optimal multi-component designs ^{[5, 4, 2]}.

Complex design tasks are typically partitioned into subtasks solved by specialized subteams ^{[9, 26, 2]}. This is both necessary and advantageous ^[7, 27]:

• Divide-and-conquer yields manageable subtasks.
• Subtasks can be executed in parallel, shortening development time.
• No single engineer can master all required fields (electronics, mechanics, etc.).
• Dedicated tools (e.g., FEM, CFD) are inherently partitioned by vendor or infrastructure.

Partitioning demands frequent communication between subteams to maintain design coherence ^[28, 4]. As an example, the Supersonic Business Jet (SSBJ) design (adapted from ^{[29, 1, 30]}) is partitioned into the subtasks shown below:

Once partitioned into coupled subsystems, optimization algorithms can efficiently explore the large design spaces characteristic of early conceptual phases ^{[5, 26, 2]}. However, engineers must formulate each subtask as a mathematical optimization problem ^[31, 6].

Adopting the notation from ^[1] (similar to ^[14]), the optimization problem of subsystem \(i\) reads:

\[\begin{aligned} {}^{i}x^{*},\;\left\{{}^{i}_{j}z^{*}\right\}_{j\in{}^{i}N} := \; & \argmin_{\substack{{}^{i}x \;\in\; {}^{i}\mathcal{X} \\ {}^{i}_{j}z \;\in\; {}^{i}_{j}\mathcal{Z} \;\forall\; j\in{}^{i}N}} \quad {}^{i}v_f\!\left({}^{i}r\right) \\ & \text{s.t.} \quad {}^{i}v_g\!\left({}^{i}r\right) \leq 0, \\ & \phantom{\text{s.t.}} \quad {}^{i}v_h\!\left({}^{i}r\right) = 0, \\ & \text{where} \quad {}^{i}r := {}^{i}r\!\left({}^{i}x,\;\left\{{}^{i}_{j}z,\;{}^{j}_{i}h\right\}_{j\in{}^{i}N}\right). \end{aligned}\]

The notation is summarized as follows:

Symbol	Description
${}^{i}$$x$	Local design variables of subsystem $i$ within bounds ${}^{i}\mathcal{X}$
${}^{i}_{j}$$z$	Shared design variables with neighbor $j$ within bounds ${}^{i}_{j}\mathcal{Z}$
${}^{i}$$N$	Index set of all neighbors $j$ coupled to subsystem $i$
${}^{i}$$v_f$	Objective function
${}^{i}$$v_g$, ${}^{i}$$v_h$	Inequality and equality constraints
${}^{i}$$r$	Response function
${}^{j}_{i}$$h$	Coupling variable provided by neighbor $j$
${}^{j}_{i}$$H$	Coupling function (mapping) from subsystem $j$ to $i$

The coupling variable is defined as:

\[{}^{j}_{i}h := {}^{j}_{i}H\!\left({}^{j}r\right),\]

where the expression \({}^{j}_{i}H\!\left({}^{j}r\right)\) is called the mapped response of subsystem \(j\) to \(i\).

The figure below translates the SSBJ partitioning into the mathematical notation above.

The SSBJ example is also used as an illustration in Tutorial > Problem Defintion and Algorithm Execution and Tutorial > Processing. Further infromation on the SSBJ can be found in the SSBJ example.