5.1. MDAO (Multidisciplinary Design Analysis and Optimization)#

Warning

This course is under construction…

5.2. What is MDAO?#

MDAO stands for Multidisciplinary Design Analysis and Optimization. It is a scientific domain developed to adress the design optimization of multidisciplinary engineering systems that:

  • involve several disciplines/components

  • involve the presence of multidisciplinary couplings (algebraic loops)

  • involve a large number of design variables and constraints (10 to 100 000)

Therefore, it is particularly interesting for the sizing and optimization of mechatronic systems.

NOTE: MDA vs MDAO: Performing an MDA means to perform a system analysis that solves the multidisciplinary couplings without varying the design variables.

5.2.1. MDAO of engineering systems: # of disciplines vs fidelity of models#

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5.2.2. MDAO of engineering systems: multidisciplinary couplings#

The design of engineering systems can imply the presence and solving of multidiscipliary couplings. A bad ordering of disciplines or computation steps can lead to the creation or addition or creation of couplings:

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5.2.3. MDAO frameworks#

All commercial modelling and optimization software (Matlab, Wolfram Mathematica…) can be used to implement MDAO techniques. However, recently Python scientific framework have been developed in order to provide a framework to implement such problem more easily and solve them more efficiently:

  • OpenMDAO (NASA)

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  • GEMSEO (IRT Saint Exupery)

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5.2.4. Gradient-based MDAO: derivate computation techniques#

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There are different possibilities to compute gradients (derivatives) using typical MDAO frameworks:

  • full analytic derivative method uses the analytic derivatives of the model analysis functions (obtained by hand) and the analytic derivatives of the total model

  • semi-analytic derivative method is similar to the full analytic except that the model analysis functions derivatives are computed numerically (e.g. Finite Difference (FD))

  • monolithic derivative method estimates directly the total system derivatives numerically (e.g. FD) and does not require any knowledge of analysis function derivatives

5.2.5. MDAO formulations#

For solving an MDAO problem multiple formulations (or architectures) can be implemented. In 2012, Martins and Lambe proposed a survey of these formulations.

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5.2.6. MDAO formulations: distributed vs monolithic (1/2)#

Distributed formulations use more than one optimizer:

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Monolithic formulations use only one optimizer:

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5.2.7. MDAO formulations: distributed vs monolithic (2/2)#

The Sellar problem is a simple and well known test case for MDO formulations:

(5.1)#\[\begin{equation} \begin{aligned} & \text{minimize} & & x_{1}^{2} + z_{2} + y_{1} + e^{-y_{2}} \\ & \text{with respect to} & & z_{1}, z_{2}, x_{1}\\ & \text{subject to} & & \frac{y_{1}}{3.16} - 1 \geq 0 & & 1 - \frac{y_{2}}{24} \geq 0 & & -10 \geq z_{1} \leq 10 & & 0 \geq z_{2} \leq 10 & & 0 \geq x_{1} \leq 10 \end{aligned} \end{equation}\]

Where \(y_{1} = z_{1}^2 + x_{1} + z_{2} - 0.2 y_{2}\) and \(y_{2}=\sqrt{y_{1}} + z_{1} + z_{2}\).

The number of function evaluation is a key metric to assess the computational cost as it enables to avoid considering the performance of the computer used.

# Function evaluations

Discipline 1

Discipline 2

IDF

60

50

MDF

222

216

CO

5647

8252

BLISS

3344

3130

BLISS-2000

818

108

In the context of sizing we are particularly interested in computation time as we are in a decision making process. Hence, we generally use monolithic formulations.

5.2.8. Monolithic formulations: High dynamic EMA case study#

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(5.2)#\[\begin{equation} T_{em} = J_{mot} \cdot A_{max}\frac{N_{red}}{p} + F_{ema} \frac{p}{N_{red}} \label{eq_1} \end{equation}\]
(5.3)#\[\begin{equation} J_{mot} = J_{mot_{ref}} \cdot \left(\frac{T_{em}}{T_{em_{ref}}}\right)^{\frac{5}{3.5}} \label{eq_2} \end{equation}\]
(5.4)#\[\begin{equation} \Omega_{mot} = \Omega_{mot_{ref}} \cdot \left(\frac{T_{em}}{T_{em_{ref}}}\right)^{-\frac{1}{3.5}} \label{eq_3} \end{equation}\]
(5.5)#\[\begin{equation} \Omega_{mot} \geq V_{max}\cdot \frac{N_{red}}{p} \label{eq_4} \end{equation}\]
(5.6)#\[\begin{equation} M_{mot} = M_{mot_{ref}} \cdot \left(\frac{T_{em}}{T_{em_{ref}}}\right)^{\frac{3}{3.5}} \label{eq_5} \end{equation}\]
(5.7)#\[\begin{equation} \begin{aligned} & \text{minimize} & & M_{mot} \\ & \text{with respect to} & & N_{red}\\ & \text{subject to} & & V_{max} \cdot \frac{N_{red}}{p} - \Omega_{mot} \leq 0 \end{aligned} \end{equation}\]

5.2.9. Monolithic formulations: MultiDisciplinary Feasible (MDF)#

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(5.8)#\[\begin{equation} \begin{aligned} & \text{minimize} & & M_{mot} \\ & \text{with respect to} & & N_{red}\\ & \text{subject to} & & V_{max} \cdot \frac{N_{red}}{p} - \Omega_{mot} \leq 0 \end{aligned} \end{equation}\]

5.2.10. Monolithic formulations: Individual Discipline Feasible (IDF)#

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5.2.11. Monolithic formulations: Hybrid#

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5.2.12. Monolithic formulations: Normalized Variable Hybrid (NVH)#

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(5.9)#\[\begin{equation} \begin{aligned} & \text{min.} & & M_{mot} \\ & \text{w.r.t} & & N_{red}, k_{os}\\ & \text{s.t.} & & V_{max} \cdot \frac{N_{red}}{p} - \Omega_{mot} \leq 0 \\ & & & J_{mot} \cdot A_{max}\frac{N_{red}}{p} + F_{ema} \frac{p}{N_{red}} - T_{em} \leq 0 \end{aligned} \end{equation}\]

5.2.13. Monolithic formulations: Performance comparison#

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The NVH formulation is an intrusive formulation as it requires to redefine the optimization problem and models. However, it provides significant reduction of computational cost.