Benchmarking of monolithic MDO formulations and derivative computation techniques using OpenMDAO

Abstract

The design optimization of coupled systems requires the implementation of multidisciplinary design optimization techniques in order to obtain consistent and optimal solutions. The associated research topics include the development of optimization algorithms, computational frameworks, and multidisciplinary design optimization formulations. This paper presents a benchmarking of the combination of monolithic formulations and derivative computation techniques. The monolithic formulations include typical literature formulations as well as new normalized variable hybrid formulation. A novel test problem is proposed which consists in the sizing of a space launcher thrust vector control electro-mechanical actuator. Solving the single multidisciplinary coupling present in this problem is complex due to the possibility to face one, two, or no solutions depending on the external load and reducer gear ratio configuration. A larger scale version of this test problem is also proposed and tested by adding a high degree of freedom point-to-point trajectory. The tests are carried out in order to obtain typical performance criteria but also some proposed additional robustness criteria such as variation of the initial conditions or the external load scale. These additional criteria are particularly relevant in an industrial engineering design context where knowledge capitalization and reuse are sought. The most significant findings are the interesting performances of the new formulation in terms of computational cost and the robustness. Furthermore, the effect of the choice of derivative computation strategy on different performance criteria is assessed for the original and larger scale problem, and thus underlines the benefits of full analytic gradient-based optimization.

Efficient sizing and optimization of multirotor drones based on scaling laws and similarity models

“…Step 1: Define a system of algebraic equations S based on estimation models and simulation models subject to inequality constraints Analytical models, Scaling laws, Datasheet regression [24,34,14] Step 2: Reduce (when possible) the number of inequality constraints First Monotonicity Principle (MP1) Section 3.2.1, [35] Step 3: Build undirected bipartite graph/design graph from S, identify Strongly Connected Components (SCC) and solve them by implementing the NVH formulation Design graphs, Symbolic computation, Graph algorithms Section 3.3, Section 3.4, [36,37,34,28] Step 4: Perform a matching and ordering of the modified system and generate the sizing procedure Graph algorithms, Symbolic computation [36,37,34] the efficiency of the ordering and is useful to visualize the models involved and the shared variables. Once the required thrust for every mission is estimated from an initial guess of the total drone mass, propeller torque and speed characteristics are estimated.…”

“…For the remaining loops, the NVH formulation approach is implemented by introducing dedicated additional constraints and design variables in an optimization process in order to solve the multidisciplinary couplings such as the mass loop in Fig. 2 [33,28].…”

“…The resolution of the multidisciplinary couplings can be achieved using different distributed or monolithic MDO formulations such as the MultiDisciplinary Feasible (MDF) of the Individual Discipline Feasible (IDF). Here a monolithic Normalized Variable Hybrid (NVH) formulation is used as it is fast and robust to scale-change due to the usage of inequality constraints, a small dimension consistency design variable and a reduced number of constraints and design variables [28]. However, it does not guarantee system consistency when the optimizer fails when compared to the MDF.…”

“…Although the implementation is somehow complex and needs a good understanding of the system to replace equations, the use of design graphs and the algorithms presented in [36,34] makes this step easier. The interesting performances, highlighted by the number of iterations, and reduced resolution time in Table 6 of this Normalized Variable Hybrid formulation is also benchmarked in terms of computational cost and robustness versus other monolithic MDO formulations in [28].…”

Trajectory and design optimization of multirotor drones with system simulation

“…The first one is FAST-OAD [7] framework which is based on OpenMDAO [8]. For this environment, the SLSQP optimizer is used and the semi-analytic finite difference approach is chosen for computing derivatives [9]. The second one is Modelon Impact [10] that uses the Modelica modelling language [11] and the Optimica language [12,13].…”