90C90 Applications of mathematical programming
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Multidisciplinary optimizations (MDOs) encompass optimization problems that combine different disciplines into a single optimization with the aim of converging towards a design that simultaneously fulfills multiple criteria. For example, considering both fluid and structural disciplines to obtain a shape that is not only aerodynamically efficient, but also respects structural constraints. Combined with CAD-based parametrizations, the optimization produces an improved, manufacturable shape. For turbomachinery applications, this method has been successfully applied using gradient-free optimization methods such as genetic algorithms, surrogate modeling, and others. While such algorithms can be easily applied without access to the source code, the number of iterations to converge is dependent on the number of design parameters. This results in high computational costs and limited design spaces. A competitive alternative is offered by gradient-based optimization algorithms combined with adjoint methods, where the computational complexity of the gradient calculation is no longer dependent on the number of design parameters, but rather on the number of outputs. Such methods have been extensively used in single-disciplinary aerodynamic optimizations using adjoint fluid solvers and CAD parametrizations. However, CAD-based MDOs leveraging adjoint methods are just beginning to emerge.
This thesis contributes to this field of research by setting up a CAD-based adjoint MDO framework for turbomachinery design using both fluid and structural disciplines. To achieve this, the von Kármán Institute’s existing CAD-based optimization framework cado is augmented by the development of a FEM-based structural solver which has been differentiated using the algorithmic differentiation tool CoDiPack of TU Kaiserslautern. While most of the code could be differentiated in a black-box fashion, special treatment is required for the iterative linear and eigenvalue solvers to ensure accuracy and reduce memory consumption. As a result, the solver has the capability of computing both stress and vibration gradients at a cost independent on the number of design parameters. For the presented application case of a radial turbine optimization, the full gradient calculation has a computational cost of approximately 3.14 times the cost of a primal run and the peak memory usage of approximately 2.76 times that of a primal run.
The FEM code leverages object-oriented design such that the same base structure can be reused for different purposes with minimal re-differentiation. This is demonstrated by considering a composite material test case where the gradients could be easily calculated with respect to an extended design space that includes material properties. Additionally, the structural solver is reused within a CAD-based mesh deformation, which propagates the structural FEM mesh gradients through to the CAD parameters. This closes the link between the CAD shape and FEM mesh. Finally, the MDO framework is applied by optimizing the aerodynamic efficiency of a radial turbine while respecting structural constraints.
Aggregation of Large-Scale Network Flow Problems with Application to Evacuation Planning at SAP
(2005)
Our initial situation is as follows: The blueprint of the ground floor of SAP’s main building the EVZ is given and the open question on how mathematic can support the evacuation’s planning process ? To model evacuation processes in advance as well as for existing buildings two models can be considered: macro- and microscopic models. Microscopic models emphasize the individual movement of evacuees. These models consider individual parameters such as walking speed, reaction time or physical abilities as well as the interaction of evacuees during the evacuation process. Because of the fact that the microscopic model requires lots of data, simulations are taken for implementation. Most of the current approaches concerning simulation are based on cellular automats. In contrast to microscopic models, macroscopic models do not consider individual parameters such as the physical abilities of the evacuees. This means that the evacuees are treated as a homogenous group for which only common characteristics are considered; an average human being is assumed. We do not have that much data as in the case of the microscopic models. Therefore, the macroscopic models are mainly based on optimization approaches. In most cases, a building or any other evacuation object is represented through a static network. A time horizon T is added, in order to be able to describe the evolution of the evacuation process over time. Connecting these two components we finally get a dynamic network. Based on this network, dynamic network flow problems are formulated, which can map evacuation processes. We focused on the macroscopic model in our thesis. Our main focus concerning the transfer from the real world problem (e.g. supporting the evacuation planning) will be the modeling of the blueprint as a dynamic network. After modeling the blueprint as a dynamic network, it will be no problem to give a formulation of a dynamic network flow problem, the so-called evacuation problem, which seeks for an optimal evacuation time. However, we have to solve a static large-scale network flow problem to derive a solution for this formulation. In order to reduce the network size, we will examine the possibility of applying aggregation to the evacuation problem. Aggregation (lat. aggregare = piling, affiliate; lat. aggregatio = accumulation, union; the act of gathering something together) was basically used to reduce the size of general large-scale linear or integer programs. The results gained for the general problem definitions were then applied to the transportation problem and the minimum cost network flow problem. We review this theory in detail and look on how results derived there can be used for the evacuation problem, too.