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Gearboxes are mechanical transmission systems that provide speed and torque conversions from a rotating power source. Being a central element of the drive train, they are relevant for the efficiency and durability of motor vehicles. In this work, we present a new approach for gearbox design: Modeling the design problem as a mixed-integer nonlinear program (MINLP) allows us to create gearbox designs from scratch for arbitrary requirements and—given enough time—to compute provably globally optimal designs for a given objective. We show how different degrees of freedom influence the runtime and present an exemplary solution.
Altherr, Lena ; Brötz, Nicolas ; Dietrich, Ingo ; Gally, Tristan ; Geßner, Felix ; Kloberdanz, Hermann ; Leise, Philipp ; Pelz, Peter Franz ; Schlemmer, Pia ; Schmitt, Andreas
Resilience as a concept has found its way into different disciplines to describe the ability of an individual or system to withstand and adapt to changes in its environment. In this paper, we provide an overview of the concept in different communities and extend it to the area of mechanical engineering. Furthermore, we present metrics to measure resilience in technical systems and illustrate them by applying them to load-carrying structures. By giving application examples from the Collaborative Research Centre (CRC) 805, we show how the concept of resilience can be used to control uncertainty during different stages of product life.
The understanding that optimized components do not automatically lead to energy-efficient systems sets the attention from the single component on the entire technical system. At TU Darmstadt, a new field of research named Technical Operations Research (TOR) has its origin. It combines mathematical and technical know-how for the optimal design of technical systems. We illustrate our optimization approach in a case study for the design of a ventilation system with the ambition to minimize the energy consumption for a temporal distribution of diverse load demands. By combining scaling laws with our optimization methods we find the optimal combination of fans and show the advantage of the use of multiple fans.
Energy-efficient components do not automatically lead to energy-efficient systems. Technical Operations Research (TOR) shifts the focus from the single component to the system as a whole and finds its optimal topology and operating strategy simultaneously. In previous works, we provided a preselected construction kit of suitable components for the algorithm. This approach may give rise to a combinatorial explosion if the preselection cannot be cut down to a reasonable number by human intuition. To reduce the number of discrete decisions, we integrate laws derived from similarity theory into the optimization model. Since the physical characteristics of a production series are similar, it can be described by affinity and scaling laws. Making use of these laws, our construction kit can be modeled more efficiently: Instead of a preselection of components, it now encompasses whole model ranges. This allows us to significantly increase the number of possible set-ups in our model. In this paper, we present how to embed this new formulation into a mixed-integer program and assess the run time via benchmarks. We present our approach on the example of a ventilation system design problem.
Planning the layout and operation of a technical system is a common task
for an engineer. Typically, the workflow is divided into consecutive stages: First,
the engineer designs the layout of the system, with the help of his experience or of
heuristic methods. Secondly, he finds a control strategy which is often optimized
by simulation. This usually results in a good operating of an unquestioned sys-
tem topology. In contrast, we apply Operations Research (OR) methods to find a
cost-optimal solution for both stages simultaneously via mixed integer program-
ming (MILP). Technical Operations Research (TOR) allows one to find a provable
global optimal solution within the model formulation. However, the modeling error
due to the abstraction of physical reality remains unknown. We address this ubiq-
uitous problem of OR methods by comparing our computational results with mea-
surements in a test rig. For a practical test case we compute a topology and control
strategy via MILP and verify that the objectives are met up to a deviation of 8.7%.
Pure analytical or experimental methods can only find a control strategy for technical systems with a fixed setup. In former contributions we presented an approach that simultaneously finds the optimal topology and the optimal open-loop control of a system via Mixed Integer Linear Programming (MILP). In order to extend this approach by a closed-loop control we present a Mixed Integer Program for a time discretized tank level control. This model is the basis for an extension by combinatorial decisions and thus for the variation of the network topology. Furthermore, one is able to appraise feasible solutions using the global optimality gap.
Nach Stand von Wissenschaft und Technik werden Komponenten hinsichtlich ihrer Eigenschaften, wie Lebensdauer oder Energieeffizienz, optimiert. Allerdings können selbst hervorragende Komponenten zu ineffizienten oder instabilen Systemen führen, wenn ihr Zusammenspiel nur unzureichend berücksichtigt wird. Eine Systembetrachtung schafft ein größeres Optimierungspotential - dem erhöhten Potential steht jedoch auch ein erhöhter Komplexitätsgrad gegenüber. Die vorliegende Arbeit ist im Rahmen des Sonderforschungsbereichs 805 entstanden, dessen Ziel die Beherrschung von Unsicherheit in Systemen des Maschinenbaus ist. Die Arbeit zeigt anhand eines realen Systems aus dem Bereich der Hydraulik, wie Unsicherheit in der Entwicklungsphase beherrscht werden kann. Hierbei ist neu, dass die durch den späteren Betrieb zu erwartende Systemdegradation eines jeden möglichen Systemvorschlags antizipiert werden kann. Dadurch können Betriebs- und Wartungskosten vorausgesagt und minimiert werden und durch eine optimale Betriebs- und Wartungsstrategie die Verfügbarkeit des Systems garantiert werden. Wesentliche Fragen bei der optimalen Auslegung des betrachteten hydrostatischen Getriebes sind dessen physikalische Modellierung, die Darstellung des Optimierungsproblems als gemischt-ganzzahliges lineares Programm, und dessen algorithmische Behandlung zur Lösungsfindung. Hierzu werden Heuristiken zum schnelleren Auffinden sinnvoller Systemtopologien vorgestellt und mittels mathematischer Dekomposition eine Bewertung des dynamischen Verschleiß- und Wartungsverlaufs möglicher Systemvorschläge vorgenommen. Die Arbeit stellt die Optimierung technischer Systeme an der Schnittstelle von Mathematik, Informatik und Ingenieurwesen sowohl gründlich als auch anschaulich und nachvollziehbar dar.
Finding a good system topology with more than a handful of components is a
highly non-trivial task. The system needs to be able to fulfil all expected load cases, but at the
same time the components should interact in an energy-efficient way. An example for a system
design problem is the layout of the drinking water supply of a residential building. It may be
reasonable to choose a design of spatially distributed pumps which are connected by pipes in at
least two dimensions. This leads to a large variety of possible system topologies. To solve such
problems in a reasonable time frame, the nonlinear technical characteristics must be modelled
as simple as possible, while still achieving a sufficiently good representation of reality. The
aim of this paper is to compare the speed and reliability of a selection of leading mathematical
programming solvers on a set of varying model formulations. This gives us empirical evidence
on what combinations of model formulations and solver packages are the means of choice with the current state of the art.
The UN sets the goal to ensure access to water and sanitation for all people by 2030. To address this goal, we present a multidisciplinary approach for designing water supply networks for slums in large cities by applying mathematical optimization. The problem is modeled as a mixed-integer linear problem (MILP) aiming to find a network describing the optimal supply infrastructure. To illustrate the approach, we apply it on a small slum cluster in Dhaka, Bangladesh.