Mathematical Research

In recent decades, the complexity of the challenges facing industry has grown increasingly, so that the concrete application of mathematical methods has become imperative in many areas. TWT has been investing in the development of mathematics since its foundation in 1986. The Mathematical Research Group, consisting of more than 30 mathematicians and people interested in mathematics, the Mathematics Circle, takes this fact into account in several ways. Through the broadly spread expertise of the staff, we offer expertise from all major areas of mathematics, such as analysis, algebra, number theory, numerics, stochastics, data analytics, and operations research. Among other things, we use this knowledge to investigate and solve fundamental mathematical problems at the highest academic level. We then use the knowledge gained in this way in the sense of technical-scientific transfer to give our customers the necessary edge in the increasingly complex engineering, physical and technical problems that will ensure their competitiveness in the industry of the future. In order to live up to this claim, we maintain a constantly growing network of leading institutes and professors around the world, who actively advise and support us in our projects. This includes former TWT colleagues, who have returned to the university.

Illustration of the symmetry of a graph of the sum of two particular exponential functions.

Number Theory

Basic mathematical research deals with abstract problems, which often at first sight have no direct application in industrial projects. The importance of basic mathematical research lies mainly in the development of new methods and the extension of existing knowledge. The full potential of these research results is often only realized later and very often in surprising and unforeseen ways. In this sense, basic mathematical research is an important contribution to future innovations.

A current project from basic research at TWT deals with the symmetry of finite sums of complex exponential functions. The exponential function is of fundamental importance in mathematics and occurs, for example, as an elementary building block in Fourier series. If one interprets the graph of a complex exponential function as a parameterized curve, the result is the unit circle. However, if one adds several different complex exponential functions, fascinating graphs arise whose symmetry depends on the generating parameters. In the current work,

PAUSINGER, Florian; VARTZIOTIS, Dimitris. On the symmetry of finite sums of exponentials. Elemente der Mathematik, 2021, 76. Jg., Nr. 2, S. 62-73, Elemente der Mathematik, 2021, 76. Jg., Nr. 2, S. 62-73,

the symmetry groups of these graphs could be determined as a function of the generating parameters.

1.2 Game Theory

Typically, game theory is applied in economics. The question of the timing and price of a product at market launch or the identification of power relationships are classic problems. In a specific assignment, for example, we have already advised our client on how to ideally allocate the award of projects to external partners, taking into account several internal stakeholders with their own priorities regarding the award. Here, we were particularly interested in the decision optimum, but also in the likely behavior patterns of the stakeholders, in order to be able to derive optimal strategies.

For some years now, game theory methods have been increasingly used in other areas, such as route planning. We are also interested in further developing game theory at the academic level and thus advancing new areas, as this published research sketch shows:

VARTZIOTIS, Dimitris; BOHNET, Doris; HIMPEL, Benjamin. Smoothing Game. arXiv preprint arXiv:2010.04956,, 2020, ( Link: ).

Resource planning

To improve meshes for finite element simulations, they need to be smoothed by moving the nodes. We want to introduce a new smoothing approach by treating each geometric element as a player in a game: a quest for the best element quality. In other words, each player has the goal of becoming as regular as possible. .

Optimization of power quality using game theory. (Link:
Demonstrator for the optimization of highly complex systems with Cubic AI.

Optimization and Graph Theory

In practice, relevant key figures for a process or model often need to be improved. We use state-of-the-art algorithms from derivative-free, global, AI-assisted or even nonlinear optimization. Regardless of whether the problem is discrete or continuous, we investigate it for solvability and propose robust algorithms. An example can be found in our product Veris® (Link: ) which, among other things, performs optimization in route planning. In this environment, we conduct research in the area of quantum algorithms, which we presented at the Digital Product Forum 2022 at Mercedes-Benz, among others:

KRAUS, Hamzeh, RIEGG, Andreas (Mercedes-Benz). Co-Autoren: Leder, M. (Mercedes-Benz AG), Vaudrevange, P., Stasinou, M., Banerjee, R., Kröker, R., Dierolf, B., Rößler, T., Fäßler, V., Keckeisen, M.. Quantum computing for a breadth-first search algorithm with application in vehicle route planning, DPF 2022, Stuttgart.

With Cubic AI  we also take the optimization of highly complex systems to a whole new level. Almost like a genie in a bottle, you can wish for the behavior of your system and the modern AI will determine the best approximation to your specification in real time, taking into account the physical limitations. Cubic AI was also presented at the Digital Product Forum 2022.

WOLF, Thomas (TWT GmbH), KRIWET, Alexander (Mercedes-Benz AG). Co-Autoren: Kohler, J., Springmann, P., Gutsche, R. (Mercedes-Benz AG); Riegler, T., Vaudrevange P., Rößler T., Fäßler V., Keckeisen M.. Digital Powertrain - Artificial Intelligence for Virtual NVH Verification of Modern Powertrains, DPF 2022, Stuttgart.

Numerical Methods

Do you work with complex physical processes? We simulate them using the latest numerical algorithms from the world of multibody simulation, fluid dynamics and others, using artificial intelligence for simulation acceleration. For one of our clients, for example, we investigated why their solver used for a high-dimensional system of equations, originated from a brake simulation, fails to find a solution in certain scenarios. The complex model, which included discontinuous equations, was adapted so that, using an appropriate solution strategy, the system of equations could be solved for physically relevant parameters.

Another example of the pioneering application of numerical methods, especially in the field of finite elements, can be found in our product GETMe, the Geometric Element Transformation Method.

GETMe stands out as most consistent in comparison with two other msh smooting methods. from D. S. H. Lo, Finite Element Mesh Generation, p. 350.

Geometric element transformation method (GETMe)

High quality meshes play a key role in many applications based on digital modeling and simulation. The finite element method is a prime example of such an approach, and it is well known that high-quality meshes can significantly improve the computational efficiency and solution accuracy of this method. As a result, much work has been invested in methods to improve mesh quality. These range from simple geometric approaches such as Laplace smoothing, which results in high computational efficiency but potentially low mesh quality, to global optimization-based methods that result in excellent mesh quality at the cost of increased computational and implementation complexity.

The geometric element transform (GETMe) method bridges the gap between these two approaches

It is based on geometric mesh element transformations, where polygonal and polyhedral elements are iteratively transformed into their regular counterparts or into elements with a given shape. GETMe combines computational efficiency similar to Laplace smoothing with effectiveness approaching global optimization methods.

Smoothing of the Europe mesh with zoom on Norway. The network elements are colored with respect to their mean quality number.

GETme as a pioneering method is now indispensable in the field of mesh smoothing and is attracting increasing attention in the relevant literature:

„A breakthrough was made when Vartziotis et al. (2008) proposed the GETMe method, which is purely a geometric process to move the nodes of a triangle so as to improve its quality. (...)

GETMe is the most interesting node-smoothing scheme as it is purely geometric in nature in transforming elements into regular forms without a direct link to shape quality measure. It is robust in removing inverted and poorly shaped elements rapidly and is consistent in diverse applications to produce quality meshes."
Daniel S.H. Lo (Finite Element Mesh Generation)

A detailed description of the mathematical theory as well as numerical studies of GETMe can be found in the book

VARTZIOTIS, Dimitris; WIPPER, Joachim. The GETMe mesh smoothing framework: A geometric way to quality finite element meshes.. CRC Press, 2018,

and the associated publications, such as

  • VARTZIOTIS, Dimitris, et al. Mesh smoothing using the geometric element transformation method. Computer Methods in Applied Mechanics and Engineering, 2008, 197. Jg., Nr. 45-48, S. 3760-3767,

  • VARTZIOTIS, Dimitris; WIPPER, Joachim; SCHWALD, Bernd. The geometric element transformation method for tetrahedral mesh smoothing. Computer Methods in Applied Mechanics and Engineering, 2009, 199. Jg., Nr. 1-4, S. 169-182,

  • VARTZIOTIS, Dimitris; WIPPER, Joachim. The geometric element transformation method for mixed mesh smoothing. Engineering with Computers, 2009, 25. Jg., Nr. 3, S. 287-301,

  • VARTZIOTIS, Dimitris; HIMPEL, Benjamin. Efficient and global optimization-based smoothing methods for mixed-volume meshes. In: Proceedings of the 22nd International Meshing Roundtable. Springer, Cham, 2014. S. 293-311,

  • VARTZIOTIS, Dimitris; BOHNET, Doris; HIMPEL, Benjamin. GETOpt mesh smoothing: Putting GETMe in the framework of global optimization-based schemes. Finite Elements in Analysis and Design, 2018, 147. Jg., S. 10-20.

Data Science

The hot topic of Data Science is also represented by us. With our TWT product ZAMRIS, we offer a quality assurance framework which, in addition to an intuitive user interface and professional quality rule management, provides machine learning algorithms for quality assurance. Thereby, an intelligent time series analysis of arbitrary data sets is performed by means of Machine Learning and Pattern Recognition. ZAMRIS can be used both as a stand-alone application and integrated into existing simulation processes, thus enabling efficient and quality-assured development even with an increasing number of variants to be examined.

he use of Data Science is also relevant in our research projects. For example, in OPsTIMALwww.opstimal.deair traffic is optimized holistically from the perspective of the aviation company. But also in other projects we use modern tools like image recognition with Convolutional Neural Networks, self-learning neural networks or state estimation with Kalman filters.

The goal of the KARLI project is to develop adaptive, responsive and level-compliant interaction in the vehicle of the future. To this end, customer-relevant AI functions are being developed in KARLI that detect driver states and design interactions for different stages on the way to an automated vehicle (automation level).

These AI functions are developed in KARLI from empirical and synthetically generated data. The data will be collected and used in KARLI in such a way that the project results are scalable to Big Data from production vehicles that will be available in the future.

Use of artificial intelligence for human-machine interaction in the KARLI project (
Geometric transformation of polygons related to higher energy superposition states in a circular system of quantum wells. (Link: )

Breaking boundaries

"Mathematics compares the most diverse phenomena and discovers the secret analogies that connect them." - Joseph Fourier, French mathematician and physicist

Loosely based on this quote, we too are constantly trying to find new connections between phenomena or models that apparently have no connection. Would you have thought, for example, that there is a direct analogy between the transformation of triangular meshes in a CAD model and the placement of so-called quantum wells in the quantum computing environment? We are exploring these and other analogies in order to apply findings from one area to another and gain more understanding of the structure inherent in the problem. If you are interested in this topic, please feel free to read the following publication:

VARTZIOTIS, Dimitris; HIMPEL, Benjamin; PFEIL, Markus. Creation of higher-energy superposition quantum states motivated by geometric transformations. arXiv preprint arXiv:1712.07963, 2017,

We propose a way to generate higher energy superposition states in a circular system of quantum wells. This is inspired by a connection to convergence results for geometric transformations of polygons with circulant Hermitian matrices.


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