Differential-Algebraic Systems Group

Research areas

  • Applied and numerical mathematics, in particular coupled systems of differential-algebraic and partial differential equations
  • Multidisciplinary projects in the area of vehicle dynamics, material sciences and fluid mechanics
  • Modeling and numerics of shape memory materials
  • Isogeometric finite elements
  • Haemodynamics and dynamics of skeletal muscles

Current research projects

Coupled analysis of active biological processes for meniscus tissue regeneration:

Project members:    Prof. Dr. Bernd Simeon and  Dr. Elise Grosjean

Project start:             02/2022

Funding:                       Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)

Project partners:      

Project contents and goals:

This project is part of SPP 2311 program (Robuste Kopplung kontinuumsbiomechanischer in silico Modelle für aktive biologische Systeme als Vorstufe klinischer Applikationen - Co-Design von Modellierung, Numerik und Nutzbarkeit).

During the last decades, mathematical modeling and simulation have become valuable tools for investigating complex biomedical systems. They significantly contribute to understand different aspects of a biological process, often allowing to extend the study results to related, mutually conditioned processes. This project is concerned with modeling, simulation and experimental validation for a prominent biomedical problem: the meniscus regeneration and involved cell and tissue-level phenomena.
Clinical studies indicate that partial and total meniscectomies lead to prevalence of premature osteoarthritis in knee joints. Therefore, substantial efforts are being made towards finding adequate regenerative tissue for meniscus replacement. Although there are some solutions described in the literature, to date the optimal substitute has not been developed. Most regenerative approaches are clinically motivated and focus rather on the practical application than on the micro- and macroscopic cellular mechanisms and the interactions with the scaffold material. The latter viewpoint is promising in the sense that it aims to understand the basic control mechanisms in cell-scaffold interactions under different environmental parameters, thus providing a selective prognosis of the most significant combinations of these parameters.

Thus, in collaboration with biologists, mathematicians, and engineers, our aim is to realize a sensitivity analysis on the meniscus regeneration simulations through differential-based methods. Moreover, the time scales of the different processes differ vastly and call for appropriate co-simulation strategies as well as model order reduction techniques. While meaningful clinical data is very difficult to obtain from in vivo meniscus tissue, this off-the-wall approach provides comprehensive underpinnings for the mathematical modeling and numerical simulation.

Scaled boundary isogeometric analysis with advanced features for trimmed objects, higher order continuity

and structural dynamics:


Project members:    Prof. Dr. Bernd Simeon and  M. Sc. Jeremias Nathanael Arf

Project start:             2020

Project partners:       Prof. Dr.-Ing. Sven Klinkel and M. Sc. Mathias Reichle (RWTH Aachen)

Funding:                       Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)

Project contents and goals:

The connection between CAD and FEM theory was established by the concept of isogeometric analysis. For the latter, underlying computational domains are often defined by means of their boundaries. This fact is the starting point for the scaled boundary isogeometric analysis (SB-IGA), which uses NURBS boundary elements and  proper scaling centers to parametrize domain patches.  In the project,  which we investigate in cooperation with the RWTH Aachen, we focus on the coupling of domain patches with an emphasis on higher global regularity. This makes it possible to utilize SB-IGA e.g. within the Kirchhoff-Love theory. Furthermore, the advantages of the scaled boundary perspective  in the context  of “trimming”, meaning the cutting of shapes in CAD, are evaluated. The theoretical considerations and findings are then applied in the field of structural dynamics.


The following picture shows a 'SB-IGA mesh of Scordelis-Lo shell' (on the left) as well as the corresponding 'z-displacement' (on the right):

Examples of past research projects


DYMARA - Ein dynamisches Manikin mit faserbasierter Modellierung der Skelettmuskulatur:

Projektmitarbeiter:      Prof. Dr. Bernd Simeon, Dr. Ing. Michael Gfrerer                                 

Projektdauer:              Dez. 2016 - Dez. 2019            

Förderung:                 BMBF - Verbundprojekt

In DYMARA kooperieren die Arbeitsgruppen von

  • Prof. Dr. Bernd Simeon (TU Kaiserslautern, Felix-Klein-Zentrum)
  • Prof. Dr.-Ing. habil. Sigrid Leyendecker (Friedrich-Alexander-Universität Erlangen-Nürnberg, Lehrstuhl für Angewandte Dynamik)
  • Dr. Michael Burger (Fraunhofer Institut für Techno- und Wirtschaftsmathematik)

mit den Praxispartnern

  • fleXstructures GmbH, Kaiserslautern
  • MaRhyThe-Systems GmbH & Co. KG, Gröbenzel

Projektinhalt und Ziele:

Das Verbundprojekt DYMARA hat die Entwicklung eines innovativen digitalen Menschmodells (Manikins) mit  detaillierter Modellierung der Skelettmuskulatur und schnellen numerischen Algorithmen zum Ziel. Mit diesem Manikin soll es möglich werden, den Menschen simulationsgestützt auf optimale Weise in sein Arbeitsumfeld zu integrieren und Ermüdungen, Erkrankungen sowie Unfälle am Arbeitsplatz zu vermeiden. Neben diesen ergonomischen Gesichtspunkten soll das Menschmodell auch zur Therapieplanung im muskulären Bereich und zur Gestaltung von Prothesen und Orthesen eingesetzt werden können. Um die Dynamik des muskuloskeletalen Systems hinreichend genau zu erfassen, wird ein Modellierungsansatz verfolgt, der auf der Methode der mechanischen Mehrkörpersysteme (MKS) basiert. Solche Modelle sind durch die Robotik inspiriert und werden bereits heute in vielen biomechanischen Anwendungsfeldern eingesetzt. Die Modellierung der Muskulatur stellt jedoch nach wie vor eine große Herausforderung dar, insbesondere wenn Aspekte wie Rechenzeit auf der einen und Berücksichtigung der anatomischen und physiologischen Gegebenheiten auf der anderen Seite zu beachten sind. Hier setzen wir mit unserem Projekt an: Ein neu zu entwickelndes eindimensionales Kontinuumsmodell, das einzelne Muskelfaserbündel realitätsnah beschreibt, soll die bisher üblichen diskreten Kraftelemente im MKS-Modell ersetzen und mit schnellen, problemangepassten numerischen Algorithmen zur Berechnung von Bewegungssequenzen und zur Steuerung des Manikins kombiniert werden.

MOTOR - Multi-ObjecTive design Optimization of fluid eneRgy machines:

Project members: Prof. Dr. Bernd Simeon and Dipl. Math. Alexander Shamanskiy

Project duration:  Sept. 2015 - Sept. 2018

Project web site:          project-motor.eu

Project partners:

  • Delft University of Technology (Netherlands)
  • Caterpillar (Sweden)
  • ESS Engineering Software Steyr GmbH (Austria)
  • Johannes Kepler University of Linz (Austria)
  • Maritime Research Institute Nederland (Netherlands)
  • Mavel (Czech Republic)
  • MTU Aero Engines AG, Munich (Germany)
  • University of West Bohemia (Czech Republic)
  • TU Dortmund University (Germany)
  • TU Kaiserslautern (Germany)

Von Karman Institute of Fluid Dynamics (Belgium)

Project contents and goals:

The MOTOR project focuses on ICT-enabled design optimization technologies for fluid energy machines
(FEMs) that transfer mechanical energy to and from the fluid, in particular for aircraft engines, ship pro-
pellers, water turbines, and screw machines. The performance of these machines essentially depends
on the shape of their geometry, which is described by functional free-form surfaces. Even small modifica-
tions have significant impact on the performance; hence the design process requires a very accurate
representation of the geometry.
Our vision is to link all computational tools involved in the chain of design, simulation and optimization to
the same representation of the geometry, thereby reducing the number of approximate conversion steps
between different representations. The improved accuracy and reliability of numerical simulations ena-
bles the design of more efficient FEMs by effective design optimization methods. MOTOR also exploits
the synergies between the design optimization technologies for the different types of FEMs that have so
far been developed independently.
MOTOR adopts a modular approach for developing novel methodologies and computational tools and
integrating them into real process chains, contributing

  • a volumetric mesh generator with exact interface matching for multi-domain geometries enabling a high-order multi-physics simulations with enhanced accuracy,
  • an isogeometric analysis simulation toolbox for CFD, CSM, and FSI problems and advanced interactive visualization toolkit for high-order solutions, and
  • automatic shape optimization based on a multi-level approach in the parameterization enabling different levels of shape variety to combine design space exploration with local searches.

The effectiveness of our approach in terms of reduced time to production and increased efficiency of theoptimally designed product will be validated by developing four proof-of-concept demonstrators with themodernized process chains.

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