Parallel Finite Element Methods for Complex Flows of Complex Fluids

The objective of this ongoing project is the continuous development and advancement of effective simulation methods for unsteady fluid flow problems and their application to real-life engineering problems. Common ground for the subprojects presented below is our in-house parallel finite element solver for multi-physics problems, including compressible and incompressible Navier-Stokes flows, linear elastic materials, non-Newtonian fluids, as well as thermal and scalar transport problems.

Project Details

Project term

May 1, 2023–April 30, 2024


RWTH Aachen University


Chair for Computational Analysis of Technical Systems

Project Manager

Maximilian Schuster

Principal Investigator

Prof. Marek Behr, Ph.D.


Dr.-Ing. Norbert Hosters
Nico Dirkes
Blanca Ferrer Fabón
Felipe González
Veronika Travnikova
Anna Ranno
Patrick Antony


The in-house FE solver XNS is written in Fortran and C and utilizes an MPI parallelized framework. The machine learning applications are employed in two in-house frameworks. One of them is built up on the Deep-XDE python package and leverages automatic differentiation integrated in TensorFlow to evaluate differential operators. In the PINN algorithm, the residuals of both the governing partial differential equations (PDEs) and boundary constraints are incorporated into the loss function which is minimized during the training of the neural network. For more complex problems we use a domain decomposition approach, where two or more neural networks are trained in parallel. The second framework is developed from scratch, using functionality provided by PyTorch to train the neural networks.


In the subproject Compressible Two-Phase Flows in Moving Domains XNS was used to simulate oil films in a compressible air flow. It was extended with a workflow to enable automatic adaptive mesh refinement using the library mmg.

In the subproject Physics-Informed Neural Networks for Flow Field Prediction in Stirred Tank Reactors, we constructed two-dimensional models capable of accurate predictions for velocity and pressure within a range of Reynolds numbers.

Within the subproject Precision Melt Engineering (SFB 1120), high-precision 2D simulations of the injection molding process have been performed. These simulations have been successfully compared to experimental results. Furthermore, to investigate the requirements for efficient, and yet accurate simulations, a mesh study has been performed.

In the subproject Multi-strand Fused Deposition Modeling Simulation, we developed a framework in our in-house fluid solver to study the multi-strand fused deposition modeling simulation. The novelty of this approach is that we use a spline-based boundary description to include the effect of previously printed strands in the simulation.

Within the subproject In-Stent Restenosis in Coronary Arteries, we coupled a multiphysics model for in-stent restenosis to hemodynamics and drug elution in stented arteries. We tested our approach on a simplified artery segment with ring stent. The growth model is implemented in the solid solver FEAP and for the fluid model we use the in-house code.


The simulations of compressible two-phase flows gave some insights to the leakage path of oil in internal combustion engines with late-post injection. The two-phase flow model will be extended to simplex based space-time finite element discretizations, and the parallel scaling behaviour on Lichtenberg II will be compared to the previously used discretization with prismatic elements. Further efforts on reducing the computational time of these simulations will include the use of simplex-based space-time finite element discretizations. Furthermore, we plan to apply the gained knowledge to 3D simulations. After having identified the most promising methods for improving the prediction accuracy of PINNs for stirred tank reactors in a 2D test case, we will proceed to extend the model to a realistic 3D geometry.

The multi-strand simulation of fused deposition modeling allowed the study of porosity, which is a crucial macroscale property in objects built by 3d printing technology. We plan to extend the study to different deposition strategies, such as aligned, skewed, and crossed layers.

In the subproject on in-stent restenosis in coronary arteries we plan to expand the model to more complex and patient-base geometries.

Additional Project Information

DFG classification: 404-03 Fluid Mechanics
Software: TensorFlow, PyTorch
Cluster: Lichtenberg


Cornelissen, A., Florescu, R. A., Reese, S., Behr, M., Ranno, A., Manjunatha, K., Schaaps, N., Böhm, C., Liehn, E. A., Zhao, L., et al. In-vivo assessment of vascular injury for the prediction of in-stent restenosis. International Journal of Cardiology, 388:131-151, (2023).

Dirkes, N., Key, F., & Behr, M. Eulerian formulation of the tensor-based morphology equations for strain-based blood damage modeling. arXiv preprint (2024).

Fabón, F. B., Alms, J., Behr, M., & Hopmann, C. High-resolution numerical simulations of polymer injection molding: Analysis of mesh size and refinement. Proceedings in Applied Mathematics and Mechanics: PAMM, (2023).

Fabón, F. B., & Behr, M. Towards the optimization of injection molding processes. 7. Young Investigators Conference ECCOMAS, Porto (Portugal), 19 Jun 2023 – 21 Jun 2023.

González, F. A., Elgeti, S., & Behr, M. The surface-reconstruction virtual-region mesh update method for problems with topology changes. International Journal for Numerical Methods in Engineering, pages 1– 18, (2023).

González, F. A., Elgeti, S., Behr, M., & Auricchio, F. A deforming-mesh finite-element approach applied to the large-translation and free-surface scenario of fused deposition modeling. International Journal for Numerical Methods in Fluids, 95(2):334–351, (2023).

Hilger, D., Sasse, J., Hopmann, C., Behr, M., & Hosters, N. Numerical analysis and reduced-order modeling of plastic profile extrusion processes. Proceedings in Applied Mathematics and Mechanics: PAMM, 23(1):e202200069, (2023).

Manjunatha, K., Ranno, A., Shi, J., Schaaps, N., Nilcham, P., Cornelissen, A., Vogt, F., Behr, M., & Reese, S. In silico reproduction of the pathophysiology of in-stent restenosis. arXiv preprint arXiv:2401.03961, (2024).

Ranno, A., Manjunatha, K., Glitz, A., Schaaps, N., Reese, S., Vogt, F., & Behr, M. In-silico analysis of hemodynamic indicators in idealized stented coronary arteries for varying stent indentation. arXiv preprint arXiv:2401.08701, (2024).

Schuster, M. R., Dirkes, N., Key, F., Elgeti, S., & Behr, M. Exploring the influence of parametrized pulsatility on left ventricular washout under LVAD support: A computational study using reduced-order models. Computer Methods in Biomechanics and Biomedical Engineering, (2024).

Spenke, T., Make, M., & Hosters, N. A Robin-Neumann scheme with quasi-newton acceleration for partitioned fluid-structure interaction. International Journal for Numerical Methods in Engineering, 124(4):979–997, (2023).

Tillmann, S. F. M., Behr, M., & Elgeti, S. N. Using Bayesian optimization for warpage compensation in injection molding. Materials Science and Engineering Technology, 55(1):13–20, (2024).

Trávníková, V., Wolff, D., Dirkes, N., Elgeti, S., von Lieres, E., & Behr, M. A model hierarchy for predicting the flow in stirred tanks with physics-informed neural networks. arXiv preprint arXiv:2403.04576, (2024).

Von Danwitz, M., Voulis, I., Hosters, N., & Behr, M. Time-continuous and time-discontinuous space-time finite elements for advection-diffusion problems. International Journal for Numerical Methods in Engineering, 124(14):3117–3144, (2023).

Wittschieber, S., Rangarajan, A., May, G., & Behr, M. Metric-based anisotropic mesh adaptation for viscoelastic flows. Computers & Mathematics with Applications, 151:67–79, (2023).


Boledi, L. Numerical Multiphysics Modeling with Space-Time Finite Elements Assessing the Performance of Thermal Melting Probes. Dissertation, RWTH Aachen University, Aachen, (2024).

González, F. A. Boundary-Conforming Finite-Element Methods applied to Fused Deposition Modeling. Dissertation, RWTH Aachen University, Aachen, (2023).

Hilger, D. Reduced-Order Modeling for Plastics Profile Extrusion. Dissertation, RWTH Aachen University, Aachen, submitted (2024).

Hube, S. Numerical Shape Optimization for Dynamic Mixing Elements in Single-Screw Extruders. Dissertation, RWTH Aachen University, Aachen, (2023).

Wittschieber, S. Robust Finite Element Methods for Viscoelastic Constitutive Laws in Log-Conformation Form. Dissertation, RWTH Aachen University, Aachen, (2023).

Wolff, D. Learning-based Approaches for the Analysis and Optimization of Profile Extrusion Dies and Bioreactors. Dissertation, RWTH Aachen University, Aachen, (2023).

Conference participations:

Behr, M., Schuster, M., Dirkes, N., Ferrer Fabon, B., Gonzalez, F., Travnikova, V., Ranno, A., Antony, P. Computational Fluids Conference, Cannes, France, 25 – 28 April 2023.

Schuster, M., Dirkes, N., Ferrer Fabon, B., Travnikova, V., Ranno, A. ECCOMAS Young Investigators Conference, Porto, Portugal, 19 – 21 June 2023.

Behr, M., Travnikova, V., Ranno, A., Antony, P. GACM Colloquium for Young Scientists 2023, Vienna, Austria, 10 – 13 September 2023.