Parallel Finite Element Methods for Complex Flows of Complex Fluids

Methods

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.
Results

Results

During the last allocated project period, substantial progress was achieved across all ten sub-projects. A coupled multiphysics model for in-stent restenosis was developed and applied to a patient-specific coronary artery, integrating cell dynamics, growth mechanics, hemodynamics, and drug elution. In injection molding process, space-time adaptive two-phase flow simulations were successfully benchmarked against COMSOL in 2D and experimentally validated in 3D, while short-fiber reinforced injection molding was advanced using a simplex space-time finite element method with adaptive mesh refinement (AMR) to improve fiber orientation predictions. Additionally, our work identified the key parameters governing the printing force in fused deposition modeling-based 4D printing. A partitioned fluid-structure interaction (FSI) framework coupling the fluid solver XNS with a solid solver was established for the seismic analysis of storage tanks. A unified fluid-structure-contact interaction workflow, supported by a newly developed 4D space-time mesher, was demonstrated for aluminum sheet rolling. In addition, a new Eulerian hemolysis model was developed and validated, and a novel approach to incorporating viscoelastic fluid models was proposed. For FSI in plastics profile extrusion dies, adaptive finite element discretizations for solids and simplex space-time meshes for incompressible Navier-Stokes flow were implemented and tested in XNS. A new workflow featuring automatic AMR based on the oil-gas interface was developed, extended to moving domains, and equipped with error estimators derived from the fluid solution. Finally, Physics-Informed Neural Networks were benchmarked as surrogate models for stirred-tank reactors, with their data efficiency evaluated against classical supervised methods.

Discussion

We expect to obtain a patient‑specific, quasi‑steady in‑silico tool that couples restenosis growth, arterial mechanics, hemodynamics and drug‑elution to optimise drug‑eluting stent implantation.

Space‑time adaptive grids will deliver fast, high‑precision predictions of the entire injection‑molding process—including polymer solidification and warpage—for industrial optimisation.

Thermoviscoelastic melt modelling will be used to design and optimise nozzles for 4D‑printing of shape‑memory polymers, improving stress development and shape‑morphing.

Scalable space‑time adaptive discretisations will enable large‑scale 3D fibre‑reinforced moulding simulations that simultaneously solve flow, level‑set, temperature and fibre‑orientation equations with error control.

A validated fluid‑structure‑interaction solver will reproduce free‑surface experiments and demonstrate robustness for large deformations such as sloshing in soft‑walled tanks.

A full 3D fluid‑structure‑contact framework will analyse lubricant‑alloy combinations in skin‑pass rolling, incorporating grid‑convergence studies and topology‑changing 4D grids.

High‑fidelity CFD combined with a practical hemolysis model will predict red‑blood‑cell damage in advanced biomedical devices, offering design‑level hemocompatibility insights.

Adaptive finite‑element methods on time‑continuous simplex space‑time meshes will provide anisotropic refinement that treats space and time uniformly, eliminating separate remeshing steps.

Adaptive mesh‑refinement workflows will be scaled to realistic 3D piston‑ring‑pack gas‑flow cases, supplying performance models for efficient industrial analysis.

Physics‑informed neural networks will be extended to turbulent 3D stirred‑tank reactors, integrating data‑driven RANS closures and quantifying their data‑efficiency versus conventional supervised networks.

Additional Project Information

DFG classification: 402-02 Mechanics
Software: XNS, pINS
Cluster: CLAIX

Publications

In-silico analysis of hemodynamic indicators in idealized stented coronary arteries for varying stent indentation,
A. M. Ranno, K. Manjunatha, A. Glitz, N. Schaaps, S. Reese, F. Vogt, M. Behr,
https://dx.doi.org/10.1080/10255842.2024.2382819, July 2024

On the Significance of Flow Vorticity for Hemolysis Modeling,
N. Dirkes, M. Behr,
https://dx.doi.org/10.14311/TPFM.2025.008, 2025

Addressing a Major Bottleneck in Computational Mechanics with Simplex Space-Time Finite Elements: Application to Free-Surface Flows,
Norbert Hosters, Blanca Ferrer-Fabón, Moritz Billen, Marek Behr,
https://dx.doi.org/10.1007/978-3-031-93213-7_13, 2025

Experimental Investigation of the Applicability of the Stress‐Based and Strain‐Based Hemolysis Models for Short‐Term Stress Peaks Typical for Rotary Blood Pumps,
Michael Lommel, Vera Charlotte Lommel, Henri Wolff, Nico Dirkes, Katharina Vellguth, Marek Behr, Ulrich Kertzscher,
https://dx.doi.org/10.1111/aor.15002, April 2025

Quantifying data needs in surrogate modeling for flow fields in two-dimensional stirred tanks with physics-informed neural networks,
Veronika Trávníková, Eric von Lieres, Marek Behr,
https://dx.doi.org/10.1063/5.0291054, November 2025

A computational model of coronary arteries with in-stent restenosis coupling hemodynamics and pharmacokinetics with growth mechanics,
Anna Ranno, Kiran Manjunatha, Thore Koritzius, Ivo Steinbrecher, Norbert Hosters, Maximilian Nachtsheim, Pakhwan Nilcham, Nicole Schaaps, Anne Turoni-Glitz, Janina Datz, Alexander Popp, Kevin Linka, Felix Vogt, Marek Behr,
https://dx.doi.org/10.1038/s41598-025-22291-w, November 2025

Tensorial models in numerical simulations of hemodynamics and hemolysis,
Nico Dirkes, Adélia Sequeira, Marek Behr, and Tomáš Bodnár, 2026

Adaptive Refinement for Multi-Phase Flow Simulation on Moving Domains,
Patrick Antony, Blanca Ferrer Fabón, Norbert Hosters & Marek Behr,
https://dx.doi.org/10.1007/978-3-032-16253-3_1, 2026

A Practical Computational Hemolysis Model Incorporating Biophysical Properties of the Red Blood Cell Membrane,
Niko Dirkes, Marek Behr,
https://dx.doi.org/10.48550/arXiv.2601.19994, January 2026

An Ising machine formulation for design updates in topology optimization of flow channels,
Yudai Suzuki, Shiori Aoki, Fabian Key, Katsuhiro Endo, Yoshiki Matsuda, Shu Tanaka, Marek Behr, Mayu Muramatsu,
https://dx.doi.org/10.1007/s00366-025-02265-2, January 2026

Printing force analysis in FDM-based 4D printing of singular lines: A numerical and experimental study,
Ferdinand Cerbe, Felipe A. González, Michael Sinapius, Marek Behr, Stefanie Elgeti,
https://dx.doi.org/10.1016/j.addma.2026.105108, February 2026

Two‐Phase Flow Simulations for Injection Molding Applications: Comparison between COMSOL Multiphysics and XNS In‐House Solver,
Blanca Ferrer Fabón, Jonathan Alms, Marek Behr, Christian Hopmann,
https://dx.doi.org/10.1002/pamm.70147, May 2026

Numerical optimization of planar nozzle shapes for fused deposition modeling,
Steffen Tillmann, Felipe A. González, Stefanie Elgeti,
https://dx.doi.org/10.1108/HFF-02-2026-0141, May 2026

Thesis:

Advances and Applications in High-Definition Simulation of Packed Bed Liquid Chromatography,
Jayghosh Subodh Rao,
PhD thesis, 2025

Modeling and simulation of blood damage in medical devices,
Stefan Thomas Haßsler,
PhD thesis, 2025