# Research

The research group works on the development of efficient numerical methods for the simulation of problems from computational science and engineering. This comprises the development of efficient algorithms, their theoretical analysis, and the implementation on large parallel computers with up to several hundreds of thousands of cores. A special focus in the applications is currently on problems from biomechanics/medicine, structural mechanics, and material science.

The research is in the field of numerical methods for partial differential equations and high performance parallel scientific computing. Selected topics are

- Highly scalable domain decomposition methods
- Parallel scalable simulations on several hundreds of thousands of cores
- Computational scale bridging and multiscale methods (FE2) in material science
- Numerical simulation of problems in structural mechanics (elasticity, plasticity, multiphase steels)
- Numerical algorithms for coupled and multifield problems
- Modeling and numerical simulation of arterial walls
- Fluid structure interaction of arterial walls and blood flow
- Modelling and numerical simulation of bone substitution materials using micromorphic models
- Scientific machine learning (SciML)

# Selected research projects

## Scientific machine learning

One of our most recent research fields is scientific machine learning (SciML). Scientific machine learning is a new, rapidly developing field of research in which techniques of scientific computing and machine learning are combined and further developed. This results in hybrid methods which are applied for the discretization of partial differential equations, the development of fast and robust solvers and new modeling techniques.

In particular, we work on:

- Domain decomposition methods using machine learning
- Flow predictions using convolutional neural networks (CNNs)
- Parameter identification using physics-informed networks (PINNs)

## Highly scalable parallel domain decomposition methods

Domain decomposition methods are an efficient approach for the solution of elliptic partial differential equations on parallel computers. Here, we understand by domain decomposition methods preconditioned iterative algorithms for the solution of large linear systems of equations obtained, either directly or by linearization, from the discretization of partial differential equations. In such methods, the domain, on which the partial differential equation has to be solved, is decomposed into a number of nonoverlapping subdomains. In each step of the iterative method and for each subdomain, a local problem is solved. This local problem is often an approximation of the partial differential equation restricted to the subdomain; here, we neglect for the moment that the boundary conditions are usually different for the local problem and the problem on the original domain. Depending on the particular domain decomposition method, the local problem is solved approximately itself or exactly, using a direct algorithm, e.g., a Gaussian elimination algorithm. For elliptic problems, also a small global problem is needed in order to obtain a parallel scalable algorithm, i.e., to exploit efficiently a growing number of processors of a parallel computer; in general one processor can obtain more than one subdomain. Robust and parallel scalable algorithms are developed for different problems including very demanding and difficult structural mechanics problems. Parallel scalability, for different algorithms, has been proven on several supercomputers and architectures for several tens and hundreds of thousands of parallel ranks and, in extreme cases, for more than a million MPI ranks.

## EXASTEEL - Bridging Scales for Multiphase Steels

The computational simulation of advanced high strength steels, incorporating phase transformation phenomena at the microscale, on the future supercomputers developed for exascale computing is considered in this project. To accomplish this goal, new ultra-scalable, robust algorithms and solvers have to be developed and incorporated into a new application software for the simulation of this three dimensional multiscale material science problem. Such algorithms must specifically be designed to allow the efficient use of the hardware. Here, a direct multiscale approach (FE2) will be combined with new, highly efficient, parallel solver algorithms. For the latter algorithms, a hybrid algorithmic approach will be taken, combining nonoverlapping parallel domain decomposition (FETI) methods with efficient, parallel multigrid preconditioners. A comprehensive performance engineering approach will be implemented to ensure a systematic optimization and parallelization process across all software layers. The envisioned scale-bridging will still require a computational power which will only be obtainable when exascale computing becomes available. The FE2 methods has shown to be scalable on the complete JUQUEEN supercomputer at Forschungszentrum Jülich (BG/Q with 458 752 cores).

## Fluid structure interaction of arterial walls and blood flow

FSI Homepage

The simulation of the physiological loading situation of arteries with moderate atherosclerotic plaque may provide additional indicators for medical doctors to estimate if the plaque is likely to rupture and if surgical intervention is required. In particular the transmural stresses are important in this context. In order to account for the correct behavior of the arterial wall to predict these stresses we apply fully coupled fluid-structure-interaction algorithms and nonlinear anisotropic hyperelastic material wall models. Stress distributions in walls of in vivo arteries (transmural stresses) are a major factor driving, e.g., the processes of arteriosclerosis and arteriogenesis which are well-known to be of a major relevance to the human health. Our attention is on fluid-structure interaction using sophisticated nonlinear structural models. Such models have been developed in the past and their parameters have been adapted to experimental data. Here, we use an anisotropic, polyconvex hyperelastic material model for the structure. The resulting coupled problems are solved using a monolithic approach based on Domain Decomposition algorithms, more precisely Overlapping Schwarz and Dirichlet-Neumann. Our project is based on a solver environment coupling the finite element software packages FEAP, the library LifeV as well as parallel domain decomposition preconditioners using fully nonlinear models for the fluid and a fully nonlinear, polyconvex, anisotropic model for the structure. Joint work with Prof. Dr. Oliver Rheinbach, Institute für Numerische Mathematik und Optimierung, TU Bergakademie Freiberg, Prof. Dr.-Ing. Jörg Schröder, Institut für Mechanik, Abt. Bauwissenschaften, Fakultät für Ingenieurwissenschaften, Universität Duisburg-Essen, Prof. Dr.-Ing. Daniel Balzani, Institut für Mechanik und Flächentragwerke, Fakultät für Bauingenieurwesen, TU Dresden, Prof. Dr. Alfio Quarteroni, EPF Lausanne, Dr. Simone Deparis, EPF Lausanne, and Prof. Dr. med. Raimund Erbel, Westdeutsches Herzzentrum, Universitätsklinik Essen.

## Numerical modeling and simulation of arterial walls

Stent surgery and balloon angioplasty are standard treatments for ateriosclerotic arteries. The treatment of arteriosclerosis by a balloon angioplasty and also the placement of a stent involves large deformations of the arterial walls. The numerical simulation of these large deformations is a challenging task which involves many computational state of the art techniques. Numerical methods for anisotropic, almost incompresible nonlinear hyperelastic elasticity models are developed including apropriate finite elment discretizations, robust nonlinear solvers and highly parallizable domain decomposition methods. Joint work with Prof. Dr.-Ing. Jörg Schröder, Institute for Mechanics, Abt. Bauwissenschaften, Fakultät für Ingenieurwissenschaften, Universität Duisburg-Essen, Prof. Dr. Oliver Rheinbach, Institut für Numerische Mathematik und Optimierung, TU Freiberg, and Prof. Dr. med. Raimund Erbel, Westdeutsches Herzzentrum, Universiätsklinik Essen.

_{Stress in an arterial wall. See D. Balzani, D. Böse, D. Brands, R. Erbel, A. Klawonn, O. Rheinbach, J. Schröder Engineering Computations, Vol. 29 No. 8, pp. 888-906 (2012).}

## Numerical methods for finite micromorphic elasticity problems

Micromorphic elasticity models are applied in the description of cellular materials, metallic foams or material inhomogeneities. They also help to model size effects in a natural way, i.e., small samples of a material behave comparatively stiffer than large samples which has important applications in the mechanics of nano devices. The micromorphic models considered here consist of a coupled system of nonlinear partial differential equations which are discretized by finite elements and are solved by a staggered algorithm. Each of the subproblems arising from this algorithm again require sophisticated numerical methods for their solution, including robust nonlinear solvers and domain decomposition techniques suitable for parallel computing. Joint work with Prof. Dr. Patrizio Neff, Lehrstuhl für Nichtlineare Analysis und Modellierung, and Dr. Stefanie Vanis, Lehrstuhl für Numerische Mathematik, Fakultät für Mathematik, Universität Duisburg-Essen, and Prof. Dr. Oliver Rheinbach, Institut für Numerische Mathematik und Optimierung, TU Freiberg.

_{A metal foam. Image from S. Vanis, O. Rheinbach, A. Klawonn, O. Prymak, M. Epple, Mat.-wiss. u. Werkstofftechn., Vol. 37, No. 6, (2006)}