### Doktorandenseminar des WIAS

## FG3: Numerische Mathematik und Wissenschaftliches
Rechnen /

*RG3: Numerical Mathematics
and Scientific Computing*/

LG5: Numerik für innovative
Halbleiter-Bauteile

*LG5: Numerics for
innovative semiconductor devices*

## Seminar Numerische Mathematik / *Numerical mathematics seminars*

aktuelles Programm / *current program*

_{Archiv}

Hybrid-/Online-Vorträge finden über ''Zoom'' statt. Der Zoom-Link wird jeweils ca. 15 Minuten vor Beginn des Gesprächs versendet. /

*The zoom link will be send about 15 minutes before the start of the talk. People who are not members of the research group 3 and who are interested in participating should contact christian.merdon@wias-berlin.de to obtain the zoom login details.*

**
Dienstag, 16. 07. 2024, 13:30 Uhr (WIAS-ESH)
**

Nishant Ranwan(IISER Thiruvananthapuram)

Existence of a Weak Solution to the Fluid-Structure Interaction Problem of Blood Flow in Coronary ArteryThis work is dedicated to proving the existence of a weak solution to the nonlinear coupled fluid-structure interaction (FSI) problem arising in modeling the blood flow through the coronary artery. Considering blood as a homogeneous, incompressible, Newtonian fluid, the $3D$ Navier--Stokes equations model the flow. We adopt the St. Venant--Kirchhoff elasticity model for the arterial wall. With the help of dynamic and kinematic boundary conditions, we define the fluid-structure coupling (two-way) at the interface. Using the operator splitting approach, we split the coupled problem into two parts: the structure sub-problem (SSP) in the Galerkin framework and the piece-wise stationary fluid sub-problem (FSP). In order to establish the existence of a weak solution, the initial step is to tackle the SSP. Following this, the fluid domain is updated through the Harmonic Extension of the structural displacement. Subsequently, the focus shifts to solving the coupled Navier--Stokes equations within the updated fluid domain.

**
Donnerstag, 06. 06. 2024, 14:00 Uhr (WIAS-ESH)
**

Steffen Maass(WIAS/TU Berlin)

Symbolic-numeric modeling of electrochemical interfaces

**
Donnerstag, 06. 06. 2024, 14:30 Uhr (WIAS-ESH)
**

Tim Siebert(WIAS/TU Berlin)

Algorithmic differentiation with a focus on higher-order derivatives

**
Dienstag, 04. 06. 2024, 13:30 Uhr (WIAS-ESH)
**

Jürgen Fuhrmann(WIAS)

The nonlinear finite Volume solver VoronoiFVM.jl: capabilities, applications and plans

**
Dienstag, 04. 06. 2024, 14:00 Uhr (WIAS-ESH)
**

Daniel Runge(WIAS)

Mass-conservative reduced basis approach for heterogeneous catalysis

**
Dienstag, 28. 05. 2024, 13:30 Uhr (WIAS-406)
**

Sarah Katz(WIAS)

Pulsatile flow in the aorta: modelling and simulationThe modelling and simulation of blood flow is one of the major applications of (computational) fluid dynamics in medicine. We present a brief overview of some past results on turbulence and viscosity modelling for aortic flow and introduce currently ongoing work on ML-accelerated simulations.

**
Dienstag, 28. 05. 2024, 14:00 Uhr (WIAS-406)
**

Francesco Romor(WIAS)

Registration-based data assimilation of aortic blood flowsReal-world applications often require the execution of expensive numerical simulations on different geometrical domains, represented by computational meshes. This is especially true when performing shape optimization or when these domains model shapes from the real world, such as anatomical organs of the human body. How we can develop surrogate models or perform data assimilation in these contexts is the main concern of this work. Even if a rich database is acquired for a specific application, sometimes it cannot be exploited completely because each quantity of interest is supported on its own computational mesh. In fact, the employment of neural network architectures tailored for this framework is still not consolidated in the literature. A solution is represented by shape registration: a reference template geometry is designed from a cohort of available geometries which can then be mapped onto it. Through registration, a correspondence between every geometry of the database of parametric PDEs’ solutions is designed.

**
Donnerstag, 23. 05. 2024, 14:00 Uhr (WIAS-ESH)
**

Alfonso Caiazzo(WIAS)

Multiscale modeling of elastic materials & Overview of applications in the assimilation of biomedical dataThe first part of the talk will describe our recent work concerning the modeling of biphasic tissues composed of an elastic matrix (2D or 3D) and fluid inclusions of co-dimension 2 (0D or 1D, respectively). The model is based on a finite element immersed method where the effect of the fluid is accounted via proper singular terms. The dimensionality reduction is achieved using a reduced Lagrange multipliers formulation of the resulting PDE. The second part of the talk is dedicated to an overview of applications (past, present, and upcoming) in quantitative biomedicine focusing on the role of mathematical modeling and scientific computing for data assimilation problems.

**
Dienstag, 21. 05. 2024, 13:30 Uhr (WIAS-ESH)
**

Christian Merdon(WIAS)

Pressure-robustness in Navier--Stokes simulations

**
Donnerstag, 16.05.2024, 14:00 Uhr (WIAS-ESH)
**

Medine Demir(WIAS)

Pressure-robust approximation of the incompressible Navier--Stokes equations in a rotating frame of referenceA pressure-robust space discretization of the incompressible Navier--Stokes equations in a rotating frame of reference is considered. The discretization employs divergence-free, $H^1$-conforming mixed finite element methods like Scott--Vogelius pairs. An error estimate for the velocity is derived that tracks the dependency of the error bound on the coefficients of the problem, in particular on the angular velocity. Numerical examples support the theoretical results.

**
Dienstag, 14. 05. 2024, 13:30 Uhr (WIAS-ESH)
**

Cristian Cárcamo(WIAS)

Equal-order finite element methods for coupled and multiphysics problemsWe discuss the well-posedness and error analysis of the Biot's Poroelastic equations in this talk. To demonstrate the solvability of the poroelastic continuous issue, we first use the well-known Fredholm Alternative. In order to improve computational efficiency and address the issues raised by the discrete inf-sup condition, we present a novel and stable stabilized numerical system that is tuned for equal polynomial order. We also perform a numerical analysis to determine the stability of solutions and offer an a priori analysis of them. Lastly, we provide a few numerical illustrations. These examples offer strong proof of the usefulness and effectiveness of the suggested numerical framework.

**
Dienstag, 14. 05. 2024, 14:00 Uhr (WIAS-ESH)
**

Marwa Zainelabdeen(WIAS)

Physics-informed neural networks for convection-dominated convection-diffusion problemsIn the convection-dominated regime, solutions of convection-diffusion problems usually possess layers, which are regions where the solution has a steep gradient. It is well known that many classical numerical discretization techniques face difficulties when approximating the solution to these problems. In recent years, physics-informed neural networks (PINNs) for approximating the solution to (initial-)boundary value problems received a lot of interest. PINNs are trained to minimize several residuals (loss functionals) of the problem in collocation points. In this talk, we introduce various loss functionals for PINNs that are especially designed for convection-dominated convection-diffusion problems. They are numerically tested and compared on two benchmark problems whose solutions possess different types of layers. We observed that certain special functionals reduce the L^2 (Ω) error compared to the standard residual loss functional. Furthermore, three types of collocation points applied to hardconstrained PINNs are compared: layer-adapted, equispaced and uniformly randomly chosen points. We observe that layer-adapted points work the best for a problem with an interior layer and the worst for a problem with boundary layers.

**
Dienstag, 07. 05. 2024, 13:30 Uhr (WIAS-ESH)
**

Dilara Abdel(WIAS)

Modeling and simulation of vacancy-assisted charge transport in innovative semiconductor devicesIn response to the climate crisis, there is a need for technological innovations to reduce the escalating CO$_2$ emissions. Two promising semiconductor technologies in this regard, perovskite-based solar cells and memristive devices based on two-dimensional layered transition metal dichalcogenide (TMDC), can potentially contribute to the expansion of renewable energy sources and the development of energy-efficient computing hardware. Within perovskite and TMDC materials, ions dislocate from their ideal position in the semiconductor crystal and leave void spaces. So far, the precise influence of these vacancies and their dynamics on device performance remain underexplored.

Therefore, this talk is dedicated to comprehensively examining the impact of vacancy-assisted charge transport in innovative semiconductor devices through a theoretical approach by modeling and simulating systems of partial differential equations. We start by deriving drift-diffusion equations using thermodynamic principles. Furthermore, we formulate drift-diffusion models to describe charge transport in perovskite solar cells and TMDC memristors. We discretize the transport equations via the finite volume method and establish the existence of discrete solutions. Our study concludes with simulations conducted with an open source software tool developed in the programming language Julia. These simulations explore the influence of volume exclusion effects on charge transport in perovskite solar cells and compare our simulation results with experimental measurements found in literature for TMDC-based memristive devices.

**
Dienstag, 07. 05. 2024, 14:00 Uhr (WIAS-ESH)
**

Timo Streckenbach(WIAS)

Alternative approach to model lattice mismatch of semiconductor nanostructuresAbout using MFC (multi freedom constraints) to connect nanostructures with different lattice constants in FEM strain simulation.

**
Donnerstag, 02. 05. 2024, 14:00 Uhr (WIAS-ESH)
**

Baptiste Moreau(WIAS)

**
Donnerstag, 02. 05. 2024, 14:30 Uhr (WIAS-ESH)
**

Ondřej Pártl(WIAS)

Optimization of geothermal energy production from fracture-controlled reservoirs via 3D numerical modeling and simulationWe develop a computation framework from scratch that allows us to conduct 3D numerical simulations of groundwater flow and heat transport in hot fractured-controlled reservoirs to find optimal placements of injection and production wells that sustainably optimize geothermal energy production.

We model the reservoirs as geologically consistent randomly generated discrete fracture networks (DFN) in which the fractures are 2D manifolds with polygonal boundary embedded in a 3D porous medium. The wells are modeled as line sources and sinks. The flow and heat transport in the DFN-matrix system are modeled by solving the balance equations for mass, momentum, and energy. The fully developed computational framework combines the finite element method with semi-implicit time-stepping and algebraic flux correction. To perform the optimization, we use various gradient-free algorithms.

I will present our latest results for several geologically and physically realistic scenarios.

**
Dienstag, 30. 04. 2024, 13:30 Uhr (WIAS-ESH)
**

Patricio Farrell(WIAS)

Numerical methods for innovative semiconductor devicesThe Leibniz group NUMSEMIC develops and numerically solves nonlinear PDE models. These models are often inspired by charge transport in innovative semiconductor devices. In particular, applications include perovskite solar cells, memristors, nanowires, quantum wells, lasers as well as doping reconstruction. To translate these applications into mathematical models, we rely on nonlinear drift-diffusion, hyperelastic material models, inverse PDE problems, localized landscape theory and atomistic coupling. Our methodologies include physics-preserving finite volume methods, data-driven techniques as well as meshfree methods.

**
Donnerstag, 25. 04. 2024, 14:00 Uhr (WIAS-ESH)
**

Holger Stephan & trainees(WIAS)

Modelling of drift diffusion problems from different perspectivesJoined work with Mihaela Karcheva, Leo Markmann, Liam Johnen und Fedor Romanov

Balance equations such as drift-diffusion equations are phenomenological equations that describe the interaction of temporal changes in extensive and spatial changes in intensive physical quantities. Typical problems with such drift-diffusion models are missing oscillations and reaching the steady state after an infinitely long time. We try to overcome these problems with atypical approaches.

**
Dienstag, 23. 04. 2024, 13:30 Uhr (WIAS-ESH)
**

Zeina Amer(WIAS)

Numerical methods for coupled drift-diffusion and Helmholtz models for laser applicationsSemiconductor lasers are pivotal components in modern technologies, spanning medical procedures, manufacturing, and autonomous systems like LiDARs. Understanding their operation and developing simulation tools are paramount for advancing such technologies. In this talk, we present a mathematical PDE model for an edge-emitting laser, combining charge transport and light propagation. The charge transport will be described by a drift-diffusion model and the light propagation by the Helmholtz equation. We discuss a coupling strategy for both models and showcase initial numerical simulations.

**
Dienstag, 23. 04. 2024, 14:00 Uhr (WIAS-ESH)
**

Yiannis Hadjimichael(WIAS)

An energy-based finite-strain model for 3D heterostructured materialsThis talk presents a mathematical model that accurately describes the intrinsic strain response of 3D heterostructures arising from lattice mismatch. Combining materials with different lattice constants induces strain, leading to the bending of such heterostructures. To validate our model, we apply it to bimetallic beams and hexagonal hetero-nanowires and perform numerical simulations using finite element methods (FEM). In order to assess the fidelity of the model and the accuracy of simulations, we compare the calculated curvature with analytically derived formulations. The outcomes of our study yield valuable insights into the behavior of strained bent heterostructures. We compare the strain profiles of wurtzite and zincblende crystal structures, shedding light on their distinct characteristics. This is particularly significant as the strain has the potential to influence piezoelectricity, the electronic band structure, and the dynamics of charge carriers.

**
Donnerstag, 18. 04. 2024, 14:00 Uhr (WIAS-ESH)
**

Volker John(WIAS/FU Berlin)

Finite element methods respecting the discrete maximum principle for convection-diffusion equationsConvection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solution of these equations satisfy under certain conditions maximum principles, which represent physical bounds of the solution. That the same bounds are respected by numerical approximations of the solution is often of utmost importance in practice. The mathematical formulation of this property, which contributes to the physical consistency of a method, is called Discrete Maximum Principle (DMP). In many applications, convection dominates diffusion by several orders of magnitude. It is well known that standard discretizations typically do not satisfy the DMP in this convection-dominated regime. In fact, in this case, it turns out to be a challenging problem to construct discretizations that, on the one hand, respect the DMP and, on the other hand, compute accurate solutions. This paper presents a survey on finite element methods, with a main focus on the convection-dominated regime, that satisfy a local or a global DMP. The concepts of the underlying numerical analysis are discussed. The survey reveals that for the steady-state problem there are only a few discretizations, all of them nonlinear, that at the same time satisfy the DMP and compute reasonably accurate solutions, e.g., algebraically stabilized schemes. Moreover, most of these discretizations have been developed in recent years, showing the enormous progress that has been achieved lately. Methods based on algebraic stabilization, nonlinear and linear ones, are currently as well the only finite element methods that combine the satisfaction of the global DMP and accurate numerical results for the evolutionary equations in the convection-dominated situation.