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 angezeigt. / The zoom link will be send to the staff of the institute 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 at this talk should contact Ulrich Wilbrandt wilbrandt@wias-berlin.de for obtaining the zoom login details.
Donnerstag, 18. 11. 2021, 14:00 Uhr (Hybrid Event im ESH)
Dr. Felipe Galarce Marin (WIAS Berlin)
Physics-based assimilation of displacements data from magnetic resonance elastographyThis work addresses the state estimation problem of reconstructing 3D displacement fields from partial 2D data acquired with Magnetic Resonance Elastography (MRE). MRE is a medical imaging technique sensitive to mechanical properties of tissues. Recent studies demonstrated its potential for the non-invasive estimation of pathological interstitial pressure, e.g., in the contexts of hydrocephalus (brain). We use a variational data assimilation framework based on a background parametric PDE (poroelasticity) for the tissue dynamics and on model order reduction (POD) of the admissible solution space. Preliminary results for the reconstruction of displacement field and characterization of the pressure field are presented.
For zoom login details please contact Ulrich Wilbrandt wilbrandt@wias-berlin.de
Donnerstag, 11. 11. 2021, 14:00 Uhr (Hybrid Event im ESH)
Ondrej Partl (WIAS Berlin)
Estimation of flow domain from velocity data based on the optimization of porous media flow parametersWe present an algorithm that estimates the partially unknown shape of a flow domain using internal velocity measurement. The intended application of such an algorithm is the problem of finding a stenosis in blood vessels in the human body.
Our algorithm looks as follows: We model the flow using a Brinkmann problem in which the presence of obstacles inside the flow domain is taken into account via a space-dependent effective permeability term; we solve this problem using a stabilized equal-order finite element method. To get the permeability term, we use a gradient-based variational optimization method that minimizes the error between the flow velocity from the Brinkman problem and the available velocity data. Finally, we apply various post-processing steps to reconstruct the true boundary of the flow domain.
We will present our latest results in 3D that are based on realistic geometries and synthetic velocity data.
Donnerstag, 15. 07. 2021, 14:00 Uhr (Online Event)
Prof. Marc-Andre Keip (Universität Stuttgart)
Multiscale analysis of instability-induced pattern transformations of composite active solidsWe discuss instability phenomena of active solids with periodic microstructures. Depending on the material, instabilities can be triggered by different means. In a first part of the talk, we analyze instability-induced pattern transformations of electroactive polymer composites and magnetorheological elastomers. In a second part, we widen the view towards the transient analysis of diffusion-driven instabilities of hydrogels, where we study pattern transformations of composite hydrogel thin films at different length scales.
Acknowledgements: Elten Polukhov, Siddharth Sriram
References
[1] G. Geymonat, S. Müller and N. Triantafyllidis. Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity. Archive for Rational Mechanics and Analysis 122:231-290 (1993)
[2] K. Bertoldi, M. Boyce, S. Deschanel, S. Prange and T. Mullin. Mechanics of deformation-triggered pattern transformations and superelastic behavior in periodic elastomeric structures. Journal of the Mechanics and Physics of Solids 56:2642-2668 (2008)
[3] C. Tipton, E. Han and T. Mullin. Magneto-elastic buckling of a soft cellular solid. Soft Matter 8:6880-6883 (2012)
[4] Wu, G., Xia, Y., Yang, S. Buckling, symmetry breaking, and cavitation in periodically micro-structured hydrogel membranes. Soft Matter 10:1392-1399 (2014)
[5] Polukhov, E., Vallicotti, D. and Keip, M.-A. Computational stability analysis of periodic electroactive polymer composites across scales. Computer Methods in Applied Mechanics and Engineering 337:165-197 (2018)
[6] Polukhov, E. and Keip, M.-A. Multiscale stability analysis of periodic magnetorheological elastomers. Mechanics of Materials, 159:103699 (2021)
[7] Sriram, S., Polukhov, E. and Keip, M.-A. Transient stability analysis of composite hydrogel structures based on a minimization-type variational formulation. International Journal of Solids and Structures, 111080 (2021)
For zoom login details please contact Alexander Linke linke@wias-berlin.de
Donnerstag, 08. 07. 2021, starting at 13:00 o'clock (Online Event)
Minisymposium
Minisymposium ''Perspectives of Gradient-Robustness''
For zoom login details please contact Jürgen Fuhrmann fuhrmann@wias-berlin.de
Donnerstag, 24. 06. 2021, 13:00 Uhr (Online Event)
Felipe Galarce Marin (WIAS Berlin)
Inverse problems in haemodynamics. Fast estimation of blood flows on non-parametric domains from medical dataThis seminar presents a work at the interface between applied mathematics and biomedical engineering. The work's main subject is the estimation of blood flows and quantities of medical interest in diagnosing certain diseases concerning the cardiovascular system. We propose a complete pipeline, providing the theoretical foundations for state estimation from medical data using reduced-order models, and addressing inter-patient variability. Extensive numerical tests are shown in realistic 3D scenarios that verify the potential impact of the work in the medical comunnity.
For zoom login details please contact Alexander Linke linke@wias-berlin.de
Donnerstag, 20. 05. 2021, 14:00 Uhr (Online Event)
Prof. Thomas Richter (Universität Magdeburg)
Deep neural networks for accelerating fluid-dynamics simulationsIn this talk we discuss the use of deep neural networks for augmenting classical finite element simulations in fluid-dynamics.
First, we present the deep neural network multigrid method (DNN-MG) [1,2] that can be considered as a multiscale scheme based on classical finite elements for the coarse scale and a deep neural network for predicting the fine scales. Embedded in a Newton-multigrid framework, the neural network is only acting locally. This allows for very small and efficient networks that require only a small amount of high fidelity data for training. At the same time it shows very promising generalization capabilities, since the network does not need any information about the problem geometry. We present first numerical results and discuss the approximation and stability properties of the augmented finite element / neural network framework.
In detail we will discuss two aspects of the DNN-MG approach: can we guarantee that the network augmented solution is able to preserve important physical structures like mass conservation of the fluid velocity field. And second, can we guarantee bounds on the approximation quality of the neural network enhanced solution?
If time permits we also give a quick overview on the efficient neural network accelerated simulation of particulate flows with non-spherical particles [3,4]. We have in mind configurations with many and small particles, which cannot be resolved in detail in a classical simulation. On the other hand, for general non-spherical particles, like blood cells or polygonal objects, there exist no analytical laws for predicting hydrodynamical coefficients like drag, lift or torque. Based on resolved simulations we train neural networks for establishing exactly such a mapping from the particle- and flow-configuration to the resulting forces. We discuss different use cases and demonstrate the accuracy and generalizability of this approach.
[1] D. Hartmann, C. Lessig, N. Margenberg and T. Richter. "A neural network multigrid solver for the Navier-Stokes equations" arxiv, 2021, https://arxiv.org/abs/2008.11520
[2] N. Margenberg, C. Lessig and T. Richter. "Structure Preservation for the Deep Neural Network Multigrid Solver" Electronic Transactions on Numerical Analysis, accepted 2021, http://numerik.uni-hd.de/~richter/pdf/https://arxiv.org/abs/2012.05290
[3] M. Minakowska, T. Richter and S. Sager. "A finite element / neural network framework for modeling suspensions of non-spherical particles. Concepts and medical applications" Vietnam Journal of Mathematics, Vol. 49, No. 1, pp. 207--235, 2021, https://link.springer.com/article/10.1007/s10013-021-00477-9
[4] H. von Wahl, T. Richter. "Using a deep neural network to predict the motion of under-resolved triangular rigid bodies in an incompressible flow" 2021, https://arxiv.org/abs/2102.11636
For zoom login details please contact Alexander Linke linke@wias-berlin.de
Donnerstag, 29. 04. 2021, starting at 09:30 o'clock (Online Event)
Workshop
FUHRI2021: Finite volUme metHods for Real-world applIcations
For zoom login details please contact Alexander Linke linke@wias-berlin.de
Dienstag, 27. 04. 2021, starting at 10:00 o'clock (Online Event)
Prof. Sashikumaar Ganesan and Prof. V. John (Indian Institute of Science Bangalore and WIAS Berlin)
Minisymposium on machine learning
For zoom login details please contact Volker John john@wias-berlin.de
Donnerstag, 15. 04. 2021, 14:00 Uhr (Online Event)
Prof. Julia Novo (Universidad Autonoma de Madrid, Spanien)
Error analysis of proper orthogonal decomposition stabilized methods for incompressible flowsProper orthogonal decomposition (POD) stabilized methods for the Navier-Stokes equations are presented. We consider two cases: the case in which the snapshots are based on a non inf-sup stable method and the case in which the snapshots are based on an inf-sup stable method. For both cases we construct approximations to the velocity and the pressure. For the first case, we analyze a method in which the snapshots are based on a stabilized scheme with equal order polynomials for the velocity and the pressure with local projection stabilization (LPS) for the gradient of the velocity and the pressure. For the POD method we add the same kind of LPS stabilization for the gradient of the velocity and the pressure as the direct method, together with grad-div stabilization. In the second case, the snapshots are based on an inf-sup stable Galerkin method with grad-div stabilization and for the POD model we also apply grad-div stabilization. In this case, since the snapshots are discretely divergence-free, the pressure can be removed from the formulation of the POD approximation to the velocity. To approximate the pressure, needed in many engineering applications, we use a supremizer pressure recovery method. Error bounds with constants independent of inverse powers of the viscosity parameter are proved for both methods. Numerical experiments show the accuracy and performance of the schemes.
For zoom login details please contact Alexander Linke linke@wias-berlin.de
Donnerstag, 25. 03. 2021, 14:00 Uhr (Online Event)
Benoit Gaudeul (Universite Lille, France)
Two entropic finite volume schemes for a Nernst--Planck--Poisson system with ion volume constraintsIn this talk, I consider a drift-diffusion system with cross-coupling through the chemical potentials comprising a model for the motion of finite size ions in liquid electrolytes. The drift term is due to the self-consistent electric field maintained by the ions and described by a Poisson equation.
I present two finite volume schemes based on different formulations of the fluxes. I will also provide a stability analysis of these schemes and an existence result for the corresponding discrete solutions. A convergence result is proposed for non-degenerate solutions. Numerical experiments show the behavior of these schemes.
This is a joint work with Jürgen Fuhrmann.
For zoom login details please contact Alexander Linke linke@wias-berlin.de
Donnerstag, 18. 03. 2021, 14:00 Uhr (Online Event)
Prof. Leo Rebholz (Clemson University, USA)
Anderson acceleration and how it speeds up convergence in fixed point iterationsAnderson acceleration (AA) is an extrapolation technique originally proposed in 1965 that recombines the most recent iterates and update steps in a fixed point iteration to improve the convergence properties of the sequence. Despite being successfully used for many years to improve nonlinear solver behavior on a wide variety of problems, a theory that explains the often-observed accelerated convergence was lacking. In this talk, we give an introduction to AA, then present a proof of AA convergence which shows that it improves the linear convergence rate based on a gain factor of an underlying optimization problem, but also introduces higher order terms in the residual error bound. We then discuss improvements to AA based on our convergence theory, and show numerical results for the algorithms applied to several application problems including Navier--Stokes, Boussinesq, and nonlinear Helmholtz systems.
For zoom login details please contact Alexander Linke linke@wias-berlin.de
Dienstag, 16. 03. 2021, 13:30 Uhr (Online Event)
Dr. Jürgen Fuhrmann (WIAS Berlin)
PDELib.jl: Towards software components for the numerical solution of partial differential equations in JuliaIn recent years, in particular since the release of version 1.0, the Julia programming language gained significant momentum in fields related to scientific computing and data science. Taking advantage of accumulated experience and know-how in language design, designed around the just-in-time compilation paradigm, and featuring first class multidimensional array handling, it allows for the implementation of complex numerical algorithms without sacrificing efficiency. In recent years, in particular since the release of version 1..0, the Julia programming language gained significant momentum in fields related to scientific computing and data science.
In the talk, we will give a short overview on features of the Julia language which renders it well suited for the implementation of solvers for complex systems of partial differential equations, including multiple dispatch allowing for the implementation of automatic differentiation, interface oriented API design and its package manager supporting reusability and reproducibility.
We will report on the successful steps towards the implementation of software components for the numerical solution of PDEs.. Focus will be on the package VoronoiFVM.jl and supporting packages.
For zoom login details please contact Alexander Linke linke@wias-berlin.de
Donnerstag, 25. 02. 2021, 14:00 Uhr (Online Event)
Vojtěch Miloš and Petr Vágner (University of Chemistry and Technology Technická, Prag and WIAS Berlin)
Modelling of YSZ|LSM|O_2 electrode with experimental validationA generalized Poisson-Nernst-Planck model of an yttria-stabilized zirconia (YSZ) electrolyte endowed with a lanthanum strontium manganite (LSM)--oxygen electrode interface is formulated with in the framework of non-equilibrium thermodynamics. The model takes into account limitations in oxide ion concentrations due to the limited availability of oxygen vacancies. The electrolyte model is coupled with a general reaction kinetic scheme capturing the oxygen reduction.
The model is fitted and compared to a large experimental dataset of electrochemical impedance spectra and cyclic voltammograms spanning temperatures from 700$^circ$ C to 850$^circ$ C. An apriori temperature dependenciies of the model parameters are introduced and the Nelder-Mead algorithm is employed to minimize a compound cost function. The compound cost function allows for different weights if the two types of experiments leading to interesting physical interpretations.
For zoom login details please contact Alexander Linke linke@wias-berlin.de
Donnerstag, 18. 02. 2021, 14:00 Uhr (Online Event)
Derk Frerichs (WIAS Berlin)
On reducing spurious oscillations in discontinuous Galerkin methods for convection-diffusion equationsAmong other desirable features, standard discontinuous Galerkin methods for discretizing steady-state convection-diffusion-reaction equations are able to produce sharp layers in the convection-dominated regime, but also show large spurious oscillations. A computationally cheap way to reduce nonphysical oscillations is to apply post-processing techniques that replace the solution in the vicinity of layers by linear or constant approximations. This talk presents known post-processing methods form the literature, and proposes several generalizations as well as novel modifications. Results for two numerical benchmark problems are shown indicating pros and cons of the methods.
For zoom login details please contact Alexander Linke linke@wias-berlin.de
Donnerstag, 11. 02. 2021, 14:00 Uhr (Online Event)
Dr. Ulrich Wilbrandt (WIAS Berlin)
Optimization of stabilization parameters for convection-dominated convection-diffusion equationsSolutions to convection-dominated convection-diffusion equations typically have spurious oscillations and other non-physical properties. To overcome these difficulties in the Finite Element context one often uses stabilizations, which are additional terms in the weak formulation. We will consider the streamline upwind Petrov-Galerkin (SUPG) and a spurious oscillations at layers diminishing (SOLD) method. These involve a user-chosen parameter on each cell and it is unknown how to choose these parameters exactly. The approach presented in this talk is to let an optimization algorithm decide on these parameters which leaves the choice of an appropriate objective to the user. We will show some objectives and their motivation and discuss a reduction of the rather larger control space.
For zoom login details please contact Alexander Linke linke@wias-berlin.de
Dienstag, 02. 02. 2021, 13:30 Uhr (Online Event)
Volker Kempf (Bundeswehr-Universität München)
Anisotropic and pressure-robust finite element discretizations for the Stokes equationsThe incompressible Stokes equations show a fundamental invariance property, meaning that the velocity solution is not affected by a change of the right hand side data in form of a gradient field. Due to relaxation of the divergence constraint most common finite element discretizations, like the Taylor-Hood or Mini-element, are however not pressure-robust, i.e. they do not reproduce this property on the discrete level. The mentioned classical methods are also in general not inf-sup stable on anisotropic meshes with potentially unbounded aspect ratio, which are used e.g. when treating singularities caused by re-entrant edges.
For zoom login details please contact Alexander Linke linke@wias-berlin.de
Donnerstag, 21. 01. 2021, 14:00 Uhr (Online Event)
Felipe Galarce (INRIA, Paris)
Inverse problems in haemodynamics -- fast estimation of blood flows from medical dataWe present a study at the interface between applied mathematics and biomedical engineering. The work's main subject is the estimation of blood flows and quantities of medical interest in diagnosing certain diseases concerning the cardiovascular system. We propose a complete pipeline, providing the theoretical foundations for state estimation from medical data using reduced-order models, and addressing inter-patient variability. Extensive numerical tests are shown in realistic 3D scenarios that verify the potential impact of the work in the medical community.
For zoom login details please contact Alexander Linke linke@wias-berlin.de