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End date: 2023 × Has files: Yes × Subject: finite element method ×

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Now showing 1 - 10 of 20
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    Numerical modelling of climate change impacts on freshwater lenses on the North Sea Island of Borkum using hydrological and geophysical methods
    (Munich : EGU, 2012) Sulzbacher, H.; Wiederhold, H.; Siemon, B.; Grinat, M.; Igel, J.; Burschil, T.; Günther, T.; Hinsby, K.
    A numerical, density dependent groundwater model is set up for the North Sea Island of Borkum to estimate climate change impacts on coastal aquifers and especially the situation of barrier islands in the Wadden Sea. The database includes information from boreholes, a seismic survey, a helicopter-borne electromagnetic (HEM) survey, monitoring of the freshwater-saltwater boundary by vertical electrode chains in two boreholes, measurements of groundwater table, pumping and slug tests, as well as water samples. Based on a statistical analysis of borehole columns, seismic sections and HEM, a hydrogeological model is set up. The groundwater model is developed using the finite-element programme FEFLOW. The density dependent groundwater model is calibrated on the basis of hydraulic, hydrological and geophysical data, in particular spatial HEM and local monitoring data. Verification runs with the calibrated model show good agreement between measured and computed hydraulic heads. A good agreement is also obtained between measured and computed density or total dissolved solids data for both the entire freshwater lens on a large scale and in the area of the well fields on a small scale. For simulating future changes in this coastal groundwater system until the end of the current century, we use the climate scenario A2, specified by the Intergovernmental Panel on Climate Change and, in particular, the data for the German North Sea coast. Simulation runs show proceeding salinisation with time beneath the well fields of the two waterworks Waterdelle and Ostland. The modelling study shows that the spreading of well fields is an appropriate protection measure against excessive salinisation of the water supply until the end of the current century.
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    Development of a numerical workflow based on μ-CT imaging for the determination of capillary pressure–saturation-specific interfacial area relationship in 2-phase flow pore-scale porous-media systems: a case study on Heletz sandstone
    (Göttingen : Copernicus Publ., 2016) Peche, Aaron; Halisch, Matthias; Bogdan Tatomir, Alexandru; Sauter, Martin
    In this case study, we present the implementation of a finite element method (FEM)-based numerical pore-scale model that is able to track and quantify the propagating fluid–fluid interfacial area on highly complex micro-computed tomography (μ-CT)-obtained geometries. Special focus is drawn to the relationship between reservoir-specific capillary pressure (pc), wetting phase saturation (Sw) and interfacial area (awn). The basis of this approach is high-resolution μ-CT images representing the geometrical characteristics of a georeservoir sample. The successfully validated 2-phase flow model is based on the Navier–Stokes equations, including the surface tension force, in order to consider capillary effects for the computation of flow and the phase-field method for the emulation of a sharp fluid–fluid interface. In combination with specialized software packages, a complex high-resolution modelling domain can be obtained. A numerical workflow based on representative elementary volume (REV)-scale pore-size distributions is introduced. This workflow aims at the successive modification of model and model set-up for simulating, such as a type of 2-phase problem on asymmetric μ-CT-based model domains. The geometrical complexity is gradually increased, starting from idealized pore geometries until complex μ-CT-based pore network domains, whereas all domains represent geostatistics of the REV-scale core sample pore-size distribution. Finally, the model can be applied to a complex μ-CT-based model domain and the pc–Sw–awn relationship can be computed.
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    Computational modelling and simulation of cancer growth and migration within a 3D heterogeneous tissue: The effects of fibre and vascular structure
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Macnamara, Cicely K.; Caiazzo, Alfonso; Ramis-Conde, Ignacio; Chaplain, Mark A.J.
    The term cancer covers a multitude of bodily diseases, broadly categorised by having cells which do not behave normally. Since cancer cells can arise from any type of cell in the body, cancers can grow in or around any tissue or organ making the disease highly complex. Our research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modeling. We present a 3D individual-based model which allows one to simulate the behaviour of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels. Each agent (a single cell, for example) is fully realised within the model and interactions are primarily governed by mechanical forces between elements. However, as well as the mechanical interactions we also consider chemical interactions, for example, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells. The current state of the art of the model allows us to simulate tumour growth around an arbitrary blood-vessel network or along the striations of fibrous tissue.
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    A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Akbas, Mine; Gallouët, Thierry; Gaßmann, Almut; Linke, Alexander; Merdon, Christian
    A novel notion for constructing a well-balanced scheme --- a gradient-robust scheme --- is introduced and a showcase application for a steady compressible, isothermal Stokes equations is presented. Gradient-robustness means that arbitrary gradient fields in the momentum balance are well-balanced by the discrete pressure gradient --- if there is enough mass in the system to compensate the force. The scheme is asymptotic-preserving in the sense that it degenerates for low Mach numbers to a recent inf-sup stable and pressure-robust discretization for the incompressible Stokes equations. The convergence of the coupled FEM-FVM scheme for the nonlinear, isothermal Stokes equations is proved by compactness arguments. Numerical examples illustrate the numerical analysis, and show that the novel approach can lead to a dramatically increased accuracy in nearly-hydrostatic low Mach number flows. Numerical examples also suggest that a straight-forward extension to barotropic situations with nonlinear equations of state is feasible.
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    Displacement and pressure reconstruction from magnetic resonance elastography images: Application to an in silico brain model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Galarce Marín, Felipe; Tabelow, Karsten; Polzehl, Jörg; Papanikas, Christos Panagiotis; Vavourakis, Vasileios; Lilaj, Ledia; Sack, Ingolf; Caiazzo, Alfonso
    This paper investigates a data assimilation approach for non-invasive quantification of intracranial pressure from partial displacement data, acquired through magnetic resonance elastography. Data assimilation is based on a parametrized-background data weak methodology, in which the state of the physical system tissue displacements and pressure fields is reconstructed from partially available data assuming an underlying poroelastic biomechanics model. For this purpose, a physics-informed manifold is built by sampling the space of parameters describing the tissue model close to their physiological ranges, to simulate the corresponding poroelastic problem, and compute a reduced basis. Displacements and pressure reconstruction is sought in a reduced space after solving a minimization problem that encompasses both the structure of the reduced-order model and the available measurements. The proposed pipeline is validated using synthetic data obtained after simulating the poroelastic mechanics on a physiological brain. The numerical experiments demonstrate that the framework can exhibit accurate joint reconstructions of both displacement and pressure fields. The methodology can be formulated for an arbitrary resolution of available displacement data from pertinent images. It can also inherently handle uncertainty on the physical parameters of the mechanical model by enlarging the physics-informed manifold accordingly. Moreover, the framework can be used to characterize, in silico, biomarkers for pathological conditions, by appropriately training the reduced-order model. A first application for the estimation of ventricular pressure as an indicator of abnormal intracranial pressure is shown in this contribution.
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    Challenges for drift-diffusion simulations of semiconductors: A comparative study of different discretization philosophies
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Farrell, Patricio; Peschka, Dirk
    We analyze and benchmark the error and the convergence order of finite difference, finite-element as well as Voronoi finite-volume discretization schemes for the drift-diffusion equations describing charge transport in bulk semiconductor devices. Three common challenges, that can corrupt the precision of numerical solutions, will be discussed: boundary layers at Ohmic contacts, discontinuties in the doping profile, and corner singularities in L-shaped domains. The influence on the order of convergence is assessed for each computational challenge and the different discretization schemes. Additionally, we provide an analysis of the inner boundary layer asymptotics near Ohmic contacts to support our observations.
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    Modelling and simulation of flame cutting for steel plates with solid phases and melting
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Arenas Jaén, Manuel J.; Hömberg, Dietmar; Lasarzik, Robert; Mikkonen, Pertti; Petzold, ThomasFlame cutting, finite element method, heat equation, phase transitions, transport equation
    The goal of this work is to describe in detail a quasi-stationary state model which can be used to deeply understand the distribution of the heat in a steel plate and the changes in the solid phases of the steel and into liquid phase during the flame cutting process. We use a 3D-model similar to previous works from Thiebaud [1] and expand it to consider phases changes, in particular, austenite formation and melting of material. Experimental data is used to validate the model and study its capabilities. Parameters defining the shape of the volumetric heat source and the power density are calibrated to achieve good agreement with temperature measurements. Similarities and differences with other models from literature are discussed.
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    Functional a posteriori error estimation for stationary reaction-convection-diffusion problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Eigel, Martin; Samrowski, Tatiana
    A functional type a posteriori error estimator for the finite element discretisation of the stationary reaction-convection-diffusion equation is derived. In case of dominant convection, the solution for this class of problems typically exhibits boundary layers and shock-front like areas with steep gradients. This renders the accurate numerical solution very demanding and appropriate techniques for the adaptive resolution of regions with large approximation errors are crucial. Functional error estimators as derived here contain no mesh-dependent constants and provide guaranteed error bounds for any conforming approximation. To evaluate the error estimator, a minimisation problem is solved which does not require any Galerkin orthogonality or any specific properties of the employed approximation space. Based on a set of numerical examples, we assess the performance of the new estimator. It is observed that it exhibits a good efficiency also with convection-dominated problem settings.
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    Optimal and robust a posteriori error estimates in L∞(L2) for the approximation of Allen-Cahn equations past singularities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Bartels, Sören; Müller, Rüdiger
    Optimal a posteriori error estimates in L∞(L2) are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming, nonconforming, mixed, and discontinuous Galerkin methods. Numerical experiments illustrate the theoretical results.
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    Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Bartels, Sören; Müller, Rüdiger
    A fully computable upper bound for the finite element approximation error of Allen-Cahn and Cahn-Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.