MEGIDDO: Computational study of the MR-DTI visualization algorithm

Pavel Strachota

Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague

Abstract:For the purpose of MR-DTI data visualization, we have developed a numerical algorithm based on a mathematical model of texture diffusion. Accompanied by data reprocessing and postprocessing procedures, this algorithm forms the cornerstone of the MEGIDDO (Medical Employment of Generating Images by Degenerate Diffusion Operator) software tool, which is briefly introduced in this contribution. Aiming at the application in the clinical environment, all components of the procedure must be tuned accordingly to find the optimal settings with respect to both visual appearance of the results and computational resources utilization. We demonstrate the results of some of the computational studies focused on numerical scheme assessment, model parameters adjustment and parallel processing benchmark.

Numerical Solution of the Gray-Scott Model

Jan Mach

Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague

Abstract:This contribution deals with numerical solution of the Gray-Scott model [GS1983, GS1984]. We introduce numerical schemes for this model based on the method of lines. To perform spatial discretization we use FDM and FEM. Resulting systems of ODEs are solved using the modified Runge-Kutta method with adaptive time-stepping. We present some of our numerical simulations and perform comparison of these schemes from the qualitative point of view.

[GS1983] P. Gray and S. K. Scott. Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability. Chem. Eng. Sci. 38:29-43 (1983)

[GS1984] P. Gray and S. K. Scott. Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A+2B->3B, B->C. Chem. Eng. Sci. 39:1087-1097 (1984).

Model of Topological Changes in Discrete Dislocation Dynamics

Petr Pauš

Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague

Abstract:This contribution deals with the numerical simulation of dislocation dynamics, their interaction, merging and other changes in the dislocation topology. The glide dislocations are represented by parametrically described curves moving in gliding planes. The simulation model is based on the numerical solution of the dislocation motion law belonging to the class of curvature driven curve dynamics. Mutual forces between dislocations are incorporated in the model. We focus on the simulation of the cross-slip of two dislocation curves where each curve evolves in a different gliding plane and after applying certain stress, the curves may merge. The simulation of the Frank-Read source of dislocations which describes how new dislocations are created is also presented. Merging and splitting of multiple (more than two) dislocation curves in persistent slip bands and their interactions in channels of the bands are also simulated.

Transport of colloids in heterogeneous porous media

Pavel Beneš

Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague

Abstract:This contribution deals with a numerical simulation of transport of colloids in heterogeneous porous media. The transport is described by the generalized convection-diffusion equation [Sun]. This equation is solved by means of the finite volume method using the operator splitting technique [Lev]: 1)the generalized convection-diffusion equation without the diffusion therm is solved explicitly using the finite volume method 2)the diffusion equation is solved implicitly by means of the finite volume method using solution from 1) as the initial condition. Some of our numerical simulations will be presented.

[Sun] N. Sun, M. Elimelech, N.-Z. Sun A novel two-dimensional model for colloid transport in physically and geochemically heterogeneous porous media. ’Journal of Contaminant Hydrology 49’, (2001), 173–199

[Lev] Leveque: The Finite Volume Methods for Hyperbolic Problems, Cambridge, 2002

Numerical Simulation of Epitaxial Crystal Growth with Elastic Effects

Hung Hoang Dieu*, Michal Beneš* and Atsushi Suzuki**

* Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague,
**CERMICS, Ecole Nationale des Ponts et Chaussees

Abstract:This contribution deals with the numerical simulation of epitaxial growth with elastic effects. The numerical scheme, which was developed to solve this problem, is based on the finite difference method. The elastic equations were solved by the finite element method. In order to verify that the phase-field model validates the reported experimental observations a number of numerical tests was performed. We showed that elastic effects strongly influence the crystal surface.

Comparison of the Lagrangean and level-set method for the Willmore flow

Tomáš Oberhuber

Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague

Abstract:We present two numerical methods for the Willmore flow of the planar curves. The Lagrangean approach works with parametrised curves. Discretisation leads to a "string" of nodes approximating the curve. To be able to compute evolution of such curve, redistribution of the nodes along the curve is necessary. There are several methods of the redistribution aim of which is to keep equidistant distribution of the nodes. The main advantage of this method is its efficiency, on the other hand it does not allow any changes in topology of the curve (merging or splitting). In this case the level-set method is good choice. It expresses the curve implicitly which increases the dimension of the problem by one. Unfortunately, it also means more expansive computations. We present numerical schemes for both methods together with comparison on several non-trivial examples and we also demonstrate experiments with topological changes obtained by the level-set method.

Numerical Simulation of Flow over a Rough Surface

Petr Bauer

Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague

Abstract:We attempt to model a 2D rough surface by computing non-stationary Navier-Stokes flow over a periodic pattern. The solution is obtained by means of finite element method (FEM). We use non-conforming Crouzeix Raviart elements for velocity and piecewise constant elements for pressure. The resulting linear system is solved by multigrid method. We present computational studies of the problem.

Numerical Simulation of Dynamic Effect in Capillarity in Heterogeneous Porous Media

Radek Fučík

Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague

Abstract:We present a one-dimensional fully-implicit numerical scheme to investigate the dynamic effect in the capillary pressure-saturation relationship used in the modelling of two-phase flow in porous media. Its validity is discussed by means of semi-analytical solutions developed by McWhorter and Sunada and by the authors. The numerical scheme is used to simulate a drainage experiment where the sand and fluid properties were known. Then, the numerical scheme is used to simulate a laboratory experiment in a homogeneous column including three major models of the dynamic effect coefficient and the respective results are presented and discussed. The presented numerical scheme can handle porous medium heterogeneity and it is used to simulate a fictitious experimental setup with two different sands. As a result, the penetration time of air phase through layered porous medium for models including dynamic effects varied between 50% to 150% compared to static models of capillary pressure-saturation relationship. Additionaly, the accumulation time of air at a material interface (i.e., delay of the air at the interface due to capillary barrier effect) is shown as a function of the ratio between air-entry pressure values of the adjacent sands emphasizing the differences between the dynamic and static capillary pressure models.

A multi-mesh finite element method for Lagrange elements of arbitrary degree

Thomas Witkowski

TU Dresden, Institut of Scientific Computing

Abstract:We consider within a finite element approach the usage of different adaptively refined meshes or different variables in systems of nonlinear, time-depended PDEs. To resolve different solution behaviours of these variables, the meshes can be independently adapted. The resulting linear systems are usually much smaller, when compared to the usage of a single mesh, and the overall computational runtime can be more than halved in such cases. Our multi-mesh method works for Lagrange finite elements of arbitrary degree and is independent of the spatial dimension. The approach is well defined, and can be implemented in existing adaptive finite element codes with minimal effort. We show computational examples in 2D and 3D ranging from dendritic growth to solid-solid phase-transitions. A further application comes from fluid dynamics where we demonstrate the applicability of the approach for solving the incompressible Navier-Stokes equations with Lagrange finite elements of the same order for velocity and pressure. The approach thus provides an easy to implement alternative to stabilized finite element schemes, if Lagrange finite elements of the same order are required.

Stabilizing fluid-fluid interfaces by nanoparticles - a Navier-Stokes-Cahn-Hilliard-Phase-Field-Crystal approach

Sebastian Aland

TU Dresden, Institut of Scientific Computing

Abstract:Bicontinuous interfacially jammed emulsion gels ('bijels') were proposed in 2005 as a hypothetical new class of soft materials in which interpenetrating, continuous domains of two immiscible fluids are maintained in a rigid state, by a jammed layer of colloidal particles at their interface. We develop a model for such a system which combines a Cahn-Hilliard- Navier-Stokes model for the macroscopic two-phase system with a surface Phase-Field-Crystal model for the microscopic colloidal system along the interface. First numerical results based on a finite element discretization will be presented.

Advected phase-field-crystal model for particles in a flowing solvent

Simon Praetorius

TU Dresden, Institut of Scientific Computing

Abstract:The phase-field-crystal (pfc) equation, that models particle interactions with a mean-field approach, will be extended by an advection term, to describe the influence of a flow-field to particle interactions. A test case of flow around a cylinder will be analysed and first results for (drag)/depletion forces at the obstacle, w.r. to the density-field obtained by pfc, will be shown.

The vorticity equation to handle divergence free flows on Riemannian manifolds

Ingo Nitschke

TU Dresden, Institut of Scientific Computing

Abstract:We start with the Navier-Stokes equation on a 2D compact and smooth Riemannian manifold M without boundaries. Then we embed M into the 3D space and derive a well-defined rotation-operator, wich maps a vector-valued function from tangentialspace to a scalar-valued function in the normalspace. By applying the rotation to Navier-Stokes equation we obtain the vorticity equation. This equation can be solved by FEM with 2D-elements in a 3D-world.