Workshop on Scientific Computing
June 2023, 2010

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Abstract:For the purpose of MRDTI 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.
Abstract:This contribution deals with numerical solution of the GrayScott 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 RungeKutta method with adaptive timestepping. 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. 38:2943 (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. 39:10871097 (1984).
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 crossslip 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 FrankRead 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.
Abstract:This contribution deals with a numerical simulation of transport of colloids in heterogeneous porous media. The transport is described by the generalized convectiondiffusion equation [Sun]. This equation is solved by means of the finite volume method using the operator splitting technique [Lev]:
1)the generalized convectiondiffusion 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 twodimensional 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
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 phasefield model validates the reported experimental observations a number of numerical tests was performed. We showed that elastic effects strongly influence the crystal surface.
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 levelset 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 nontrivial examples and we also demonstrate experiments with topological changes obtained by the levelset method.
Abstract:We attempt to model a 2D rough surface by computing nonstationary NavierStokes flow over a periodic pattern. The solution is obtained by means of finite element method (FEM). We use nonconforming 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.
Abstract:We present a onedimensional fullyimplicit numerical scheme to investigate the dynamic effect in the capillary pressuresaturation relationship used in the modelling of twophase flow in porous media. Its validity is discussed by means of semianalytical 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 pressuresaturation 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 airentry pressure values of the adjacent sands emphasizing the differences between the dynamic and static capillary pressure models.
Abstract:We consider within a finite element approach the usage of different adaptively refined meshes or different variables in systems of nonlinear, timedepended 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 multimesh 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 solidsolid phasetransitions. A further application comes from fluid dynamics where we demonstrate the applicability of the approach for solving the incompressible NavierStokes 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.
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 CahnHilliard NavierStokes model for the macroscopic twophase system with a surface PhaseFieldCrystal model for the microscopic colloidal system along the interface. First numerical results based on a finite element discretization will be presented.
Abstract:The phasefieldcrystal (pfc) equation, that models particle interactions with a meanfield approach, will be extended by an advection term, to describe the influence of a flowfield 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 densityfield obtained by pfc, will be shown.
Abstract:We start with the NavierStokes equation on a 2D compact and smooth Riemannian manifold M without boundaries. Then we embed M into the 3D space and derive a welldefined rotationoperator, wich maps a vectorvalued function from tangentialspace to a scalarvalued function in the normalspace. By applying the rotation to NavierStokes equation we obtain the vorticity equation. This equation can be solved by FEM with 2Delements in a 3Dworld.