Thomas Witkowski

Technische Universität Dresden, Dresden, Saxony, Germany

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Publications (5)8.69 Total impact

  • Thomas Witkowski, Rainer Backofen, Axel Voigt
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    ABSTRACT: A theoretical explanation of the existence of lipid rafts in cell membranes remains a topic of lively debate. Large, micrometer sized rafts are readily observed in artificial membranes and can be explained using thermodynamic models for phase separation and coarsening. In live cells such domains are not observed and various models are proposed to describe why the systems do not coarsen. We review these attempts critically and show within a phase field approach that membrane bound proteins have the potential to explain the different behaviour observed in vitro and in vivo. Large scale simulations are performed to compute scaling laws and size distribution functions under the influence of membrane bound proteins and to observe a significant slow down of the domain coarsening at longer times and a breakdown of the self-similarity of the size-distribution function.
    Physical Chemistry Chemical Physics 07/2012; 14(42):14509-15. · 4.20 Impact Factor
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    ABSTRACT: We consider the classical problem to find optimal distributions of interacting particles on a sphere by solving an evolution problem for a particle density. The higher order surface partial differential equation is an approximation of a surface dynamic density functional theory. We motivate the approach phenomenologically and sketch a derivation of the model starting from an interatomic potential. Different numerical approaches are discussed to solve the evolution problem: (a) an implicit approach to describe the surface using a phase-field description, (b) a parametric finite element approach, and (c) a spectral method based on non-equispaced fast Fourier transforms on the sphere. Results for computed minimal energy configurations are discussed for various particle numbers and are compared with known rigorous asymptotic results. Furthermore extensions to other more complex and evolving surfaces are mentioned.
    SIAM Journal on Multiscale Modeling and Simulation 01/2011; 9:314-334. · 1.56 Impact Factor
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    Thomas Witkowski, Axel Voigt
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    ABSTRACT: We consider within a finite element approach the usage of different adaptively refined meshes for 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.
    05/2010;
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    Rainer Backofen, Axel Voigt, Thomas Witkowski
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    ABSTRACT: We present a dynamic model to study ordering of particles on arbitrary curved surfaces. Thereby the particles are represented as maxima in a density field and a surface partial differential equation for the density field is solved to the minimal energy configuration. We study annihilation of dislocations within the ordered system and premelting along grain-boundary scars. The obtained minimal energy configurations on a sphere are compared with existing results and scaling laws are computed for the number of excess dislocations as a function of system size.
    Physical Review E 02/2010; 81(2 Pt 2):025701. · 2.31 Impact Factor
  • Axel Voigt, Thomas Witkowski
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    ABSTRACT: We present a hybrid OpenMP/MPI parallelization of the finite element method that is suitable to make use of modern high performance computers. These are usually built from a large bulk of multi-core systems connected by a fast network. Our parallelization method is based firstly on domain decomposition to divide the large problem into small chunks. Each of them is then solved on a multi-core system using parallel assembling, solution and error estimation. To make domain decomposition for both, the large problem and the smaller sub-problems, sufficiently fast we make use of a hierarchical mesh structure. The partitioning is done on a coarser mesh level, resulting in a very fast method that shows good computational balancing results. Numerical experiments show that both parallelization methods achieve good scalability in computing solution of nonlinear, time dependent, higher order partial differential equations (PDEs) on large domains. The parallelization is realized in the adaptive finite element software AMDiS.
    Kybernetika -Praha- 01/2010; 2(2). · 0.62 Impact Factor