Abstract: We present a three-dimensional geometry model with tetrakaidecahedra for the biphasic model stratum corneum (SC) membrane ΩSC consisting of corneocytes embedded in a lipid matrix. Two practical domains for ΩSC are realized: the simple model SC-membrane ΩsSC and a realistic model SC-membrane ΩrSC with dimensions for abdominal human SC. The new geometry model uses tetrakaidecahedra as basic units. It is possible to assemble the tetrakaidecahedra one upon the other and side by side without gaps in a densest packing and with minimal area for all required interfaces. Geometric characteristics such as length, depth, height and angles of the corneocytes as well as the thickness of the lipid channels can be chosen arbitrarily. Furthermore, we are able to control the shift of the corneocytes and our concept allows to assemble many corneocytes in rows, columns and layers all embedded in a lipid matrix. With the aid of this concept the non-steady-state problem of drug diffusion within a three-dimensional, biphasic model SC-membrane, such as ΩsSC or ΩrSC, having homogeneous lipid and corneocyte phases is solved numerically with a multigrid method. The numerical computations are done with our simulation system UG. Our method for solving the diffusion problem is validated with homogeneous model SC-membranes with varying size of corneocytes and lipid channels, different numbers of corneocytes, and corneocyte alignment. Several time-dependent drug concentration profiles within the heterogeneous model SC-membranes are calculated and graphically shown for different values of relative corneocyte permeability ɛ = Dcor/Dlip.

Abstract: We present an algorithm that constructs parametrizations of boundary and interface surfaces automatically. Starting with high-resolution triangulated surfaces describing the computational domains, we iteratively simplify the surfaces yielding a coarse approximation of the boundaries with the same topological type. While simplifying we construct a function that is defined on the coarse surface and whose image is the original surface. This function allows access to the correct shape and surface normals of the original surface as well as to any kind of data defined on it. Such information can be used by geometric multigrid solvers doing adaptive mesh refinement. Our algorithm runs stable on all types of input surfaces, including those that describe domains consisting of several materials. We have used our method with success in different fields and we discuss examples from structural mechanics and biomechanics.

Abstract: We introduce semi-implicit complementary volume numerical scheme for solving the level setformulation of Riemannian mean curvature flow problem arising in image segmentation, edge detection, missing boundary completion and subjective contour extraction. The scheme is robust and efficient since it is linear, and it is stable in L_∞ and weighted W 1,1 sense without any restriction on a time step. The computational results related to medical image segmentation with partly missing boundaries and subjective contours extraction are presented.

Abstract: Traditional thin plate splines use radial basis functions and require the solution of a dense linear system of equations whose size is proportional to the number of data points. Instead of radial basis functions we present a method based on the use of polynomials with local support defined on finite element grids. This method is more efficient when dealing with large data sets as the resulting system of equations is sparse and its size depends only on the number of nodes in the finite element grid. Theory is developed for general d-dimensional data sets and model problems are presented in 3D to study the convergence behaviour.

Abstract: The Bank–Holst adaptive meshing paradigm is an efficient approach for parallel adaptive meshing of elliptic partial differential equations. It is designed to keep communication costs low and to take advantage of existing sequential adaptive software. While in principle the procedure could be used in any parallel environment, it was mainly conceived for use on small Beowulf clusters with a relatively small number of processors and a slow communication network. A typical calculation on such a machine might involve, say p = 32 processors, an adaptive fine mesh with a few million vertices, and use 2–3 min of computational time. In this work we, discuss a variant of the original scheme that could be used in situations where a much larger number of processors, say p > 100 is available. In this case the problem size could be much larger, say 10–100 million, with still a low to moderate computation time.

Abstract: In this article we present the numerical techniques for and the results of simulations of crystal growth and breakage in a stirred crystallizer. The population dynamics of the crystals is fully coupled with the flow in the tank. Our simulation involves the parallel computations and the adaptive refinement of the grid.

Abstract: We present a new numerical technique to approximate solutions to unsteady free surface flows modelled by the two-dimensional shallow water equations. The method we propose in this paper consists of an Eulerian–Lagrangian splitting of the equations along the characteristic curves. The Lagrangian stage of the splitting is treated by a non-oscillatory modified method of characteristics, while the Eulerian stage is approximated by an implicit time integration scheme using finite element method for spatial discretization. The combined two stages lead to a Lagrange–Galerkin method which is robust, second order accurate, and simple to implement for problems on complex geometry. Numerical results are shown for several test problems with different ranges of difficulty.

Abstract: The construction of streamlines is one of the most common methods for visualising fluid motion. Streamlines can be computed from the intersection of two nonparallel stream surfaces, which are iso-surfaces of dual stream functions. Stream surfaces are also useful to isolate part of the flow domain for detailed study. This paper introduces a technique for calculating dual stream functions for momentum fields that are defined analytically and depend on only two variables. For axi-symmetric flows, one of the dual stream functions is the well-known Stokes stream function. The analysis reduces the problem from the solution of partial differential equations to the solution of two ordinary differential equations. Example applications include the Moffat [17] vortex bubble, for which new solutions are presented.

Abstract: Structured adaptive mesh refinement (SAMR) techniques can provide accurate and cost- effective solutions to realistic scientific and engineering simulations modeling complex physical phenomena. However, the adaptive nature and inherent space–time heterogeneity of SAMR applications result in significant runtime management challenges. Moreover, certain SAMR applications involving reactive flows exhibit pointwise varying workloads and cannot be addressed by traditional parallelization approaches, which assume homogeneous loads. This paper presents hierarchical partitioning, bin-packing based load balancing, and Dispatch structured partitioning strategies to manage the spatiotemporal and computational heterogeneity in SAMR applications. Experimental evaluation of these schemes using 3-D Richtmyer–Meshkov compressible turbulence and 2-D reactive-diffusion kernels demonstrates the improvement in overall performance.

Abstract: The paper presents a new method for generating isotropic realizations of a random field with varying resolution scales and local block averages in one, two, or three dimensions. The random field has a lognormal distribution as found for the hydraulic conductivity in Darcy's flow in hydrology. We also present a fast method for generating anisotropic realizations of aGaussian random field using superposition of harmonic modes. Both methods are implemented in a software package which is presented. The realizations of the rescaled random field are made up ofupscaled local averages of the underlying random field which are consistentwith the level of resolution and which can be conditioned. The approach is motivated by the need to represent engineering properties aslocal averages and to be able to easily condition the realizations to incorporate known data or to change the resolution within sub-regions. The numerical results show that the method is very efficient computationally.

Abstract: Two-dimensional (2D) energy equation coupled with three temperatures such as electron, ion and photon is widely used to approximately describe the evolution of radiation energy across multiple materials and to study the exchange of energy among electrons, ions and photons for numerical research on laser-driven implosion of a fuel capsule in inertial confinement fusion experiments. The numerical solution of such equations is always fascinating because of its strongly nonlinear phenomena and strongly discontinuous interfaces. Using the UG framework, this paper successfully solves such equations on 2D unstructured grids with a fully implicit finite volume discretization scheme and parallel adaptive multigrid. Significant numerical results using 32 processors are given and analyzed.

Abstract: A method to simulate compressible flows at all speed on unstructured grids is presented. Special effort is set on low Mach number flows both in two and three dimensions. Weakly compressible flows can be treated by applying an asymptotic analysis to the equations. The behaviour of a multigrid method solving the linearized system is shown. Furthermore adaptive grid refinement and parallelization is used.

Abstract: Based on an asymptotic analysis of the compressible Navier–Stokes equations, the basic concept of the Multiple Pressure Variables (MPV) approach is to split up the pressure into three different terms representing global background effects, acoustic waves and the incompressible pressure, respectively. Special attention is payed to aeroacoustic phenomena and the numerical treatment of the generation and spreading of sound waves. To this aim, further perturbation analysis is done about the incompressible limit solution. This gives hints how to extract acoustic waves responsible for the noise generation from the flow field.

Abstract: We present a new numerical method for the identification of the most important metastable states of a system with complicated dynamical behavior from time series information. The approach is based on the representation of the effective dynamics of the full system by a Markov jump process between metastable states, and the dynamics within each of these metastable states by rather simple stochastic differential equations (SDEs). Its algorithmic realization exploits the concept of hidden Markov models (HMMs) with output behavior given by SDEs. A first complete algorithm including an explicit Euler---Maruyama-based likelihood estimator has already been presented in Horenko et al. (MMS, 2006a). Herein, we present a semi-implicit exponential estimator that, in contrast to the Euler---Maruyama-based estimator, also allows for reliable parameter optimization for time series where the time steps between single observations are large. The performance of the resulting method is demonstrated for some generic examples, in detail compared to the Euler---Maruyama-based estimator, and finally applied to time series originating from a 100 ns B-DNA molecular dynamics simulation.

Abstract: This work deals with all aspects of the numerical simulation of nonlinear time-periodic eddy current problems, ranging from the description of the nonlinearity to an efficient solution procedure. Due to the periodicity of the solution, we suggest a truncated Fourier series expansion, i.e. a so-called multiharmonic ansatz, instead of a costly time-stepping scheme. Linearization is done by a Newton iteration, where the preconditioning of the linearized problems is a special issue: since the matrices are non-symmetric, we need a special adaptation of a multigrid preconditioner to our problem. Eddy current problems comprise another difficulty that complicates the numerical simulation, namely the formation of extremely thin boundary layers. This challenge is handled by means of adaptive mesh refinement.

Abstract: We present a method that has been developed for the efficient numerical simulation of two-phase incompressible flows. For capturing the interface between the phases the level set technique is applied. The continuous model consists of the incompressible Navier–Stokes equations coupled with an advection equation for the level set function. The effect of surface tension is modeled by a localized force term at the interface (so-called continuum surface force approach). For spatial discretization of velocity, pressure and the level set function conforming finite elements on a hierarchy of nested tetrahedral grids are used. In the finite element setting we can apply a special technique to the localized force term, which is based on a partial integration rule for the Laplace–Beltrami operator. Due to this approach the second order derivatives coming from the curvature can be eliminated. For the time discretization we apply a variant of the fractional step θ-scheme. The discrete saddle point problems that occur in each time step are solved using an inexact Uzawa method combined with multigrid techniques. For reparametrization of the level set function a new variant of the fast marching method is introduced. A special feature of the solver is that it combines the level set method with finite element discretization, Laplace–Beltrami partial integration, multilevel local refinement and multigrid solution techniques. All these components of the solver are described. Results of numerical experiments are presented.

Abstract: We present efficient and robust multigrid methods for the solution of large, nonlinear, non-smooth systems as resulting from implicit time discretization of vector-valued Allen-Cahn equations with isotropic interfacial energy and logarithmic potential. The algorithms are shown to be robust in the sense that convergence is preserved for arbitrary values of temperature, including the deep quench limit. Numerical experiments indicate that the convergence speed as well is independent of temperature.

Abstract: We propose a linear finite-element discretization of Dirichlet problems for static Hamilton–Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local variational principle. It generalizes several approaches known in the literature and allows for a simple and transparent convergence theory. In this paper the resulting system of nonlinear equations is solved by an adaptive Gauss–Seidel iteration that is easily implemented and quite effective as a couple of numerical experiments show.

Abstract: We consider the numerical solution of optimization problems for systems of partial differential equations with constraints on the state and design variables as they arise in the optimal design of the shape and the topology of continuum mechanical structures. After discretization the resulting nonlinear programming problems are solved by an “all-at-once” approach featuring the numerical solution of the state equations as an integral part of the optimization routine. In particular, we focus on primal-dual Newton methods combined with interior-point techniques for an appropriate handling of the inequality constraints. Special emphasis is given on the efficient solution of the primal-dual system that results from the application of Newton’s method to the Karush–Kuhn–Tucker conditions where we take advantage of the special block structure of the primal-dual Hessian. Applications include structural optimization of microcellular biomorphic ceramics by homogenization modeling, the shape optimization of electrorheological devices, and the topology optimization of high power electromotors.