Numerical solution of the generalised Poisson equation on parallel computers

Zur numerischen Lösung elliptischer Differentialgleichungen gibt es bereits viele bestens erprobte Algorithmen. Allerdings sind beim wissenschaftlichen Rechnen oftmals die Anforderungen an Arbeitsspeicher und Rechenleistung zu hoch, als dass sie von einem einzelnen Rechenkern erfüllt werden könnten. Daher muss die Rechenarbeit auf mehrere Kerne aufgeteilt werden, wofür spezielle Algorithmen notwendig sind. In dieser Diplomarbeit wird die Schur-Komplement-Methode vorgestellt, mit deren Hilfe das lineare Gleichungssystem, dass sich aus der Diskretisierung eines elliptischen Operators ergibt, parallel gelöst werden kann. Darüber hinaus wird anhand von zwei Beispielen die Bedeutung von elliptischen Operatoren in der Astrophysik gezeigt.There are a lot of well known algorithms to solve elliptic partial differential equations numerically. But for many applications, the computational domain and the memory requirements are too large for one single processing element (PE). The computational work must be done by several PE's and therefore the need of parallel algorithms arises. In this work the Schur Complement Method is presented with allows to solve the linear system corresponding to the discretisation of an elliptic operator in parallel. Furthermore, some examples of elliptic equations in Astrophyiscs are shown

Structure of the solar photosphere studied from the radiation hydrodynamics code ANTARES

The ANTARES radiation hydrodynamics code is capable of simulating the solar granulation in detail unequaled by direct observation. We introduce a state-of-the-art numerical tool to the solar physics community and demonstrate its applicability to model the solar granulation. The code is based on the weighted essentially non-oscillatory finite volume method and by its implementation of local mesh refinement is also capable of simulating turbulent fluids. While the ANTARES code already provides promising insights into small-scale dynamical processes occurring in the quiet-Sun photosphere, it will soon be capable of modeling the latter in the scope of radiation magnetohydrodynamics. In this first preliminary study we focus on the vertical photospheric stratification by examining a 3-D model photosphere with an evolution time much larger than the dynamical timescales of the solar granulation and of particular large horizontal extent corresponding to $25\!" \!\! \times \, 25\!"$ on the solar surface to smooth out horizontal spatial inhomogeneities separately for up- and downflows. The highly resolved Cartesian grid thereby covers $\sim 4~\mathrm{Mm}$ of the upper convection zone and the adjacent photosphere. Correlation analysis, both local and two-point, provides a suitable means to probe the photospheric structure and thereby to identify several layers of characteristic dynamics: The thermal convection zone is found to reach some ten kilometers above the solar surface, while convectively overshooting gas penetrates even higher into the low photosphere. An $\approx 145\,\mathrm{km}$ wide transition layer separates the convective from the oscillatory layers in the higher photosphere.Comment: Accepted for publication in Astrophysics and Space Science; 18 pages, 12 figures, 2 tables; typos correcte

Simulations of stellar convection, pulsation and semiconvection

We report on modelling in stellar astrophysics with the ANTARES code. First, we describe properties of turbulence in solar granulation as seen in high-resolution calculations. Then, we turn to the first 2D model of pulsation-convection interaction in a cepheid. We discuss properties of the outer and the HEII ionization zone. Thirdly, we report on our work regarding models of semiconvection in the context of stellar physics.Comment: Astrophysical Dynamics: From Stars to Galaxies. IAU Symposium 27

Detection of small convective patterns in observations and simulations

Recent results from high resolution solar granulation observations indicate the existence of a population of small granular cells on scales below 600 km in diameter, located in the intergranular lanes. We studied a set of Hinode SOT images and high resolution radiation hydrodynamics simulations in order to analyze small granular cells and to study their physical properties. An automated image segmentation algorithm specifically adapted to high resolution simulations for the identification of granules was developed. The algorithm was also used to analyze and compare physical quantities provided by the simulation and the observations. We found that small granules make a distinct contribution to the total area of granules. Both in observations and simulations, small granular cells exhibit on average lower intensities and vertical velocities

Efficient Characterization and Modelling of the Nonlinear Behaviour of LFT for Crash Simulations

Modeling the nonlinear material behaviour of long fiber reinforced thermoplastics (LFT) presents a challenging task since local inhomogeneities and nonlinear effects must be taken into account also on the microscale. We present a computational method with which we can predict the nonlinear material response of a composite material using only standard DMA measurements on the pure polymer matrix material. The material models considered include plasticity, damage, viscoelasticity, and viscoplasticity as described in [1]. These models can be combined similar to the model from [2] and extended to the composite by assigning linear elastic properties to the fibers. The mechanical response of the composite is computed using an FFT-based technique [3]. The geometry of the composite, in particular the fiber orientation, can be characterized using injection molding simulations or micro CT scans. We create virtual models of the composite using the algorithm of [4]. We show that with this method, the material behaviour of the composite can be predicted while the experimental complexity needed for the material characterization is low

A Low Mach Number Solver: Enhancing Applicability

In astrophysics and meteorology there exist numerous situations where flows exhibit small velocities compared to the sound speed. To overcome the stringent timestep restrictions posed by the predominantly used explicit methods for integration in time the Euler (or Navier-Stokes) equations are usually replaced by modified versions. In astrophysics this is nearly exclusively the anelastic approximation. Kwatra et al. have proposed a method with favourable time-step properties integrating the original equations (and thus allowing, for example, also the treatment of shocks). We describe the extension of the method to the Navier-Stokes and two-component equations. - However, when applying the extended method to problems in convection and double diffusive convection (semiconvection) we ran into numerical difficulties. We describe our procedure for stabilizing the method. We also investigate the behaviour of Kwatra et al.'s method for very low Mach numbers (down to Ma = 0.001) and point out its very favourable properties in this realm for situations where the explicit counterpart of this method returns absolutely unusable results. Furthermore, we show that the method strongly scales over 3 orders of magnitude of processor cores and is limited only by the specific network structure of the high performance computer we use.Comment: author's accepted version at Elsevier,JCP; 42 pages, 14 figure

River models for transport of matter and heat

Insufficient quality of water-supply may become the imiting factor for growth and development particularly in highly industrialized regions. A deterioration of water quality may result, aside from natural sources, from excessive municipal or industrial waste-water or cooling water discharges. Assessment of the environmental impact of such discharges requires adequate knowledge of the mixing and transport processes to which the introduced substances are subjected. Such phenomena are investigated in hydraulic models. A decisive distinction between mixing- and conventional models is the requirement to simulate turbulent transport processes correctly, which necessitates an exact reproduction of the local velocity distribution in the model. This results in more sophisticated requirements for model similarity. The main task of mixing models including intake- or outlet structures is usually the determination of the effluent concentration field in the water body, which is a prerequisite for the evaluation of possible negative consequences on the river ecology and on other water users located downstream. The model experiment gives answers to the question of how changes in the design of the outlet structure can influence the mixing pattern. Such questions are primarily important for large rivers and reservoirs, where incomplete mixing and stratification are likely to occur. For small rivers or creeks, the main problem is longitudinal dispersion, since cross-sectional mixing is quickly achieved because of the large ratio of effluent- to river flow rate. Similarly, the effect of very small waste-water discharges is easily evaluated as long as they do not produce a noticeable disturbance of the river flow