Boosting the accuracy of SPH techniques: Newtonian and special-relativistic tests

We study the impact of different discretization choices on the accuracy of SPH and we explore them in a large number of Newtonian and special-relativistic benchmark tests. As a first improvement, we explore a gradient prescription that requires the (analytical) inversion of a small matrix. For a regular particle distribution this improves gradient accuracies by approximately ten orders of magnitude and the SPH formulations with this gradient outperform the standard approach in all benchmark tests. Second, we demonstrate that a simple change of the kernel function can substantially increase the accuracy of an SPH scheme. While the "standard" cubic spline kernel generally performs poorly, the best overall performance is found for a high-order Wendland kernel which allows for only very little velocity noise and enforces a very regular particle distribution, even in highly dynamical tests. Third, we explore new SPH volume elements that enhance the treatment of fluid instabilities and, last, but not least, we design new dissipation triggers. They switch on near shocks and in regions where the flow --without dissipation-- starts to become noisy. The resulting new SPH formulation yields excellent results even in challenging tests where standard techniques fail completely.Comment: accepted for publication in MNRA

Keskosa interpolatsioonil põhinevad meetodid nõrgalt singulaarsete integraalvõrrandite lahendamiseks

Paljud keemia, polümeeride füüsika, matemaatilise füüsika jt teadusalade probleemid on formuleeritavad integraalvõrrandite kujul ning nende probleemide käsitlus taandub integraalvõrrandite lahendamisele või kvalitatiivsele uurimisele. Integraalvõrrandeid, mida saab täpselt lahendada, on suhteliselt vähe, seega on väga olulised meetodid võrrandite numbriliseks lahendamiseks. Käesolevas doktoritöös pakume välja kaks kõrget järku numbrilist meetodit, mis ei kasuta lineaarse teist liiki singulaarsustega Fredholmi integraalvõrrandi lahendamiseks ebaühtlast võrku. Need meetodid on kollokatsioonimeetod ja korrutise integreerimise meetod. Nimetatud meetodid põhinevad keskosa interpolatsioonil polünoomidega ühtlasel võrgul ja silendaval muutujate vahetusel. Lõigu keskosas on interpolatsioonivea hinnang ligikaudu 2m korda täpsem kui kogu lõigul. Lisaks on interpolatsiooniprotsess ühtlasel võrgul lõigu keskosas m-i kasvades stabiilne. Muutujate vahetuse abil parendame me võrrandi täpse lahendi käitumist. Doktoritöös on kirjeldatud toodud meetodite koondumist ja koondumiskiirustThere are a number of problems from many different fields, for example chemistry, physics of polymers and mathematical physics, which are directly formulated in terms of integral equations; and there are problems that are represented in terms of differential equations with auxiliary conditions, but which can be reduced to integral equations. There are relatively few integral equations which can be solved exactly, hence, numerical schemes are required for dealing with these equations in a proper manner. In this thesis we propose two new classes of high order numerical methods, which do not need graded grids for solving linear Fredholm integral equations of the second kind with singularities. The methods are developed by means of the 'central part' interpolation by polynomials on the uniform grid and smoothing change of variables. In the central parts of the interval, the estimates of interpolation error are approximately 2m times more precise than on the whole interval. In the central parts of the interval, the interpolation process on the uniform grid also has good stability properties as m increases. With the help of a change of variables we improve the boundary behaviour of the exact solution of the problem. The convergence and the convergence order of methods is studied

The rich cluster of galaxies ABCG~85. IV. Emission line galaxies, luminosity function and dynamical properties

This paper is the fourth of a series dealing with the cluster of galaxies ABCG 85. Using our two extensive photometric and spectroscopic catalogues (with 4232 and 551 galaxies respectively), we discuss here three topics derived from optical data. First, we present the properties of emission line versus non-emission line galaxies, showing that their spatial distributions somewhat differ; emission line galaxies tend to be more concentrated in the south region where groups appear to be falling onto the main cluster, in agreement with the hypothesis (presented in our previous paper) that this infall may create a shock which can heat the X-ray emitting gas and also enhance star formation in galaxies. Then, we analyze the luminosity function in the R band, which shows the presence of a dip similar to that observed in other clusters at comparable absolute magnitudes; this result is interpreted as due to comparable distributions of spirals, ellipticals and dwarfs in these various clusters. Finally, we present the dynamical analysis of the cluster using parametric and non-parametric methods and compare the dynamical mass profiles obtained from the X-ray and optical data.Comment: accepted for publication in A&

Modelling of wreckage trajectories following in-flight breakup

The study of ballistic trajectories is well-­established, with work dating back to the 1940s. More recently, ballistic trajectory analysis has been used by air accident investigators in an attempt to understand events leading up to an accident, with notable examples including the investigation into the Lockerbie bombing. Building on the history of a previous model, this thesis offers an enhanced model for the calculation of ballistic trajectories incorporating altitude dependence for wind, density and gravity. Attempts to solve the model analytically were unsuccessful, and therefore, the model was solved numerically using an implicit scheme, to deal with the inherent stiffness of the equations, and an extrapolation technique. Cubic splines were used to accurately represent the wind proUile in an analytical way. The numerical solution was veriUied against a simpliUied analytical case and results are presented for two simulated breakup cases. Four key parameters were then varied to provide information about the sensitivity of Uinal wreckage location to variation in system parameters. The results indicate that for simulated large aircraft breakups, low ballistic coefUicient items are most heavily affected by breakup altitude, wind magnitude and wind angle whereas large ballistic coefUicient items are most heavily affected by breakup velocity, although to a much lesser extent (around 15% of the distance of low ballistic coefUicient). For small aircraft breakups, wind angle and breakup altitude have the largest effect on low ballistic coefUicient items, with velocity and altitude affecting high ballistic coefUicient items to a larger extent (around 50% of low ballistic coefUicient items). The results will allow investigators to understand better the factors which affect items of differing ballistic coefUicient in different situations. The project as a whole provides a new solution engine for the trajectory problem which can form the basis of a new software package for investigators

Applicability of spline collocation to cordial volterra equations

We study the applicability of the standard spline collocation method, on a uniform grid, to linear Volterra integral equations of the second kind with the so-called cordial operators; these operators are noncompact and the applicability of the collocation method becomes crucial in the convergence analysis. In particular, piecewise constant, piecewise linear and piecewise quadratic collocation methods are applicable under wide, quite acceptable conditions. For higher order spline collocation, it is more complicated to carry out an analytical study of the applicability of the method; however, a numerical check is rather simple and this is illustrated by some numerical examples

Vlasov versus N-body: the H\'enon sphere

We perform a detailed comparison of the phase-space density traced by the particle distribution in Gadget simulations to the result obtained with a spherical Vlasov solver using the splitting algorithm. The systems considered are apodized H\'enon spheres with two values of the virial ratio, R ~ 0.1 and 0.5. After checking that spherical symmetry is well preserved by the N-body simulations, visual and quantitative comparisons are performed. In particular we introduce new statistics, correlators and entropic estimators, based on the likelihood of whether N-body simulations actually trace randomly the Vlasov phase-space density. When taking into account the limits of both the N-body and the Vlasov codes, namely collective effects due to the particle shot noise in the first case and diffusion and possible nonlinear instabilities due to finite resolution of the phase-space grid in the second case, we find a spectacular agreement between both methods, even in regions of phase-space where nontrivial physical instabilities develop. However, in the colder case, R=0.1, it was not possible to prove actual numerical convergence of the N-body results after a number of dynamical times, even with N=10$^8$ particles.Comment: 19 pages, 11 figures, MNRAS, in pres

Meshless hydrodynamic simulations of young supernova remnants

The majority of massive stars end their lives by ejecting their outer envelopes in a corecollapse supernova explosion. The collision of their ejecta with the surrounding circumstellar medium results in the formation of supernova remnants that have been detected at all wavelengths, from radio to gamma-rays. For several dozen supernova remnants, very-long-baseline radio interferometers have spatially resolved the interaction region and directly measured the expansion rates of the shocked gas; many show evidence of the interaction of supernova ejecta with the dense slow winds characteristic of the red supergiant progenitors. Understanding the dynamics and morphology of the interaction region, particularly in young supernova remnants leads to estimates of the total mass of the circumstellar medium, as well as its density distribution around the star given the value of the wind velocity. Here we studied the interaction of the supernova ejecta with different circumstellar environments to investigate the hydrodynamic evolution of young supernova remnants in the SedovTaylor phase. We used the massively parallel, multi-physics magneto-hydrodynamics (MHD) and gravity code, GIZMO, for our simulations. We chose GIZMO for its flexibility in allowing the user to choose different methods to solve the fluid equations, i.e., new Lagrangian Godunovtype schemes, e.g., Meshless Finite Volume (MFV) and Meshless Finite Mass (MFM), as well as various flavors of smoothed particle hydrodynamics (SPH), or Eulerian fixed-grid schemes. Since the majority of previous studies used the latter, we focused on an extensive comparison of all the meshless methods in solving the Sedov-Taylor blastwave test, a problem for which there is an exact solution. For our given compute resources, we found the parameters (e.g., smoothing length, number of neighbours, artificial viscosity, and particle resolution) for each meshless method that gave the best agreement with the exact solution. We then carried out 2D and 3D simulations of the hydrodynamic interaction of the supernova ejecta with varying density profiles assumed for the circumstellar medium, namely: a 1/r 2 density profile, for a typical, spherically symmetric red supergiant stellar wind, and an axisymmetric torus profile, inspired by the observation of a dense, dusty torus of the circumstellar material around the red supergiant, WOH G64 (Ohnaka et al., 2008). Radially assembled Hierarchical Equal Area isoLatitude Pixelization (HEALPix) shells were used to set-up the initial density and velocity profiles for the ejecta, which is marked by a flat inner core and a steeply declining outer edge. The Weighted Voronoi Tessellation code was used to produce the 1/r 2 and axisymmetric torus density distributions. We showed that the growth of Richtmyer-Meshkov instabilities in the 2D and 3D 1/r 2 profiles are visible as early as 20 yrs into the evolution of the remnant and become increasingly unstable up to 100 yr. While 2D simulations of 1/r 2 profiles show the presence of the Richtmyer-Meshkov instabilities in the hot shell of a contact discontinuity, in 3D we see large bubbles and filamentary structure of the instabilities. Our results for the numerical approaches to simulating the systems for the 1/r 2 density cases were broadly consistent with previous studies in the literature where stationary grids were used. Two scenarios with different torus-cavity density contrasts were considered in which we found that the instability rolls along the half-opening angle takes ∼ 40 yr to develop in the axisymmetric torus with smooth density drop, whereas the axisymmetric torus with steep density drop does not develop any instability rolls up to the end of the simulation. We concluded with a discussion of the implications of our models for the morphology of supernova remnants and their expected levels of multi-wavelength emission

Computer-Aided Geometry Modeling

Techniques in computer-aided geometry modeling and their application are addressed. Mathematical modeling, solid geometry models, management of geometric data, development of geometry standards, and interactive and graphic procedures are discussed. The applications include aeronautical and aerospace structures design, fluid flow modeling, and gas turbine design