Radial basis function (RBF) methods for partial differential equations (PDEs), either in bulk domains, on surfaces, or in a combination of the formers, arise in a wide range of practical applications. This thesis proposes numerical approaches of RBF-based meshless techniques to solve these three kinds of PDEs on stationary and nonstationary surfaces and domains. In Chapter 1, we introduce the background of RBF methods, some basic concepts, and error estimates for RBF interpolation. We then provide some preliminaries for manifolds, restricted RBFs on manifolds, and some convergence properties of RBF interpolation. Finally, implicit-explicit time stepping schemes are briefly presented. In Chapter 2, we propose methods to implement meshless collocation approaches intrinsically to solve elliptic PDEs on smooth, closed, connected, and complete Riemannian manifolds with arbitrary codimensions. Our methods are based on strong-form collocations with oversampling and least-squares minimizations, which can be implemented either analytically or approximately. By restricting global kernels to the manifold, our methods resemble their easy-to-implement domain-type analogies, that is, Kansa methods. Our main theoretical contribution is a robust convergence analysis under some standard smoothness assumptions for high-order convergence. We simulate reaction-diffusion equations to generate Turing patterns and solve shallow water problems on manifolds. In Chapter 3, we consider convective-diffusion problems that model surfactants or heat transport along moving surfaces. We propose two time-space algorithms by combining the methods of lines and kernel-based meshless collocation techniques intrinsic to surfaces. We use a low-order time discretization for fair comparison, and higher-order schemes in time are possible. The proposed methods can achieve second-order convergence. They use either analytic or approximated spatial discretization of the surface operators, which do not require regeneration of point clouds at each temporal iteration. Thus, they are alternatively applied to handle models on two types of evolving surfaces, which are defined as prescribed motions and governed by geometric evolution laws, respectively. We present numerical examples on various evolving surfaces for the performance of our algorithms and apply the approximated one to merging surfaces. In Chapter 4, a kernel-based meshless method is developed to solve coupled second-order elliptic PDEs in bulk domains and on surfaces, subject to Robin boundary conditions. It combines a least-squares kernel-based collocation method with a surface-type intrinsic approach. We can thus use each pair for discrete point sets, RBF kernels (globally and restrictedly), trial spaces, and some essential assumptions, to search for least-squares solutions in bulks and on surfaces, respectively. We first analyze error estimates for a domain-type Robin-boundary problem. Based on this analysis and the existing results for surface PDEs, we discuss the theoretical requirements for the Sobolev kernels used. We then select the orders of smoothness for the kernels in bulks and on surfaces. Finally, several numerical experiments are demonstrated to test the robustness of the coupled method in terms of accuracy and convergence rates under different settings.

The analysis of the convergence behavior of the multilevel methods is in the literature typically carried out under the assumption that the problem on the coarsest level is solved exactly. The aim of this thesis is to present a description of the multilevel methods which allows inexact solve on the coarsest level and to revisit selected results presented in literature using these weaker assumptions. In particular, we focus on the derivation of the uniform bound on the rate of convergence. Moreover, we discuss the possible dependence of the convergence behavior on the mesh size of the initial triangulation. 41

As part of our Spitzer Spirals, Bridges, and Tails project to help understand the effects of galaxy interactions on star formation, we analyze Galaxy Evolution Explorer UV, Southeastern Association for Research in Astronomy optical, and Spitzer IR images of the interacting galaxy pair Arp 82 (NGC 2535/6) and compare to a numerical simulation of the interaction. We investigate the multiwavelength properties of several individual star-forming complexes (clumps). Using optical and UV colors, EW(Hα), and population synthesis models we constrain the ages of the clumps and find that the median clump age is ∼ Myr. The clumps have masses ranging from a few × 106 to 109 M⊙;. In general, the clumps in the tidal features have ages similar to those in the spiral region, but are less massive. The clumps provide 33%, 36%, and 70% of the far-UV, 8.0 μm, and 24 μm emission, respectively. The 8 and 24 μm luminosities are used to estimate the far-IR luminosities and the star formation rates of the clumps. The total clump star formation rate is ∼2.0 ± 0.8 M⊙ yr -1, while the entire Arp 82 system is forming stars at a rate of ·4.9 ± 2.0 M· yr-1. We find, for the first time, stars in the H I arc to the southeast of the NGC 2535 disk. Population synthesis models indicate that all of the observed populations have young to intermediate ages. We conclude that, although the gas disks and some old stars may have formed early on, the progenitors may have been of late-type or low surface brightness, and the evolution of these galaxies seems to have halted until the recent encounter.

We study the polarization produced by scattering from dust in a bow shock-shaped region of enhanced density surrounding a stellar source, using the Monte Carlo radiative transfer code SLIP. Bow shocks are structures formed by the interaction of the winds of fast-moving stars with the interstellar medium. Our previous study focused on the polarization produced in these structures by electron scattering; we showed that polarization is highly dependent on inclination angle and that multiple scattering changes the shape and degree of polarization. In contrast to electron scattering, dust scattering is wavelength-dependent, which changes the polarization behaviour. Here, we explore different dust particle sizes and compositions and generate polarized spectral energy distributions for each case. We find that the polarization spectral energy distribution behaviour depends on the dust composition and grain size. Including dust emission leads to polarization changes with temperature at higher optical depth in ways that are sensitive to the orientation of the bow shock. In various scenarios and under certain assumptions, our simulations can constrain the optical depth and dust properties of resolved and unresolved bow shock-shaped scattering regions. Constraints on optical depth can provide estimates of local interstellar medium density for observed bow shocks. We also study the impact of dust grains filling the region between the star and bow shock. We see that as the density of dust between the star and bow shock increases, the resulting polarization is suppressed for all the optical depth regimes.

We use a 3D Monte Carlo radiative transfer code to study the polarimetric and photometric variability from stationary corotating interaction regions (CIR) in the wind of massive stars. Our CIRs are approximated by Archimedean spirals of higher (or lower) density formed in a spherical wind originating from the star and we also made allowance for a bright Gaussian spot at the base of the CIR. Comparing results from our code to previous analytical calculations in the optically thin case, we find differences which we attribute mainly to a better estimation of the total unpolarized flux reaching the observer. In the optically thick case, the differences with the analytical calculations are much larger, as multiple scattering introduces additional complexities including occultation effects. The addition of a Gaussian spot does not alter the shape of the polarization curve significantly but does create a small excess in polarization. On the other hand, the effect can be larger on the light curve and can become dominant over the resulting CIR, depending on the spot parameters and density of the wind.

No description available.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23705号 / 理博第4795号 / 新制||理||1686(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 嶺重 慎, 准教授 前田 啓一, 准教授 野上 大作 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Current methods for solving thennoelasticity problems involve using finite element analysis, boundary element analysis, or other meshed-type methods to determine the deflections under an imposed temperature/stress field. This thesis will detail a new approach using meshless methods to solve these types of thermoelasticity problems in which the solution is independent of boundary and internal meshing. With the rapidly increasing availability and performance of computer workstations and clusters, the major time requirement for solving a thermoelasticity model is no longer the computation time, but rather the problem setup. Defining the required mesh for a complex geometry can be extremely complicated and time consuming, and new methods are desired that can reduce this model setup time. The proposed meshless methods completely eliminate the need for a mesh, and thus, eliminate the need for complicated meshing procedures. Although the savings gain due to eliminating the meshing process would be more than sufficient to warrant further study, the localized meshless method can also be comparable in computational speed to more traditional finite element solvers when analyzing complex problems. The reduction of both setup and computational time makes the meshless approach an ideal method of solving coupled thermoelasticity problems. Through the development of these methods it can be determined whether they are feasible as potential replacements for more traditional solution methods. More specifically, two methods will be covered in depth from the development to the implementation. The first method covered will be the global meshless method and the second will be the improved localized method. Although they both produce similar results in terms of accuracy, the localized method greatly improves upon the stability and computation time of the global method.

Context. Since its rendezvous with comet 67P/Churyumov-Gerasimenko (67P), the Rosetta spacecraft has provided invaluable information contributing to our understanding of the cometary environment. On board, the VIRTIS and ROSINA instruments can both measure gas parameters in the rarefied cometary atmosphere, the so-called coma, and provide complementary results with remote sensing and in situ measurement techniques, respectively. The data from both ROSINA and VIRTIS instruments suggest that the source regions of H2O and CO2 are not uniformly distributed over the surface of the nucleus even after accounting for the changing solar illumination of the irregularly shaped rotating nucleus. The source regions of H2O and CO2 are also relatively different from one another. Aims. The use of a combination of a formal numerical data inversion method with a fully kinetic coma model is a way to correlate and interpret the information provided by these two instruments to fully understand the volatile environment and activity of comet 67P. Methods. In this work, the nonuniformity of the outgassing activity at the surface of the nucleus is described by spherical harmonics and constrained by ROSINA-DFMS data. This activity distribution is coupled with the local illumination to describe the inner boundary conditions of a 3D direct simulation Monte-Carlo (DSMC) approach using the Adaptive Mesh Particle Simulator (AMPS) code applied to the H2O and CO2 coma of comet 67P. Results. We obtain activity distribution of H2O and CO2 showing a dominant source of H2O in the Hapi region, while more CO2 is produced in the southern hemisphere. The resulting model outputs are analyzed and compared with VIRTIS-M/-H and ROSINADFMS measurements, showing much better agreement between model and data than a simpler model assuming a uniform surface activity. The evolution of the H2O and CO2 production rates with heliocentric distance are derived accurately from the coma model showing agreement between the observations from the different instruments and ground-based observations. Conclusions. We derive the activity distributions for H2O and CO2 at the surface of the nucleus described in spherical harmonics, which we couple to the local solar illumination to constitute the boundary conditions of our coma model. The model presented reproduces the coma observations made by the ROSINA and VIRTIS instruments on board the Rosetta spacecraft showing our understanding of the physics of 67P’s coma. This model can be used for further data analyses, such as dust modeling, in a future work.

We introduce a new suite of radiation- hydrodynamical simulations of galaxy formation and reionization called Aurora. The Aurora simulations make use of a spatially adaptive radiative transfer technique that lets us accurately capture the small- scale structure in the gas at the resolution of the hydrodynamics, in cosmological volumes. In addition to ionizing radiation, Aurora includes galactic winds driven by star formation and the enrichment of the universe with metals synthesized in the stars. Our reference simulation uses 2 x 512(3) dark matter and gas particles in a box of size 25 h(-1) comoving Mpc with a force softening scale of at most 0.28 h(-1) kpc. It is accompanied by simulations in larger and smaller boxes and at higher and lower resolution, employing up to 2 x 1024(3) particles, to investigate numerical convergence. All simulations are calibrated to yield simulated star formation rate functions in close agreement with observational constraints at redshift z = 7 and to achieve reionization at z approximate to 8.3, which is consistent with the observed optical depth to reionization. We focus on the design and calibration of the simulations and present some first results. The median stellar metallicities of low- mass galaxies at z = 6 are consistent with the metallicities of dwarf galaxies in the Local Group, which are believed to have formed most of their stars at high redshifts. After reionization, the mean photoionization rate decreases systematically with increasing resolution. This coincides with a systematic increase in the abundance of neutral hydrogen absorbers in the intergalactic medium.