Conformal map has application in a lot of areas of science, e.g., fluid flow, heat conduction, solidification, electromagnetic, etc. Especially conformal map applied to elasticity theory can provide most simple and useful solution. But finding of conformal map for custom domain is not trivial problem. We used a numerical method for building a conformal map to solve torsion problem. In addition it was considered an infinite system method to solve the same problem. Results are compared.

This thesis concentrates on the algorithmic aspects of Method of Moments (MoM) and Locally Corrected Nyström (LCN) numerical methods in electromagnetics. The data dependency in each step of the algorithm is analyzed to implement a parallel version that can harness the powerful processing power of a General Purpose Graphics Processing Unit (GPGPU). The GPGPU programming model provided by NVIDIA's Compute Unified Device Architecture (CUDA) is described to learn the software tools at hand enabling us to implement C code on the GPGPU. Various optimizations such as the partial update at every iteration, inter-block synchronization and using shared memory enable us to achieve an overall speedup of approximately 10. The study also brings out the strengths and weaknesses in implementing different methods such as Crout's LU decomposition and triangular matrix inversion on a GPGPU architecture. The results suggest future directions of study in different algorithms and their effectiveness on a parallel processor environment. The performance data collected show how different features of the GPGPU architecture can be enhanced to yield higher speedup.

<p>A mathematical model for a consolidation process of a highlyconcentrated, flocculated suspension is developed.Thesuspension is treated as a mixture of a fluid and solidparticles by an Eulerian two-phase fluid model.W e characterizethe suspension by constitutive relations correlating thestresses, interaction forces, and inter-particle forces toconcentration and velocity gradients.This results in threeempirically determined material functions: a hystereticpermeability, a non-Newtonian viscosity and a non-reversibleparticle interaction pressure.P arameters in the models arefitted to experimental data.</p><p>A simulation program using finite difference methods both intime and space is applied to one and two dimensional testcases.Numer ical experiments are performed to study the effectof different viscosity and permeability models. The effect ofshear on consolidation rate is studied and it is significantwhen the permeability hysteresis model is employed.</p>

Due to its adaptable nature in a broad range of problem domains, Smoothed Particle Hydrodynamics (SPH) is a popular numerical technique for computing solutions in astrophysics. This dissertation discusses the SPH technique and assesses its capabilities for reproducing steady-state spherically-symmetric accretion flow. The accretion scenario is of great interest for its applicability in a diverse array of astrophysical phenomena and, under certain assumptions, it also provides an accepted analytical solution against which the numerical method can be validated. After deriving the necessary equations from astrophysical fluid dynamics, giving a detailed review of solving the steady-state spherical accretion problem, and developing the SPH methodology, this work suggests solutions to the issues that must be overcome in order to successfully employ the SPH methodology to reproduce steady-state spherical accretion flow. Several techniques for setting initial data are addressed, resolution requirements are illustrated, inner and outer boundary conditions are discussed, and artificial dissipation parameters and methodologies are explored. / text

This thesis is a computational study of circumstellar gas disks, with a special focus on modeling techniques and on numerical methods not only as scientific tools but also as a target of study. In particular, in-depth discussions are included on the main numerical strategy used, namely the moving-mesh method for astrophysical hydrodynamics. In this work, the moving-mesh approach is used to simulate circumstellar disks for the first time. / Astronomy

My dissertation focuses on a relatively new field of study: the region immediately around galaxies known as the circumgalactic medium (CGM). The CGM holds vast quantities of mass and metals, yet its connection to galaxies is not well understood. My work uses cosmological hydrodynamic simulations and comparisons to data from Hubble's Cosmic Origins Spectrograph (COS) to understand the CGM's connection to galaxy evolution, gas accretion, outflows, star formation, and baryon cycling. This includes studies of the CGM's extent and physical conditions; the cause and nature of outflows; gas dynamics, including the first comprehensive study of tracers of inflowing and outflowing gas at low redshift (z=0.25); and direct comparison of theoretical results to observational data. Chapter 1 introduces my research and show its connection to galaxy evolution. Chapter 2 investigates hydrogen and metal line absorption around low-redshift galaxies in cosmological hydrodynamic simulations. This chapter studies different models for stellar outflows, physical conditions, and dependencies on halo mass. Chapter 3 examines the flow of gas into, out of, and around galaxies using a novel particle tracking technique. This chapter examines the baryon cycle in detail for our preferred model of stellar outflows. Chapter 4 compares our model results, including two separate prescriptions for outflows, with data from COS. We contrast these wind models, showing how they cycle baryons differently, and show degeneracies in observational diagnostics. In Chapter 5, I summarize and discuss plans for future research in this field, and how it can be more fully leveraged to understand galaxy evolution.

An initial-value problem consists of an ordinary differential equation subject to an initial condition. The right-hand side of the differential equation can be interpreted as additively split when it is comprised of the sum of two or more contributing factors. For instance, the right-hand sides of initial-value problems derived from advection-diffusion-reaction equations are comprised of the sum of terms emanating from three distinct physical processes: advection, diffusion, and reaction. In some cases, solutions to initial-value problems can be calculated analytically, but when an analytic solution is unknown or nonexistent, methods of numerical integration are used to calculate solutions. The runtime performance of numerical methods is problem dependent; therefore, one must choose an appropriate numerical method to achieve favourable performance, according to characteristics of the problem. Additive methods of numerical integration apply distinct methods to the distinct contributing factors of an additively split problem. Treating the contributing factors with methods that are known to perform well on them individually has the potential to yield an additive method that outperforms single methods applied to the entire (unsplit) problem. Splittings of the right-hand side can be physics-based, i.e., based on physical characteristics of the problem, such as advection, diffusion, or reaction terms. Splittings can also be based on linearization, called Jacobian splitting in this thesis, where the linearized part of the problem is treated with one method and the rest of the problem is treated with another. A comparison of these splitting techniques is performed by applying a set of additive methods to a test suite of problems. Many common non-additive methods are also included to serve as a performance baseline. To perform this numerical study, a problem-solving environment was developed to evaluate permutations of problems, methods, and their associated parameters. The test suite is comprised of several distinct advection-diffusion-reaction equations that have been chosen to represent a wide range of common problem characteristics. When solving split problems in the test suite, it is found that additive Runge–Kutta methods of orders three, four, and five using Jacobian splitting generally outperform those same methods using physics-based splitting. These results provide evidence that Jacobian splitting is an effective approach when solving such initial-value problems in practice.

Dauguma praktinių mokslinių, įvairių taikymų uždavinių yra spendžiami skaitiniais metodais. Šie metodai aktualūs ir mokymo aspektu. Įvairiose aukštosiose mokyklose studentai, studijuodami skaitinių metodų teoriją, sprendžia praktines užduotis, taigi tenka arba patiems programuodami algoritmus, arba naudoti kompiuterines programas. Šiame darbe nagrinėjami universalių kompiuterinių matematinių sistemų – MATLAB IR MATHCAD, kaip vieno svarbiausių skaitiniais metodais sprendžiamų uždavinių priemonės. Taip pat nagrinėjamas Excel programos panaudojimas netiesinių lygčių sprendimui skaitiniais metodais. Darbe išnagrinėti netiesinių lygčių sprendimo metodai, tokie kaip paprastųj����� iteracijų, pusiaukirtos, Niutono ir kirstinių metodai, jų realizacija kiekvienoje iš nurodytų programų. Pateikiant kiekvieną iš šių temų yra trumpai aprašyti patys metodai, lyginamos kiekvienos kompiuterinės programos galimybės spręsti uždavinius pagal šiuos metodus, jų funkcijų aprašymas, taikymai ir sprendimo pavyzdžiai. Darbe pristatyta sukurta kompiuterinė mokymo priemonė, skirta netiesinių lygčių sprendimo skaitiniais metodais įsisavinimui. / The most practical, scientific and various applied tasks are solved by using numerical methods. These methods are urgent in teaching aspect, too. The students of various higher schools are solving practical tasks when studying theory of numerical methods, therefore they have to programme algorithms by themselves or use computer programs. In this work there are analyzed universal computer algebra systems – Matlab and Mathcad – as the one of the most important measures of solving tasks by using numerical methods. Also there is analyzed the use of Excel program for the reason how to solve the nonlinear equation by using numerical methods. In the work there are analyzed methods of nonlinear equations solving, such as simple iterative, bisection, Newton and secant methods and their realization in each referred program.. There are described the very methods when presenting every those themes and comparised the possibilities of each computer program for the reason how to solve the tasks by using these methods. There is given description of methods functions, application and examples of solving. In the work there is presented created computer training appliance for the reason of mastering with nonlinear equation solving by using numerical methods.

In the dissertation mathematical modelling problems in the design of electrical cables and cable fibres in modern vehicles, and of the heating of metals or semiconductors by ultra short (pico- or femtosecond) laser pulses are investigated. The problems are described by systems of differential equations and are solved by applying numerical methods. The methodology of problems being solved includes the following mathematical modelling steps: description of formulated problems using mathematical models, selection of model parameters, development and analysis of numerical algorithms (analysis of approximation errors, solution stability, convergence and accuracy), implementation of algorithms, application of parallel algorithms, comparison of mathematical experiments with results obtained in real experiments. The following main objectives are formulated for this thesis: to create mathematical models of the heat exchange in cable fibres and numerical algorithms, which will enable the virtual simulation of temperature distribution in electrical cables and optimization of geometric parameters of cables; to create mathematical models of laser impact on metals and their numerical algorithms allowing the modelling of material heating and removal processes. The thesis consists of the following main sections: introduction, three chapters, conclusion chapter, bibliography chapter, a list of the author's publications on the dissertation topic. The introductory section of the thesis... [to full text] / Disertacijoje nagrinėjami elektros kabelių ir kabelių pluoštų projektavimo šiuolaikiniuose automobiliuose, metalų arba puslaidininkių kaitinimo ultratrumpais (piko- arba femtosekundiniais) lazerio impulsais matematinio modeliavimo uždaviniai. Nagrinėjami uždaviniai aprašomi diferencialinėmis lygtimis ir sprendžiami skaitiniais metodais. Nagrinėjamų uždavinių metodiką apima šie matematinio modeliavimo etapai: suformuluotų uždavinių aprašymas matematiniais modeliais, modelių parametrų parinkimas, skaitinių algoritmų sudarymas ir tyrimas (aproksimacijos paklaidų, sprendinio stabilumo, konvergavimo ir tikslumo analizė), algoritmų realizavimas, lygiagrečiųjų algoritmų taikymas, skaičiavimo eksperimentų rezultatų palyginimas su realaus eksperimento rezultatais. Disertacijoje suformuluoti šie pagrindiniai darbo tikslai: sukurti šilumos mainų kabelių pluoštuose matematinius modelius ir skaitinės analizės algoritmus, kurie virtualiojo eksperimento būdu leistų modeliuoti temperatūros pasiskirstymą elektros kabeliuose ir optimizuoti geometrinius laidų parametrus; sukurti lazerio poveikio metalui matematinius modelius ir jų skaitinės analizės algoritmus, leidžiančius modeliuoti medžiagos kaitinimo ir pašalinimo procesus. Disertaciją sudaro įvadas, trys skyriai, rezultatų apibendrinimas, naudotos literatūros ir autorės publikacijų disertacijos tema sąrašas. Įvadiniame skyriuje aptariama tiriamoji problema, darbo aktualumas, aprašomas tyrimų objektas, formuluojami darbo tikslai ir... [toliau žr. visą tekstą]

For a uniquely defined subset of phase space, solutions of non-linear, coupled reaction-diffusion equations may converge to heterogeneous steady states, organic in appearance. Hence, many theoretical models for pattern formation, as in the theory of morphogenesis, include the mechanics of reaction-diffusion equations. The standard method of simulation for such pattern formation models does not facilitate reproducibility of results, or the verification of convergence to a solution of the problem via the method of mesh refinement. In this thesis we explore a new methodology circumventing the aforementioned issues, which is independent of the choice of programming language. While the new method allows more control over solutions, the user is required to make more choices, which may or may not have a determining effect on the nature of resulting patterns. In an attempt to quantify the extent of the possible effects, we study heterogeneous steady states for two well known reaction-diffusion models, the Gierer-Meinhardt model and the Schnakenberg model. / Alexander Graham Bell Canada Graduate Scholarship provides financial support to high calibre scholars who are engaged in master's or doctoral programs in the natural sciences or engineering. / Natural Sciences and Engineering Research Council of Canada