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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

Numerical Solution of Moment Equations Using the Discontinuous-Galerkin Hancock Method

Miri, Seyedalireza 11 January 2019 (has links)
Moment methods from the kinetic theory of gases exist as an alternative to the Navier-Stokes model. Models in this family are described by first-order hyperbolic PDEs with local relaxation. They provide a natural treatment for non-equilibrium effects and expand the regime for which the model is physically applicable past the Navier-Stokes level (when the continuum assumption breaks down). Discontinuous-Galerkin (DG) methods are very well suited for distributed parallel solution of first-order PDEs. This is because the optimal locality of the method minimizes needed communication between computational processes. One highly efficient, coupled space-time DG method that achieves third-order accuracy in both space and time while using only linear elements is the discontinuous-Galerkin Hancock (DGH) scheme, which was specifically designed for the efficient solution of PDEs resulting from moment closures. Third-order accuracy is obtained through the use of a technique originally proposed by Hancock. The combination of moment methods with the DGH discretization leads to a very efficient numerical treatment for viscous compressible gas flows that is accurate both in and out of local thermodynamic equilibrium. This thesis describe the first-ever implementation of this scheme for the solution of moment equations on large-scale distributed-memory computers. This implementation uses solution-directed automatic mesh refinement to increase accuracy while reducing cost. A linear hyperbolic-relaxation equation is used to verify the order of accuracy of the scheme. Next a supersonic compressible Euler case is used to demonstrate the mesh refinement as well as the scheme’s ability to capture sharp discontinuities. Third, a moment-closure is then used to compute a viscous mixing layer. This serves to demonstrate the ability of the first-order PDEs and the DG scheme to efficiently compute viscous solutions. A moment-closure is used to compute the solution for Stokes flow past a circular cylinder. This case reinforces the hyperbolic PDEs’ ability to accurately predict viscous phenomena. As this case is very low speed, it also demonstrates the numerical technique’s ability to accurately solve problems that are ill-conditioned due to the extremely low Mach number. Finally, the parallel efficiency of the scheme is evaluated on Canada’s largest supercomputer. It may be surprising to some that viscous flow behaviour can be accurately predicted by first-order PDEs. However, the applicability of hyperbolic moment methods to both continuum and non-equilibrium gas flows is now well established. Such a first-order treatment brings many physical and computational advantages to gas flow prediction.
2

Application of the discontinuous Galerkin time domain method in the simulation of the optical properties of dielectric particles

Tang, Guanglin 2010 May 1900 (has links)
A Discontinuous Galerkin Time Domain method (DGTD), using a fourth order Runge-Kutta time-stepping of Maxwell's equations, was applied to the simulation of the optical properties of dielectric particles in two-dimensional (2-D) geometry. As examples of the numerical implementation of this method, the single-scattering properties of 2D circular and hexagonal particles are presented. In the case of circular particles, the scattering phase matrix was computed using the DGTD method and compared with the exact solution. For hexagonal particles, the DGTD method was used to compute single-scattering properties of randomly oriented 2-D hexagonal ice crystals, and results were compared with those calculated using a geometric optics method. Both shortwave (visible) and longwave (infrared) cases are considered, with particle size parameters 50 and 100. Ice in shortwave and longwave cases is absorptive and non-absorptive, respectively. The comparisons between DG solutions and the exact solutions in computing the optical properties of circular ice crystals reveal the applicability of the DG method to calculations of both absorptive and non-absorptive particles. In the hexagonal case scattering results are also presented as a function of both incident and scattering angles, revealing structure apparently not reported before. Using the geometric optics method we are able to interpret this structure in terms of contributions from varying numbers of internal reflections within the crystal.
3

Discontinuous Galerkin Multiscale Methods for Elliptic Problems

Elfverson, Daniel January 2010 (has links)
In this paper a continuous Galerkin multiscale method (CGMM) and a discontinuous Galerkin multiscale method (DGMM) are proposed, both based on the variational multiscale method for solving partial differential equations numerically. The solution is decoupled into a coarse and a fine scale contribution, where the fine-scale contribution is computed on patches with localized right hand side. Numerical experiments are presented where exponential decay of the error is observed when increasing the size of the patches for both CGMM and DGMM. DGMM gives much better accuracy when the same size of the patches are used.
4

The Discontinuous Galerkin Method Applied to Problems in Electromagnetism

Connor, Dale January 2012 (has links)
The discontinuous Galerkin method (DGM) is applied to a number of problems in computational electromagnetics. This is achieved by obtaining numerical solutions to Maxwell's equations using the DGM. The aim of these simulations is to highlight the strengths of the method while showing its resilience in handling problems other schemes may not be able to accurately model. Although no method will ever be the best choice for every problem in electromagnetics, the discontinuous Galerkin method is able to accurately approximate any problem, although the computational costs can make the scheme impractical for some. Like other time domain schemes, the DGM becomes inefficient on large domains where the solution contains small wavelengths. We demonstrate that all of the different types of boundary conditions in electromagnetic wave propagation can be implemented into the DGM. Reflection and transmission boundaries fit easily into the framework, whereas perfect absorption requires a more advanced technique known as the perfectly matched layer. We begin by simulating mirrors with several different geometries, and analyze how the DGM method performs, and how it offers a more complete evaluation of the behavior in this problem than some other methods. Since Maxwell's equations describe the macroscopic features of electromagnetics, our simulations are able to capture the wave features of electromagnetics, such as interference and diffraction. We demonstrate this by accurately modelling Young's double slit experiment, a classic experiment which features well understood interference and diffraction phenomena. We also extend the basic electromagnetic wave propagation simulations to include situations where the waves travel into new media. The formulation of the DGM for Maxwell's equations allows the numerical solutions to accurately resolve the features at the interface of two media as predicted by the Fresnel coefficients. This allows the DGM to model lenses and other sources of refraction. We predict that the DGM will become an increasingly valuable method for computational electromagnetics because of its wide range of applicability as well as the lack of undesirable features in the numerical solutions. Furthermore, the only limiting factor for applying DGM, its computational cost, will become less influential as computing power continues to increase, allowing us to apply the DGM to an increasing set of applications.
5

The Discontinuous Galerkin Method Applied to Problems in Electromagnetism

Connor, Dale January 2012 (has links)
The discontinuous Galerkin method (DGM) is applied to a number of problems in computational electromagnetics. This is achieved by obtaining numerical solutions to Maxwell's equations using the DGM. The aim of these simulations is to highlight the strengths of the method while showing its resilience in handling problems other schemes may not be able to accurately model. Although no method will ever be the best choice for every problem in electromagnetics, the discontinuous Galerkin method is able to accurately approximate any problem, although the computational costs can make the scheme impractical for some. Like other time domain schemes, the DGM becomes inefficient on large domains where the solution contains small wavelengths. We demonstrate that all of the different types of boundary conditions in electromagnetic wave propagation can be implemented into the DGM. Reflection and transmission boundaries fit easily into the framework, whereas perfect absorption requires a more advanced technique known as the perfectly matched layer. We begin by simulating mirrors with several different geometries, and analyze how the DGM method performs, and how it offers a more complete evaluation of the behavior in this problem than some other methods. Since Maxwell's equations describe the macroscopic features of electromagnetics, our simulations are able to capture the wave features of electromagnetics, such as interference and diffraction. We demonstrate this by accurately modelling Young's double slit experiment, a classic experiment which features well understood interference and diffraction phenomena. We also extend the basic electromagnetic wave propagation simulations to include situations where the waves travel into new media. The formulation of the DGM for Maxwell's equations allows the numerical solutions to accurately resolve the features at the interface of two media as predicted by the Fresnel coefficients. This allows the DGM to model lenses and other sources of refraction. We predict that the DGM will become an increasingly valuable method for computational electromagnetics because of its wide range of applicability as well as the lack of undesirable features in the numerical solutions. Furthermore, the only limiting factor for applying DGM, its computational cost, will become less influential as computing power continues to increase, allowing us to apply the DGM to an increasing set of applications.
6

Shock Capturing with Discontinuous Galerkin Method

Nguyen, Vinh Tan, Khoo, Boo Cheong, Peraire, Jaime, Persson, Per-Olof 01 1900 (has links)
Shock capturing has been a challenge for computational fluid dynamicists over the years. This article deals with discontinuous Galerkin method to solve the hyperbolic equations in which solutions may develop discontinuities in finite time. The high order discontinuous Galerkin method combining the basis of finite volume and finite element methods has shown a lot of attractive features for a wide range of applications. Various techniques proposed in the literature to deal with discontinuities basically reduce the order of interpolation in the region around these discontinuities. The accuracy of the scheme therefore may be degraded in the vicinity of the shock. The proposed method resolves the discontinuities presented in the solution by applying viscosity into the shock-containing elements. The discontinuity is spread over a distance and is well approximated in the space of interpolation functions. The technique of adding viscosity to the system and the indicator based on the expansion coefficients of the solution are presented. A number of numerical examples in one and two dimensions is carried out to show the capability of the scheme for shock capturing. / Singapore-MIT Alliance (SMA)
7

Discontinuous Galerkin Method for Hyperbolic Conservation Laws

Mousikou, Ioanna 11 November 2016 (has links)
Hyperbolic conservation laws form a special class of partial differential equations. They describe phenomena that involve conserved quantities and their solutions show discontinuities which reflect the formation of shock waves. We consider one-dimensional systems of hyperbolic conservation laws and produce approximations using finite difference, finite volume and finite element methods. Due to stability issues of classical finite element methods for hyperbolic conservation laws, we study the discontinuous Galerkin method, which was recently introduced. The method involves completely discontinuous basis functions across each element and it can be considered as a combination of finite volume and finite element methods. We illustrate the implementation of discontinuous Galerkin method using Legendre polynomials, in case of scalar equations and in case of quasi-linear systems, and we review important theoretical results about stability and convergence of the method. The applications of finite volume and discontinuous Galerkin methods to linear and non-linear scalar equations, as well as to the system of elastodynamics, are exhibited.
8

New Transport Capabilities and Timesteppers for a Discontinuous Galerkin Wave Model

Sebian, Rachel A. 19 September 2016 (has links)
No description available.
9

Discontinuous Galerkin methods for resolving non linear and dispersive near shore waves

Panda, Nishant 23 October 2014 (has links)
Near shore hydrodynamics has been an important research area dealing with coastal processes. The nearshore coastal region is the region between the shoreline and a fictive offshore limit which usually is defined as the limit where the depth becomes so large that it no longer influences the waves. This spatially limited but highly energetic zone is where water waves shoal, break and transmit energy to the shoreline and are governed by highly dispersive and non-linear effects. An accurate understanding of this phenomena is extremely useful, especially in emergency situations during hurricanes and storms. While the shallow water assumption is valid in regions where the characteristic wavelength exceeds a typical depth by orders of magnitude, Boussinesq-type equations have been used to model near-shore wave motion. Unfortunately these equations are complex system of coupled non-linear and dispersive differential equations that have made the developement of numerical approximations extremely challenging. In this dissertation, a local discontinuous Galerkin method for Boussinesq-Green Naghdi Equations is presented and validated against experimental results. Currently Green-Naghdi equations have many variants. We develop a numerical method in one horizontal dimension for the Green-Naghdi equations based on rotational characteristics in the velocity field. Stability criterion is also established for the linearized Green-Naghdi equations and a careful proof of linear stability of the numerical method is carried out. Verification is done against a linearized standing wave problem in flat bathymetry and h,p (denoted by K in this thesis) error rates are plotted. The numerical method is validated with experimental data from dispersive and non-linear test cases. / text
10

Compatible Subdomain Level Isotropic/Anisotropic Discontinuous Galerkin Time Domain (DGTD) Method for Multiscale Simulation

Ren, Qiang January 2015 (has links)
<p>Domain decomposition method provides a solution for the very large electromagnetic</p><p>system which are impossible for single domain methods. Discontinuous Galerkin</p><p>(DG) method can be viewed as an extreme version of the domain decomposition,</p><p>i.e., each element is regarded as one subdomain. The whole system is solved element</p><p>by element, thus the inversion of the large global system matrix is no longer necessary,</p><p>and much larger system can be solved with the DG method compared to the</p><p>continuous Galerkin (CG) method.</p><p>In this work, the DG method is implemented on a subdomain level, that is, each subdomain contains multiple elements. The numerical flux only applies on the</p><p>interfaces between adjacent subdomains. The subodmain level DG method divides</p><p>the original large global system into a few smaller ones, which are easier to solve,</p><p>and it also provides the possibility of parallelization. Compared to the conventional</p><p>element level DG method, the subdomain level DG has the advantage of less total</p><p>DoFs and fexibility in interface choice. In addition, the implicit time stepping is </p><p>relatively much easier for the subdomain level DG, and the total CPU time can be</p><p>much less for the electrically small or multiscale problems.</p><p>The hybrid of elements are employed to reduce the total DoF of the system.</p><p>Low-order tetrahedrons are used to catch the geometry ne parts and high-order</p><p>hexahedrons are used to discretize the homogeneous and/or geometry coarse parts.</p><p>In addition, the non-conformal mesh not only allow dierent kinds of elements but</p><p>also sharp change of the element size, therefore the DoF can be further decreased.</p><p>The DGTD method in this research is based on the EB scheme to replace the</p><p>previous EH scheme. Dierent from the requirement of mixed order basis functions</p><p>for the led variables E and H in the EH scheme, the EB scheme can suppress the</p><p>spurious modes with same order of basis functions for E and B. One order lower in</p><p>the basis functions in B brings great benets because the DoFs can be signicantly</p><p>reduced, especially for the tetrahedrons parts.</p><p>With the basis functions for both E and B, the EB scheme upwind </p><p>ux and</p><p>EB scheme Maxwellian PML, the eigen-analysis and numerical results shows the</p><p>eectiveness of the proposed DGTD method, and multiscale problems are solved</p><p>eciently combined with the implicit-explicit hybrid time stepping scheme and multiple</p><p>kinds of elements.</p><p>The EB scheme DGTD method is further developed to allow arbitrary anisotropic</p><p>media via new anisotropic EB scheme upwind </p><p>ux and anisotropic EB scheme</p><p>Maxwellian PML. The anisotropic M-PML is long time stable and absorb the outgoing</p><p>wave eectively. A new TF/SF boundary condition is brought forward to</p><p>simulate the half space case. The negative refraction in YVO4 bicrystal is simulated</p><p>with the anisotropic DGTD and half space TF/SF condition for the rst time with</p><p>numerical methods.</p> / Dissertation

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