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Johansson, August. "Duality-based adaptive finite element methods with application to time-dependent problems." Doctoral thesis, Umeå : Institutionen för matematik och matematisk statistik, Umeå universitet, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-33872.
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Marcati, Carlo. "Discontinuous hp finite element methods for elliptic eigenvalue problems with singular potentials : with applications to quantum chemistry." Thesis, Sorbonne université, 2018. http://www.theses.fr/2018SORUS349.
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Dans cette thèse, on étudie des problèmes aux valeurs propres elliptiques avec des potentiels singuliers, motivés par plusieurs modèles en physique et en chimie quantique, et on propose une méthode des éléments finis de type hp discontinus (dG) adaptée pour l’approximation des modes propres. Dans ces modèles, arrivent naturellement des potentiels singuliers (associés à l’interaction entre noyaux et électrons). Notre analyse commence par une étude de la régularité elliptique dans des espaces de Sobolev à poids. On montre comment un opérateur elliptique avec potentiel singulier est un isomorphisme entre espaces de Sobolev à poids non homogènes et que l’on peut développer des bornes de type analytique à poids sur les solutions des problèmes aux valeurs propres associés aux opérateurs. La méthode hp/dG graduée qu’on utilise converge ainsi de façon exponentielle. On poursuit en considérant une classe de problèmes non linéaires représentatifs des applications. On montre que, sous certaines conditions, la méthode hp/dG graduée converge et que, si la non linéarité est de type polynomiale, on obtient les mêmes estimations de type analytique que dans le cas linéaire. De plus, on étudie la convergence de la valeur propre pour voir sous quelles conditions la vitesse de convergence est améliorée par rapport à celle des vecteurs propres. Pour tous les cas considérés, on effectue des tests numériques, qui ont pour objectif à la fois de valider les résultats théoriques, mais aussi d’évaluer le rôle des sources d’erreur non considérées dans l’analyse et d’aider dans la conception de méthode hp/dG graduée pour des problèmes plus complexesIn this thesis, we study elliptic eigenvalue problems with singular potentials, motivated by several models in physics and quantum chemistry, and we propose a discontinuous Galerkin hp finite element method for their solution. In these models, singular potentials occur naturally (associated with the interaction between nuclei and electrons). Our analysis starts from elliptic regularity in non homogeneous weighted Sobolev spaces. We show that elliptic operators with singular potential are isomorphisms in those spaces and that we can derive weighted analytic type estimates on the solutions to the linear eigenvalue problems. The isotropically graded hp method provides therefore approximations that converge with exponential rate to the solution of those eigenproblems. We then consider a wide class of nonlinear eigenvalue problems, and prove the convergence of numerical solutions obtained with the symmetric interior penalty discontinuous Galerkin method. Furthermore, when the non linearity is polynomial, we show that we can obtain the same analytic type estimates as in the linear case, thus the numerical approximation converges exponentially. We also analyze under what conditions the eigenvalue converges at an increased rate compared to the eigenfunctions. For both the linear and nonlinear case, we perform numerical tests whose objective is both to validate the theoretical results, but also evaluate the role of sources of errors not considered previously in the analysis, and to help in the design of hp/dG graded methods for more complex problems
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Gürkan, Ceren. "Extended hybridizable discontinuous Galerkin method." Doctoral thesis, Universitat Politècnica de Catalunya, 2018. http://hdl.handle.net/10803/664035.
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This thesis proposes a new numerical technique: the eXtended Hybridizable Discontinuous Galerkin (X-HDG) Method, to efficiently solve problems including moving boundaries and interfaces. It aims to outperform available methods and improve the results by inheriting favored properties of Discontinuous Galerkin (HDG) together with an explicit interface definition. X-HDG combines the Hybridizable HDG method with an eXtended Finite Element (X-FEM) philosophy, with a level set description of the interface, to form an hp convergent, high order unfitted numerical method. HDG outperforms other Discontinuous Galerkin (DG) methods for problems involving self-adjoint operators, due to its hybridization and superconvergence properties. The hybridization process drastically reduces the number of degrees of freedom in the discrete problem, similarly to static condensation in the context of high-order Continuous Galerkin (CG). On other hand, HDG is based on a mixed formulation that, differently to CG or other DG methods, is stable even when all variables (primal unknowns and derivatives) are approximated with polynomials of the same degree k. As a result, convergence of order k+1 in the L2 norm is proved not only for the primal unknown, but also for its derivatives. Therefore, a simple element-by-element postprocess of the derivatives leads to a superconvergent approximation of the primal variables, with convergence of order k+2 in the L2 norm. X-HDG inherits these favored properties of HDG in front of CG and DG methods; moreover, thanks to the level set description of interfaces, costly remeshing is avoided when dealing with moving interfaces. This work demonstrates that X-HDG keeps the optimal and superconvergence of HDG with no need of mesh fitting to the interface.
In Chapters 2 and 3, the X-HDG method is derived and implemented to solve the steady-state Laplace equation on a domain where the interface separates a single material from the void and where the interface separates two different materials. The accuracy and the convergence of X-HDG is tested over examples with manufactured solutions and it is shown that X-HDG outperforms the previous proposals by demonstrating high order optimum and super convergence, together with reduced system size thanks to its hybrid nature, without mesh fitting.
In Chapters 4 and 5, the X-HDG method is derived and implemented to solve Stokes interface problem for void and bimaterial interfaces. With X-HDG, high order convergence is demonstrated over unfitted meshes for incompressible flow problems.
X-HDG for moving interfaces is studied in Chapter 6. A transient Laplace problem is considered, where the time dependent term is discretized using the backward Euler method. A collapsing circle example together with two-phase Stefan problem are analyzed in numerical examples section. It is demonstrated that X-HDG offers high-order optimal convergence for time-dependent problems. Moreover, with Stefan problem, using a polynomial degree k, a more accurate approximation of interface position is demonstrated against X-FEM, thanks to k+1 convergent gradient approximation of X-HDG. Yet again, results obtained by previous proposals are improved.Esta tesis propone una nueva técnica numérica: eXtended Hybridizable Discontinuous Galerkin (X-HDG), para resolver eficazmente problemas incluyendo fronteras en movimiento e interfaces. Su objetivo es superar las limitaciones de los métodos disponibles y mejorar los resultados, heredando propiedades del método Hybridizable Discontinuous Galerkin method (HDG), junto con una definición de interfaz explícita. X-HDG combina el método HDG con la filosofía de eXtended Finite Element method (X-FEM), con una descripción level-set de la interfaz, para obtener un método numérico hp convergente de orden superior sin ajuste de la malla a la interfaz o frontera. HDG supera a otros métodos de DG para los problemas implícitos con operadores autoadjuntos, debido a sus propiedades de hibridación y superconvergencia. El proceso de hibridación reduce drásticamente el número de grados de libertad en el problema discreto, similar a la condensación estática en el contexto de Continuous Galerkin (CG) de alto orden. Por otro lado, HDG se basa en una formulación mixta que, a diferencia de CG u otros métodos DG, es estable incluso cuando todas las variables (incógnitas primitivas y derivadas) se aproximan con polinomios del mismo grado k. Como resultado, la convergencia de orden k + 1 en la norma L2 se demuestra no sólo para la incógnita primal sino también para sus derivadas. Por lo tanto, un simple post-proceso elemento-a-elemento de las derivadas conduce a una aproximación superconvergente de las variables primales, con convergencia de orden k+2 en la norma L2. X-HDG hereda estas propiedades. Por otro lado, gracias a la descripción level-set de la interfaz, se evita caro remallado tratando las interfaces móviles. Este trabajo demuestra que X-HDG mantiene la convergencia óptima y la superconvergencia de HDG sin la necesidad de ajustar la malla a la interfaz. En los capítulos 2 y 3, se deduce e implementa el método X-HDG para resolver la ecuación de Laplace estacionaria en un dominio donde la interfaz separa un solo material del vacío y donde la interfaz separa dos materiales diferentes. La precisión y convergencia de X-HDG se prueba con ejemplos de soluciones fabricadas y se demuestra que X-HDG supera las propuestas anteriores mostrando convergencia óptima y superconvergencia de alto orden, junto con una reducción del tamaño del sistema gracias a su naturaleza híbrida, pero sin ajuste de la malla. En los capítulos 4 y 5, el método X-HDG se desarrolla e implementa para resolver el problema de interfaz de Stokes para interfaces vacías y bimateriales. Con X-HDG, de nuevo se muestra una convergencia de alto orden en mallas no adaptadas, para problemas de flujo incompresible. X-HDG para interfaces móviles se discute en el Capítulo 6. Se considera un problema térmico transitorio, donde el término dependiente del tiempo es discretizado usando el método de backward Euler. Un ejemplo de una interfaz circulas que se reduce, junto con el problema de Stefan de dos fases, se discute en la sección de ejemplos numéricos. Se demuestra que X-HDG ofrece un alto grado de convergencia óptima para problemas dependientes del tiempo. Además, con el problema de Stefan, usando un grado polinomial k, se demuestra una aproximación más exacta de la posición de la interfaz contra X-FEM, gracias a la aproximación del gradiente convergente k + 1 de X-HDG. Una vez más, se mejoran los resultados obtenidos por las propuestas anteriores
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Kaufmann, Willem. "Extended Hydrodynamics Using the Discontinuous-Galerkin Hancock Method." Thesis, Université d'Ottawa / University of Ottawa, 2021. http://hdl.handle.net/10393/42672.
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Moment methods derived from the kinetic theory of gases can be used for the prediction of continuum and non-equilibrium flows and offer numerical advantages over other methods, such as the Navier-Stokes model. Models developed in this fashion are described by first-order hyperbolic partial differential equations (PDEs) with stiff local relaxation source terms.
The application of discontinuous-Galerkin (DG) methods for the solution of such models has many benefits. Of particular interest is the third-order accurate, coupled space-time discontinuous-Galerkin Hancock (DGH) method. This scheme is accurate, as well as highly efficient on large-scale distributed-memory computers.
The current study outlines a general implementation of the DGH method used for the parallel solution of moment methods in one, two, and three dimensions on modern distributed clusters. An algorithm for adaptive mesh refinement (AMR) was developed alongside the implementation of the scheme, and is used to achieve even higher accuracy and efficiency.
Many different first-order hyperbolic and hyperbolic-relaxation PDEs are solved to demonstrate the robustness of the scheme. First, a linear convection-relaxation equation is solved to verify the order of accuracy of the scheme in three dimensions. Next, some classical compressible Euler problems are solved in one, two, and three dimensions to demonstrate the scheme's ability to capture discontinuities and strong shocks, as well as the efficacy of the implemented AMR. A special case, Ringleb's flow, is also solved in two-dimensions to verify the order of accuracy of the scheme for non-linear PDEs on curved meshes. Following this, the shallow water equations are solved in two dimensions. Afterwards, the ten-moment (Gaussian) closure is applied to two-dimensional Stokes flow past a cylinder, showing the abilities of both the closure and scheme to accurately compute classical viscous solutions. Finally, the one-dimensional fourteen-moment closure is solved.
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Toprakseven, Suayip. "Error Analysis of Extended Discontinuous Galerkin (XdG) Method." University of Cincinnati / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1418733307.
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Elfverson, Daniel. "Discontinuous Galerkin Multiscale Methods for Elliptic Problems." Thesis, Uppsala universitet, Institutionen för informationsteknologi, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-138960.
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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.
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Galbraith, Marshall C. "A Discontinuous Galerkin Chimera Overset Solver." University of Cincinnati / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1384427339.
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Lui, Ho Man. "Runge-Kutta Discontinuous Galerkin method for the Boltzmann equation." Thesis, Massachusetts Institute of Technology, 2006. http://hdl.handle.net/1721.1/39215.
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Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2006.Includes bibliographical references (p. 85-87).
In this thesis we investigate the ability of the Runge-Kutta Discontinuous Galerkin (RKDG) method to provide accurate and efficient solutions of the Boltzmann equation. Solutions of the Boltzmann equation are desirable in connection to small scale science and technology because when characteristic flow length scales become of the order of, or smaller than, the molecular mean free path, the Navier-Stokes description fails. The prevalent Boltzmann solution method is a stochastic particle simulation scheme known as Direct Simulation Monte Carlo (DSMC). Unfortunately, DSMC is not very effective in low speed flows (typical of small scale devices of interest) because of the high statistical uncertainty associated with the statistical sampling of macroscopic quantities employed by this method. This work complements the recent development of an efficient low noise method for calculating the collision integral of the Boltzmann equation, by providing a high-order discretization method for the advection operator balancing the collision integral in the Boltzmann equation. One of the most attractive features of the RKDG method is its ability to combine high-order accuracy, both in physical space and time, with the ability to capture discontinuous solutions.
(cont.) The validity of this claim is thoroughly investigated in this thesis. It is shown that, for a model collisionless Boltzmann equation, high-order accuracy can be achieved for continuous solutions; whereas for discontinuous solutions, the RKDG method, with or without the application of a slope limiter such as a viscosity limiter, displays high-order accuracy away from the vicinity of the discontinuity. Given these results, we developed a RKDG solution method for the Boltzmann equation by formulating the collision integral as a source term in the advection equation. Solutions of the Boltzmann equation, in the form of mean velocity and shear stress, are obtained for a number of characteristic flow length scales and compared to DSMC solutions. With a small number of elements and a low order of approximation in physical space, the RKDG method achieves similar results to the DSMC method. When the characteristic flow length scale is small compared to the mean free path (i.e. when the effect of collisions is small), oscillations are present in the mean velocity and shear stress profiles when a coarse velocity space discretization is used. With a finer velocity space discretization, the oscillations are reduced, but the method becomes approximately five times more computationally expensive.
(cont.) We show that these oscillations (due to the presence of propagating discontinuities in the distribution function) can be removed using a viscosity limiter at significantly smaller computational cost.
by Ho Man Lui.
S.M.
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Bala, Chandran Ram. "Development of discontinuous Galerkin method for nonlocal linear elasticity." Thesis, Massachusetts Institute of Technology, 2007. http://hdl.handle.net/1721.1/41730.
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Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007.Includes bibliographical references (p. 75-81).
A number of constitutive theories have arisen describing materials which, by nature, exhibit a non-local response. The formulation of boundary value problems, in this case, leads to a system of equations involving higher-order derivatives which, in turn, results in requirements of continuity of the solution of higher order. Discontinuous Galerkin methods are particularly attractive toward this end, as they provide a means to naturally enforce higher interelement continuity in a weak manner without the need of modifying the finite element interpolation. In this work, a discontinuous Galerkin formulation for boundary value problems in small strain, non-local linear elasticity is proposed. The underlying theory corresponds to the phenomenological strain-gradient theory developed by Fleck and Hutchinson within the Toupin-Mindlin framework. The single-field displacement method obtained enables the discretization of the boundary value problem with a conventional continuous interpolation inside each finite element, whereas the higher-order interelement continuity is enforced in a weak manner. The proposed method is shown to be consistent and stable both theoretically and with suitable numerical examples.
by Ram Bala Chandran.
S.M.
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Ekström, Sven-Erik. "A vertex-centered discontinuous Galerkin method for flow problems." Licentiate thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-284321.
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The understanding of flow problems, and finding their solution, has been important for most of human history, from the design of aqueducts to boats and airplanes. The use of physical miniature models and wind tunnels were, and still are, useful tools for design, but with the development of computers, an increasingly large part of the design process is assisted by computational fluid dynamics (CFD). Many industrial CFD codes have their origins in the 1980s and 1990s, when the low order finite volume method (FVM) was prevalent. Discontinuous Galerkin methods (DGM) have, since the turn of the century, been seen as the successor of these methods, since it is potentially of arbitrarily high order. In its lowest order form DGM is equivalent to FVM. However, many existing codes are not compatible with standard DGM and would need a complete rewrite to obtain the advantages of the higher order. This thesis shows how to extend existing vertex-centered and edge-based FVM codes to higher order, using a special kind of DGM discretization, which is different from the standard cell-centered type. Two model problems are examined to show the necessary data structures that need to be constructed, the order of accuracy for the method, and the use of an hp-adaptation scheme to resolve a developing shock. Then the method is further developed to solve the steady Euler equations, within the existing industrial Edge code, using acceleration techniques such as local time stepping and multigrid. With the ever increasing need for more efficient and accurate solvers and algorithms in CFD, the modified DGM presented in this thesis could be used to help and accelerate the adoption of high order methods in industry.
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Wukie, Nathan A. "A Discontinuous Galerkin Method for Turbomachinery and Acoustics Applications." University of Cincinnati / OhioLINK, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1543840344167045.
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COLOMBO, Alessandro (ORCID:0000-0002-6527-8148). "An agglomeration-based discontinuous Galerkin method for compressible flows." Doctoral thesis, Università degli studi di Bergamo, 2011. http://hdl.handle.net/10446/886.
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This thesis investigates the flexibility associated to Discontinuous Galerkin (DG) discretization on very general meshes obtained by means of agglomeration techniques. The work begins with a brief overview of the main tools that have been extended or specifically developed to deal with arbitrarily shaped elements in the DG context. Then two different implementations of the BRMPS scheme introduced by Bassi, Rebay, Mariotti, Pedinotti and Savini in [16] for the DG discretization of the Laplace operator on arbitrarily shaped elements have been presented. The validation of the scheme on a Poisson problem shows that the discrete polynomial space preserves optimal convergence properties.
The discretization of the second order differential operator has been directly extended to the Navier-Stokes equations and the Reynolds Averaged Navier-Stokes (RANS) equations coupled with the k-w turbulence model of Wilcox [54]. In this regard, an implicit time integration strategy has been considered and assessed on classical validation test cases for the compressible f uid dynamics.
Then a simple alternative approach to high-order mesh generation is presented. Indeed, once a standard fine grid able to provide an accurate domain discretization has been produced by means of standard low-order grid generation tools, a computational mesh suitable for the desired accuracy and computationally affordable can be obtained via agglomeration while keeping the boundary resolution of the fine grid. The effectiveness of this approach in representing the geometry of the domain is numerically assessed both on a Poisson model problem and on challenging inviscid and viscous test cases.
Finally, the freedom in simply defining the topology of agglomerated meshes leads to a nonstandard approach to h-adaptivity that exploits adaptive agglomeration coarsening of a properly fine underlying grid. The effectiveness of this approach has been assessed on test cases involving both error-based and fl ow feature-based simple estimators.
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COLOMBO, Alessandro (ORCID:0000-0002-6527-8148). "An agglomeration-based discontinuous Galerkin method for compressible flows." Doctoral thesis, Università degli studi di Bergamo, 2011. http://hdl.handle.net/10446/222124.
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This thesis investigates the flexibility associated to Discontinuous Galerkin (DG) discretization on very general meshes obtained by means of agglomeration techniques. The work begins with a brief overview of the main tools that have been extended or specifically developed to deal with arbitrarily shaped elements in the DG context. Then two different implementations of the BRMPS scheme introduced by Bassi, Rebay, Mariotti, Pedinotti and Savini in [16] for the DG discretization of the Laplace operator on arbitrarily shaped elements have been presented. The validation of the scheme on a Poisson problem shows that the discrete polynomial space preserves optimal convergence properties. The discretization of the second order differential operator has been directly extended to the Navier-Stokes equations and the Reynolds Averaged Navier-Stokes (RANS) equations coupled with the k-w turbulence model of Wilcox [54]. In this regard, an implicit time integration strategy has been considered and assessed on classical validation test cases for the compressible f uid dynamics. Then a simple alternative approach to high-order mesh generation is presented. Indeed, once a standard fine grid able to provide an accurate domain discretization has been produced by means of standard low-order grid generation tools, a computational mesh suitable for the desired accuracy and computationally affordable can be obtained via agglomeration while keeping the boundary resolution of the fine grid. The effectiveness of this approach in representing the geometry of the domain is numerically assessed both on a Poisson model problem and on challenging inviscid and viscous test cases. Finally, the freedom in simply defining the topology of agglomerated meshes leads to a nonstandard approach to h-adaptivity that exploits adaptive agglomeration coarsening of a properly fine underlying grid. The effectiveness of this approach has been assessed on test cases involving both error-based and fl ow feature-based simple estimators.
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Mukhamedov, Farukh. "High performance computing for the discontinuous Galerkin methods." Thesis, Brunel University, 2018. http://bura.brunel.ac.uk/handle/2438/16769.
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Discontinuous Galerkin methods form a class of numerical methods to find a solution of partial differential equations by combining features of finite element and finite volume methods. Methods are defined using a weak form of a particular model problem, allowing for discontinuities in the discrete trial and test spaces. Using a discontinuous discrete space mesh provides proper flexibility and a compact discretisation pattern, allowing a multidomain and multiphysics simulation. Discontinuous Galerkin methods with a higher approximation polynomial order, the socalled p-version, performs better in terms of convergence rate, compared with the low order h-version with smaller element sizes and bigger mesh. However, the condition number of the Galerkin system grows subsequently. This causes surge in the amount of required storage, computational complexity and in the time required for computation. We use the following three approaches to keep the advantages and eliminate the disadvantages. The first approach will be a specific choice of basis functions which we call C1 polynomials. These ensure that the majority of integrals over the edge of the mesh elements disappears. This reduces the total number of non-zero elements in the resulting system. This decreases the computational complexity without loss in precision. This approach does not affect the number of iterations required by chosen Conjugate Gradients method when compared to the other choice of basis functions. It actually decreases the total number of algebraic operations performed. The second approach is the introduction of suitable preconditioners. In our case, the Additive two-layer Schwarz method, developed in [4], for the iterative Conjugate Gradients method is considered. This directly affects the spectral condition number of the system matrix and decreases the number of iterations required for the computation. This approach, however, increases the total number of algebraic operations and might require more operational time. To tackle the rise in the number of algebraic operations, we introduced a modified Additive two-layer non-overlapping Schwarz method with a Multigrid process. This using a fixed low-order approximation polynomial degree on a coarse grid. We show that this approach is spectrally equivalent to the first preconditioner, and requires less time for computation. The third approach is a development of an efficient mathematical framework for distributed data structure. This allows a high performance, massively parallel, implementation of the discontinuous Galerkin method. We demonstrate that it is possible to exploit properties of the system matrix and C1 polynomials as basis functions to optimize the parallel structures. The previously mentioned parallel data structure allows us to parallelize at the same time both the matrix-vector multiplication routines for the Conjugate Gradients method, as well as the preconditioner routines on the solver level. This minimizes the transfer ratio amongst the distributed system. Finally, we combined all three approaches and created a framework, which allowed us to successfully implement all of the above.
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Javadzadeh, Moghtader Mostafa. "High-order hybridizable discontinuous Galerkin method for viscous compressible flows." Doctoral thesis, Universitat Politècnica de Catalunya, 2016. http://hdl.handle.net/10803/404125.
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Computational Fluid Dynamics (CFD) is an essential tool for engineering design and analysis, especially in applications like aerospace, automotive and energy industries. Nowadays most commercial codes are based on Finite Volume (FV) methods, which are second order accurate, and simulation of viscous compressible flow around complex geometries is still very expensive due to large number of low-order elements required. One the other hand, some sophisticated physical phenomena, like aeroacoustics, vortex dominated flows and turbulence, need very high resolution methods to obtain accurate results. High-order methods with their low spatial discretization errors, are a possible remedy for shortcomings of the current CFD solvers.
Discontinuous Galerkin (DG) methods have emerged as a successful approach for non-linear hyperbolic problems and are widely regarded very promising for next generation CFD solvers. Their efficiency for high-order discretization makes them suitable for advanced physical models like DES and LES, while their stability in convection dominated regimes is also a merit of them. The compactness of DG methods, facilitate the parallelization and their element-by-element discontinuous nature is also helpful for adaptivity.
This PhD thesis focuses on the development of an efficient and robust high-order Hybridizable Discontinuous Galerkin (HDG) Finite Element Method (FEM) for compressible viscous flow computations. HDG method is a new class of DG family which enjoys from merits of DG but has significantly less globally coupled unknowns compared to other DG methods. Its features makes HDG a possible candidate to be investigated as next generation high-order tools for CFD applications.
The first part of this thesis recalls the basics of high-order HDG method. It is presented for the two-dimensional linear convection-diffusion equation, and its accuracy and features are investigated. Then, the method is used to solve compressible viscous flow problems modelled by non-linear compressible Navier-Stokes equations; and finally a new linearized HDG formulation is proposed and implemented for that problem, all using high-order approximations. The accuracy and efficiency of high-order HDG method to tackle viscous compressible flow problems is investigated, and both steady and unsteady solvers are developed for this purpose.
The second part is the core of this thesis, proposing a novel shock-capturing method for HDG solution of viscous compressible flow problems, in the presence of shock waves. The main idea is to utilize the stabilization of numerical fluxes, via a discontinuous space of approximation inside the elements, to diminish or remove the oscillations in the vicinity of discontinuity. This discontinuous nodal basis functions, leads to a modified weak form of the HDG local problem in the stabilized elements. First, the method is applied to convection-diffusion problems with Bassi-Rebay and LDG fluxes inside the elements, and then, the strategy is extended to the compressible Navier-Stokes equations using LDG and Lax-Friedrichs fluxes. Various numerical examples, for both convection-diffusion and compressible Navier-Stokes equations, demonstrate the ability of the proposed method, to capture shocks in the solution, and its excellent performance in eliminating oscillations is the vicinity of shocks to obtain a spurious-free high-order solution.Dinámica de Fluidos Computacional (CFD) es una herramienta esencial para el diseño y análisis en ingeniería, especialmente en aplicaciones de ingeniería aeroespacial, automoción o energía, entre otros. Hoy en día, la mayoría de los códigos comerciales se basan en el método de Volúmenes Finitos (FV), con precisión de segundo orden. Sin embargo, la simulación del flujo compresible y viscoso alrededor de geometrías complejas mediante estos métodos es todavía muy cara, debido al gran número de elementos de orden bajo requeridos. Algunos fenómenos físicos sofisticados, por ejemplo en aeroacústica, presentan vórtices y turbulencias, y necesitan métodos de muy alta resolución para obtener resultados precisos. Los métodos de alto orden, con bajos errores de discretización espacial, pueden superar las deficiencias de los actuales códigos de CFD. Los métodos Galerkin discontinuos (DG) han surgido como un enfoque exitoso para problemas hiperbólicos no lineales, y son ampliamente considerados muy prometedores para la próxima generación de códigos de CFD. Su eficiencia de alto orden los hace adecuados para modelos físicos avanzados como DES (Direct Numerial Simulation) y LES (Large Eddy Simulation), mientras que su estabilidad en problemas de convención dominante es también un mérito de ellos. La compacidad de los métodos DG facilita la paralelización, y su naturaleza discontinua es también útil para la adaptabilidad. Esta tesis doctoral se centra en el desarrollo de un método de alto orden, eficiente y robusto, basado en el método de elementos finitos Hybridizable Discontinuous Galerkin (HDG), para cálculos de flujo viscoso y compresible. HDG es un método novedoso, con los méritos de los métodos DG, pero con significativamente menos grados de libertad a nivel global en comparación con otros métodos discontinuos. Sus características hacen de HDG un candidato prometedor a ser investigado como una herramienta de alto orden de próxima generación para aplicaciones de CFD. La primera parte de esta tesis, recuerda los fundamentos del método HDG. Se presenta la aplicación del método para la ecuación de convección-difusión lineal en dos dimensiones, y se investiga su precisión y sus características. Posteriormente, el método se utiliza para resolver problemas de flujo viscoso compresible modelados por las ecuaciones de Navier-Stokes compresibles no lineales. Por último, se propone una nueva formulación HDG linealizada de alto orden y se implementa para este tipo de problemas. También se estudia su precisión y su eficiencia para problemas estacionarios y transitorios. La segunda parte es el núcleo de esta tesis. Se propone un nuevo método de captura de choque para la solución HDG de problemas de compresibles y viscosos, en presencia de choques o frentes verticales pronunciados. La idea principal es utilizar la estabilización que proporcionan los flujos numéricos, considerando un espacio discontinuo de aproximación en interior de los elementos, para disminuir o eliminar las oscilaciones en la proximidad de la discontinuidad o el frente. Las funciones de base nodales discontinuas, requieren una forma débil modificada del problema local de HDG en los elementos estabilizados. En primer lugar, el método se aplica a problemas de convección-difusión, con flujos numéricos de Bassi-Rebay y de LDG (Local Discontinuous Galerkin) dentro de los elementos. A continuación, la estrategia se extiende a las ecuaciones de Navier-Stokes compresibles utilizando flujos numéricos de LDG y de Lax-Friedrichs. Finalmente, varios ejemplos numéricos, tanto para convección-difusió, como para las ecuaciones de Navier-Stokes compresibles, demuestran la capacidad del método propuesto para capturar los choques o frentes verticales en la solución. Su excelente rendimiento, elimina o atenúa significativamente las oscilaciones alrededor de los choques, obteniendo una solución estable.
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Özdemir, Hüseyin. "High-order discontinuous Galerkin method on hexahedral elements for aeroacoustics." Enschede : University of Twente [Host], 2006. http://doc.utwente.nl/57867.
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Lindley, Jorge Vicente Malik. "A discontinuous Galerkin finite element method for quasi-geostrophic frontogenesis." Thesis, University of Warwick, 2017. http://wrap.warwick.ac.uk/102632/.
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In this thesis, a mixed continuous and discontinuous Galerkin finite element method is developed for the three-dimensional quasi-geostrophic equations, and is used to investigate the role that weather front formation plays in the transfer of energy to small scales that would produce a k. 5=3 energy spectrum as observed in the atmosphere. The quasi-geostrophic equations are used for computational efficiency and are found to be sufficient for producing simple fronts. Discontinuous Galerkin finite elements are used for the potential vorticity as continuous Galerkin methods perform poorly with advection dominated problems. The less dynamical vertical direction is discretised with finite difference to simplify the finite element method in the horizontal. Streamfunction boundary values are derived for free-slip boundary conditions in the three-dimensional model. The scheme is verified with numerical tests and is shown to converge at optimal rates until free-slip boundaries are introduced. Conservation of energy and enstrophy are shown numerically. Using the numerical method, a channel model simulation suggests that the bend up of fronts produced by a meandering zonal jet could be a viable mechanism for producing a k.5=3 regime.
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Burleson, John Taylor. "Numerical Simulations of Viscoelastic Flows Using the Discontinuous Galerkin Method." Thesis, Virginia Tech, 2021. http://hdl.handle.net/10919/104869.
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In this work, we develop a method for solving viscoelastic fluid flows using the Navier-Stokes equations coupled with the Oldroyd-B model. We solve the Navier-Stokes equations in skew-symmetric form using the mixed finite element method, and we solve the Oldroyd-B model using the discontinuous Galerkin method. The Crank-Nicolson scheme is used for the temporal discretization of the Navier-Stokes equations in order to achieve a second-order accuracy in time, while the optimal third-order total-variation diminishing Runge-Kutta scheme is used for the temporal discretization of the Oldroyd-B equation. The overall accuracy in time is therefore limited to second-order due to the Crank-Nicolson scheme; however, a third-order Runge-Kutta scheme is implemented for greater stability over lower order Runge-Kutta schemes. We test our numerical method using the 2D cavity flow benchmark problem and compare results generated with those found in literature while discussing the influence of mesh refinement on suppressing oscillations in the polymer stress.Master of Science
Viscoelastic fluids are a type of non-Newtonian fluid of great importance to the study of fluid flows. Such fluids exhibit both viscous and elastic behaviors. We develop a numerical method to solve the partial differential equations governing viscoelastic fluid flows using various finite element methods. Our method is then validated using previous numerical results in literature.
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Biotto, Cristian. "A discontinuous Galerkin method for the solution of compressible flows." Thesis, Imperial College London, 2011. http://hdl.handle.net/10044/1/6413.
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This thesis presents a methodology for the numerical solution of one-dimensional (1D) and two-dimensional (2D) compressible flows via a discontinuous Galerkin (DG) formulation. The 1D Euler equations are used to assess the performance and stability of the discretisation. The explicit time restriction is derived and it is established that the optimal polynomial degree, p, in terms of efficiency and accuracy of the simulation is p = 5. Since the method is characterised by minimal diffusion, it is particularly well suited for the simulation of the pressure wave generated by train entering a tunnel. A novel treatment of the area-averaged Euler equations is proposed to eliminate oscillations generated by the projection of a moving area on a fixed mesh and the computational results are validated against experimental data. Attention is then focussed on the development of a 2D DG method implemented using the high-order library Nektar++. An Euler and a laminar Navier–Stokes solvers are presented and benchmark tests are used to assess their accuracy and performance. An artificial diffusion term is implemented to stabilise the solution of the Euler equations in transonic flow with discontinuities. To speed up the convergence of the explicit method, a new automatic polynomial adaptive procedure (p-adaption) and a new zonal solver are proposed. The p-adaptive procedure uses a discontinuity sensor, originally developed as an artificial diffusion sensor, to assign appropriate polynomial degrees to each element of the domain. The zonal solver uses a modification of a method for matching viscous subdomains to set the interface conditions between viscous and inviscid subdomains that ensures stability of the flow computation. Both the p-adaption and the zonal solver maintain the high-order accuracy of the DG method while reducing the computational cost of the simulation.
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Shelton, Andrew Brian. "A multi-resolution discontinuous galerkin method for unsteady compressible flows." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/24715.
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Thesis (Ph.D.)--Aerospace Engineering, Georgia Institute of Technology, 2009.Committee Chair: Smith, Marilyn; Committee Co-Chair: Zhou, Hao-Min; Committee Member: Dieci, Luca; Committee Member: Menon, Suresh; Committee Member: Ruffin, Stephen
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Huynh, Dan-Nha. "Nonlinear optical phenomena within the discontinuous Galerkin time-domain method." Doctoral thesis, Humboldt-Universität zu Berlin, 2018. http://dx.doi.org/10.18452/19396.
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Diese Arbeit befasst sich mit der theoretischen Beschreibung nichtlinearer optischer Phänomene in Hinblick auf das (numerische) unstetige Galerkin-Zeitraumverfahren. Insbesondere werden zwei Materialmodelle behandelt: das hydrodynamische Modell für Metalle und das Modell für Raman-aktive Materialien. Im ersten Teil der Arbeit wird das hydordynamische Modell für Metalle unter Verwendung eines störungstheoretischen Ansatzes behandelt. Insbesondere wird dieser Ansatz genutzt, um die nichtlinearen optischen Effekte, Erzeugung zweiter Harmonischer und Summenfrequenzerzeugung, mit Hilfe des unstetigen Galerkin-Verfahrens zu studieren. In diesem Zusammenhang wird demonstriert, wie das optische Signal zweiter Ordnung von Nanoantennen optimiert werden kann. Hierzu wird ein hier erarbeitetes Schema für die Abstimmung des eingestrahten Lichtes angewandt. Zudem führt eine intelligente Wahl des Antennendesigns zu einem optimierten Signal. Im zweiten Teil dieser Arbeit wird das Modell für Raman-aktive Dielektrika behandelt. Genauer wird die nichtlineare Antwort dritter Ordnung für stimulierte Raman-Streuung hergeleitet. Diese wird dazu genutzt, um ein System aus Hilfsdifferentialgleichungen für das unstetige Galerkin-Verfahren zu konstruieren. Die Ergebnisse des erweiterten numerischen Verfahrens werden im Anschluss gezeigt und diskutiert.This thesis is concerned with the theoretical description of nonlinear optical phenomena with regards to the (numerical) discontinuous Galerkin time-domain (DGTD) method. It deals with two different material models: the hydrodynamic model for metals and the model for Raman-active dielectrics. In the first part, we review the hydrodynamic model for metals, where we apply a perturbative approach to the model. We use this approach to calculate the second-order nonlinear optical effects of second-harmonic generation and sum-frequency generation using the DGTD method. In this context, we will see how to optimize the second-order response of plasmonic nanoantennas by applying a deliberate tuning scheme for the optical excitations as well as by choosing an intelligent nanoantenna design. In the second part, we examine the material model for Raman-active dielectrics. In particular, we see how to derive the third-order nonlinear response by which one can describe the process of stimulated Raman scattering. We show how to incorporate this third-order response into the DGTD scheme yielding a novel set of auxiliary differential equations. Finally, we demonstrate the workings of the modified numerical scheme.
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Jayasinghe, Yashod Savithru. "An adaptive space-time discontinuous Galerkin method for reservoir flows." Thesis, Massachusetts Institute of Technology, 2018.
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This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Thesis: Ph. D., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2018
Cataloged from student-submitted PDF version of thesis.
Includes bibliographical references (pages 205-216).
Numerical simulation has become a vital tool for predicting engineering quantities of interest in reservoir flows. However, the general lack of autonomy and reliability prevents most numerical methods from being used to their full potential in engineering analysis. This thesis presents work towards the development of an efficient and robust numerical framework for solving reservoir flow problems in a fully-automated manner. In particular, a space-time discontinuous Galerkin (DG) finite element method is used to achieve a high-order discretization on a fully unstructured space-time mesh, instead of a conventional time-marching approach. Anisotropic mesh adaptation is performed to reduce the error of a specified output of interest, by using a posteriori error estimates from the dual weighted residual method to drive a metric-based mesh optimization algorithm.
An analysis of the adjoint equations, boundary conditions and solutions of the Buckley-Leverett and two-phase flow equations is presented, with the objective of developing a theoretical understanding of the adjoint behaviors of porous media models. The intuition developed from this analysis is useful for understanding mesh adaptation behaviors in more complex flow problems. This work also presents a new bottom-hole pressure well model for reservoir simulation, which relates the volumetric flow rate of the well to the reservoir pressure through a distributed source term that is independent of the discretization. Unlike Peaceman-type models which require the definition of an equivalent well-bore radius dependent on local grid length scales, this distributed well model is directly applicable to general discretizations on unstructured meshes.
We show that a standard DG diffusive flux discretization of the two-phase flow equations in mass conservation form results in an unstable semi-discrete system in the advection-dominant limit, and hence propose modifications to linearly stabilize the discretization. Further, an artificial viscosity method is presented for the Buckley-Leverett and two-phase flow equations, as a means of mitigating Gibbs oscillations in high-order discretizations and ensuring convergence to physical solutions. Finally, the proposed adaptive solution framework is demonstrated on compressible two-phase flow problems in homogeneous and heterogeneous reservoirs. Comparisons with conventional time-marching methods show that the adaptive space-time DG method is significantly more efficient at predicting output quantities of interest, in terms of degrees-of-freedom required, execution time and parallel scalability.
by Yashod Savithru Jayasinghe.
Ph. D.
Ph.D. Massachusetts Institute of Technology, Department of Aeronautics and Astronautics
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Wang, Siyang. "Finite Difference and Discontinuous Galerkin Methods for Wave Equations." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-320614.
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Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. The finite difference method is conceptually simple and easy to implement, but has difficulties in handling complex geometries of the computational domain. We construct high order finite difference methods for wave propagation in heterogeneous media with complex geometries. In addition, we derive error estimates to a class of finite difference operators applied to the acoustic wave equation. The discontinuous Galerkin method is flexible with complex geometries. Moreover, the discontinuous nature between elements makes the method suitable for multiphysics problems. We use an energy based discontinuous Galerkin method to solve a coupled acoustic-elastic problem.
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Voonna, Kiran. "Development of discontinuous galerkin method for 1-D inviscid burgers equation." ScholarWorks@UNO, 2003. http://louisdl.louislibraries.org/u?/NOD,75.
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Thesis (M.S.)--University of New Orleans, 2003.Title from electronic submission form. "A thesis ... in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mechanical Engineering"--Thesis t.p. Vita. Includes bibliographical references.
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Chaurasia, Hemant Kumar. "A time-spectral hybridizable discontinuous Galerkin method for periodic flow problems." Thesis, Massachusetts Institute of Technology, 2014. http://hdl.handle.net/1721.1/90647.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2014.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 110-120).
Numerical simulations of time-periodic flows are an essential design tool for a wide range of engineered systems, including jet engines, wind turbines and flapping wings. Conventional solvers for time-periodic flows are limited in accuracy and efficiency by the low-order Finite Volume and time-marching methods they typically employ. These methods introduce significant numerical dissipation in the simulated flow, and can require hundreds of timesteps to describe a periodic flow with only a few harmonic modes. However, recent developments in high-order methods and Fourier-based time discretizations present an opportunity to greatly improve computational performance. This thesis presents a novel Time-Spectral Hybridizable Discontinuous Galerkin (HDG) method for periodic flow problems, together with applications to flow through cascades and rotor/stator assemblies in aeronautical turbomachinery. The present work combines a Fourier-based Time-Spectral discretization in time with an HDG discretization in space, realizing the dual benefits of spectral accuracy in time and high-order accuracy in space. Low numerical dissipation and favorable stability properties are inherited from the high-order HDG method, together with a reduced number of globally coupled degrees of freedom compared to other DG methods. HDG provides a natural framework for treating boundary conditions, which is exploited in the development of a new high-order sliding mesh interface coupling technique for multiple-row turbomachinery problems. A regularization of the Spalart-Allmaras turbulence model is also employed to ensure numerical stability of unsteady flow solutions obtained with high-order methods. Turning to the temporal discretization, the Time-Spectral method enables direct solution of a periodic flow state, bypasses initial transient behavior, and can often deliver substantial savings in computational cost compared to implicit time-marching. An important driver of computational efficiency is the ability to select and resolve only the most important frequencies of a periodic problem, such as the blade-passing frequencies in turbomachinery flows. To this end, the present work introduces an adaptive frequency selection technique, using the Time-Spectral residual to form an inexpensive error indicator. Having selected a set of frequencies, the accuracy of the Time-Spectral solution is greatly improved by using optimally selected collocation points in time. For multi-domain problems such as turbomachinery flows, an anti-aliasing filter is also needed to avoid errors in the transfer of the solution across the sliding interface. All of these aspects contribute to the Adaptive Time-Spectral HDG method developed in this thesis. Performance characteristics of the method are demonstrated through applications to periodic ordinary differential equations, a convection problem, laminar flow over a pitching airfoil, and turbulent flow through a range of single- and multiple-row turbomachinery configurations. For a 2:1 rotor/stator flow problem, the Adaptive Time-Spectral HDG method correctly identifies the relevant frequencies in each blade row. This leads to an accurate periodic flow solution with greatly reduced computational cost, when compared to sequentially selected frequencies or a time-marching solution. For comparable accuracy in prediction of rotor loading, the Adaptive Time- Spectral HDG method incurs 3 times lower computational cost (CPU time) than time-marching, and for prediction of only the 1st harmonic amplitude, these savings rise to a factor of 200. Finally, in three-row compressor flow simulations, a high-order HDG method is shown to achieve significantly greater accuracy than a lower-order method with the same computational cost. For example, considering error in the amplitude of the 1st harmonic mode of total rotor loading, a p = 1 computation results in 20% error, in contrast to only 1% error in a p = 4 solution with comparable cost. This highlights the benefits that can be obtained from higher-order methods in the context of turbomachinery flow problems.
by Hemant Kumar Chaurasia.
Ph. D.
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Moura, Rodrigo Costa. "A high-order unstructured discontinuous galerkin finite element method for aerodynamics." Instituto Tecnológico de Aeronáutica, 2012. http://www.bd.bibl.ita.br/tde_busca/arquivo.php?codArquivo=2158.
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The present thesis discuss in a didactic and detailed way the high-order scheme known as the Discontinuous Galerkin (DG) method, with special focus on applications in aerodynamics. The theoretical formulation of the method is presented in one and two dimensions with great depth, being properly discussed issues of convergence, basis functions, interelement communication, boundary conditions, shock treatment, as well as inviscid and viscous numerical fluxes. As part of this effort, a parallel computer code was developed to simulate the Euler equations of gas dynamics in two dimensions with general boundary conditions over unstructured meshes of triangles. Numerical simulations are addressed in order to demonstrate the characteristics of the Discontinuous Galerkin scheme, as well as to validate the developed solver. It is worth mentioning that the present work can be regarded as new within the Brazilian scientific community and, as such, may be of great importance concerning the introduction of the DG method for Brazilian CFD researchers and practitioners.
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Miri, Seyedalireza. "Numerical Solution of Moment Equations Using the Discontinuous-Galerkin Hancock Method." Thesis, Université d'Ottawa / University of Ottawa, 2019. http://hdl.handle.net/10393/38678.
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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.
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Gomez, carrero Quriaky. "Discontinuous Galerkin Modeling of Wave Propagation in Damaged Materials." Thesis, Sorbonne Paris Cité, 2017. http://www.theses.fr/2017USPCD054/document.
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Dans cette thèse on utilise une méthode de Galerkin discontinue (GD) pour modéliser la propagation des ondes dans un matériau endommagé. Deux modèles différents pour la description de l’endommagement ont été considérés. Dans la première partie de la thèse on utilise un modèle d’endommagent assez général, basé sur une modélisation micromécanique. Pour ce modèle on établit un critère de stabilité basé sur une densité critique de fissuration. On développe aussi une méthode numérique GD capable de capturer les instabilités au niveau microscopique. On construit une solution exacte pour analyser la précision de la méthode proposée.Plusieurs résultats numériques vont permettre d’analyser la propagation des ondes dans les configurations planes et anti-planes. Dans la deuxième parte de la thèse on étudie la propagation des ondes dans un milieux fissuré (microfissures en contact avec frottement). La méthode numérique développée utilise une technique GD et la méthode du Lagrangien augmenté. En utilisant cette méthode on a pu calculer numériquement la vitesse de propagation moyenne dans un matériau endommagé. On a pu comparer les résultats obtenus avec les formules analytiques obtenues avec des approches micromécaniques. Finalement, on a utilisé les calculs numériques pour étudier la propagation des ondes après un impact sur une plaque céramique pour les deux modèles mécaniques considérésA discontinuous Galerkin (DG) technique for modeling wave propagation in damaged (brittle) materials is developed in this thesis. Two different types of mechanical models for describing the damaged materials are considered. In the first part of the thesis general micro-mechanics based damage models were used. A critical crack density parameter, which distinguishes between stable and unstable behaviors, wascomputed. A new DG-numerical scheme able to capture the instabilities and a micro-scale time step were proposed. An exact solution is constructed and the accuracy of the numerical scheme was analyzed. The wave propagation in one dimensional and anti-plane configuration was analyzed through several numerical computations. In the second part of the thesis the wave propagation in cracked materials with a nonlinear micro-structure (micro-cracks in frictional contact) was investigated. The numerical scheme developed makes use of a DG-method and an augmented Lagrangian technique. The effective wave velocity in a damaged material, obtained by a numerical upscaling homogenization method, was compared with analytical formula of effective elasticity theory. The wave propagation (speed, amplitude and pulse length) in micro-cracked materials in complex configurations was studied. Finally, numerical computations of blast wave propagation,for the both models, illustrate the role played by the micro-cracks orientation and by the friction
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Naddei, Fabio. "Adaptive Large Eddy Simulations based on discontinuous Galerkin methods." Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLX060/document.
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L'objectif principal de ce travail est d'améliorer la précision et l'efficacité des modèles LES au moyen des méthodes Galerkine discontinues (DG). Deux thématiques principales ont été étudiées: les stratégies d'adaptation spatiale et les modèles LES pour les méthodes d'ordre élevé.Concernant le premier thème, dans le cadre des méthodes DG la résolution spatiale peut être efficacement adaptée en modifiant localement soit le maillage (adaptation-h) soit le degré polynômial de la solution (adaptation-p). L'adaptation automatique de la résolution nécessite l'estimation des erreurs pour analyser la qualité de la solution locale et les exigences de résolution. L'efficacité de différentes stratégies de la littérature est comparée en effectuant des simulations h- et p-adaptatives.Sur la base de cette étude comparative, des algorithmes statiques et dynamiques p-adaptatifs pour la simulation des écoulements instationnaires sont ensuite développés et analysés. Les simulations numériques réalisées montrent que les algorithmes proposés peuvent réduire le coût de calcul des simulations des écoulements transitoires et statistiquement stationnaires.Un nouvel estimateur d'erreur est ensuite proposé. Il est local, car n'exige que des informations de l'élément et de ses voisins directs, et peut être calculé en cours de simulation pour un coût limité. Il est démontré que l'algorithme statique p-adaptatif basé sur cet estimateur d'erreur peut être utilisé pour améliorer la précision des simulations LES sur des écoulements turbulents statistiquement stationnaires.Concernant le second thème, une nouvelle méthode, consistante avec la discrétisation DG, est développée pour l'analyse a-priori des modèles DG-LES à partir des données DNS. Elle permet d'identifier le transfert d'énergie idéal entre les échelles résolues et non résolues. Cette méthode est appliquée à l'analyse de l'approche VMS (Variational Multiscale). Il est démontré que pour les résolutions fines, l'approche DG-VMS est capable de reproduire le transfert d'énergie idéal. Cependant, pour les résolutions grossières, typique de la LES à nombres de Reynolds élevés, un meilleur accord peut être obtenu en utilisant un modèle mixte Smagorinsky-VMSThe main goal of this work is to improve the accuracy and computational efficiency of Large Eddy Simulations (LES) by means of discontinuous Galerkin (DG) methods. To this end, two main research topics have been investigated: resolution adaptation strategies and LES models for high-order methods.As regards the first topic, in the framework of DG methods the spatial resolution can be efficiently adapted by modifying either the local mesh size (h-adaptation) or the degree of the polynomial representation of the solution (p-adaptation).The automatic resolution adaptation requires the definition of an error estimation strategy to analyse the local solution quality and resolution requirements.The efficiency of several strategies derived from the literature are compared by performing p- and h-adaptive simulations. Based on this comparative study a suitable error indicator for the adaptive scale-resolving simulations is selected.Both static and dynamic p-adaptive algorithms for the simulation of unsteady flows are then developed and analysed. It is demonstrated by numerical simulations that the proposed algorithms can provide a reduction of the computational cost for the simulation of both transient and statistically steady flows.A novel error estimation strategy is then introduced. It is local, requiring only information from the element and direct neighbours, and can be computed at run-time with limited overhead. It is shown that the static p-adaptive algorithm based on this error estimator can be employed to improve the accuracy for LES of statistically steady turbulent flows.As regards the second topic, a novel framework consistent with the DG discretization is developed for the a-priori analysis of DG-LES models from DNS databases. It allows to identify the ideal energy transfer mechanism between resolved and unresolved scales.This approach is applied for the analysis of the DG Variational Multiscale (VMS) approach. It is shown that, for fine resolutions, the DG-VMS approach is able to replicate the ideal energy transfer mechanism.However, for coarse resolutions, typical of LES at high Reynolds numbers, a more accurate agreement is obtained by a mixed Smagorinsky-VMS model
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Spindeldreher, Stefan. "The discontinuous Galerkin method applied on the equations of ideal relativistic hydrodynamics." [S.l. : s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=965285502.
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Gempesaw, Daniel. "A multi-resolution discontinuous Galerkin method for rapid simulation of thermal systems." Thesis, Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/42775.
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Efficient, accurate numerical simulation of coupled heat transfer and fluid dynamics systems continues to be a challenge. Direct numerical simulation (DNS) packages like FLU- ENT exist and are sufficient for design and predicting flow in a static system, but in larger systems where input parameters can change rapidly, the cost of DNS increases prohibitively. Major obstacles include handling the scales of the system accurately - some applications span multiple orders of magnitude in both the spatial and temporal dimensions, making an accurate simulation very costly. There is a need for a simulation method that returns accurate results of multi-scale systems in real time. To address these challenges, the Multi- Resolution Discontinuous Galerkin (MRDG) method has been shown to have advantages over other reduced order methods. Using multi-wavelets as the local approximation space provides an inherently efficient method of data compression, while the unique features of the Discontinuous Galerkin method make it well suited to composition with wavelet theory. This research further exhibits the viability of the MRDG as a new approach to efficient, accurate thermal system simulations. The development and execution of the algorithm will be detailed, and several examples of the utility of the MRDG will be included. Comparison between the MRDG and the "vanilla" DG method will also be featured as justification of the advantages of the MRDG method.
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Kretzschmar, Fritz [Verfasser], Thomas [Akademischer Betreuer] Weiland, and Herbert [Akademischer Betreuer] Egger. "The Discontinuous Galerkin Trefftz Method / Fritz Kretzschmar. Betreuer: Thomas Weiland ; Herbert Egger." Darmstadt : Universitäts- und Landesbibliothek Darmstadt, 2015. http://d-nb.info/1112044590/34.
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Israeli, Emily Renee. "Simulations of a passively actuated oscillating airfoil using a discontinuous Galerkin method." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/45892.
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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2008.Includes bibliographical references (p. 87-89).
Natural flappers, such as birds and bats, effectively maneuver in transitional, low Reynolds number flow, outperforming any current small engineered flapping vehicle. Thus, engineers are inspired to investigate the flapping dynamics present in nature to further understand the non-traditional flow aerodynamics in which they operate. Undeniably the success of biological flapping flight is the exploitation of fluid structure interaction response i.e. wing mechanics, deformation, and morphing. Even though all these features are encountered in nature, it is important to note that natural flappers have not just adapted to optimize their aerodynamic behavior, they also have evolved due to biological constraints. Therefore, in bio-inspired design one carefully uses the insight gained from understanding natural flappers. Here, a 2-D simulation of a pitching and heaving foil attempts to indicate flapping parameter specifics that generate an efficient, thrust producing flapper. The simulations are performed using a high-order Discontinuous Galerkin finite element solver for the compressible Navier Stokes equations. A brief investigation of a simple problem in which pitch and heave of a foil are prescribed highlights the necessity to use an inexpensive lower fidelity model to narrow down the large design space to a manageable region of interest. A torsional spring is placed at the foil's leading edge to passively modulate the pitch while the foil is harmonically heaved.
(cont.) This model gives the foil passive structural compliance that automatically determines the pitch. The two-way fluid structure interaction thus results from the simultaneous resolution of the fluid and moment equations. This thesis explores the pitch profile and force generation characteristics of the spring-driven, oscillating foil. The passive strategy is found to enhance the propulsive efficiency and thrust production of the flappers specifically in cases where separation is encountered. Furthermore, the passive spring system performs like an ideal actuator that enables the oscillating foil to extract energy from the fluid motion without additional power input. Thus, this is the optimal mechanism to drive the foil dynamics for efficient flight with kinematic flexibility.
by Emily Renee Israeli.
S.M.
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Xiao, Yilong. "A Discontinuous Galerkin Finite Element Method Solution of One-Dimensional Richards’ Equation." The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1461255638.
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Tripathi, Bharat. "Discontinuous Galerkin Method for Propagation of Acoustical Shock Waves in Complex Geometry." Thesis, Paris 6, 2015. http://www.theses.fr/2015PA066344/document.
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Un nouveau code de simulation numérique pour la propagation des ondes de choc acoustiques dans des géométries complexes a été développé. Le point de départ a été la méthode de Galerkin discontinu qui utilise des maillages non structurés (ici des éléments triangulaires), particulièrement adaptés aux géométries complexes. Cependant, cette discrétisation conduit à l'apparition d'oscillation de Gibbs. Pour pallier ce problème, nous avons choisi d'introduire de la viscosité artificielle au voisinage des chocs. Cela a nécessité le développement de trois outils originaux : (i) un nouveau détecteur de choc sensible aux ondes de chocs acoustiques sur des maillages non structurés, (ii) un nouveau terme de viscosité artificielle dans les équations de l'acoustique non linéaire défini élément par élément et (iii) un nouveau terme permettant de régler le niveau de viscosité locale à partir du raidissement des fronts d'onde. Le code de calcul a été utilisé pour étudier deux configurations différentes. La première concerne la réflexion d'ondes de choc acoustiques sur des surfaces rigides. Différents régimes de réflexion ont alors été observés allant, de la réflexion classique de Snell Descartes jusqu'à celui dit de réflexion faible de Von Neumann. La deuxième configuration était consacrée à la focalisation d'ondes de choc acoustiques produites par un transducteur à haute intensité (comme ceux utilisés en HIFU). Un soin particulier a été pris pour étudier le calcul de l'intensité et pour étudier l'interaction entre les ondes de choc et des obstacles placés dans la région du foyerA new numerical solver for the propagation of acoustical shock waves in complex geometry has been developed. This is done starting from the discontinuous Galerkin method. This method is based on unstructured mesh (triangular elements here), and so, naturally it is well-adapted for complex geometries. Nevertheless, the discretization induces Gibbs oscillations. To manage this problem, we choose to introduce some artificial viscosity only in the vicinity of the shocks. This necessitates the development of three original tools. First of all, a new shock sensor for unstructured mesh sensitive to acoustical shock waves has been designed. It senses where the local artificial viscosity has to be introduced thanks to a reformulation of a new element centred smooth artificial viscosity term in the equations. Finally, the amount of viscosity is computed by the introduction of an original notion of gradient factor linked to the steepening of the waveform. The numerical solver has been used to investigate two different physical situations. The first one is the nonlinear reflection of acoustical shock waves on rigid surfaces. Different regimes of reflection have been observed ranging from the linear Snell Descartes reflection to the weak von Neumann case. The second configuration deals with the focusing of shock waves produced by high intensity transducers (like in HIFU). Special attention has been given to the careful computation of intensity and to the interaction between the shock waves and obstacles in the region of the focus
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Chen, Zhiyun. "Coupled space-time discontinuous Galerkin method for dynamical modeling in porous media." Aachen Shaker, 2008. http://d-nb.info/994883005/04.
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Hellwig, Friederike. "Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods." Doctoral thesis, Humboldt-Universität zu Berlin, 2019. http://dx.doi.org/10.18452/20034.
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Die vorliegende Arbeit "Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods" beweist optimale Konvergenzraten für vier diskontinuierliche Petrov-Galerkin (dPG) Finite-Elemente-Methoden für das Poisson-Modell-Problem für genügend feine Anfangstriangulierung. Sie zeigt dazu die Äquivalenz dieser vier Methoden zu zwei anderen Klassen von Methoden, den reduzierten gemischten Methoden und den verallgemeinerten Least-Squares-Methoden. Die erste Klasse benutzt ein gemischtes System aus konformen Courant- und nichtkonformen Crouzeix-Raviart-Finite-Elemente-Funktionen. Die zweite Klasse verallgemeinert die Standard-Least-Squares-Methoden durch eine Mittelpunktsquadratur und Gewichtsfunktionen.
Diese Arbeit verallgemeinert ein Resultat aus [Carstensen, Bringmann, Hellwig, Wriggers 2018], indem die vier dPG-Methoden simultan als Spezialfälle dieser zwei Klassen charakterisiert werden. Sie entwickelt alternative Fehlerschätzer für beide Methoden und beweist deren Zuverlässigkeit und Effizienz.
Ein Hauptresultat der Arbeit ist der Beweis optimaler Konvergenzraten der adaptiven Methoden durch Beweis der Axiome aus [Carstensen, Feischl, Page, Praetorius 2014]. Daraus folgen dann insbesondere die optimalen Konvergenzraten der vier dPG-Methoden.
Numerische Experimente bestätigen diese optimalen Konvergenzraten für beide Klassen von Methoden. Außerdem ergänzen sie die Theorie durch ausführliche Vergleiche beider Methoden untereinander und mit den äquivalenten dPG-Methoden.The thesis "Adaptive Discontinuous Petrov-Galerkin Finite-Element-Methods" proves optimal convergence rates for four lowest-order discontinuous Petrov-Galerkin methods for the Poisson model problem for a sufficiently small initial mesh-size in two different ways by equivalences to two other non-standard classes of finite element methods, the reduced mixed and the weighted Least-Squares method. The first is a mixed system of equations with first-order conforming Courant and nonconforming Crouzeix-Raviart functions. The second is a generalized Least-Squares formulation with a midpoint quadrature rule and weight functions. The thesis generalizes a result on the primal discontinuous Petrov-Galerkin method from [Carstensen, Bringmann, Hellwig, Wriggers 2018] and characterizes all four discontinuous Petrov-Galerkin methods simultaneously as particular instances of these methods. It establishes alternative reliable and efficient error estimators for both methods. A main accomplishment of this thesis is the proof of optimal convergence rates of the adaptive schemes in the axiomatic framework [Carstensen, Feischl, Page, Praetorius 2014]. The optimal convergence rates of the four discontinuous Petrov-Galerkin methods then follow as special cases from this rate-optimality. Numerical experiments verify the optimal convergence rates of both types of methods for different choices of parameters. Moreover, they complement the theory by a thorough comparison of both methods among each other and with their equivalent discontinuous Petrov-Galerkin schemes.
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Yucel, Hamdullah. "Adaptive Discontinuous Galerkin Methods For Convectiondominated Optimal Control Problems." Phd thesis, METU, 2012. http://etd.lib.metu.edu.tr/upload/12614523/index.pdf.
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Many real-life applications such as the shape optimization of technological devices, the identification
of parameters in environmental processes and flow control problems lead to optimization
problems governed by systems of convection diusion partial dierential equations
(PDEs). When convection dominates diusion, the solutions of these PDEs typically exhibit
layers on small regions where the solution has large gradients. Hence, it requires special numerical
techniques, which take into account the structure of the convection. The integration
of discretization and optimization is important for the overall eciency of the solution process.
Discontinuous Galerkin (DG) methods became recently as an alternative to the finite
dierence, finite volume and continuous finite element methods for solving wave dominated
problems like convection diusion equations since they possess higher accuracy.
This thesis will focus on analysis and application of DG methods for linear-quadratic convection
dominated optimal control problems. Because of the inconsistencies of the standard stabilized
methods such as streamline upwind Petrov Galerkin (SUPG) on convection diusion
optimal control problems, the discretize-then-optimize and the optimize-then-discretize do not commute. However, the upwind symmetric interior penalty Galerkin (SIPG) method leads to
the same discrete optimality systems. The other DG methods such as nonsymmetric interior
penalty Galerkin (NIPG) and incomplete interior penalty Galerkin (IIPG) method also yield
the same discrete optimality systems when penalization constant is taken large enough. We
will study a posteriori error estimates of the upwind SIPG method for the distributed unconstrained
and control constrained optimal control problems. In convection dominated optimal
control problems with boundary and/or interior layers, the oscillations are propagated downwind
and upwind direction in the interior domain, due the opposite sign of convection terms in
state and adjoint equations. Hence, we will use residual based a posteriori error estimators to
reduce these oscillations around the boundary and/or interior layers. Finally, theoretical analysis
will be confirmed by several numerical examples with and without control constraints
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Clément, Jean-Baptiste. "Simulation numérique des écoulements en milieu poreux non-saturés par une méthode de Galerkine discontinue adaptative : application aux plages sableuses." Electronic Thesis or Diss., Toulon, 2021. http://www.theses.fr/2021TOUL0022.
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Les écoulements en milieux poreux non-saturés sont modélisés par l'équation de Richards qui est une équation non-linéaire parabolique dégénérée. Ses limites et les défis que soulèvent sa résolution numérique sont présentés. L'obtention de résultats robustes, précis et efficaces est difficile en particulier à cause des fronts de saturation raides et dynamiques induits par les propriétés hydrauliques non-linéaires. L'équation de Richards est discrétisée par une méthode de Galerkine discontinue en espace et des formules de différentiation rétrograde en temps. Le schéma numérique résultant est conservatif, d'ordre élevé et très flexible. Ainsi, des conditions aux limites complexes sont facilement intégrées comme la condition de suintement ou un forçage dynamique. De plus, une stratégie adaptative est proposée. Un pas de temps adaptatif rend la convergence non-linéaire robuste et un raffinement de maillage adaptatif basée sur des blocs est utilisée pour atteindre la précision requise efficacement. Un indicateur d'erreur a posteriori approprié aide le maillage à capturer les fronts de saturation raides qui sont également mieux approximés par une discontinuité introduite dans la solution grâce à une méthode de Galerkine discontinue pondérée. L'approche est validée par divers cas-tests et un benchmark 2D. Les simulations numériques sont comparées à des expériences de laboratoire de recharge/drainage de nappe et une expérience à grande échelle d'humidification, suite à la mise en eau du barrage multi-matériaux de La Verne. Ce cas exigeant montre les potentialités de la stratégie développée dans cette thèse. Enfin, des applications sont menées pour simuler les écoulements souterrains sous la zone de jet de rive de plages sableuses en comparaison avec des observations expérimentalesFlows in unsaturated porous media are modelled by the Richards' equation which is a degenerate parabolic nonlinear equation. Its limitations and the challenges raised by its numerical solution are laid out. Getting robust, accurate and cost-effective results is difficult in particular because of moving sharp wetting fronts due to the nonlinear hydraulic properties. Richards' equation is discretized by a discontinuous Galerkin method in space and backward differentiation formulas in time. The resulting numerical scheme is conservative, high-order and very flexible. Thereby, complex boundary conditions are included easily such as seepage condition or dynamic forcing. Moreover, an adaptive strategy is proposed. Adaptive time stepping makes nonlinear convergence robust and a block-based adaptive mesh refinement is used to reach required accuracy cost-effectively. A suitable a posteriori error indicator helps the mesh to capture sharp wetting fronts which are also better approximated by a discontinuity introduced in the solution thanks to a weighted discontinuous Galerkin method. The approach is checked through various test-cases and a 2D benchmark. Numerical simulations are compared with laboratory experiments of water table recharge/drainage and a largescale experiment of wetting, following reservoir impoundment of the multi-materials La Verne dam. This demanding case shows the potentiality of the strategy developed in this thesis. Finally, applications are handled to simulate groundwater flows under the swash zone of sandy beaches in comparison with experimental observations
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Luo, Luqing. "A Reconstructed Discontinuous Galerkin Method for the Compressible Euler Equations on Arbitrary Grids." NCSU, 2010. http://www.lib.ncsu.edu/theses/available/etd-12032009-162626/.
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A reconstruction-based discontinuous Galerkin (RDG) method is presented for the solution of the compressible Euler equations on arbitrary grids. By taking advantage of handily available and yet invaluable information, namely the derivatives, in the context of the discontinuous Galerkin methods, a polynomial solution of one degree higher is reconstructed using a least-squares method. The stencils used in the reconstruction involve only the von Neumann neighborhood (face-neighboring cells) and are compact and consistent with the underlying DG method. The resulting RDG method can be regarded as an improvement of a recovery-based DG method, in the sense that it shares the same nice features, such as high accuracy and efficiency, and yet overcomes some of its shortcomings such as a lack of flexibility, compactness, and robustness. The developed RDG method is used to compute a variety of flow problems on arbitrary meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical results indicate that this RDG method is third-order accurate at a cost slightly higher than its underlying second-order DG method, at the same time providing a better performance than the third order DG method, in terms of both computing costs and storage requirements.
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Renaud, Adrien. "The Discontinuous Galerkin Material Point Method : Application to hyperbolic problems in solid mechanics." Thesis, Ecole centrale de Nantes, 2018. http://www.theses.fr/2018ECDN0058/document.
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Dans cette thèse, la Méthode des Points Matériels (MPM) est étendue à l’approximation de Galerkin Discontinue (DG) et appliquée aux problèmes hyperboliques en mécanique des solides. La méthode résultante (DGMPM) a pour objectif de suivre précisément les ondes dans des solides subissant de fortes déformations et dont les modèles constitutifs dépendent de l’histoire du chargement. A la croisée des méthodes de types éléments finis et volumes finis, la DGMPM s’appuie sur une grille de calcul arbitraire dans laquelle des flux sont calculés au moyen de solveurs de Riemann approximés sur les arêtes entre les éléments. L’intérêt de ce type de solveurs est qu’ils permettent l’introduction de la structure caractéristique des solutions des équations aux dérivées partielles hyperboliques directement dans le schéma numérique. Les analyses de stabilité et de convergence ainsi que l’illustration de la méthode sur des simulations de problèmes unidimensionnels et bidimensionnels montrent que le schéma numérique permet d’améliorer le suivi des ondes par rapport à la MPM. Par ailleurs, un deuxième objectif poursuivi dans cette thèse consiste à caractériser la réponse des solides élastoplastiques à des sollicitations dynamiques en deux dimensions en vue d’améliorer la résolution numérique de ces problèmes. Bien qu’un certain nombre de travaux aient déjà été menés dans cette direction, les problèmes étudiés se limitent à des cas particuliers. Un cadre unifié pour l’étude de la propagation d’ondes simples dans les solides élastoplastiques en déformations et contraintes plane est proposé dans cette thèse. Les trajets de chargement suivis à l’intérieur de ces ondes simples sont de plus analysésIn this thesis, the material point method (MPM) is extended to the discontinuous Galerkin approximation (DG) and applied to hyperbolic problems in solid mechanics. The resulting method (DGMPM) aims at accurately following waves in finite-deforming solids whose constitutive models may depend on the loading history. Merging finite volumes and finite elements methods, the DGMPM takes advantage of an arbitrary computational grid in which fluxes are evaluated at element faces by means of approximate Riemann solvers. This class of solvers enables the introduction of the characteristic structure of the solutions of hyperbolic partial differential equations within the numerical scheme. Convergence and stability analyses, along with one and two-dimensional numerical simulations,demonstrate that this approach enhances the MPM ability to track waves. On the other hand, a second purpose has been followed: it consists in identifying the response of two-dimensional elastoplastic solids to dynamic step-loadings in order to improve numerical results on these problems. Although some studies investigated similar questions, only particular cases have been treated. Thus,a generic framework for the study of the propagation of simple waves in elastic-plastic solids under plane stress and plane strain problems is proposed in this thesis. The loading paths followed inside those simple waves are further analyzed
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Föll, Fabian [Verfasser]. "A Discontinuous Galerkin Method for Transcritical Multicomponent Flows with Phase Transition / Fabian Föll." München : Verlag Dr. Hut, 2021. http://d-nb.info/1232847895/34.
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Dosopoulos, Stylianos. "Interior Penalty Discontinuous Galerkin Finite Element Method for the Time-Domain Maxwell's Equations." The Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1337787922.
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Sun, Weizhou. "LOCAL DISCONTINUOUS GALERKIN METHOD FOR KHOKHLOV-ZABOLOTSKAYA-KUZNETZOV EQUATION AND IMPROVED BOUSSINESQ EQUATION." The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1480327264817905.
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Cai, Zhemin. "A High-order Discontinuous Galerkin Method for Simulating Incompressible Fluid-Thermal-Structural Problems." Thesis, The University of Sydney, 2018. http://hdl.handle.net/2123/20961.
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The use of discontinuous Galerkin (DG) methods to solve fluid thermal structure interaction problems in numerical modelling is known to offer several advantages. In particular, DG methods provide the flexibility of using different approximations in different elements, which makes the methods ideal for hp-adaptivity. The first objective of this thesis is to present a framework for the computation of fluid thermal structure interaction problems within both the single and multi-solid domain using DG methods on unstructured grids. The full solver consists of four main components: the incompressible fluid solver, the conjugate heat transfer solver, the linear elastic solver and the fluid to structure interaction solver. Based on an earlier developed DG solver for the incompressible Navier-Stokes equation, the fluid advection-diffusion equation, the Boussinesq term, the solid heat equation and the linear elastic equation are introduced using an explicit DG formulation. A Dirichlet-Neumann partitioning strategy has been implemented to achieve the data exchange process via the numerical flux of interface quadrature points in the fluid-solid interface. Formal h and p convergence studies employing the method of manufactured solutions demonstrate that the expected order of accuracy is achieved. Computational effort is documented in detail demonstrating precisely that for all cases the highest order accurate algorithm has several magnitudes lower error than lower-order schemes for a given computational effort. Secondly, this thesis has proposed a detailed compact thermoelectric cooler (TEC) modelling method based on an existing black box like compact TEC model. Close comparisons validate that both the detailed and the black box like compact model are accurate enough to simulate the conduction only case. When air convection is required to carry out a system-level thermal management optimization, the detailed compact modelling method is more reliable.
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Zeifang, Jonas [Verfasser]. "A Discontinuous Galerkin Method for Droplet Dynamics in Weakly Compressible Flows / Jonas Zeifang." München : Verlag Dr. Hut, 2020. http://d-nb.info/1219321540/34.
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Nappi, Angela. "Development and Application of a Discontinuous Galerkin-based Wave Prediction Model." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1385998191.
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Moon, Kihyo. "Immersed Discontinuous Galerkin Methods for Acoustic Wave Propagation in Inhomogeneous Media." Diss., Virginia Tech, 2016. http://hdl.handle.net/10919/70906.
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We present immersed discontinuous Galerkin finite element methods for one and two dimensional acoustic wave propagation problems in inhomogeneous media where elements are allowed to be cut by the material interface. The proposed methods use the standard discontinuous Galerkin finite element formulation with polynomial approximation on elements that contain one fluid while on interface elements containing more than one fluid they use specially-built piecewise polynomial shape functions that satisfy appropriate interface jump conditions. The finite element spaces on interface elements satisfy physical interface conditions from the acoustic problem in addition to extended conditions derived from the system of partial differential equations. Additional curl-free and consistency conditions are added to generate bilinear and biquadratic piecewise shape functions for two dimensional problems. We established the existence and uniqueness of one dimensional immersed finite element shape functions and existence of two dimensional bilinear immersed finite element shape functions for the velocity.
The proposed methods are tested on one dimensional problems and are extended to two dimensional problems where the problem is defined on a domain split by an interface into two different media. Our methods exhibit optimal $O(h^{p+1})$ convergence rates for one and two dimensional problems. However it is observed that one of the proposed methods is not stable for two dimensional interface problems with high contrast media such as water/air. We performed an analysis to prove that our immersed Petrov-Galerkin method is stable for interface problems with high jumps across the interface. Local time-stepping and parallel algorithms are used to speed up computation.
Several realistic interface problems such as ether/glycerol, water/methyl-alcohol and water/air with a circular interface are solved to show the stability and robustness of our methods.Ph. D.
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Hoffmann, Malte [Verfasser]. "An Explicit Discontinuous Galerkin Method for Parallel Compressible Two-Phase Flow Simulations / Malte Hoffmann." München : Verlag Dr. Hut, 2017. http://d-nb.info/1149580283/34.
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Morel, Guillaume. "Asymptotic-preserving and well-balanced schemes for transport models using Trefftz discontinuous Galerkin method." Thesis, Sorbonne université, 2018. http://www.theses.fr/2018SORUS556.
Full textAbstract:
Cette thèse traite de l’étude et de l’analyse d’un schéma de type Trefftz Galerkin discontinu (TDG) pour un problème modèle de transport avec relaxation linéaire. Nous montrons que la méthode TDG fournie naturellement des discrétisations bien équilibrées et asymptotic-preserving puisque des solutions exactes, éventuellement non polynomiales, sont utilisées localement dans les fonctions de base. En particulier, la formulation de la méthode du TDG est donnée dans le cas général des systèmes de Friedrichs. En pratique, une attention particulière est consacrée à l’approximation PN de l’équation de transport. Pour ce modèle bidimensionnel, des fonctions de base polynomiales et exponentielles sont construites et la convergence du schéma est étudiée. Les exemples numériques sur les modèles P1 et P3 montrent que la méthode TDG surpasse la méthode Galerkin discontinue standard pour certains tests avec termes source raides. En particulier, la méthode TDG permet d’obtenir des schémas efficaces pour capturer les couches limites et la limite de diffusion de l’équation de transportSome solutions to the transport equation admit a diffusion limit and boundary layers which may be very costly to approximate with naive numerical methods. To address these issues, a possible approach is to consider well-balanced (WB) and asymptotic-preserving (AP) schemes. Such schemes are known, in some cases, to greatly improve the numerical solution on coarse meshes. This thesis deals with the study and analysis of a Trefftz Discontinuous Galerkin (TDG) scheme for a model problem of transport with linear relaxation. We show that natural well-balanced and asymptotic-preserving discretization are provided by the TDG method since exact solutions, possibly non-polynomials, are used locally in the basis functions. In particular, the formulation of the TDG method for the general case of Friedrichs systems is given. For the practical examples, a special attention is devoted to the PN approximation of the transport equation. For this two dimensional model, polynomial and exponential basis functions are constructed and the convergence of the scheme is studied. Numerical examples on the P1 and P3 models show that the TDG method outperforms the standard discontinuous Galerkin method when considering stiff coefficients. In particular, the TDG method leads to efficient schemes to capture boundary layers and the diffusion limit of the transport equation