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Ozisik, Sevtap. "Fully Computable Convergence Analysis Of Discontinous Galerkin Finite Element Approximation With An Arbitrary Number Of Levels Of Hanging Nodes." Phd thesis, METU, 2012. http://etd.lib.metu.edu.tr/upload/12614345/index.pdf.
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In this thesis, we analyze an adaptive discontinuous finite element method for symmetric
second order linear elliptic operators. Moreover, we obtain a fully computable convergence
analysis on the broken energy seminorm in first order symmetric interior penalty discontin-
uous Galerkin finite element approximations of this problem. The method is formulated on
nonconforming meshes made of triangular elements with first order polynomial in two di-
mension. We use an estimator which is completely free of unknown constants and provide a
guaranteed numerical bound on the broken energy norm of the error. This estimator is also
shown to provide a lower bound for the broken energy seminorm of the error up to a constant
and higher order data oscillation terms. Consequently, the estimator yields fully reliable,
quantitative error control along with efficiency.
As a second problem, explicit expression for constants of the inverse inequality are given in
1D, 2D and 3D. Increasing mathematical analysis of finite element methods is motivating the
inclusion of mesh dependent terms in new classes of methods for a variety of applications.
Several inequalities of functional analysis are often employed in convergence proofs. Inverse
estimates have been used extensively in the analysis of finite element methods. It is char-
acterized as tools for the error analysis and practical design of finite element methods with
terms that depend on the mesh parameter. Sharp estimates of the constants of this inequality
is provided in this thesis.
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Nytra, Jan. "Řešení problémů akustiky pomocí nespojité Galerkinovy metody." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2015. http://www.nusl.cz/ntk/nusl-232174.
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Parciální diferenciální rovnice hrají důležitou v inženýrských aplikacích. Často je možné tyto rovnice řešit pouze přibližně, tj. numericky. Z toho důvodu vzniklo množství diskretizačních metod pro řešení těchto rovnic. Uvedená nespojitá Galerkinova metoda se zdá jako velmi obecná metoda pro řešení těchto rovnic, především pak pro hyperbolické systémy. Naším cílem je řešit úlohy aeroakustiky, přičemž šíření akustických vln je popsáno pomocí linearizovaných Eulerových rovnic. A jelikož se jedná o hyperbolický systém, byla vybrána právě nespojitá Galerkinova metoda. Mezi nejdůležitější aspekty této metody patří schopnost pracovat s geometricky složitými oblastmi, možnost dosáhnout metody vysokého řádu a dále lokální charakter toho schématu umožnuje efektivní paralelizaci výpočtu. Nejprve uvedeme nespojitou Galerkinovu metodu v obecném pojetí pro jedno- a dvoudimenzionalní úlohy. Algoritmus následně otestujeme pro řešení rovnice advekce, která byla zvolena jako modelový případ hyperbolické rovnice. Metoda nakonec bude testována na řadě verifikačních úloh, které byly formulovány pro testování metod pro výpočetní aeroakustiku, včetně oveření okrajových podmínek, které, stejně jako v případě teorie proudění tekutin, jsou nedílnou součástí výpočetní aeroakustiky.
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Bonnasse-Gahot, Marie. "Simulation de la propagation d'ondes élastiques en domaine fréquentiel par des méthodes Galerkine discontinues." Thesis, Nice, 2015. http://www.theses.fr/2015NICE4125/document.
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Le contexte scientifique de cette thèse est l'imagerie sismique dont le but est de reconstituer la structure du sous-sol de la Terre. Comme le forage a un coût assez élevé, l'industrie pétrolière s'intéresse à des méthodes capables de reconstituer les images de la structure terrestre interne avant de le faire. La technique d'imagerie sismique la plus utilisée est la technique de sismique-réflexion qui est basée sur le modèle de l'équation d'ondes. L'imagerie sismique est un problème inverse qui requiert de résoudre un grand nombre de problèmes directs. Dans ce contexte, nous nous intéressons dans cette thèse à la résolution du problème direct en régime harmonique, soit à la résolution des équations d'Helmholtz. L'objectif principal est de proposer et de développer un nouveau type de solveur élément fini (EF) caractérisé par un opérateur discret de taille réduite (comparée à la taille des solveurs déjà existants) sans pour autant altérer la précision de la solution numérique. Nous considérons les méthodes de Galerkine discontinues (DG). Comme les méthodes DG classiques sont plus coûteuses que les méthodes EF continues si l'on considère un même problème à cause d'un grand nombre de degrés de liberté couplés, résultat des approximations discontinues, nous développons une nouvelle classe de méthode DG réduisant ce problème : la méthode DG hybride (HDG). Pour valider l'efficacité de la méthode HDG proposée, nous comparons les résultats obtenus avec ceux obtenus avec une méthode DG basée sur des flux décentrés en 2D. Comme l'industrie pétrolière s'intéresse au traitement de données réelles, nous développons ensuite la méthode HDG pour les équations élastiques d'Helmholtz 3DThe scientific context of this thesis is seismic imaging which aims at recovering the structure of the earth. As the drilling is expensive, the petroleum industry is interested by methods able to reconstruct images of the internal structures of the earth before the drilling. The most used seismic imaging method in petroleum industry is the seismic-reflection technique which uses a wave equation model. Seismic imaging is an inverse problem which requires to solve a large number of forward problems. In this context, we are interested in this thesis in the modeling part, i.e. the resolution of the forward problem, assuming a time-harmonic regime, leading to the so-called Helmholtz equations. The main objective is to propose and develop a new finite element (FE) type solver characterized by a reduced-size discrete operator (as compared to existing such solvers) without hampering the accuracy of the numerical solution. We consider the family of discontinuous Galerkin (DG) methods. However, as classical DG methods are much more expensive than continuous FE methods when considering steady-like problems, because of an increased number of coupled degrees of freedom as a result of the discontinuity of the approximation, we develop a new form of DG method that specifically address this issue: the hybridizable DG (HDG) method. To validate the efficiency of the proposed HDG method, we compare the results that we obtain with those of a classical upwind flux-based DG method in a 2D framework. Then, as petroleum industry is interested in the treatment of real data, we develop the HDG method for the 3D elastic Helmholtz equations
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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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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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Elfverson, Daniel. "On discontinuous Galerkin multiscale methods." Licentiate thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-200260.
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In this thesis a new multiscale method, the discontinuous Galerkin multiscale method, is proposed. The method uses localized fine scale computations to correct a global coarse scale equation and thereby takes the fine scale features into account. We show a priori error bounds for convection dominated diffusion-convection-reaction problems with variable coefficients. We also present a posteriori error bound in the case of no convection or reaction and present an adaptive algorithm which tunes the method parameters automatically. We also present extensive numerical experiments which verify our analytical findings.
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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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Casoni, Rero Eva. "Shock capturing for discontinuous Galerkin methods." Doctoral thesis, Universitat Politècnica de Catalunya, 2011. http://hdl.handle.net/10803/51571.
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Aquesta tesi doctoral proposa formulacions de Galerkin Discontinu (DG) d’alt ordre per la captura de shocks, obtenint alhora
solucions altament precises per problemes de flux compressible.
En les últimes dècades, la investigació en els mètodes de DG ha estat en constant creixement. L'èxit dels mètodes DG en
problemes hiperbòlics ha conduit el seu desenvolupament en lleis de conservació no lineals i problemes de convecció dominant.
Entre els avantatges dels mètodes DG, destaquen la seva estabilitat inherent i les propietats locals de conservació. D'altra banda,
els mètodes DG estan especialment dissenyats per l’ús aproximacions d'ordre superior. De fet, en els últims anys s'ha demostrat
que la resolució de problemes de convecció dominant ja no es restringeix només a elements d'ordre inferior. De fet, es necessiten
models numèrics d'alta precisió per aconseguir prediccions altament fiables dins la dinàmica de fluids computacional (CFD). En
aquest context es presenten i discuteixen dos tècniques de captura de shocks.
En primer lloc, es presenta una tècnica novedosa i senzilla basada en la introducció d'una nova base de funcions de forma. Aquesta
base té la capacitat de canviar a nivell local entre una interpolació contínua o discontínua, depenent de la suavitat de la funció que
es vol aproximar. En presència de xocs, les discontinuïtats introduïdes dins l’element permeten incloure l'estabilització necessària
gràcies a l’ús dels fluxos numèrics, i alhora exploten les propietats intrínsiques del mètodes DG. En conseqüència, es poden utilitzar
malles grolleres amb elements d’ordre superior. Amb aquestes discretitzacions i, utilitzant el mètode proposats, els xocs queden
continguts a l’interior de l’element i per tant, és possible evitar l’ús de tècniques de refinament adaptatiu de la malla, alhora que es
manté la localitat i compacitat dels esquemes DG.
En segon lloc, es proposa una tècnica clàssica i, aparentment simple: la introducció de la viscositat artificial. Primerament es realitza
un estudi detallat per al cas unidimensional. S’obté una viscositat d’alta precisió que escala segons el valor hk amb 1 ≤ k ≤ p i essent
h la mida de l’element. En conseqüència, s’obté un xoc amb amplitud del mateix ordre. Seguidament, l'estudi de la viscositat
unidimensional obtenida s'extén al cas multidimensional per a malles triangulars. L'extensió es basa en la projecció de la viscositat
unidimensional en unes determinades direccions espacials dins l’element. Es demostra de manera consistent que la viscositat
introduïda és, com a molt, del mateix ordre que la resolució donada per la discretització espacial, és a dir, h/p. El mètode és
especialment eficient per aproximacions de Galerkin discontinu d’alt ordre, per exemple p≥ 3.
Les dues metodologies es validen mitjançant una àmplia selecció d’exemples numèrics. En alguns exemples, els mètodes
proposats permeten una reducció en el nombre de graus de llibertat necessaris per capturar xocs acuradament de fins i tot un ordre
de magnitud, en comparació amb mètodes estàndar de refinament adaptatiu amb aproximacions de baix ordre.This thesis proposes shock-capturing methods for high-order Discontinuous Galerkin (DG) formulations providing highly accurate solutions for compressible flows. In the last decades, research in DG methods has been very active. The success of DG in hyperbolic problems has driven many studies for nonlinear conservation laws and convection-dominated problems. Among all the advantages of DG, their inherent stability and local conservation properties are relevant. Moreover, DG methods are naturally suited for high-order approximations. Actually, in recent years it has been shown that convection-dominated problems are no longer restricted to low-order elements. In fact, highly accurate numerical models for High-Fidelity predictions in CFD are necessary. Under this rationale, two shock-capturing techniques are presented and discussed. First, a novel and simple technique based on on the introduction of a new basis of shape functions is presented. It has the ability to change locally between a continuous or discontinuous interpolation depending on the smoothness of the approximated function. In the presence of shocks, the new discontinuities inside an element introduce the required stabilization thanks to the numerical fluxes, thus exploiting DG inherent properties. Large high-order elements can therefore be used and shocks are captured within a single element, avoiding adaptive mesh refinement and preserving the locality and compactness of the DG scheme. Second, a classical and, apparently simple, technique is advocated: the introduction of artificial viscosity. First, a one-dimensional study is perfomed. Viscosity of the order O(hk) with 1≤ k≤ p is obtained, hence inducing a shock width of the same order. Second, the study extends the accurate one-dimensional viscosity to triangular multidimensional meshes. The extension is based on the projection of the one-dimensional viscosity into some characteristic spatial directions within the elements. It is consistently shown that the introduced viscosity scales, at most, withthe DG resolutions length scales, h/p. The method is especially reliable for highorder DG approximations, say p≥3. A wide range of different numerical tests validate both methodologies. In some examples the proposed methods allow to reduce by an order of magnitude the number of degrees of freedom necessary to accurately capture the shocks, compared to standard low order h-adaptive approaches.
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Dong, Zhaonan. "Discontinuous Galerkin methods on polytopic meshes." Thesis, University of Leicester, 2017. http://hdl.handle.net/2381/39140.
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This thesis is concerned with the analysis and implementation of the hp-version interior penalty discontinuous Galerkin finite element method (DGFEM) on computational meshes consisting of general polygonal/polyhedral (polytopic) elements. Two model problems are considered: general advection-diffusion-reaction boundary value problems and time dependent parabolic problems. New hp-version a priori error bounds are derived based on a specific choice of the interior penalty parameter which allows for edge/face-degeneration as well as an arbitrary number of faces and hanging nodes per element. The proposed method employs elemental polynomial bases of total degree p (Pp- bases) defined in the physical coordinate system, without requiring mapping from a given reference or canonical frame. A series of numerical experiments highlighting the performance of the proposed DGFEM are presented. In particular, we study the competitiveness of the p-version DGFEM employing a Pp-basis on both polytopic and tensor-product elements with a (standard) DGFEM and FEM employing a (mapped) Qp-basis. Moreover, a careful theoretical analysis of optimal convergence rate in p for Pp-basis is derived for several commonly used projectors, which leads to sharp bounds of exponential convergence with respect to degrees of freedom (dof) for the Pp-basis.
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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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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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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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Gryngarten, Leandro Damian. "Multi-phase flows using discontinuous Galerkin methods." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/45824.
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This thesis is concerned with the development of numerical techniques to simulate compressible multi-phase flows, in particular a high-accuracy numerical approach with mesh adaptivity. The Discontinuous Galerkin (DG) method was chosen as the framework for this work for being characterized for its high-order of accuracy -thus low numerical diffusion- and being compatible with mesh adaptivity due to its locality. A DG solver named DiGGIT (Discontinuous Galerkin at the Georgia Institute of Technology) has been developed and several aspects of the method have been studied. The Local Discontinuous Galerkin (LDG) method -an extension of DG for equations with high-order derivatives- was extended to solve multiphase flows using Diffused Interface Methods (DIM). This multi-phase model includes the convection of the volume fraction, which is treated as a Hamilton-Jacobi equation. This is the first study, to the author's knowledge, in which the volume fraction of a DIM is solved using the DG and the LDG methods. The formulation is independent of the Equation of State (EOS) and it can differ for each phase. This allows for a more accurate representation of the different fluids by using cubic EOSs, like the Peng-Robinson and the van der Waals models. Surface tension is modeled with a new numerical technique appropriate for LDG. Spurious oscillations due to surface tension are common to all the capturing schemes, and this new approach presents oscillations comparable in magnitude to the most common schemes. The moment limiter (ML) was generalized for non-uniform grids with hanging nodes that result from adaptive mesh refinement (AMR). The effect of characteristic, primitive, or conservative decomposition in the limiting stage was studied. The characteristic option cannot be used with the ML in multi-dimensions. In general, primitive variable decomposition is a better option than with conservative variables, particularly for multiphase flows, since the former type of decomposition reduces the numerical oscillations at material discontinuities. An additional limiting technique was introduced for DIM to preserve positivity while minimizing the numerical diffusion, which is especially important at the interface. The accuracy-preserving total variation diminishing (AP-TVD) marker for ``troubled-cell' detection, which uses an averaged-derivative basis, was modified to use the Legendre polynomial basis. Given that the latest basis is generally used for DG, the new approach avoids transforming to the averaged-derivative basis, what results in a more efficient technique.
Furthermore, a new error estimator was proposed to determine where to refine or coarsen the grid. This estimator was compared against other estimator used in the literature and it showed an improved performance. In order to provide equal order of accuracy in time as in space, the commonly used 3rd-order TVD Runge-Kutta (RK) scheme in the DG method was replaced in some cases by the Spectral Deferred Correction (SDC) technique. High orders in time were shown to only be required when the error in time is significant. For instance, convection-dominated compressible flows require for stability a time step much smaller than is required for accuracy, so in such cases 3rd-order TVD RK resulted to be more efficient than SDC with higher orders.
All these new capabilities were included in DiGGIT and have provided a generalized approach capable of solving sub- and super-critical flows at sub- and super-sonic speeds, using a high-order scheme in space and time, and with AMR.
Canonical test cases are presented to verify and validate the formulation in one, two, and three dimensions. Finally, the solver is applied to practical applications. Shock-bubble interaction is studied and the effect of the different thermodynamic closures is assessed. Interaction between single-drops and a wall is simulated. Sticking and the onset of splashing are observed. In addition, the solver is used to simulate turbulent flows, where the high-order of accuracy clearly shows its benefits. Finally, the methodology is challenged with the simulation of a liquid jet in cross flow.
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Kanschat, Guido. "Discontinuous Galerkin methods for viscous incompressible fl." Wiesbaden Dt. Univ.-Verl, 2004. http://dx.doi.org/10.1007/978-3-8350-5519-3.
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Kanschat, Guido. "Discontinuous Galerkin methods for viscous incompressible flow." Wiesbaden : Deutscher Universitäts-Verlag, 2008. http://dx.doi.org/10.1007/978-3-8350-5519-3.
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Kanschat, Guido. "Discontinuous Galerkin methods for viscous incompressible flow." Wiesbaden : Dt. Univ.-Verl, 2004. http://www.myilibrary.com?id=134464.
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Sabawi, Younis Abid. "Adaptive discontinuous Galerkin methods for interface problems." Thesis, University of Leicester, 2017. http://hdl.handle.net/2381/39386.
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The aim of this thesis is to derive adaptive methods for discontinuous Galerkin approximations for both elliptic and parabolic interface problems. The derivation of adaptive method, is usually based on a posteriori error estimates. To this end, we present a residual-type a posteriori error estimator for interior penalty discontinuous Galerkin (dG) methods for an elliptic interface problem involving possibly curved interfaces, with flux-balancing interface conditions, e.g., modelling mass transfer of solutes through semi-permeable membranes. The method allows for extremely general curved element shapes employed to resolve the interface geometry exactly. Respective upper and lower bounds of the error in the respective dG-energy norm with respect to the estimator are proven. The a posteriori error bounds are subsequently used to prove a basic a priori convergence result. Moreover, a contraction property for a standard adaptive algorithm utilising these a posteriori bounds, with a bulk refinement criterion is also shown, thereby proving that the a posteriori bounds can lead to a convergent adaptive algorithm subject to some mesh restrictions. This work is also concerned with the derivation of a new L1∞(L2)-norm a posteriori error bound for the fully discrete adaptive approximation for non-linear interface parabolic problems. More specifically, the time discretization uses the backward Euler Galerkin method and the space discretization uses the interior penalty discontinuous Galerkin finite element method. The key idea in our analysis is to adapt the elliptic reconstruction technique, introduced by Makridakis and Nochetto [48], enabling us to use the a posteriori error estimators derived for elliptic interface models and to obtain optimal order in both L1∞(L2) and L1∞(L2) + L2(H¹) norms. The effectiveness of all the error estimators and the proposed algorithms is confirmed through a series of numerical experiments.
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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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Merton, Simon Richard. "Discontinuous Galerkin methods for computational radiation transport." Thesis, Imperial College London, 2012. http://hdl.handle.net/10044/1/9906.
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This Thesis demonstrates advanced new discretisation technologies that improve the accuracy and stability of the discontinuous Galerkin finite element method applied to the Boltzmann Transport Equation, describing the advective transport of neutral particles such as photons and neutrons within a domain. The discontinuous Galerkin method in its standard form is susceptible to oscillation detrimental to the solution. The discretisation schemes presented in this Thesis enhance the basic form with linear and non-linear Petrov Galerkin methods that remove these oscillations. The new schemes are complemented by an adjoint-based error recovery technique that improves the standard solution when applied to goal-based functional and eigenvalue problems. The chapters in this Thesis have been structured to be submitted individually for journal publication, and are arranged as follows. Chapter 1 outlines the Thesis and contains a brief literature review. Chapter 2 introduces the underlying space-angle discretisation method used in the work, and discusses a series of potential modifications to the standard discontinuous Galerkin method. These differ in how the upwinding is performed on the element boundary, and comprise an upwind-average method, a Petrov-Galerkin method that removes oscillation by adding artificial diffusion internal to an element and a more sophisticated Petrov-Galerkin scheme that adds dissipation in the coupling between each element. These schemes are tested in one-dimension and Taylor analysis of their convergence rate is included. The chapter concludes with selection of one of the schemes to be developed in the next part of the Thesis. Chapter 3 develops the selected method extending it to multi-dimensions. The result is a new discontinuous Petrov-Galerkin method that is residual based and removes unwanted oscillation from the transport solution by adding numerical dissipation internal to an element. The method uses a common length scale in the upwind term for all elements. This is not always satisfactory, however, as it gives the same magnitude and type of dissipation everywhere in the domain. The chapter concludes by recommending some form of non-linearity be included to address this issue. Chapter 4 adds non-linearity to the scheme. This projects the streamline direction, in which the dissipation acts, onto the solution gradient direction. It defines locally the optimal amount of dissipation needed in the discretisation. The non-linear scheme is tested on a variety of steady-state and time-dependent transport problems. Chapters 5, 6 and 7 develop an adjoint-based error measure to complement the scheme in functional and eigenvalue problems. This is done by deriving an approximation to the error in the the bulk functional or eigenvalue, and then removing it from the calculated value in a post-process defect iteration. This is shown to dramatically accelerate mesh convergence of the goal-based functional or eigenvalue. Chapter 8 concludes the Thesis with recommendations for a further plan of work.
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Hadi, Justin. "Discontinuous Galerkin methods for hyperbolic conservation laws." Thesis, Imperial College London, 2012. http://hdl.handle.net/10044/1/10563.
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New numerical methods are developed for single phase compressible gas flow and two phase gas/liquid flow in the framework of the discontinuous Galerkin finite element method (DGFEM) and applied to Riemann problems. A residual based diffusion scheme inspired by the streamline upwind Petrov-Galerkin method (SUPG) of Brookes and Hughes [15] is applied to the Euler equations of gas dynamics and the single pressure incompressible liquid/compressible gas flow system of Toumi and Kumbaro [137]. A Roe [119] based approximate Riemann solver is applied. To minimise unstable overshoots, diffusivity is added in the direction of the gradient of the solution as opposed to the direction of the streamlines in SUPG for the continuous finite element method (CFEM). The methods are tested on Cartesian meshes with scalar advection problems, the computationally challenging Sod shock tube and Lax Riemann problems, explosion problems in gas dynamics and the water faucet test and explosion problems in two phase flow. An extension to two dimensions and comparisons to existing methods are made. A framework for the well posedness of two phase flow equations is posited and virtual mass terms are added to the two phase flow equations of Toumi and Kumbaro to ensure hyperbolicity. A viscous path based Roe solver for DGFEM is applied mirroring the method of Toumi and Kumbaro in a framework for discontinuous solutions.
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Madhavan, Pravin. "Analysis of discontinuous Galerkin methods on surfaces." Thesis, University of Warwick, 2014. http://wrap.warwick.ac.uk/66157/.
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In this thesis, we extend the discontinuous Galerkin framework to surface partial differential equations. This is done by deriving both a priori and a posteriori error estimates for model elliptic problems posed on compact smooth and oriented surfaces in R3, and investigating both theoretical estimates and several generalisations numerically.
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Hall, Edward John Cumes. "Anisotropic adaptive refinement for discontinuous Galerkin methods." Thesis, University of Leicester, 2007. http://hdl.handle.net/2381/30536.
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We consider both the a priori and a posteriori error analysis and hp-adaptation strategies for discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes with anisotropically enriched polynomial degrees. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution, exploiting duality based arguments.;The a priori error analysis is carried out in two settings. In the first, full orientation of elements is allowed but only (possibly high-order) isotropic polynomial degrees considered; our analysis, therefore, extends previous results, where only finite element spaces comprising piecewise linear polynomials were considered, by utilizing techniques from tensor analysis. In the second case, anisotropic polynomial degrees are allowed, but the elements are assumed to be axiparallel; we thus apply previously known interpolation error results to the goal-oriented setting.;Based on our a posteriori error bound we first design and implement an adaptive anisotropic h-refinement algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement, chosen on a competitive basis requiring the solution of local problems. The superiority of the proposed algorithm in comparison with a standard h-isotropic mesh refinement algorithm and a Hessian based h-anisotropic adaptive procedure is illustrated by a series of numerical experiments. We then describe a fully hp -adaptive algorithm, once again using a competitive refinement approach, which, numerical experiments reveal, offers considerable improvements over both a standard hp-isotropic refinement algorithm and an h-anisotropic/p-isotropic adaptive procedure.
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SINGH, ONKAR DEEP. "ITERATIVE SOLVERS FOR DISCONTINUOUS GALERKIN FINITE ELEMENT METHODS." University of Cincinnati / OhioLINK, 2004. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1093023928.
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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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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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Of, Günther, Gregory J. Rodin, Olaf Steinbach, and Matthias Taus. "Coupling Methods for Interior Penalty Discontinuous Galerkin Finite Element Methods and Boundary Element Methods." Universitätsbibliothek Chemnitz, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-96885.
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This paper presents three new coupling methods for interior penalty discontinuous Galerkin finite element methods and boundary element methods. The new methods allow one to use discontinuous basis functions on the interface between the subdomains represented by the finite element and boundary element methods. This feature is particularly important when discontinuous Galerkin finite element methods are used. Error and stability analysis is presented for some of the methods. Numerical examples suggest that all three methods exhibit very similar convergence properties, consistent with available theoretical results.
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Altmann, Christoph [Verfasser]. "Explicit Discontinuous Galerkin Methods for Magnetohydrodynamics / Christoph Altmann." München : Verlag Dr. Hut, 2012. http://d-nb.info/1029400253/34.
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Villardi, de Montlaur Adeline de. "High-order discontinuous Galerkin methods for incompressible flows." Doctoral thesis, Universitat Politècnica de Catalunya, 2009. http://hdl.handle.net/10803/5928.
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Aquesta tesi doctoral proposa formulacions de Galerkin discontinu (DG) d'alt ordre per fluxos viscosos incompressibles. Es desenvolupa un nou mètode de DG amb penalti interior (IPM-DG), que condueix a una forma feble simètrica i coerciva pel terme de difusió, i que permet assolir una aproximació espacial d'alt ordre. Aquest mètode s'aplica per resoldre les equacions de Stokes i Navier-Stokes. L'espai d'aproximació de la velocitat es descompon dins de cada element en una part solenoidal i una altra irrotacional, de manera que es pot dividir la forma dèbil IPM-DG en dos problemes desacoblats. El primer permet el càlcul de les velocitats i de les pressions híbrides, mentre que el segon calcula les pressions en l'interior dels elements. Aquest desacoblament permet una reducció important del número de graus de llibertat tant per velocitat com per pressió. S'introdueix també un paràmetre extra de penalti resultant en una formulació DG alternativa per calcular les velocitats solenoidales, on les pressions no apareixen. Les pressions es poden calcular com un post-procés de la solució de les velocitats. Es contemplen altres formulacions DG, com per exemple el mètode Compact Discontinuous Galerkin, i es comparen al mètode IPM-DG.
Es proposen mètodes implícits de Runge-Kutta d'alt ordre per problemes transitoris incompressibles, permetent obtenir esquemes incondicionalment estables i amb alt ordre de precisió temporal. Les equacions de Navier-Stokes incompressibles transitòries s'interpreten com un sistema de Equacions Algebraiques Diferencials, és a dir, un sistema d'equacions diferencials ordinàries corresponent a la equació de conservació del moment, més les restriccions algebraiques corresponent a la condició d'incompressibilitat.
Mitjançant exemples numèrics es mostra l'aplicabilitat de les metodologies proposades i es comparen la seva eficiència i precisió.
This PhD thesis proposes divergence-free Discontinuous Galerkin formulations providing high orders of accuracy for incompressible viscous flows.
A new Interior Penalty Discontinuous Galerkin (IPM-DG) formulation is developed, leading to a symmetric and coercive bilinear weak form for the diffusion term, and achieving high-order spatial approximations. It is applied to the solution of the Stokes and Navier-Stokes equations. The velocity approximation space is decomposed in every element into a solenoidal part and an irrotational part. This allows to split the IPM weak form in two uncoupled problems. The first one solves for velocity and hybrid pressure, and the second one allows the evaluation of pressures in the interior of the elements. This results in an important reduction of the total number of degrees of freedom for both velocity and pressure.
The introduction of an extra penalty parameter leads to an alternative DG formulation for the computation of solenoidal velocities with no presence of pressure terms. Pressure can then be computed as a post-process of the velocity solution. Other DG formulations, such as the Compact Discontinuous Galerkin method, are contemplated and compared to IPM-DG.
High-order Implicit Runge-Kutta methods are then proposed to solve transient incompressible problems, allowing to obtain unconditionally stable schemes with high orders of accuracy in time. For this purpose, the unsteady incompressible Navier-Stokes equations are interpreted as a system of Differential Algebraic Equations, that is, a system of ordinary differential equations corresponding to the conservation of momentum equation, plus algebraic constraints corresponding to the incompressibility condition.
Numerical examples demonstrate the applicability of the proposed methodologies and compare their efficiency and accuracy.
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Virtanen, Juha Mikael. "Adaptive discontinuous Galerkin methods for fourth order problems." Thesis, University of Leicester, 2010. http://hdl.handle.net/2381/10091.
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This work is concerned with the derivation of adaptive methods for discontinuous Galerkin approximations of linear fourth order elliptic and parabolic partial differential equations. Adaptive methods are usually based on a posteriori error estimates. To this end, a new residual-based a posteriori error estimator for discontinuous Galerkin approximations to the biharmonic equation with essential boundary conditions is presented. The estimator is shown to be both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm, under minimal regularity assumptions. The reliability bound is based on a new recovery operator, which maps discontinuous finite element spaces to conforming finite element spaces (of two polynomial degrees higher), consisting of triangular or quadrilateral Hsieh-Clough-Tocher macroelements. The efficiency bound is based on bubble function techniques. The performance of the estimator within an h-adaptive mesh refinement procedure is validated through a series of numerical examples, verifying also its asymptotic exactness. Some remarks on the question of proof of convergence of adaptive algorithms for discontinuous Galerkin for fourth order elliptic problems are also presented. Furthermore, we derive a new energy-norm a posteriori error bound for an implicit Euler time-stepping method combined with spatial discontinuous Galerkin scheme for linear fourth order parabolic problems. A key tool in the analysis is the elliptic reconstruction technique. A new challenge, compared to the case of conforming finite element methods for parabolic problems, is the control of the evolution of the error due to non-conformity. Based on the error estimators, we derive an adaptive numerical method and discuss its practical implementation and illustrate its performance in a series of numerical experiments.
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Metcalfe, Stephen Arthur. "Adaptive discontinuous Galerkin methods for nonlinear parabolic problems." Thesis, University of Leicester, 2015. http://hdl.handle.net/2381/32041.
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This work is devoted to the study of a posteriori error estimation and adaptivity in parabolic problems with a particular focus on spatial discontinuous Galerkin (dG) discretisations. We begin by deriving an a posteriori error estimator for a linear non-stationary convection-diffusion problem that is discretised with a backward Euler dG method. An adaptive algorithm is then proposed to utilise the error estimator. The effectiveness of both the error estimator and the proposed algorithm is shown through a series of numerical experiments. Moving on to nonlinear problems, we investigate the numerical approximation of blow-up. To begin this study, we first look at the numerical approximation of blow-up in nonlinear ODEs through standard time stepping schemes. We then derive an a posteriori error estimator for an implicit-explicit (IMEX) dG discretisation of a semilinear parabolic PDE with quadratic nonlinearity. An adaptive algorithm is proposed that uses the error estimator to approach the blow-up time. The adaptive algorithm is then applied in a series of test cases to gauge the effectiveness of the error estimator. Finally, we consider the adaptive numerical approximation of a nonlinear interface problem that is used to model the mass transfer of solutes through semi-permiable membranes. An a posteriori error estimator is proposed for the IMEX dG discretisation of the model and its effectiveness tested through a series of numerical experiments.
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Choe, Kyu Y. "The discontinuous finite element method with the Taylor-Galerkin approach for nonlinear hyperbolic conservation laws /." Thesis, Connect to this title online; UW restricted, 1991. http://hdl.handle.net/1773/9977.
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Jakobsson, Håkan. "Discontinous Galerkin Methods for Coupled Flow and Transport problems." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-51335.
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We investigate the use of discontinuous Galerkin finite element methods in a mul- tiphysics setting involving coupled flow and transport in porous media. We solve an elliptic equation for the fluid pressure using Nitsche’s method and an approxima- tion, Σ, of the exact convection field σ will be constructed by interpolation onto the lowest-order Raviart-Thomas space of functions. We sequentially solve the transport equation, with the convection field Σ, for the fluid saturation by use of the lowest order discontinuous Galerkin method. We also supply numerical evidence of the importance of local conservation in this setting, and furthermore propose a line of argument indicating that if Σ is constructed using conservative fluxes, the modeling error σ − Σ may not have a great impact on the total error in certain quantities of interest.
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Pascal, Lucas. "Acoustique modale et stabilité linéaire par une méthode numérique avancée : Cas d'un conduit traité acoustiquement en présence d'un écoulement." Thesis, Toulouse, ISAE, 2013. http://www.theses.fr/2013ESAE0032/document.
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Ce travail de thèse s’inscrit dans l’effort de réduction des nuisances sonores dues à la soufflante d’unréacteur double-flux à l’aide de matériaux absorbants acoustiques, appelés communément «liners». Afind’optimiser ces traitements acoustiques, il convient d’étudier en détail la physique de la propagationacoustique en présence de liner. De plus, il s’agit d’améliorer la compréhension des instabilités hydrodynamiquespouvant se développer sur un liner sous des conditions particulières et possiblement génératricesde bruit. Ce travail de thèse a consisté à développer un code de calcul en formulation Galerkin discontinuepour l’analyse modale et la stabilité dans un conduit traité acoustiquement, code qui a été appliqué à desconfigurations réalistes, en considérant une section transverse ou longitudinale d’un conduit. Les étudesmodales réalisées dans la section transverse ont apporté des informations sur la propagation acoustiquedans une nacelle de turbofan avec des discontinuités du traitement acoustique («splices»), ainsi que dansle banc B2A de l’ONERA. Les calculs dans la section longitudinale ont nécessité l’implantation de conditionsaux limites PML pour tronquer le domaine de calcul, ainsi que d’une condition aux limites sur leliner, modélisée en domaine temporel à partir d’une extension de travaux existants dans la littérature.Avec ces outils, le code a permis de mettre en évidence une dynamique de type amplificateur de bruit dueau développement d’une instabilité hydrodynamique sur le liner en présence d’écoulement cisaillé ainsiqu’un rayonnement acoustique en amont et en aval du conduit dû à cette instabilitéThe current work deals with the reduction of aircraft engine fan noise using acoustic lining. In orderto optimise these liners, it is necessary to deeply understand the physics of acoustic wave propagation in lined ducts and to have a better knowledge of the hydrodynamic instabilities existing under particular conditions and likely to radiate noise. This work is about the development of a discontinuous Galerkin solver for modal and stability analysis in lined flow duct and the application of this solver to realistic configurations by considering the transverse or longitudinal section of a duct. The modal studies in the transverse section brought informations on acoustic propagation in a turbofan nacelle with lining discontinuities (“splices”) and in the B2A bench of ONERA. The computation in the longitudinal section of a duct required the implementation of PML boundary conditions in order to truncate the computational domain and of a boundary condition at the lined wall, modeled in temporal domain by the enhancement of a method published in the literature. With these features, the application of the solver highlighted a noise amplifier dynamics caused by the development of a hydrodynamic instability on the liner with sheared flow and a noise radiation mechanism upstream and downstream the lined section
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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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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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Dobrev, Veselin Asenov. "Preconditioning of discontinuous Galerkin methods for second order elliptic problems." [College Station, Tex. : Texas A&M University, 2007. http://hdl.handle.net/1969.1/ETD-TAMU-2531.
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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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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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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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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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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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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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Wei, Xiaoxi. "Mixed discontinuous Galerkin finite element methods for incompressible magnetohydrodynamics." Thesis, University of British Columbia, 2011. http://hdl.handle.net/2429/34789.
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We develop and analyze mixed discontinuous Galerkin finite element methods for the numerical approximation
of incompressible magnetohydrodynamics problems.
Incompressible magnetohydrodynamics is the area of physics that is
concerned with the behaviour of electrically conducting,
resistive, incompressible and viscous fluids in the presence of
electromagnetic fields. It is modelled
by a system of nonlinear partial differential equations, which couples the
Navier-Stokes equations with the Maxwell equations.
In the first part of this thesis, we introduce an interior penalty discontinuous
Galerkin method for the numerical approximation of a linearized incompressible magnetohydrodynamics
problem. The fluid
unknowns are discretized with the discontinuous
℘k-℘k-1 element pair, whereas the magnetic
variables are approximated by discontinuous ℘k-℘k+1 elements. Under minimal regularity
assumptions, we carry out a complete a priori error analysis and
prove that the energy norm error is optimally convergent in the
mesh size in general polyhedral domains, thus guaranteeing the
numerical resolution of the strongest magnetic singularities in
non-convex domains.
In the second part of this thesis, we propose and analyze a new mixed discontinuous Galerkin finite element method for the approximation of a fully nonlinear
incompressible magnetohydrodynamics model. The velocity field is
now discretized by divergence-conforming Brezzi-Douglas-Marini
elements, and the magnetic field by curl-conforming
Nédélec elements. In addition to correctly capturing magnetic
singularities, the method yields exactly divergence-free
velocity approximations, and is thus energy-stable. We show that the energy norm error is
convergent in the mesh size in possibly non-convex polyhedra, and
derive slightly suboptimal a priori error estimates under minimal regularity and small data
assumptions.
Finally, in the third part of this thesis, we present two extensions of our discretization techniques
to time-dependent incompressible
magnetohydrodynamics problems and to Stokes problems with nonstandard
boundary conditions.
All our discretizations and theoretical results are computationally
validated through comprehensive sets of numerical experiments.
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Georgoulis, Emmanuil H. "Discontinuous Galerkin methods on shape-regular and anisotropic meshes." Thesis, University of Oxford, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.270366.
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Jensen, Max. "Discontinuous Galerkin methods for Friedrichs systems with irregular solutions." Thesis, University of Oxford, 2005. http://sro.sussex.ac.uk/45497/.
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This work is concerned with the numerical solution of Friedrichs systems by discontinuous Galerkin finite element methods (DGFEMs). Friedrichs systems are boundary value problems with symmetric, positive, linear first-order partial differential operators and allow the unified treatment of a wide range of elliptic, parabolic, hyperbolic and mixed-type equations. We do not assume that the exact solution of a Friedrichs system belongs to a Sobolev space, but only require that it is contained in the associated graph space, which amounts to differentiability in the characteristic direction. We show that the numerical approximations to the solution of a Friedrichs system by the DGFEM converge in the energy norm under hierarchical h- and p- refinement. We introduce a new compatibility condition for the boundary data, from which we can deduce, for instance, the validity of the integration-by-parts formula. Consequently, we can admit domains with corners and allow changes of the inertial type of the boundary, which corresponds in special cases to the componentwise transition from in- to outflow boundaries. To establish the convergence result we consider in equal parts the theory of graph spaces, Friedrichs systems and DGFEMs. Based on the density of smooth functions in graph spaces over Lipschitz domains, we study trace and extension operators and also investigate the eigensystem associated with the differential operator. We pay particular attention to regularity properties of the traces, that limit the applicability of energy integral methods, which are the theoretical underpinning of Friedrichs systems. We provide a general framework for Friedrichs systems which incorporates a wide range of singular boundary conditions. Assuming the aforementioned compatibility condition we deduce well-posedness of admissible Friedrichs systems and the stability of the DGFEM. In a separate study we prove hp-optimality of least-squares stabilised DGFEMs.