Academic literature on the topic 'Hybridizable discontinuous Galerkin method'

AbstractThe embedded discontinuous Galerkin (EDG) method by Cockburn, Gopalakrishnan and Lazarov [B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second-order elliptic problems, SIAM J. Numer. Anal. 47 2009, 2, 1319–1365] is obtained from the hybridizable discontinuous Galerkin method by changing the space of the Lagrangian multiplier from discontinuous functions to continuous ones, and adopts piecewise polynomials of equal degrees on simplex meshes for all variables. In this paper, we analyze a new EDG method for second-order elliptic problems on polygonal/polyhedral meshes. By using piecewise polynomials of degrees {k+1}, {k+1}, k ({k\geq 0}) to approximate the potential, numerical trace and flux, respectively, the new method is shown to yield optimal convergence rates for both the potential and flux approximations. Numerical experiments are provided to confirm the theoretical results.

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.<br>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

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.<br>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.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2014.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 110-120).<br>Numerical simulations of time-periodic flows are an essential design tool for a wide range of engineered systems, including jet engines, wind turbines and flapping wings. Conventional solvers for time-periodic flows are limited in accuracy and efficiency by the low-order Finite Volume and time-marching methods they typically employ. These methods introduce significant numerical dissipation in the simulated flow, and can require hundreds of timesteps to describe a periodic flow with only a few harmonic modes. However, recent developments in high-order methods and Fourier-based time discretizations present an opportunity to greatly improve computational performance. This thesis presents a novel Time-Spectral Hybridizable Discontinuous Galerkin (HDG) method for periodic flow problems, together with applications to flow through cascades and rotor/stator assemblies in aeronautical turbomachinery. The present work combines a Fourier-based Time-Spectral discretization in time with an HDG discretization in space, realizing the dual benefits of spectral accuracy in time and high-order accuracy in space. Low numerical dissipation and favorable stability properties are inherited from the high-order HDG method, together with a reduced number of globally coupled degrees of freedom compared to other DG methods. HDG provides a natural framework for treating boundary conditions, which is exploited in the development of a new high-order sliding mesh interface coupling technique for multiple-row turbomachinery problems. A regularization of the Spalart-Allmaras turbulence model is also employed to ensure numerical stability of unsteady flow solutions obtained with high-order methods. Turning to the temporal discretization, the Time-Spectral method enables direct solution of a periodic flow state, bypasses initial transient behavior, and can often deliver substantial savings in computational cost compared to implicit time-marching. An important driver of computational efficiency is the ability to select and resolve only the most important frequencies of a periodic problem, such as the blade-passing frequencies in turbomachinery flows. To this end, the present work introduces an adaptive frequency selection technique, using the Time-Spectral residual to form an inexpensive error indicator. Having selected a set of frequencies, the accuracy of the Time-Spectral solution is greatly improved by using optimally selected collocation points in time. For multi-domain problems such as turbomachinery flows, an anti-aliasing filter is also needed to avoid errors in the transfer of the solution across the sliding interface. All of these aspects contribute to the Adaptive Time-Spectral HDG method developed in this thesis. Performance characteristics of the method are demonstrated through applications to periodic ordinary differential equations, a convection problem, laminar flow over a pitching airfoil, and turbulent flow through a range of single- and multiple-row turbomachinery configurations. For a 2:1 rotor/stator flow problem, the Adaptive Time-Spectral HDG method correctly identifies the relevant frequencies in each blade row. This leads to an accurate periodic flow solution with greatly reduced computational cost, when compared to sequentially selected frequencies or a time-marching solution. For comparable accuracy in prediction of rotor loading, the Adaptive Time- Spectral HDG method incurs 3 times lower computational cost (CPU time) than time-marching, and for prediction of only the 1st harmonic amplitude, these savings rise to a factor of 200. Finally, in three-row compressor flow simulations, a high-order HDG method is shown to achieve significantly greater accuracy than a lower-order method with the same computational cost. For example, considering error in the amplitude of the 1st harmonic mode of total rotor loading, a p = 1 computation results in 20% error, in contrast to only 1% error in a p = 4 solution with comparable cost. This highlights the benefits that can be obtained from higher-order methods in the context of turbomachinery flow problems.<br>by Hemant Kumar Chaurasia.<br>Ph. D.

Thesis: S.M., Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2017.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 85-89).<br>In this work novel high-order hybridizable discontinuous Galerkin (HDG) projection methods are further developed for ocean dynamics and geophysical fluid predictions. We investigate the effects of the HDG stabilization parameter for both the momentum equation as well as tracer diffusion. We also make a correction to our singularity treatment algorithm for nailing down a numerically consistent and unique solution to the pressure Poisson equation with homogeneous Neumann boundary conditions everywhere along the boundary. Extensive numerical results using physically realistic ocean flows are presented to verify the HDG projection methods, including the formation of internal wave beams over a shallow but abrupt seamount, the generation of internal solitary waves from stratified oscillatory flow over steep topography, and the circulation of bottom gravity currents down a slope. Additionally, we investigate the implementation of open boundary conditions for finite element methods and present results in the context of our ocean simulations. Through this work we present the hybridizable discontinuous Galerkin projection methods as a viable and competitive alternative for large-scale, realistic ocean modeling.<br>by Johnathan Hiep Vo.<br>S.M.

This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.<br>Thesis: S.M., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2019<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 93-99).<br>This work focuses on developing efficient and robust implementation methods for hybridizable discontinuous Galerkin (HDG) schemes for fluid and ocean dynamics. In the first part, we compare choices in weak formulations and their numerical consequences. We address details in making the leap from the mathematical formulation to the implementation, including the different spaces and mappings, discretization of the integral operators, boundary conditions, and assembly of the linear systems. We provide a flexible mapping procedure amenable to both quadrature-free and quadrature-based discretizations, and compare the accuracy of the two on different problem geometries. We verify the quadrature-free approach, demonstrating that optimal orders of convergence can be obtained, even on non-affine and curvilinear geometries. The second part of the work investigates the scalability of HDG schemes, identifying memory and time-to-solution bottlenecks. The form of the quadrature-free integral operators is exploited to develop a novel and efficient matrix-free approach to solving the global linear system that arises from HDG discretizations. Additional manipulations to improve numerical robustness are discussed. To mitigate the complexity of the implementation, we provide an automated and computationally efficient verification procedure for the HDG methodologies discussed, using a hierarchical approach to provide diagnostic information and isolate problems. Finally, challenges related to the effective visualization of high-order, discontinuous HDG-FEM data for fluid and ocean applications are illustrated and strategies are provided to address them.<br>by Corbin Foucart.<br>S.M.<br>S.M. Massachusetts Institute of Technology, Department of Mechanical Engineering

This thesis proposes a coupled continuous and hybridizable discontinuous Galerkin formulation to solve conjugate heat transfer problems. This model is then used to find the thermal response of Glass Fiber Reinforced Polymer (GFRP) tubular cross-section under fire. The first step of this thesis is to compare the computational efficiency of high-order Continuous Galerkin (CG) and Hybridizable Discontinuous Galerkin (HDG) methods for incompressible fluid flow problems in low Reynolds number regimes. Only 2-D examples and direct solvers are considered in the present work. A thoroughly comparison in terms of CPU time and accuracy for both discretization methods is made under the same platform. Various results presented suggests that HDG can be more efficient than CG when the CPU time, for a given degree, is considered. The stability of HDG and CG is studied using a manufactured solution that produces a sharp boundary layer, confirming that HDG provides smooth converged solutions in the presence of sharp fronts whereas, CG failed to converge due to the presence of numerical oscillations. Following, the solution of the coupled Navier-Stokes/convection-diffusion problem, using Boussinesq approximation, is formulated within the HDG framework and analysed using numerical experiments and benchmark problems. A coupling strategy between HDG and CG methods is proposed in the framework of second-order elliptic operators. The coupled formulation is implemented and its convergence properties are established numerically by using manufactured solutions. Finally, the proposed coupled formulation between HDG and CG for heat equation is combined with the coupled Navier--Stokes/convection diffusion equations to formulate a new CG-HDG model for solving conjugate heat transfer problems. Benchmark examples are solved using the proposed model and validated with literature values. The final part of the thesis applies the proposed CG-HDG coupled formulation to predict the thermal response of the GFRP tubular cross-section. The radiosity equation that governs the internal radiation is added to the CG-HDG coupled model. Estimates of the discretization errors are computed in order to establish the confidence intervals for quantities of interest. Results with the geometry having curved corners in the cavity are presented and shown to be within the estimated uncertainty intervals. CPU times for the linear solver suggests that the proposed CG-HDG model is more efficient than CG-CG model in all the cases considered.<br>Neste trabalho é proposta uma formulação para acoplar os modelos continuous e hybridizable discontinuous Galerkin a fim de analisar problemas conjugados de transferência de calor. Este modelo é então usado para estudar a resposta térmica de perfis pultrudidos de secção tubular em polímero reforçado com fibras de vidro (GFRP) sob a acção do fogo. O primeiro passo desta tese é comparar a eficiência computacional dos métodos Continuous Galerkin (CG) e Hybridizable Discontinuous Galerkin (HDG) de elevada ordem para problemas de escoamento de fluidos incompressíveis para valores reduzidos do número Reynolds. Apenas exemplos bidimensionais e métodos directos são considerados no presente trabalho. Uma comparação exaustiva em termos de tempo de CPU e precisão para ambos os métodos de discretização é efectuada sob uma plataforma comum. Os resultados apresentados sugerem que, em termos do tempo de CPU requerido, o HDG pode ser mais eficiente que o CG, para um determinado grau. A estabilidade do HDG e CG é estudada usando uma solução fabricada que produz uma abrupta descontinuidade, confirmando que o HDG fornece soluções convergentes e suaves na presença de descontinuidades, enquanto o CG não conseguiu convergir devido à presença de oscilações numéricas. Em seguida, a solução do problema acoplado Navier-Stokes/convecção-difusão, utilizando a aproximação de Boussinesq, é formulada no contexto HDG e analisada usando soluções de referência. Uma estratégia de acoplamento entre os métodos HDG e CG é proposta no âmbito de operadores elípticos de segunda ordem. A formulação acoplada é implementada e suas propriedades de convergência são estabelecidas numericamente usando soluções fabricadas. Finalmente, a formulação acoplada proposta entre HDG e CG para a equação do calor é combinada com as equações acopladas de Navier-Stokes/convecção-difusão para formular um novo modelo de CG-HDG para resolver problemas de transferência de calor conjugado. Exemplos de referência são resolvidos usando o modelo proposto e validados com valores de literatura. A parte final da tese aplica a formulação proposta CG-HDG acoplada para prever a resposta térmica de uma secção transversal tubular de GFRP. A equação de radiosidade que governa a radiação interna é adicionada ao modelo acoplado CG-HDG. Os erros de discretização são calculados para estabelecer os intervalos de confiança para quantidades de interesse. Resultados considerando a geometria circular dos cantos da cavidade são apresentados. Estes estão dentro do intervalo de incerteza estimado. Os tempos de CPU requeridos para resolver os sistemas de equações lineares sugerem que o modelo proposto CG-HDG é mais eficiente do que o modelo CG-CG em todos os casos considerados.<br>En esta tesis se propone una formulación acoplada del método de los elementos finitos clásico (CG) y el método Hybridizable Discontinuous Galerkin (HDG) para la a solución de problemas térmicos conjugados. El modelo se utiliza para determinar la respuesta al fuego de Polímeros Reforzados con Fibras de Vidrio (GFRP) con sección tubular. El primer paso de la tesis es la comparación de la eficiencia computacional de CG y HDG de alto orden para problemas de flujo incompresible para número de Reynolds (Re) bajo. Se consideran sólo ejemplos 2D y métodos de resolución de sistemas lineales directos. Se presenta una comparación en términos de tiempo de CPU y precisión en la solución para ambas discretizaciones, bajo la misma plataforma de implementación. Los resultados sugieren que HDG puede ser más eficiente computacionalmente que CG en tiempo de CPU, para un grado fijado. La estabilidad de HDG y CG para Re alto se estudia con una solución manufacturada que produce un frente pronunciado, confirmando que HDG proporciona soluciones convergidas suaves en presencia de frentes verticales, en casos en que las oscilaciones numéricas de CG no permiten llegar a convergencia. A continuación, se plantea la solución del problema acoplado Navier-Stokes/convección-difusión, con la aproximación de Boussinesq, en el contexto del método HDG, y se analiza con experimentos numéricos. Se propone una formulación acoplada HDG-CG para la ecuación del calor. Se comprueban numéricamente las propiedades de convergencia del método propuesto. Finalmente, se combina la formulación acoplada propuesta para la ecuación del calor con el acoplamiento con la ecuaciones de Navier-Stokes en el dominio del fluido, creando una nueva formulación CG-HDG para problemas térmicos conjugados. Se consideran tests clásicos para validar los resultados comparando con la literatura existente. La parte final de la tesis aplica la formulación acoplada CG-HDG propuesta a la predicción de la respuesta térmica de secciones tubulares de GFRP, incluyendo radiosidad interna en el modelo. Se calculan estimas de los errores de discretización para determinar intervalos de confianza para las cantidades de interés. Se presentan resultados con geometría con esquinas curvas en la cavidad mostrando resultados dentro de los intervalos de incertidumbre estimados. El tiempo de CPU para la resolución de sistemas sugiere que el modelo CG-HDG propuesto es más eficiente que el clásico método CG-CG en todos los casos considerados.<br>This thesis proposes a coupled continuous and hybridizable discontinuous Galerkin formulation to solve conjugate heat transfer problems. This model is then used to find the thermal response of Glass Fiber Reinforced Polymer (GFRP) tubular cross-section under fire. The first step of this thesis is to compare the computational efficiency of high-order Continuous Galerkin (CG) and Hybridizable Discontinuous Galerkin (HDG) methods for incompressible fluid flow problems in low Reynolds number regimes. Only 2-D examples and direct solvers are considered in the present work. A thoroughly comparison in terms of CPU time and accuracy for both discretization methods is made under the same platform. Various results presented suggests that HDG can be more efficient than CG when the CPU time, for a given degree, is considered. The stability of HDG and CG is studied using a manufactured solution that produces a sharp boundary layer, confirming that HDG provides smooth converged solutions in the presence of sharp fronts whereas, CG failed to converge due to the presence of numerical oscillations. Following, the solution of the coupled Navier–Stokes/convection-diffusion problem, using Boussinesq approximation, is formulated within the HDG framework and analysed using numerical experiments and benchmark problems. A coupling strategy between HDG and CG methods is proposed in the framework of second-order elliptic operators. The coupled formulation is implemented and its convergence properties are established numerically by using manufactured solutions. Finally, the proposed coupled formulation between HDG and CG for heat equation is combined with the coupled Navier–Stokes/convection diffusion equations to formulate a new CG-HDG model for solving conjugate heat transfer problems. Benchmark examples are solved using the proposed model and validated with literature values. The final part of the thesis applies the proposed CG-HDG coupled formulation to predict the thermal response of the GFRP tubular cross-section. The radiosity equation that governs the internal radiation is added to the CG-HDG coupled model. Estimates of the discretization errors are computed in order to establish the confidence intervals for quantities of interest. Results with the geometry having curved corners in the cavity are presented and shown to be within the estimated uncertainty intervals. CPU times for the linear solver suggests that the proposed CG-HDG model is more efficient than CG-CG model in all the cases considered<br>Neste trabalho é proposta uma formulação para acoplar os modelos continuous e hybridizable discontinuous Galerkin a fim de analisar problemas conjugados de transferência de calor. Este modelo é então usado para estudar a resposta térmica de perfis pultrudidos de secção tubular em polímero reforçado com fibras de vidro (GFRP) sob a acção do fogo. O primeiro passo desta tese é comparar a eficiência computacional dos métodos continuous Galerkin (CG) e Hybridizable Discontinuous Galerkin (HDG) de elevada ordem para problemas de escoamento de fluidos incompressíveis para valores reduzidos do número Reynolds. Apenas exemplos bidimensionais e métodos directos são considerados no presente trabalho. Uma comparação exaustiva em termos de tempo de CPU e precisão para ambos os métodos de discretização é efectuada sob uma plataforma comum. Os resultados apresentados sugerem que, em termos do tempo de CPU requerido, o HDG pode ser mais eficiente que o CG, para um determinado grau. A estabilidade do HDG e CG é estudada usando uma solução fabricada que produz uma abrupta descontinuidade, confirmando que o HDG fornece soluções convergentes e suaves na presença de descontinuidades, enquanto o CG não conseguiu convergir devido à presença de oscilações numéricas. Em seguida, a solução do problema acoplado Navier-Stokes/convecção-difusão, utilizando a aproximação de Boussinesq, é formulada no contexto HDG e analisada usando soluções de referência. Uma estratégia de acoplamento entre os métodos HDG e CG é proposta no âmbito de operadores elípticos de segunda ordem. A formulação acoplada é implementada e suas propriedades de convergência são estabelecidas numericamente usando soluções fabricadas. Finalmente, a formulação acoplada proposta entre HDG e CG para a equação do calor é combinada com as equações acopladas de Navier-Stokes/convecção-difusão para formular um novo modelo de CG-HDG para resolver problemas de transferência de calor conjugado. Exemplos de referência são resolvidos usando o modelo proposto e validados com valores de literatura. A parte final da tese aplica a formulação proposta CG-HDG acoplada para prever a resposta térmica de uma secção transversal tubular de GFRP. A equação de radiosidade que governa a radiação interna é adicionada ao modelo acoplado CG-HDG. Os erros de discretização são calculados para estabelecer os intervalos de confiança para quantidades de interesse. Resultados considerando a geometria circular dos cantos da cavidade são apresentados. Estes estão dentro do intervalo de incerteza estimado. Os tempos de CPU requeridos para resolver os sistemas de equações lineares sugerem que o modelo proposto CG-HDG é mais eficiente do que o modelo CG-CG em todos os casos considerados.<br>En esta tesis se propone una formulación acoplada del método de los elementos finitos clásico (CG) y el método Hybridizable Discontinuous Galerkin (HDG) para la a solución de problemas térmicos conjugados. El modelo se utiliza para determinar la respuesta al fuego de Polímeros Reforzados con Fibras de Vidrio (GFRP) con sección tubular. El primer paso de la tesis es la comparación de la eficiencia computacional de CG y HDG de alto orden para problemas de flujo incompresible para número de Reynolds (Re) bajo. Se consideran sólo ejemplos 2D y métodos de resolución de sistemas lineales directos. Se presenta una comparación en términos de tiempo de CPU y precisión en la solución para ambas discretizaciones, bajo la misma plataforma de implementación. Los resultados sugieren que HDG puede ser más eficiente computacionalmente que CG en tiempo de CPU, para un grado fijado. La estabilidad de HDG y CG para Re alto se estudia con una solución manufacturada que produce un frente pronunciado, confirmando que HDG proporciona soluciones convergidas suaves en presencia de frentes verticales, en casos en que las oscilaciones numéricas de CG no permiten llegar a convergencia. A continuación, se plantea la solución del problema acoplado Navier-Stokes/conveccióndifusión, con la aproximación de Boussinesq, en el contexto del método HDG, y se analiza con experimentos numéricos. Se propone una formulación acoplada HDG-CG para la ecuación del calor. Se comprueban numéricamente las propiedades de convergencia del método propuesto. Finalmente, se combina la formulación acoplada propuesta para la ecuación del calor con el acoplamiento con la ecuaciones de Navier-Stokes en el dominio del fluido, creando una nueva formulación CG-HDG para problemas térmicos conjugados. Se consideran ejemplos clásicos para validar los resultados comparando con la literatura existente. La parte final de la tesis aplica la formulación acoplada CG-HDG propuesta a la predicción de la respuesta térmica de secciones tubulares de GFRP, incluyendo radiosidad interna en el modelo. Se calculan estimas de los errores de discretización para determinar intervalos de confianza para las cantidades de interés. Se presentan resultados con geometría con esquinas curvas en la cavidad mostrando resultados dentro de los intervalos de incertidumbre estimados. El tiempo de CPU para la resolución de sistemas sugiere que el modelo CG-HDG propuesto es más eficiente que el clásico método CG-CG en todos los casos considerados.

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 3D<br>The 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

Simulations of incompressible flows are performed on a daily basis to solve problems of practical and industrial interest in several fields of engineering, including automotive, aeronautical, mechanical and biomedical applications. Although finite volume (FV) methods are still the preferred choice by the industry due to their efficiency and robustness, sensitivity to mesh quality and limited accuracy represent two main bottlenecks of these approaches. This is especially critical in the context of transient phenomena, in which FV methods show excessive numerical diffusion. In this context, there has been a growing interest towards high-order discretisation strategies in last decades. In this PhD thesis, a high-order adaptive hybidisable discontinuous Galerkin (HDG) method is proposed for the approximation of steady and unsteady laminar incompressible Navier-Stokes equations. Voigt notation for symmetric second-order tensors is exploited to devise an HDG method for the Cauchy formulation of the momentum equation with optimal convergence properties, even when low-order polynomial degrees of approximation are considered. In addition, a postprocessing strategy accounting for rigid translational and rotational modes is proposed to construct an element-by-element superconvergent velocity field. The discrepancy between the computed and postprocessed velocities is utilised to define a local error indicator to drive degree adaptivity procedures and accurately capture localised features of the flow. The resulting HDG solver is thus extended to the case of transient problems via high-order time integration schemes, namely the explicit singly diagonal implicit Runge-Kutta (ESDIRK) schemes. In this context, the embedded explicit step is exploited to define an inexpensive estimate of the temporal error to devise an efficient timestep control strategy. Finally, in order to efficiently solve the global problem arising from the HDG discretisation, a preconditioned iterative solver is proposed. This is critical in the context of high-order approximations in three-dimensional domains leading to large-scale problems, especially in transient simulations. A block diagonal preconditioner coupled with an inexpensive approximation of the Schur complement of the matrix is proposed to reduce the computational cost of the overall HDG solver. Extensive numerical validation of two and three-dimensional steady and unsteady benchmark tests of viscous laminar incompressible flows is performed to validate the proposed methodology.<br>Simulaciones de flujo incompresible se emplean a diario para resolver problemas de interés práctico e industrial en varios campos de la ingeniería, p.ej. en aplicaciones automovilísticas, aeronáuticas, mecánicas y biomédicas. Aunque los métodos de volúmenes finitos (FV) siguen siendo la opción preferida por la industria debido a su eficiencia y robustez, la sensibilidad a la calidad de la malla y la baja precisión representan dos limitaciones importantes para estas técnicas. Estas limitaciones son todavía más críticas en el contexto de simulaciones de fenómenos transitorios, donde los FV están penalizados por su excesiva difusión numérica. En este contexto, las estrategias de discretización de alto orden han ganado una popularidad creciente en las últimas décadas para problemas transitorios dónde se necesitan soluciones precisas. Esta tesis propone un método de Galerkin discontinuo híbrido (HDG), de alto orden y adaptativo para la aproximación de las ecuaciones de Navier-Stokes incomprensible laminar, en el caso estacionario y transitorio en el entorno de aplicaciones ingenieriles. Para ello, la notación de Voigt para tensores simétricos de segundo orden (habituales en mecánica de los medios continuos) permite introducir un método HDG para la formulación de Cauchy de la ecuación de momento. La novedad de este resultado reside en la convergencia óptima alcanzada por el método, incluso para aproximaciones de orden polinómico bajo. Además, se desarrolla una estrategia de post-proceso local para construir elemento a elemento un campo de velocidad súper-convergente, tomando en cuenta los modos rígidos de traslación y rotación. La discrepancia entre el campo de velocidad calculado y el súper-convergente, obtenido a través del post-proceso, permite definir un indicador del error local. De esta forma, se desarrolla una estrategia para realizar adecuar elemento a elemento el grado de la aproximación polinómica y así mejorar la precisión adaptándose a las características localizadas del flujo. Seguidamente, se extiende el método HDG propuesto al tratamiento de problemas dependientes del tiempo. Más concretamente, se consideran los esquemas de integración temporal de alto orden explicit singly diagonal implicit Runge-Kutta (ESDIRK). En este contexto, se utiliza el paso explícito embedded para calcular una estimación computacionalmente eficiente del error temporal y definir una estrategia de adaptividad del paso de tiempo. Finalmente, se desarrolla un precondicionador adaptado a la estrategia HDG que acelera la convergencia del método iterativo empleado y, de esta forma, obtener resoluciones eficaces del problema global surgido de la discretización HDG. Es importante resaltar la importancia de una herramienta de resolución eficiente para problemas de gran escala en el contexto de aproximaciones de alto orden y en dominios tridimensionales. Estas herramientas se hacen aún más criticas en simulaciones transitorias. Más concretamente, se proponen un precondicionador diagonal por bloques y una aproximación eficiente del complemento Schur de la matriz para reducir el coste computacional del método HDG. Para validar la metodología propuesta, se realizan varias simulaciones numéricas de flujo incompresible laminar viscoso, para problemas estacionarios y transitorios, en dos y tres dimensiones.

Thesis: S.M., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2018.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 51-52).<br>Finite element methods, specifically Hybridizable Discontinuous Galerkin (HDG), are used in many applications. One choice made when implementing HDG for a specific problem is whether time integration should be performed implicitly or explicitly. Both approaches have their advantages but, for some problems, a combination of these methods is a better choice than either on their own. Thus, an implicit-explicit (IMEX) scheme that splits the computational domain into implicit and explicit regions based on the domain geometry is considered in this thesis. This allows for stability throughout the domain and exploits the advantages each scheme has to offer. A study of the convergence and properties of this implementation of the IMEX method is presented along with comparisons to the individual methods.<br>by Lauren Nicole Kolkman.<br>S.M.

This dissertation presents high-order hybridisable discontinuous Galerkin (HDG) formulations coupled with implicit Runge-Kutta (RK) methods for the simulation of one-phase flow and two-phase flow problems. High-order-methods can reduce the computational cost while obtaining more accurate solutions with less dissipation and dispersion errors than low order methods. HDG is an unstructured, high-order accurate, and stable method. The stability is imposed using a single parameter. In addition, it is a conservative method at the element level, which is an important feature when solving PDEs in a conservative form. Moreover, a hybridization procedure can be applied to reduce the size of the global linear system. To keep the stability and accuracy advantages in transient problems, we couple the HDG method with high-order implicit RK schemes. The first contribution is a stable high-order HDG formulation coupled with DIRK schemes for slightly compressible one-phase flow problem. We obtain an analytical expression for the stabilization parameter using the Engquist-Osher monotone flux scheme. The selection of the stabilization parameter is crucial to ensure the stability and to obtain the high-order properties of the method. We introduce the stabilization parameter in the Newton’s solver since we analytically compute its derivatives. The second contribution is a high-order HDG formulation coupled with DIRK schemes for immiscible and incompressible two-phase flow problem. We set the water pressure and oil saturation as the main unknowns, which leads to a coupled system of two non-linear PDEs. To solve the resulting non-linear problem, we use a fix-point iterative method that alternatively solves the saturation and the pressure unknowns implicitly at each RK stage until convergence is achieved. The proposed fix-point method is memory-efficient because the saturation and the pressure are not solved at the same time. The third contribution is a discretization scheme for the two-phase flow problem with the same spatial and temporal order of convergence. High-order spatial discretization combined with low-order temporal discretizations may lead to arbitrary small time steps to obtain a low enough temporal error. Moreover, high-order stable DIRK schemes need a high number of stages above fourth-order. Thus, the computational cost can be severely hampered because a non-linear problem has to be solved at each RK stage. Thus, we couple the HDG formulation with high-order fully implicit RK schemes. These schemes can be unconditionally stable and achieve high-order temporal accuracy with few stages. Therefore, arbitrary large time steps can be used without hampering the temporal accuracy. We rewrite the non-linear system to reduce the memory footprint. Thus, we achieve a better sparsity pattern of the Jacobian matrix and less coupling between stages. Furthermore, we have adapted the previous fix-point iterative method. We first compute the saturation at all the stages by solving a single non-linear system using the Newton-Raphson method. Next, we solve the pressure equation sequentially at each RK stage, since it does not couple the unknowns at different stages. The last contribution is an efficient shock-capturing method for the immiscible and incompressible two-phase flow problem to reduce the spurious oscillations that may appear in the high-order approximations of the saturation. We introduce local artificial viscosity only in the saturation equation since only the saturation variable is non-smooth. To this end, we propose a shock sensor computed from the saturation and the post-processed saturation of the HDG method. This shock sensor is computationally efficient since the post-processed saturation is computed in an element-wise manner. Our methodology allows tracking the sharp fronts as they evolve since the shock sensor is computed at all RK stages.<br>Esta tesis presenta formulaciones de Galerkin discontinuo hibridizable de alto orden (HDG) acopladas con métodos implícitos de Runge-Kutta (RK) para la simulación de flujo monofásico y bifásico. Los métodos de alto orden pueden reducir el coste computacional mientras se obtienen soluciones más precisas con menos errores de disipación y dispersión que los de bajo orden. HDG es un método no estructurado, con precisión de alto orden y estable. La estabilidad se impone utilizando un solo parámetro. Además, es un método localmente conservativo, lo cual es importante al resolver EDPs de forma conservativa. Además, se pueden usar técnicas de hibridización para reducir el tamaño del sistema lineal global. Para mantener las ventajas de estabilidad y precisión en problemas transitorios, combinamos el método HDG con esquemas RK implícitos de alto orden. La primera contribución es una formulación HDG estable de alto orden con esquemas DIRK para problemas de flujo monofásico ligeramente compresible. Obtenemos una expresión analítica para el parámetro de estabilización utilizando el esquema de flujo monótono Engquist-Osher. La selección del parámetro de estabilización garantiza la estabilidad y las propiedades de alto orden del método. Introducimos el parámetro de estabilización en el método de Newton debido que calculamos analíticamente sus derivadas. La segunda contribución es una formulación HDG de alto orden con esquemas DIRK para problemas de flujo bifásico inmiscible e incompresible. Usamos la presión del agua y la saturación de petróleo como incógnitas principales, con lo que se obtiene un sistema acoplado de dos EDPs no lineales. Para resolver el problema no lineal, usamos un método iterativo de punto fijo que resuelve alternativamente la saturación y la presión implícitamente en cada etapa del RK hasta converger. Este método es eficiente en memoria porque la saturación y la presión no se resuelven a la vez. La tercera contribución es un esquema de discretización para el problema del flujo bifásico con el mismo orden de convergencia espacial y temporal. La discretización espacial de alto orden junto con discretizaciones temporales de bajo orden puede requerir pasos de tiempo arbitrariamente pequeños para obtener un error temporal suficientemente bajo. Además, los esquemas de DIRK estables de alto orden necesitan una gran cantidad de etapas a partir del cuarto orden. Por ello, el coste computacional puede verse gravemente afectado porque se debe resolver un problema no lineal en cada etapa del RK. Por lo tanto, combinamos la formulación HDG con esquemas RK totalmente implícitos de alto orden. Estos esquemas pueden ser incondicionalmente estables y lograr una precisión temporal de alto orden con pocas etapas. Por ello, se pueden utilizar pasos de tiempo arbitrariamente grandes sin perjudicar la precisión temporal. Reescribimos el sistema no lineal para reducir el requerimiento de memoria. De este modo, logramos un mejor patrón de llenado de la jacobiana y un menor acoplamiento entre etapas. Además, hemos adaptado el método iterativo de punto fijo anterior. Primero calculamos la saturación en todas las etapas resolviendo un solo sistema no lineal utilizando el método Newton-Raphson. Posteriormente, resolvemos la ecuación de presión secuencialmente en cada etapa del RK, ya que no combina las incógnitas en diferentes etapas. La última contribución es un método eficiente de captura de choque para el problema de flujo bifásico para reducir las oscilaciones espurias que pueden aparecer en las aproximaciones de la saturación. Introducimos viscosidad artificial localmente solo en la ecuación de saturación, ya que sólo la saturación no es suave. Por ello, calculamos un sensor de choque con la saturación y la saturación postprocesada del método HDG. Este sensor es eficiente ya que la saturación postprocesada se calcula a nivel elemental. Nuestra metodología permite seguir la evolución de los frentes, porque el sensor se calcula en cada etapa