Dissertations / Theses on the topic 'Discontinuous spectral element method'
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De, Grazia Daniele. "Three-dimensional discontinuous spectral/hp element methods for compressible flows." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/40416.
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In this thesis we analyse and develop two high-order schemes which belong to the class of discontinuous spectral/hp element methods focusing on compressible aerodynamic studies and, more specifically, on boundary-layer flows. We investigate the discontinuous Galerkin method and the flux reconstruction approach providing a detailed analysis of the connections between these methods. The connections found enable a better understanding of the broader class of discontinuous spectral/hp element methods. From this perspective it was evident that some of the issues of the discontinuous Galerkin method are also encountered in the flux reconstruction approach, and in particular, the aliasing errors of the two schemes are identical. The techniques applied in the more famous discontinuous Galerkin method for tackling these errors can be also extended to the flux reconstruction approach. We present two dealiasing strategies based on the concept of consistent integration of the nonlinear terms. The first is a localised approach which targets in each element the nonlinearities arising in the problem, while the second is a more global approach which involves a higher quadrature of the overall right-hand side of the discretised equation(s). Both the strategies have been observed to be effective in enhancing the robustness of the schemes considered. We finally present the direct numerical simulation of a high-speed subsonic boundary-layer flow past a three-dimensional roughness element, achieved by means of the compressible aerodynamic solver developed. This type of analyses have been widely performed in the past with approximated theories. Only recently, has DNS been used due to the improvement of numerical techniques and an increase in computational resources for similar studies in low-speed subsonic, supersonic and hypersonic regimes. This thesis takes a first step to close the gap between the results for a high-speed subsonic regime and the results in supersonic and hypersonic regimes.
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Citrain, Aurélien. "Hybrid finite element methods for seismic wave simulation : coupling of discontinuous Galerkin and spectral element discretizations." Thesis, Normandie, 2019. http://www.theses.fr/2019NORMIR28.
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Pour résoudre des équations d’ondes posées dans des milieux hétérogènes avec des éléments finis et un coût numérique raisonnable, nous couplons la méthode Discontinue de Galerkine (DGm) avec des éléments finis spectraux (SEm). Nous utilisons des maillages hybrides composés de tétraèdres et d’hexaèdres structurés. Le couplage est réalisé en partant d’une formulation DG mixte primale posée dans un maillage hybride composé d’un macro-élément hexaédrique et d’un sous-maillage composé de tétraèdres. La SEm est appliquée dans le macro-élément découpé en hexaèdres structurés et le couplage est assuré par les flux numériques de la DGm appliqués sur les faces internes du macro-élément communes avec le maillage tétraédrique. La stabilité de la méthode couplée est démontrée quand l’intégration en temps est effectuée avec un schéma Saute-Mouton. Les performances de la méthode couplée sont étudiées numériquement et on montre que le couplage permet de réduire les coûts numériques avec un très bon niveau de précision. On montre aussi que la formulation couplée peut stabiliser la méthode DG appliquée dans des domaines incluant des couches parfaitement adaptéesTo solve wave equations in heterogeneous media with finite elements with a reasonable numerical cost, we couple the Discontinuous Galerkin method (DGm) with Spectral Elements method (SEm). We use hybrid meshes composed of tetrahedra and structured hexahedra. The coupling is carried out starting from a mixed-primal DG formulation applied on a hybrid mesh composed of a hexahedral macro-element and a sub-mesh composed of tetrahedra. The SEm is applied in the macro-element paved with structured hexahedrons and the coupling is ensured by the DGm numerical fluxes applied on the internal faces of the macro-element common with the tetrahedral mesh. The stability of the coupled method is demonstrated when time integration is performed with a Leap-Frog scheme. The performance of the coupled method is studied numerically and it is shown that the coupling reduces numerical costs while keeping a high level of accuracy. It is also shown that the coupled formulation can stabilize the DGm applied in areas that include Perfectly Matched Layers
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Mengaldo, Gianmarco. "Discontinuous spectral/hp element methods : development, analysis and applications to compressible flows." Thesis, Imperial College London, 2015. http://hdl.handle.net/10044/1/28678.
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This thesis is concerned with the development and analysis of discontinuous spectral/hp element methods and their applications to compressible aerodynamics with special focus on boundary-layer flows. In this thesis, we provide a detailed analysis on the connections between the discontinuous Galerkin method and the flux reconstruction approach for multidimensional nonlinear systems of conservation laws on irregular meshes (i.e. meshes with deformed and/or curved elements). The results help a better understanding of the broader class of discontinuous spectral/hp element methods and allow the direct applications to the flux reconstruction approach of the existing and more established techniques used in the discontinuous Galerkin community for tackling various issues of this class of schemes, including their aliasing problems. From this perspective, we present two dealiasing strategies based on the concept of consistent integration of the nonlinear terms (also referred to as over-integration of the linear terms). The first is a localised approach and it targets in each element the nonlinearities arising in the problem, while the second is a more global approach which involves a higher quadrature of the overall right-hand side of the discretised equation(s). The two dealiasing strategies have been observed to be effective in enhancing the numerical stability of both schemes, the flux reconstruction and the discontinuous Galerkin approaches. We finally present the direct numerical simulation of a high-speed subsonic flow past a roughness element, achieved by means of the discontinuous spectral/hp element methods developed. These results were successively compared to some data obtained from the asymptotic triple-deck theory. This work, besides demonstrating that the class of schemes analysed and developed is attractive for such aerodynamic problems, also addresses the lack of comparisons between theoretical models and numerical simulations.
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Claus, Susanne. "Numerical simulation of complex viscoelastic flows using discontinuous galerkin spectral/hp element methods." Thesis, Cardiff University, 2013. http://orca.cf.ac.uk/46909/.
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Viscoelastic flows are characterised by fast spatial and temporal variations in the solution featuring thin stress boundary near walls and stress concentrations in the vicinity of geometrical singularities. Resolving these fast variations of the fields in space and time is important for two reasons: (i) they affect the quantity of interest of the computation (e.g. drag force); and (ii) they are commonly believed to be associated with the numerical breakdown of the computation. Traditional discretisation methods such as finite differences or low-order finite elements require a large number of degrees of freedom to resolve these variations. Spectral methods enable this issue to be resolved by defining spatial expansions that are able to represent such variations with a smaller number of degrees of freedom. However, such methods are limited in terms of geometric flexibility. Recently, the spectral/hp element method (Karniadakis and Sherwin, 2005) has been developed in order to guarantee both spectral convergence, and geometric flexibility by allowing the use of quadrilateral and triangular elements. Our work is the first attempt to apply this method to viscoelastic free surface flows in arbitrary complex geometries. The conservation equations are solved in combination with the Oldroyd-B or Giesekus constitutive equation using the DEVSS-G/DG formulation. The combination of this formulation with a spectral element method is novel. A continuous approximation is employed for the velocity and discontinuous approximations for pressure, velocity gradient and polymeric stress. The conservation equations are discretised using the Galerkin method and the constitutive equation using a discontinuous Galerkin method to increase the stability of the approximation. The viscoelastic free surface is traced using an arbitrary Lagrangian Eulerian method. The performance of our scheme is demonstrated on the time-dependent Poiseuille flow in a channel, the flow around a cylinder and the die-swell problem.
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Wintermeyer, Niklas [Verfasser], and Gregor [Gutachter] Gassner. "A novel entropy stable discontinuous Galerkin spectral element method for the shallow water equations on GPUs / Niklas Wintermeyer ; Gutachter: Gregor Gassner." Köln : Universitäts- und Stadtbibliothek Köln, 2019. http://d-nb.info/1182533183/34.
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Vangelatos, Serena [Verfasser]. "On the Efficiency of Implicit Discontinuous Galerkin Spectral Element Methods for the Unsteady Compressible Navier-Stokes Equations / Serena Vangelatos." München : Verlag Dr. Hut, 2020. http://d-nb.info/1222352222/34.
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Vangelatos, Serena [Verfasser], and Claus-Dieter [Akademischer Betreuer] Munz. "On the efficiency of implicit discontinuous Galerkin spectral element methods for the unsteady compressible Navier-Stokes equations / Serena Vangelatos ; Betreuer: Claus-Dieter Munz." Stuttgart : Universitätsbibliothek der Universität Stuttgart, 2020. http://d-nb.info/1206184051/34.
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Sert, Cuneyt. "Nonconforming formulations with spectral element methods." Diss., Texas A&M University, 2003. http://hdl.handle.net/1969.1/1268.
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A spectral element algorithm for solution of the incompressible Navier-Stokes and heat transfer equations is developed, with an emphasis on extending the classical conforming Galerkin formulations to nonconforming spectral elements. The new algorithm employs both the Constrained Approximation Method (CAM), and the Mortar Element Method (MEM) for p-and h-type nonconforming elements. Detailed descriptions, and formulation steps for both methods, as well as the performance comparisons between CAM and MEM, are presented. This study fills an important gap in the literature by providing a detailed explanation for treatment of p-and h-type nonconforming interfaces. A comparative eigenvalue spectrum analysis of diffusion and convection operators is provided for CAM and MEM. Effects of consistency errors due to the nonconforming formulations on the convergence of steady and time dependent problems are studied in detail. Incompressible flow solvers that can utilize these nonconforming formulations on both p- and h-type nonconforming grids are developed and validated. Engineering use of the developed solvers are demonstrated by detailed parametric analyses of oscillatory flow forced convection heat transfer in two-dimensional channels.
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Chaurasia, Hemant Kumar. "A time-spectral hybridizable discontinuous Galerkin method for periodic flow problems." Thesis, Massachusetts Institute of Technology, 2014. http://hdl.handle.net/1721.1/90647.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2014.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 110-120).
Numerical simulations of time-periodic flows are an essential design tool for a wide range of engineered systems, including jet engines, wind turbines and flapping wings. Conventional solvers for time-periodic flows are limited in accuracy and efficiency by the low-order Finite Volume and time-marching methods they typically employ. These methods introduce significant numerical dissipation in the simulated flow, and can require hundreds of timesteps to describe a periodic flow with only a few harmonic modes. However, recent developments in high-order methods and Fourier-based time discretizations present an opportunity to greatly improve computational performance. This thesis presents a novel Time-Spectral Hybridizable Discontinuous Galerkin (HDG) method for periodic flow problems, together with applications to flow through cascades and rotor/stator assemblies in aeronautical turbomachinery. The present work combines a Fourier-based Time-Spectral discretization in time with an HDG discretization in space, realizing the dual benefits of spectral accuracy in time and high-order accuracy in space. Low numerical dissipation and favorable stability properties are inherited from the high-order HDG method, together with a reduced number of globally coupled degrees of freedom compared to other DG methods. HDG provides a natural framework for treating boundary conditions, which is exploited in the development of a new high-order sliding mesh interface coupling technique for multiple-row turbomachinery problems. A regularization of the Spalart-Allmaras turbulence model is also employed to ensure numerical stability of unsteady flow solutions obtained with high-order methods. Turning to the temporal discretization, the Time-Spectral method enables direct solution of a periodic flow state, bypasses initial transient behavior, and can often deliver substantial savings in computational cost compared to implicit time-marching. An important driver of computational efficiency is the ability to select and resolve only the most important frequencies of a periodic problem, such as the blade-passing frequencies in turbomachinery flows. To this end, the present work introduces an adaptive frequency selection technique, using the Time-Spectral residual to form an inexpensive error indicator. Having selected a set of frequencies, the accuracy of the Time-Spectral solution is greatly improved by using optimally selected collocation points in time. For multi-domain problems such as turbomachinery flows, an anti-aliasing filter is also needed to avoid errors in the transfer of the solution across the sliding interface. All of these aspects contribute to the Adaptive Time-Spectral HDG method developed in this thesis. Performance characteristics of the method are demonstrated through applications to periodic ordinary differential equations, a convection problem, laminar flow over a pitching airfoil, and turbulent flow through a range of single- and multiple-row turbomachinery configurations. For a 2:1 rotor/stator flow problem, the Adaptive Time-Spectral HDG method correctly identifies the relevant frequencies in each blade row. This leads to an accurate periodic flow solution with greatly reduced computational cost, when compared to sequentially selected frequencies or a time-marching solution. For comparable accuracy in prediction of rotor loading, the Adaptive Time- Spectral HDG method incurs 3 times lower computational cost (CPU time) than time-marching, and for prediction of only the 1st harmonic amplitude, these savings rise to a factor of 200. Finally, in three-row compressor flow simulations, a high-order HDG method is shown to achieve significantly greater accuracy than a lower-order method with the same computational cost. For example, considering error in the amplitude of the 1st harmonic mode of total rotor loading, a p = 1 computation results in 20% error, in contrast to only 1% error in a p = 4 solution with comparable cost. This highlights the benefits that can be obtained from higher-order methods in the context of turbomachinery flow problems.
by Hemant Kumar Chaurasia.
Ph. D.
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Bao, Weiyu. "Modelling excavations in discontinuous rock using the distinct element method." Thesis, University of Southampton, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.431928.
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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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Toolabi, Milad. "Dynamic extended finite element method (XFEM) analysis of discontinuous media." Thesis, Imperial College London, 2015. http://hdl.handle.net/10044/1/44180.
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The extended finite element method (XFEM) is found promising in approximating solutions to locally non-smooth features such as jumps, kinks, high gradients, inclusions, or cracks in solid mechanics problems. The XFEM uses the properties of the partition of unity finite element method (PUFEM) to represent the discontinuities without the corresponding finite element mesh requirements. In the present thesis numerical simulations of statically and dynamically loaded heterogeneous beams, heterogeneous plates and two-dimensional cracked media of isotropic and orthotropic constitutive behaviour are performed using XFEM. The examples are chosen such that they represent strong and weak discontinuities, static and dynamic loading conditions, anisotropy and isotropy and strain-rate dependent and independent behaviours. At first, the Timoshenko beam element is studied by adopting the Hellinger-Reissner (HR) functional with the out-of-plane displacement and through-thickness shear strain as degrees of freedom. Heterogeneous beams are considered and the mixed formulation has been combined with XFEM thus mixed enrichment functions are used. The results from the proposed mixed formulation of XFEM correlate well with analytical solutions and Finite Element Method (FEM) and show higher rates of convergence. Thus the proposed method is shear-locking free and computationally more efficient compared to its conventional counterparts. The study is then extended to a heterogeneous Mindlin-Reissner plate with out-of-plane shear assumed constant through length of the element and with a quadratic distribution through the thickness. In all cases the zero shear on traction-free surfaces at the top and bottom are satisfied. These cases involve weak discontinuity. Then a two-dimensional orthotropic medium with an edge crack is considered and the static and dynamic J-integrals and stress intensity factors (SIF's) are calculated. This is achieved by fully (reproducing elements) or partially (blending elements) enriching the elements in the vicinity of the crack tip or body. The enrichment type is restricted to extrinsic mesh-based topological local enrichment in the current work. A constitutive model for strain-rate dependent moduli and Poisson ratios (viscoelasticity) is formulated. The same problem is studied using the viscoelastic constitutive material model implemented in ABAQUS through an implicit user defined material subroutine (UMAT). The results from XFEM correlate well with those of the finite element method (FEM). It is shown that there is an increase in the value of maximum J-integral when the material exhibits strain rate sensitivity.
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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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Jeannequin, Nicolas. "Nodal Spectral Element Method Applied To Electrocardiological Models." Thesis, University of Oxford, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.526068.
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Meng, Sha. "A spectral element method for viscoelastic fluid flow." Thesis, De Montfort University, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.369907.
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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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Moura, Rodrigo Costa. "A high-order unstructured discontinuous galerkin finite element method for aerodynamics." Instituto Tecnológico de Aeronáutica, 2012. http://www.bd.bibl.ita.br/tde_busca/arquivo.php?codArquivo=2158.
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The present thesis discuss in a didactic and detailed way the high-order scheme known as the Discontinuous Galerkin (DG) method, with special focus on applications in aerodynamics. The theoretical formulation of the method is presented in one and two dimensions with great depth, being properly discussed issues of convergence, basis functions, interelement communication, boundary conditions, shock treatment, as well as inviscid and viscous numerical fluxes. As part of this effort, a parallel computer code was developed to simulate the Euler equations of gas dynamics in two dimensions with general boundary conditions over unstructured meshes of triangles. Numerical simulations are addressed in order to demonstrate the characteristics of the Discontinuous Galerkin scheme, as well as to validate the developed solver. It is worth mentioning that the present work can be regarded as new within the Brazilian scientific community and, as such, may be of great importance concerning the introduction of the DG method for Brazilian CFD researchers and practitioners.
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Zheng, Yu. "Numerical simulation of droplet deformation using spectral element method." Thesis, Cardiff University, 2007. http://orca.cf.ac.uk/54648/.
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Spectral element methods are developed for solving two-phase flow problems of relevance to the power generation industry. In particular, an algorithm is presented for predicting the deformation of a Newtonian droplet accelerating from rest in a gas flow field. The dimensionless governing equations are written in terms of four dimensionless groups: Reynolds number, Weber number, and the viscosity and density ratios of the two fluids. The arbitrary Lagrangian-Eulerian (ALE) formulation is used to account for the movement of the mesh. Spectral element approximations are used to ensure a high degree of spatial accuracy. The computational domain is decomposed into two regions, one of which remains fixed in time while the other, located in the vicinity of the droplet, is allowed to deform within the ALE frame work. Transfinite mapping techniques are used to map the physical elements onto the computational element. Edges of elements on the free surface are described using an isoparametric mapping and blended into the elemental mapping. Surface tension is treated implicitly and naturally within the weak formulation of the problem. In addition to the movement of the mesh, the computation associated with each time step comprises an explicit treatment of the convection term and an implicit treatment of the linear terms. The generalized Stokes problem generated in the latter is solved using a nested preconditioned conjugate gradient method. The initial code development and the accuracy of the spectral element approximation is validated for the problem of flow over a solid sphere confined in a cylinder or in a uniform ambience. The problem of gas flow over a Newtonian droplet with different fluid properties is then investigated and the effects of changes in the values of the dimensionless groups on the deformation of the droplet is analysed. Finally, the Cross law is introduced to model the viscosity of non-Newtonian fluids and the problem of gas flow over a blood drop is simulated.
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Kocal, Arman. "Three Dimensional Numerical Modelling Of Discontinuous Rocks By Using Distinct Element Method." Phd thesis, METU, 2008. http://etd.lib.metu.edu.tr/upload/12610005/index.pdf.
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Shear strength characterization of discontinuities is an important concept for slope design in discontinuous rocks. This study presents the development of a methodology for implementing Barton-Bandis empirical shear strength failure criterion in three dimensional distinct element code, 3DEC, and verification of this methodology.
Normal and shear deformation characteristics of discontinuities and their relations to the discontinuity surface characteristics have been reviewed in detail.
First, a C++ dynamic link library (DLL) file was coded and embedded into 3DEC for modelling the Barton-Bandis shear strength criterion. Then, a numerically developed direct shear test model was used to verify the normal and shear deformation behaviour with respect to empirical results of the Barton-Bandis shear strength criterion.
A three dimensional simple discontinuous rock slope was modelled in 3DEC based on Barton-Bandis shear strength criterion. The slope model was first utilized by Mohr-Coulomb failure criterion. Then, with the use of the new model developed here, the effects of the discontinuity surface properties on shear strength were introduced to the slope problem.
Applicability of the developed model was verified by three large scale real case studies from different open pit lignite mines of Turkish Coal Enterprises (TKi), namely Bursa Lignites Establishment (BLi) &ndash2 cases and Ç
an Lignite Establishment (Ç
Li). The results with the new model option, which allows users to use important discontinuity surface properties like joint roughness coefficient and joint wall compressive strength, compared well with results of previous studies using Mohr-Coulomb failure criterion.
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Xiao, Yilong. "A Discontinuous Galerkin Finite Element Method Solution of One-Dimensional Richards’ Equation." The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1461255638.
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Toutip, Wattana. "The dual reciprocity boundary element method for linear and non-linear problems." Thesis, University of Hertfordshire, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.369302.
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A problem encountered in the boundary element method is the difficulty caused by corners and/or discontinuous boundary conditions. An existing code using standard linear continuous elements is modified to overcome such problems using the multiple node method with an auxiliary boundary collocation approach. Another code is implemented applying the gradient approach as an alternative to handle such problems. Laplace problems posed on variety of domain shapes have been introduced to test the programs. For Poisson problems the programs have been developed using a transformation to a Laplace problem. This method cannot be applied to solve Poissontype equations. The dual reciprocity boundary element method (DRM) which is a generalised way to avoid domain integrals is introduced to solve such equations. The gradient approach to handle corner problems is co-opted in the program using DRM. The program is modified to solve non-linear problems using an iterative method. Newton's method is applied in the program to enhance the accuracy of the results and reduce the number of iterations. The program is further developed to solve coupled Poisson-type equations and such a formulation is considered for the biharmonic problems. A coupled pair of non-linear equations describing the ohmic heating problem is also investigated. Where appropriate results are compared with those from reference solutions or exact solutions. v
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Ju, Xiaozhu. "A discontinuous transport methodology for solidification modelling." Thesis, University of Manchester, 2015. https://www.research.manchester.ac.uk/portal/en/theses/a-discontinuous-transport-methodology-for-solidification-modelling(32f86a90-05f6-45a1-938f-357e93398d39).html.
Full textAbstract:
Phase change in solidification and melting can be described with the aid of discontinuous functions. The aim of this project is to establish effective methodologies for the solution of discontinuous phase-change problems. The classic capacitance method, which distributes the effect of any discontinuity present over a finite region (typically an element), can suffer from inaccurate energy transport. Improvement is possible with the application of the classic non-physical enthalpy method. However, this approach is known to suffer with the imposition of material velocity, which gives rise to negative thermal capacitance providing a source of error and instability. In order to improve on the performance of the capacitance method and the classic non-physical enthalpy method, this research introduces a series of new non-physical variables. Firstly, a new non-physical enthalpy is defined via the weak form of the energy transport equation. The classical non-physical enthalpy was defined using a temporal integral term. In the new definition, the non-physical enthalpy involves both a temporal and an advection term, which is shown to avoid the generation of negative capacitance and improve the stability of advection heat transfer in numerical methods. Secondly, control volume analysis is performed on weighted and unweighted forms of the governing energy equation involving non-physical enthalpy. The analysis is shown to reveal non-physical source terms that facilitate the removal of phase-change discontinuities. Thirdly, it is demonstrated in the thesis how a non-physical heat source must be introduced into the governing non-physical transport equation to remove discontinuities arising from non-physical terms related to advection. To demonstrate the accuracy and stability of the new method, it is implemented in the finite element method for both one-dimensional linear rod elements and two dimensional triangular elements. Update techniques and root finding methods, such as the predictor-corrector method, the secant method and the homotopy method, are applied to solve the non-linear system of equations, which are constructed with the new theory. Results returned from the one-dimensional numerical experiments are compared with exact solutions, which show reasonable accuracy. Numerical experiments for isothermal solidification with advection-diffusion in both one and two dimensions demonstrate the feasibility of the new methodology.
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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.
Full textAbstract:
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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Dosopoulos, Stylianos. "Interior Penalty Discontinuous Galerkin Finite Element Method for the Time-Domain Maxwell's Equations." The Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1337787922.
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Burleson, John Taylor. "Numerical Simulations of Viscoelastic Flows Using the Discontinuous Galerkin Method." Thesis, Virginia Tech, 2021. http://hdl.handle.net/10919/104869.
Full textAbstract:
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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Massey, Thomas Christopher. "A Flexible Galerkin Finite Element Method with an A Posteriori Discontinuous Finite Element Error Estimation for Hyperbolic Problems." Diss., Virginia Tech, 2002. http://hdl.handle.net/10919/28245.
Full textAbstract:
A Flexible Galerkin Finite Element Method (FGM) is a hybrid class of finite element methods that combine the usual continuous Galerkin method with the now popular discontinuous Galerkin method (DGM). A detailed description of the formulation of the FGM on a hyperbolic partial differential equation, as well as the data structures used in the FGM algorithm is presented. Some hp-convergence results and computational cost are included. Additionally, an a posteriori error estimate for the DGM applied to a two-dimensional hyperbolic partial differential equation is constructed. Several examples, both linear and nonlinear, indicating the effectiveness of the error estimate are included.Ph. D.
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Harkin, Anthony. "A Legendre spectral element method for the rotational Navier-Stokes equations." Thesis, Massachusetts Institute of Technology, 1995. http://hdl.handle.net/1721.1/37551.
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Wessel, Richard Allen Jr. "Spectral element method for numerical simulation of unsteady laminar diffusion flames." Case Western Reserve University School of Graduate Studies / OhioLINK, 1993. http://rave.ohiolink.edu/etdc/view?acc_num=case1057076472.
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Ohlsson, Johan. "Spectral-element simulations of separated turbulent internal flows." Licentiate thesis, Stockholm : Skolan för teknikvetenskap, Kungliga Tekniska högskolan, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-11643.
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Klenow, Bradley. "Finite and Spectral Element Methods for Modeling Far-Field Underwater Explosion Effects on Ships." Diss., Virginia Tech, 2009. http://hdl.handle.net/10919/37648.
Full textAbstract:
The far-field underwater explosion (UNDEX) problem is a complicated problem dominated by two phenomena: the shock wave traveling through the fluid and the cavitation in the fluid. Both of these phenomena have a significant effect on the loading of ship structures subjected to UNDEX.
An approach to numerically modeling these effects in the fluid and coupling to a structural model is using cavitating acoustic finite elements (CAFE) and more recently cavitating acoustic spectral elements (CASE). The use of spectral elements in CASE has shown to offer the greater accuracy and reduced computational expense when compared to traditional finite elements. However, spectral elements also increase spurious oscillations in both the fluid and structural response.
This dissertation investigates the application of CAFE, CASE, and a possible improvement to CAFE in the form of a finite element flux-corrected transport algorithm, to the far-field UNDEX problem by solving a set of simplified UNDEX problems. Specifically we examine the effect of increased oscillations on structural response and the effect of errors in cavitation capture on the structural response which have not been thoroughly explored in previous work.
The main contributions of this work are a demonstration of the problem dependency of increased oscillations in the structural response when applying the CASE methodology, the demonstration of how the sensitivity of errors in the structural response changes with changes in the structural model, a detailed explanation of how error in cavitation capture influences the structural response, and a demonstration of the need to accurately capture the shape and magnitude of cavitation regions in the fluid in order to obtain accurate structural response results.Ph. D.
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Bailey, Teresa S. "The piecewise linear discontinuous finite element method applied to the RZ and XYZ transport equations." Diss., Texas A&M University, 2008. http://hdl.handle.net/1969.1/85942.
Full textAbstract:
In this dissertation we discuss the development, implementation, analysis and testing of
the Piecewise Linear Discontinuous Finite Element Method (PWLD) applied to the
particle transport equation in two-dimensional cylindrical (RZ) and three-dimensional
Cartesian (XYZ) geometries. We have designed this method to be applicable to
radiative-transfer problems in radiation-hydrodynamics systems for arbitrary polygonal
and polyhedral meshes. For RZ geometry, we have implemented this method in the
Capsaicin radiative-transfer code being developed at Los Alamos National Laboratory.
In XYZ geometry, we have implemented the method in the Parallel Deterministic
Transport code being developed at Texas A&M University.
We discuss the importance of the thick diffusion limit for radiative-transfer problems,
and perform a thick diffusion-limit analysis on our discretized system for both
geometries. This analysis predicts that the PWLD method will perform well in this limit
for many problems of physical interest with arbitrary polygonal and polyhedral cells.
Finally, we run a series of test problems to determine some useful properties of the
method and verify the results of our thick diffusion limit analysis.
Finally, we test our method on a variety of test problems and show that it compares
favorably to existing methods. With these test problems, we also show that our method
performs well in the thick diffusion limit as predicted by our analysis. Based on
PWLD's solid finite-element foundation, the desirable properties it shows under analysis, and the excellent performance it demonstrates on test problems even with
highly distorted spatial grids, we conclude that it is an excellent candidate for radiativetransfer
problems that need a robust method that performs well in thick diffusive
problems or on distorted grids.
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Vadsola, Mayank. "High-Order Spectral Element Method Simulation of Flow Past a 30P30N Three-Element High Lift Wing." Thesis, Université d'Ottawa / University of Ottawa, 2020. http://hdl.handle.net/10393/40964.
Full textAbstract:
The purpose of a multi-element high lift device is to increase lift dramatically
while controlling the stall limit. The fluid flow over a multi-element high lift device
has been explored widely both experimentally and numerically at high Reynolds
numbers (O(10^6 )). The numerical simulations use turbulence models and hence
details of the flow are not yet available. Low Reynolds number (O(10^4 )) flows
over high lift devices have not been explored until recently. These lower Reynolds
number flows have applications in the development of small aerial vehicles. The
present work discusses both two-dimensional and three-dimensional direct numer-
ical simulations of fluid flow over a 30P30N three-element high lift system using a
high-order spectral element method code, Nek5000, that solves the incompressible
Navier-Stokes equations. The intricate geometry of the multi-element device poses
a challenge for the high-order spectral element method. We study the complex
flow physics in the slat cove region and the wake/shear layer interaction over a
30P30N three-element high lift device. The targeted cases are at Reynolds num-
bers based on stowed chord lengths (Rec ) of 8.32 × 10^3 , 1.27 × 10^4 , and 1.83 × 10^4 at angle of attack of 4. A critical interval for Rec has previously been found
between 1.27 × 10^4 and 1.38 × 10^4 in experiments. This divides the flow into
two types: when Rec is below the critical interval, no roll-up is observed in the
slat cove and Görtler vortices dominate the slat wake; however when the Rec is
above the critical interval, a roll-up is observed in the slat cove and co-existence
of streamwise and spanwise vortices is confirmed in the slat wake. We confirm
the presence of the critical interval from the simulations performed at three values of Rec . Lift and drag analysis is provided along with pressure coefficient plots
for each element of the multi-element airfoil. Different vortical structures are also
identified in the transition of flow from two dimensions to three dimensions. The
relevant validation is performed with the available experimental data.
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Kim, Bo Hung. "A graphical preprocessing interface for non-conforming spectral element solvers." [College Station, Tex. : Texas A&M University, 2006. http://hdl.handle.net/1969.1/ETD-TAMU-1819.
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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.
Full textAbstract:
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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Taghaddosi, Farzad. "3D parallel computations of turbofan noise propagation using a spectral element method." Thesis, McGill University, 2006. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=102733.
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A three-dimensional code has been developed for the simulation of tone noise generated by turbofan engine inlets using computational aeroacoustics. The governing equations are the linearized Euler equations, which are further simplified to a set of equations in terms of acoustic potential, using the irrotational flow assumption, and subsequently solved in the frequency domain.Due to the special nature of acoustic wave propagation, the spatial discretization is performed using a spectral element method, where a tensor product of the nth-degree polynomials based on Chebyshev orthogonal functions is used to approximate variations within hexahedral elements. Non-reflecting boundary conditions are imposed at the far-field using a damping layer concept. This is done by augmenting the continuity equation with an additional term without modifying the governing equations as in PML methods.
Solution of the linear system of equations for the acoustic problem is based on the Schur complement method, which is a nonoverlapping domain decomposition technique. The Schur matrix is first solved using a matrix-free iterative method, whose convergence is accelerated with a novel local preconditioner. The solution in the entire domain is then obtained by finding solutions in smaller subdomains.
The 3D code also contains a mean flow solver based on the full potential equation in order to take into account the effects of flow variations around the nacelle on the scattering of the radiated sound field.
All aspects of numerical simulations, including building and assembling the coefficient matrices, implementation of the Schur complement method, and solution of the system of equations for both the acoustic and mean flow problems are performed on multiprocessors in parallel using the resources of the CLUMEQ Supercomputer Center. A large number of test cases are presented, ranging in size from 100 000-2 000 000 unknowns for which, depending on the size of the problem, between 8-48 CPU's are used.
The developed code is demonstrated to be robust and efficient in simulating acoustic propagation for a large number of problems, with an excellent parallel performance.
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Bolis, Alessandro. "Fourier spectral/hp element method : investigation of time-stepping and parallelisation strategies." Thesis, Imperial College London, 2013. http://hdl.handle.net/10044/1/25140.
Full textAbstract:
As computer hardware has evolved, the time required to perform numerical simulations has reduced, allowing investigations of a wide range of new problems. This thesis focuses on algorithm optimization, to minimize run-time, when solving the incompressible Navier-Stokes equations. Aspects affecting performance related to the discretisation and algorithm parallelization are investigated in the context of high-order methods. The roles played by numerical approximations and computational strategies are highlighted and it is recognized that a versatile implementation provides additional benefits, allowing an ad-hoc selection of techniques to fit the needs of heterogeneous computing environments. We initially describe the building blocks of a spectral/hp element and pure spectral method and how they can be encapsulated and combined to create a 3D discretisation, the Fourier spectral/hp element method. Time-stepping strategies are also described and encapsulated in a flexible framework based on the General Linear Method. After implementing and validating an incompressible Navier-Stokes solver, two canonical turbulent flows are analyzed. Afterward a 2D hyperbolic equation is considered to investigate the efficiency of low- and high-order methods when discretising the spatial and temporal derivatives. We perform parametric studies, monitoring accuracy and CPU-time for different numerical approximations. We identify optimal discretisations, demonstrating that high-order methods are the computationally fastest approach to attain a desired accuracy for this problem. Following the same philosophy, we investigate the benefits of using a hybrid parallel implementation. The message passing model is introduced to parallelize different kernels of an incompressible Navier-Stokes solver. Monitoring the parallel performance of these strategies the most efficient approach is highlighted. We also demonstrate that hybrid parallel solutions can be used to significantly extend the strong scalability limit and support greater parallelism.
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Cui, Xiaoming. "Discontinuous finite/boundary element method for radiative heat transfer with application in laser cancer therapy." Online access for everyone, 2005. http://www.dissertations.wsu.edu/Dissertations/Fall2005/x%5Fcui%5F121805.pdf.
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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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Quattrochi, Douglas J. (Douglas John). "Hypersonic heat transfer and anisotropic visualization with a higher order discontinuous Galerkin finite element method." Thesis, Massachusetts Institute of Technology, 2006. http://hdl.handle.net/1721.1/35567.
Full textAbstract:
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2006.Includes bibliographical references (leaves 83-89).
Higher order discretizations of the Navier-Stokes equations promise greater accuracy than conventional computational aerodynamics methods. In particular, the discontinuous Galerkin (DG) finite element method has O(hP+l) design accuracy and allows for subcell resolution of shocks. This work furthers the DG finite element method in two ways. First, it demonstrates the results of DG when used to predict heat transfer to a cylinder in a hypersonic flow. The strong shock is captured with a Laplacian artificial viscosity term. On average, the results are in agreement with an existing hypersonic benchmark. Second, this work improves the visualization of the higher order polynomial solutions generated by DG with an adaptive display algorithm. The new algorithm results in more efficient displays of higher order solutions, including the hypersonic flow solutions generated here.
by Douglas J. Quattrochi.
S.M.
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Sevilla, Cárdenas Rubén. "NURBS-Enhanced Finite Element Method (NEFEM)." Doctoral thesis, Universitat Politècnica de Catalunya, 2009. http://hdl.handle.net/10803/5857.
Full textAbstract:
Aquesta tesi proposa una millora del clàssic mètode dels elements finits (finite element method, FEM) per a un tractament eficient de dominis amb contorns corbs: el denominat NURBS-enhanced finite element method (NEFEM). Aquesta millora permet descriure de manera exacta la geometría mitjançant la seva representació del contorn CAD amb non-uniform rational B-splines (NURBS), mentre que la solució s'aproxima amb la interpolació polinòmica estàndard. Per tant, en la major part del domini, la interpolació i la integració numèrica són estàndard, retenint les propietats de convergència clàssiques del FEM i facilitant l'acoblament amb els elements interiors. Només es requereixen estratègies específiques per realitzar la interpolació i la integració numèrica en elements afectats per la descripció del contorn mitjançant NURBS.La implementació i aplicació de NEFEM a problemes que requereixen una descripció acurada del contorn són, també, objectius prioritaris d'aquesta tesi. Per exemple, la solució numèrica de les equacions de Maxwell és molt sensible a la descripció geomètrica. Es presenta l'aplicació de NEFEM a problemes d'scattering d'ones electromagnètiques amb una formulació de Galerkin discontinu. S'investiga l'habilitat de NEFEM per obtenir solucions precises amb malles grolleres i aproximacions d'alt ordre, i s'exploren les possibilitats de les anomenades malles NEFEM, amb elements que contenen singularitats dintre d'una cara o aresta d'un element. Utilitzant NEFEM, la mida de la malla no està controlada per la complexitat de la geometria. Això implica una dràstica diferència en la mida dels elements i, per tant, suposa un gran estalvi tant des del punt de vista de requeriments de memòria com de cost computacional. Per tant, NEFEM és una eina poderosa per la simulació de problemes tridimensionals a gran escala amb geometries complexes. D'altra banda, la simulació de problemes d'scattering d'ones electromagnètiques requereix mecanismes per aconseguir una absorció eficient de les ones scattered. En aquesta tesi es discuteixen, optimitzen i comparen dues tècniques en el context de mètodes de Galerkin discontinu amb aproximacions d'alt ordre.
La resolució numèrica de les equacions d'Euler de la dinàmica de gasos és també molt sensible a la representació geomètrica. Quan es considera una formulació de Galerkin discontinu i elements isoparamètrics lineals, una producció espúria d'entropia pot evitar la convergència cap a la solució correcta. Amb NEFEM, l'acurada imposició de la condició de contorn en contorns impenetrables proporciona resultats precisos inclús amb una aproximació lineal de la solució. A més, la representació exacta del contorn permet una imposició adequada de les condicions de contorn amb malles grolleres i graus d'interpolació alts. Una propietat atractiva de la implementació proposada és que moltes de les rutines usuals en un codi d'elements finits poden ser aprofitades, per exemple rutines per realitzar el càlcul de les matrius elementals, assemblatge, etc. Només és necessari implementar noves rutines per calcular les quadratures numèriques en elements corbs i emmagatzemar el valor de les funciones de forma en els punts d'integració. S'han proposat vàries tècniques d'elements finits corbs a la literatura. En aquesta tesi, es compara NEFEM amb altres tècniques populars d'elements finits corbs (isoparamètics, cartesians i p-FEM), des de tres punts de vista diferents: aspectes teòrics, implementació i eficiència numèrica. En els exemples numèrics, NEFEM és, com a mínim, un ordre de magnitud més precís comparat amb altres tècniques. A més, per una precisió desitjada NEFEM és també més eficient: necessita un 50% dels graus de llibertat que fan servir els elements isoparamètrics o p-FEM per aconseguir la mateixa precisió. Per tant, l'ús de NEFEM és altament recomanable en presència de contorns corbs i/o quan el contorn té detalls geomètrics complexes.
This thesis proposes an improvement of the classical finite element method (FEM) for an efficient treatment of curved boundaries: the NURBSenhanced FEM (NEFEM). It is able to exactly represent the geometry by means of the usual CAD boundary representation with non-uniform rational Bsplines (NURBS), while the solution is approximated with a standard piecewise polynomial interpolation. Therefore, in the vast majority of the domain, interpolation and numerical integration are standard, preserving the classical finite element (FE) convergence properties, and allowing a seamless coupling with standard FEs on the domain interior. Specifically designed polynomial interpolation and numerical integration are designed only for those elements affected by the NURBS boundary representation.
The implementation and application of NEFEM to problems demanding an accurate boundary representation are also primary goals of this thesis. For instance, the numerical solution of Maxwell's equations is highly sensitive to geometry description. The application of NEFEM to electromagnetic scattering problems using a discontinuous Galerkin formulation is presented. The ability of NEFEM to compute an accurate solution with coarse meshes and high-order approximations is investigated, and the possibilities of NEFEM meshes, with elements containing edge or corner singularities, are explored. With NEFEM, the mesh size is no longer subsidiary to geometry complexity, and depends only on the accuracy requirements on the solution, whereas standard FEs require mesh refinement to properly capture the geometry. This implies a drastic difference in mesh size that results in drastic memory savings, and also important savings in computational cost. Thus, NEFEM is a powerful tool for large-scale scattering simulations with complex geometries in three dimensions. Another key issue in the numerical solution of electromagnetic scattering problems is using a mechanism to perform the absorption of outgoing waves. Two perfectly matched layers are discussed, optimized and compared in a high-order discontinuous Galerkin framework.
The numerical solution of Euler equations of gas dynamics is also very sensitive to geometry description. Using a discontinuous Galerkin formulation and linear isoparametric elements, a spurious entropy production may prevent convergence to the correct solution. With NEFEM, the exact imposition of the solid wall boundary condition provides accurate results even with a linear approximation of the solution. Furthermore, the exact boundary representation allows using coarse meshes, but ensuring the proper implementation of the solid wall boundary condition. An attractive feature of the proposed implementation is that the usual routines of a standard FE code can be directly used, namely routines for the computation of elemental matrices and vectors, assembly, etc. It is only necessary to implement new routines for the computation of numerical quadratures in curved elements and to store the value of shape functions at integration points.
Several curved FE techniques have been proposed in the literature. In this thesis, NEFEM is compared with some popular curved FE techniques (namely isoparametric FEs, cartesian FEs and p-FEM), from three different perspectives: theoretical aspects, implementation and performance. In every example shown, NEFEM is at least one order of magnitude more accurate compared to other techniques. Moreover, for a desired accuracy NEFEM is also computationally more efficient. In some examples, NEFEM needs only 50% of the number of degrees of freedom required by isoparametric FEs or p-FEM. Thus, the use of NEFEM is strongly recommended in the presence of curved boundaries and/or when the boundary of the domain has complex geometric details.
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Storn, Johannes. "Topics in Least-Squares and Discontinuous Petrov-Galerkin Finite Element Analysis." Doctoral thesis, Humboldt-Universität zu Berlin, 2019. http://dx.doi.org/10.18452/20141.
Full textAbstract:
Aufgrund der fundamentalen Bedeutung partieller Differentialgleichungen zur Beschreibung von Phänomenen in angewandten Wissenschaften ist deren Analyse ein Kerngebiet der Mathematik. Durch Computer lassen sich die Lösungen für eine Vielzahl dieser Gleichungen näherungsweise bestimmen. Die dabei verwendeten numerischen Verfahren sollen auf möglichst exakte Approximationen führen und deren Genauigkeit verifizieren. Die Least-Squares Finite-Elemente-Methode (LSFEM) und die unstetige Petrov-Galerkin (DPG) Methode sind solche Verfahren. Sie werden in dieser Dissertation untersucht.
Der erste Teil der Arbeit untersucht die Genauigkeit der mittels LSFEM berechneten Näherungen. Dazu werden Eigenschaften der zugrundeliegenden Differentialgleichungen mit den Eigenschaften der LSFEM kombiniert. Dies zeigt, dass die Abweichung der berechneten Näherung von der exakten Lösung einem berechenbaren Residuum asymptotisch entspricht. Ferner wird ein Verfahren zu Berechnung einer garantierten oberen Fehlerschranke eingeführt. Während etablierte Fehlerschätzer den Fehler signifikant überschätzt, zeigen numerische Experimente eine äußerst geringe Überschätzung des Fehlers mittels der neuen Fehlerschranke.
Die Analyse der Fehlerschranken für das Stokes-Problem offenbart ein Beziehung der LSFEM und der LBB Konstanten. Diese Konstante ist entscheidend für die Existenz und Stabilität von Lösungen in der Strömungslehre. Der zweite Teil der Arbeit nutzt diese Beziehung und entwickelt ein auf der LSFEM basierendes Verfahren zur numerischen Berechnung der LBB Konstanten.
Der dritte Teil der Arbeit untersucht die DPG Methode. Dabei werden existierende Anwendungen der DPG Methode zusammengefasst und analysiert. Diese Analyse zeigt, dass sich die DPG Methode als eine leicht gestörte LSFEM interpretieren lässt. Diese Interpretation erlaubt die Anwendung der Resultate aus dem ersten Teil der Arbeit und ermöglicht dadurch eine genauere Untersuchung existierender und die Entwicklung neuer DPG Methoden.The analysis of partial differential equations is a core area in mathematics due to the fundamental role of partial differential equations in the description of phenomena in applied sciences. Computers can approximate the solutions to these equations for many problems. They use numerical schemes which should provide good approximations and verify the accuracy. The least-squares finite element method (LSFEM) and the discontinuous Petrov-Galerkin (DPG) method satisfy these requirements. This thesis investigates these two schemes. The first part of this thesis explores the accuracy of solutions to the LSFEM. It combines properties of the underlying partial differential equation with properties of the LSFEM and so proves the asymptotic equality of the error and a computable residual. Moreover, this thesis introduces an novel scheme for the computation of guaranteed upper error bounds. While the established error estimator leads to a significant overestimation of the error, numerical experiments indicate a tiny overestimation with the novel bound. The investigation of error bounds for the Stokes problem visualizes a relation of the LSFEM and the Ladyzhenskaya-Babuška-Brezzi (LBB) constant. This constant is a key in the existence and stability of solution to problems in fluid dynamics. The second part of this thesis utilizes this relation to design a competitive numerical scheme for the computation of the LBB constant. The third part of this thesis investigates the DPG method. It analyses an abstract framework which compiles existing applications of the DPG method. The analysis relates the DPG method with a slightly perturbed LSFEM. Hence, the results from the first part of this thesis extend to the DPG method. This enables a precise investigation of existing and the design of novel DPG schemes.
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Lu, James 1977. "An a posteriori error control framework for adaptive precision optimization using discontinuous Galerkin finite element method." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/34134.
Full textAbstract:
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2005.Includes bibliographical references (leaves 169-178).
Introduction: Aerodynamic design optimization has seen significant development over the past decade. Adjoint-based shape design for elliptic systems was first proposed by Pironneau and applied to transonic flow by Jameson . A review of the aerodynamic shape optimization literature and a large list of references is given in. Over the years much technology has been developed, allowing engineers to contemplate applying optimization methods to a wide variety of problems. In the context of structured grids, adjoint-based applications include multipoint, multi-objective airfoil design using compressible Navier-Stokes equations and 3D multipoint design of aircraft configurations using inviscid Euler equations. There have also been significant effort in applying adjoint methods to the unstructured grid setting. In this context, Newman et al., Elliot and Peraire were among the first to develop discrete adjoint approaches for the inviscid Euler equations.
by James Ching-Chi
Ph.D.
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Oliver, Todd A. 1980. "A high-order, adaptive, discontinuous Galerkin finite element method for the Reynolds-Averaged Navier-Stokes equations." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/46818.
Full textAbstract:
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 175-182).
This thesis presents high-order, discontinuous Galerkin (DG) discretizations of the Reynolds-Averaged Navier-Stokes (RANS) equations and an output-based error estimation and mesh adaptation algorithm for these discretizations. In particular, DG discretizations of the RANS equations with the Spalart-Allmaras (SA) turbulence model are examined. The dual consistency of multiple DG discretizations of the RANS-SA system is analyzed. The approach of simply weighting gradient dependent source terms by a test function and integrating is shown to be dual inconsistent. A dual consistency correction for this discretization is derived. The analysis also demonstrates that discretizations based on the popular mixed formulation, where dependence on the state gradient is handled by introducing additional state variables, are generally asymptotically dual consistent. Numerical results are presented to confirm the results of the analysis. The output error estimation and output-based adaptation algorithms used here are extensions of methods previously developed in the finite volume and finite element communities. In particular, the methods are extended for application on the curved, highly anisotropic meshes required for boundary conforming, high-order RANS simulations. Two methods for generating such curved meshes are demonstrated. One relies on a user-defined global mapping of the physical domain to a straight meshing domain. The other uses a linear elasticity node movement scheme to add curvature to an initially linear mesh. Finally, to improve the robustness of the adaptation process, an "unsteady" algorithm, where the mesh is adapted at each time step, is presented. The goal of the unsteady procedure is to allow mesh adaptation prior to converging a steady state solution, not to obtain a time-accurate solution of an unsteady problem. Thus, an estimate of the error due to spatial discretization in the output of interest averaged over the current time step is developed. This error estimate is then used to drive an h-adaptation algorithm. Adaptation results demonstrate that the high-order discretizations are more efficient than the second-order method in terms of degrees of freedom required to achieve a desired error tolerance. Furthermore, using the unsteady adaptation process, adaptive RANS simulations may be started from extremely coarse meshes, significantly decreasing the mesh generation burden to the user.
by Todd A. Oliver.
Ph.D.
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Oliver, Todd A. "A High-Order, Adaptive, Discontinuous Galerkin Finite Element Method for the Reynolds-Averaged Navier-Stokes Equations." Ft. Belvoir : Defense Technical Information Center, 2008. http://handle.dtic.mil/100.2/ADA488912.
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Piatkowski, Stephan-Marian [Verfasser], and Peter [Akademischer Betreuer] Bastian. "A Spectral Discontinuous Galerkin method for incompressible flow with Applications to turbulence / Stephan-Marian Piatkowski ; Betreuer: Peter Bastian." Heidelberg : Universitätsbibliothek Heidelberg, 2019. http://d-nb.info/1191760529/34.
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Birgersson, Fredrik. "Prediction of random vibration using spectral methods." Doctoral thesis, KTH, Aeronautical and Vehicle Engineering, 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3694.
Full textAbstract:
Much of the vibration in fast moving vehicles is caused bydistributed random excitation, such as turbulent flow and roadroughness. Piping systems transporting fast flowing fluid isanother example, where distributed random excitation will causeunwanted vibration. In order to reduce these vibrations andalso the noise they cause, it is important to have accurate andcomputationally efficient prediction methods available.
The aim of this thesis is to present such a method. Thefirst step towards this end was to extend an existing spectralfinite element method (SFEM) to handle excitation of planetravelling pressure waves. Once the elementary response tothese waves is known, the response to arbitrary homogeneousrandom excitation can be found.
One example of random excitation is turbulent boundary layer(TBL) excitation. From measurements a new modified Chase modelwas developed that allowed for a satisfactory prediction ofboth the measured wall pressure field and the vibrationresponse of a turbulence excited plate. In order to model morecomplicated structures, a new spectral super element method(SSEM) was formulated. It is based on a waveguide formulation,handles all kinds of boundaries and its elements are easily putinto an assembly with conventional finite elements.
Finally, the work to model fluid-structure interaction withanother wave based method is presented. Similar to the previousmethods it seems to be computationally more efficient thanconventional finite elements.
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Nappi, Angela. "Development and Application of a Discontinuous Galerkin-based Wave Prediction Model." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1385998191.
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Alpert, David N. "Enriched Space-Time Finite Element Methods for Structural Dynamics Applications." University of Cincinnati / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1377870451.
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Barter, Garrett E. (Garrett Ehud) 1979. "Shock capturing with PDE-based artificial viscosity for an adaptive, higher-order discontinuous Galerkin finite element method." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/44931.
Full textAbstract:
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 135-143).
The accurate simulation of supersonic and hypersonic flows is well suited to higher-order (p > 1), adaptive computational fluid dynamics (CFD). Since these cases involve flow velocities greater than the speed of sound, an appropriate shock capturing for higher-order, adaptive methods is necessary. Artificial viscosity can be combined with a higher-order discontinuous Galerkin finite element discretization to resolve a shock layer within a single cell. However, when a nonsmooth artificial viscosity model is employed with an otherwise higher-order approximation, element-to-element variations induce oscillations in state gradients and pollute the downstream flow. To alleviate these difficulties, this work proposes a new, higher-order, state based artificial viscosity with an associated governing partial differential equation (PDE). In the governing PDE, the shock sensor acts as a forcing term, driving the artificial viscosity to a non-zero value where it is necessary. The decay rate of the higher-order solution modes and edge-based jumps are both shown to be reliable shock indicators. This new approach leads to a smooth, higher-order representation of the artificial viscosity that evolves in time with the solution. For applications involving the Navier-Stokes equations, an artificial dissipation operator that preserves total enthalpy is introduced. The combination of higher-order, PDE-based artificial viscosity and enthalpy-preserving dissipation operator is shown to overcome the disadvantages of the non-smooth artificial viscosity. The PDE-based artificial viscosity can be used in conjunction with an automated grid adaptation framework that minimizes the error of an output functional. Higher-order solutions are shown to reach strict engineering tolerances with fewer degrees of freedom.
(cont.) The benefit in computational efficiency for higher-order solutions is less dramatic in the vicinity of the shock where errors scale with O(h/p). This includes the near-field pressure signals necessary for sonic boom prediction. When applied to heat transfer prediction on unstructured meshes in hypersonic flows, the PDE-based artificial viscosity is less susceptible to errors introduced by poor shock-grid alignment. Surface heating can also drive the output-based grid adaptation framework to arrive at the same heat transfer distribution as a well-designed structured mesh.
by Garrett Ehud Barter.
Ph.D.
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Akdag, Osman. "Incompressible Flow Simulations Using Least Squares Spectral Element Method On Adaptively Refined Triangular Grids." Master's thesis, METU, 2012. http://etd.lib.metu.edu.tr/upload/12614944/index.pdf.
Full textAbstract:
The main purpose of this study is to develop a flow solver that employs triangular grids to solve two-dimensional, viscous, laminar, steady, incompressible flows. The flow solver is based on Least Squares Spectral Element Method (LSSEM). It has p-type adaptive mesh refinement/coarsening capability and supports p-type nonconforming element interfaces. To validate the developed flow solver several benchmark problems are studied and successful results are obtained. The performances of two different triangular nodal distributions, namely Lobatto distribution and Fekete distribution, are compared in terms of accuracy and implementation complexity. Accuracies provided by triangular and quadrilateral grids of equal computational size are compared. Adaptive mesh refinement studies are conducted using three different error indicators, including a novel one based on elemental mass loss. Effect of modifying the least-squares functional by multiplying the continuity equation by a weight factor is investigated in regards to mass conservation.