Proper generalized decomposition solutions within a domain decomposition strategy

Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in‐plane–out‐of‐plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user‐defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution

VoroCrust: Voronoi Meshing Without Clipping

Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrarily curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably-correct algorithm for conforming polyhedral Voronoi meshing for non-convex and non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while preserving all sharp features in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf. Supplemental materials available on https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd

Hybridizable compatible finite element discretizations for numerical weather prediction: implementation and analysis

There is a current explosion of interest in new numerical methods for atmospheric modeling. A driving force behind this is the need to be able to simulate, with high efficiency, large-scale geophysical flows on increasingly more parallel computer systems. Many current operational models, including that of the UK Met Office, depend on orthogonal meshes, such as the latitude-longitude grid. This facilitates the development of finite difference discretizations with favorable numerical properties. However, such methods suffer from the ``pole problem," which prohibits the model to make efficient use of a large number of computing processors due to excessive concentration of grid-points at the poles. Recently developed finite element discretizations, known as ``compatible" finite elements, avoid this issue while maintaining the key numerical properties essential for accurate geophysical simulations. Moreover, these properties can be obtained on arbitrary, non-orthogonal meshes. However, the efficient solution of the resulting discrete systems depend on transforming the mixed velocity-pressure (or velocity-pressure-buoyancy) system into an elliptic problem for the pressure. This is not so straightforward within the compatible finite element framework due to inter-element coupling. This thesis supports the proposition that systems arising from compatible finite element discretizations can be solved efficiently using a technique known as ``hybridization." Hybridization removes inter-element coupling while maintaining the desired numerical properties. This permits the construction of sparse, elliptic problems, for which fast solver algorithms are known, using localized algebra. We first introduce the technique for compatible finite element discretizations of simplified atmospheric models. We then develop a general software abstraction for the rapid implementation and composition of hybridization methods, with an emphasis on preconditioning. Finally, we extend the technique for a new compatible method for the full, compressible atmospheric equations used in operational models.Open Acces

Comparison and coupling of continuous and hybridizable discontinuous Galerkin methods : application to multi-physics problems

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.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.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.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 consideredNeste 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.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.Postprint (published version

Fully coupled CEM/CFD modelling of microwave heating in a porous medium

Computational results for the microwave heating of a porous material are presented in this paper. Coupled finite difference time domain and finite volume methods are used to solve equations that describe the electromagnetic field and heat and mass transfer in porous media. These equations are nonlinearly coupled through the dielectric properties which depend both on temperature and moisture content. By investigating the resonant behaviour in two-dimensional microwave cavities, the FD-TD scheme is validated. Validation of the microwave power distribution in 3-D microwave enclosures is compared with other numerical results available. 3-D temperature distribution in a biomaterial is validated against experimental results. Results using the proposed fully coupled approach are discussed and analyzed. The model is able to reflect the evolution of both temperature and moisture fields as well as energy penetration as the moisture in the porous medium evaporates. Moisture movement results from internal pressure gradients produced by the internal heating and phase change. The model is validated by comparison to some published results for simpler problems

Desenvolvimento de um simulador substituto de reservatório multiescala acoplado com geomecânica

Orientador: Philippe Remy Bernard DevlooTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica e Instituto de GeociênciasResumo: Os softwares de simulação de reservatórios são utilizados como ferramentas para o entendimento dos reservatórios de petróleo e eventualmente, para diagnosticar anomalias operacionais. O aumento da potência computacional permite aos engenheiros de reservatórios desenvolver modelos geológicos mais realistas, refinados e com uma grande quantidade de dados de entrada. Alguns exemplos são os modelos multi-físicos que acoplam efeitos geomecânicos, térmicos, geoquímicos e modelos que incluem múltiplas escalas inerentes aos modelos de campo completo. Estes modelos são geralmente caros, porque o cálculo direto de um modelo geocelular refinado gera enormes sistemas lineares de equações. Quando é considerado o efeito da deformação geomecânica com o fluxo de fluido através de meios porosos, um sistema muito grande de equações associado com a elasticidade é acoplado a um sistema igualmente grande de equações, associadas ao fluxo de fluido e ao transporte de massa. Portanto, a maioria das simulações são realizadas sem considerar o acoplamento geomecânico. Essas simulações ignoram fenômenos físicos que podem ter sérios impactos ambientais, como ativação de falhas, subsidência e outros. Neste trabalho desenvolve-se um inovador método multiescala que permite diretamente simular um modelo geocelular fino em uma maneira econômica. Um modelo substituto também foi desenvolvido para simular a deformação geomecânica acoplada ao modelo de fluido. O objetivo é obter aproximações do problema multifísico não linear descrito pelas equações multifásicas poroelásticas. Para atingir esse objetivo, diferentes tecnologias de elementos finitos são integradas dentro de um simulador de reservatórios, resolvendo problemas que incluem um modelo geocelular com diferentes escalas, acoplado a um modelo substituto de deformação geomecânica. O modelo matemático é escrito em uma forma adequada para a estrutura de elementos finitos do NeoPZ. Em cada passo de tempo, a aproximação é obtida como uma sequência de problemas elásticos, de Darcy e de transporte. Cada componente nesta sequência é tratado por um esquema numérico diferente e / ou espaço de aproximação; em primeiro lugar, um modelo substituto, inspirado na teoria das inclusões poroelásticas, é usado para o cálculo da deformação geomecânica das rochas; em segundo lugar, utiliza-se um método multi-escala baseado na aproximação mista de equações multifásicas; em terceiro lugar, para a convecção das fases, uma aproximação mista multi-escala do campo de velocidade de Darcy é usada, em conjunto com um esquema de upwind de primeira ordem. O potencial da abordagem numérica é demonstrado através de vários exemplos bidimensionais e tridimensionais, em que os reservatórios são simulados usando malhas não estruturadas. Todas as simulações foram executadas usando estruturas computacionais de baixo custoAbstract: Reservoir simulation softwares are used as a tool to understand the behavior of petroleum reservoirs and, eventually, to diagnose operating anomalies. The increased computational power allows reservoir engineers to develop more realistic geological models, that are very refined and have a large amount of input data. As an example, multi-physics models couple geomechanical, thermal, geochemical effects and include multiple scales inherent to full field models. These models are generally costly, because the direct calculation of a refined geocellular model, generates huge linear systems of equations. When coupling the geomechanical deformation with fluid flow through porous media, a very large system of equations associated with elasticity, is coupled to an equally large system of equations, which is associated with fluid flow and mass transport. Therefore, most simulations are performed without considering the geomechanical coupling. These simulations ignore physical phenomena that can have serious environmental impacts such as fault activation, land subsidence and others. In this work an innovative multiscale method is developed, allowing the direct simulation of a fine geocellular model in a cost-effective way. A surrogate model has also been developed for simulating the geomechanical deformation coupled to the fluid model. The goal is obtain approximations for the nolinear multiphysic problem decribed by the multiphase poroelastic equations. In order to attain this goal, different finite element technologies are integrated within a reservoir simulator, solving problems that include a geocellular model with different scales, coupled with a surrogate model of geomechanical deformation. The mathematical model is written in a form suitable for the NeoPZ finite element framework. At each timestep, the approximation is obtained as a sequence of elastic, Darcy's and transport problems. Each component in this sequence is treated by a different numerical scheme and/or approximation space; first, a surrogate model, inspired on the theory of poroelastic inclusions, is used for the calculation of the geomechanical deformation of rocks; second, a multiscale method based on mixed approximation of multiphase equations is used; third, for the convection of the phases, a mixed multiscale approximation of the Darcy's velocity field is used together with a first-order upwind scheme. The potential of the numerical approach is demonstrated through several bi-dimensional and three-dimensional examples, in which reservoirs are simulated using unstructured meshes. All simulations have been executed using low cost computational structuresDoutoradoExplotaçãoDoutor em Ciências e Engenharia de Petróle

A scalable hp-adaptive finite element software with applications in fiber optics

In this dissertation, we present a scalable parallel version of hp3D—a finite element (FE) software for analysis and discretization of complex three-dimensional multiphysics applications. The developed software supports hybrid MPI/OpenMP parallelism for large-scale computation on modern manycore architectures. The focus of the effort lies on the development and optimization of the parallel software infrastructure underlying all distributed computation. We discuss the challenges of designing efficient data structures for isotropic and anisotropic hp-adaptive meshes with tetrahedral, hexahedral, prismatic, and pyramidal elements supporting discretization of the exact sequence energy spaces. While the code supports standard Galerkin methods, special emphasis is given to systems arising from discretization with the discontinuous Petrov–Galerkin (DPG) method. The method guarantees discrete stability by employing locally optimal test functions, and it has a built-in error indicator which we exploit to guide mesh adaptivity. In addition to interfacing with third-party packages for various tasks, we have developed our own tools including a parallel nested dissection solver suitable for scalable FE computation of waveguide geometries. We present weak-scaling results with up to 24576 CPU cores and numerical simulations with more than one billion degrees of freedom. The new software capabilities enable solution of challenging wave propagation problems with important applications in acoustics, elastodynamics, and electromagnetics. These applications are difficult to solve in the high-frequency regime because the FE discretization suffers from significant numerical pollution errors that increase with the wavenumber. It is critical to control these errors to obtain a stable and accurate method. We study the pollution effect for waveguide problems with more than 8000 wavelengths in the context of robust DPG FE discretizations for the time-harmonic Maxwell equations. We also discuss adaptive refinement strategies for multi-mode fiber waveguides where the propagating transverse modes must be resolved sufficiently. Our study shows the applicability of the DPG error indicator to this class of problems. Finally, we present both modeling and computational advancements to a unique three-dimensional DPG FE model for the simulation of laser amplification in a fiber amplifier. Fiber laser amplifiers are of interest in communication technology, medical applications, military defense capabilities, and various other fields. Silica fiber amplifiers can achieve high-power operation with great efficiency. At high optical intensities, multi-mode amplifiers suffer from undesired thermal coupling effects which pose a major obstacle in power-scaling of such devices. Our nonlinear 3D vectorial model is based on the time-harmonic Maxwell equations, and it incorporates both amplification via an active dopant and thermal effects via coupling with the heat equation. The model supports co-, counter-, and bi-directional pumping configurations, as well as inhomogeneous and anisotropic material properties. The high-fidelity simulation comes at the cost of a high-order FE discretization with many degrees of freedom per wavelength. To make the computation more feasible, we have developed a novel longitudinal model rescaling, using artificial material parameters with the goal of preserving certain quantities of interest. Numerical tests demonstrate the applicability and utility of this scaled model in the simulation of an ytterbium-doped, step-index fiber amplifier that experiences laser amplification and heating.Computational Science, Engineering, and Mathematic

Anisotropic Adaptivity and Subgrid Scale Modelling for the Solution of the Neutron Transport Equation with an Emphasis on Shielding Applications

This thesis demonstrates advanced new discretisation and adaptive meshing technologies that improve the accuracy and stability of using finite element discretisations applied to the Boltzmann transport equation (BTE). This equation describes the advective transport of neutral particles such as neutrons and photons within a domain. The BTE is difficult to solve, due to its large phase space (three dimensions of space, two of angle and one each of energy and time) and the presence of non-physical oscillations in many situations. This work explores the use of a finite element method that combines the advantages of the two schemes: the discontinuous and continuous Galerkin methods. The new discretisation uses multiscale (subgrid) finite elements that work locally within each element in the finite element mesh in addition to a global, continuous, formulation. The use of higher order functions that describe the variation of the angular flux over each element is also explored using these subgrid finite element schemes. In addition to the spatial discretisation, methods have also been developed to optimise the finite element mesh in order to reduce resulting errors in the solution over the domain, or locally in situations where there is a goal of specific interest (such as a dose in a detector region). The chapters of this thesis have been structured to be submitted individually for journal publication, and are arranged as follows. Chapter 1 introduces the reader to motivation behind the research contained within this thesis. Chapter 2 introduces the forms of the BTE that are used within this thesis. Chapter 3 provides the methods that are used, together with examples, of the validation and verification of the software that was developed as a result of this work, the transport code RADIANT. Chapter 4 introduces the inner element subgrid scale finite element discretisation of the BTE that forms the basis of the discretisations within RADIANT and explores its convergence and computational times on a set of benchmark problems. Chapter 5 develops the error metrics that are used to optimise the mesh in order to reduce the discretisation error within a finite element mesh using anisotropic adaptivity that can use elongated elements that accurately resolves computational demanding regions, such as in the presence of shocks. The work of this chapter is then extended in Chapter 6 that forms error metrics for goal based adaptivity to minimise the error in a detector response. Finally, conclusions from this thesis and suggestions for future work that may be explored are discussed in Chapter 7.Open Acces