ON THE ROLE OF INVOLUTIONS IN THE DISCONTINUOUS GALERKIN DISCRETIZATION OF MAXWELL AND MAGNETOHYDRODYNAMIC SYSTEMS

The role of involutions in energy stability of the discontinuous Galerkin (DG) discretization of Maxwell and magnetohydrodynamic (MHD) systems is examined. Important differences are identified in the symmetrization of the Maxwell and MHD systems that impact the construction of energy stable discretizations using the DG method. Specifically, general sufficient conditions to be imposed on the DG numerical flux and approximation space are given so that energy stability is retained These sufficient conditions reveal the favorable energy consequence of imposing continuity in the normal component of the magnetic induction field at interelement boundaries for MHD discretizations. Counterintuitively, this condition is not required for stability of Maxwell discretizations using the discontinuous Galerkin method

A first order hyperbolic framework for large strain computational solid dynamics: An upwind cell centred Total Lagrangian scheme

This paper builds on recent work developed by the authors for the numerical analysis of large strain solid dynamics, by introducing an upwind cell centred hexahedral Finite Volume framework implemented within the open source code OpenFOAM [http://www.openfoam.com/http://www.openfoam.com/]. In Lee, Gil and Bonet [1], a first order hyperbolic system of conservation laws was introduced in terms of the linear momentum and the deformation gradient tensor of the system, leading to excellent behaviour in two dimensional bending dominated nearly incompressible scenarios. The main aim of this paper is the extension of this algorithm into three dimensions, its tailor-made implementation into OpenFOAM and the enhancement of the formulation with three key novelties. First, the introduction of two different strategies in order to ensure the satisfaction of the underlying involutions of the system, that is, that the deformation gradient tensor must be curl-free throughout the deformation process. Second, the use of a discrete angular momentum projection algorithm and a monolithic Total Variation Diminishing Runge-Kutta time integrator combined in order to guarantee the conservation of angular momentum. Third, and for comparison purposes, an adapted Total Lagrangian version of the Hyperelastic-GLACE nodal scheme of Kluth and Despr´es [2] is presented. A series of challenging numerical examples are examined in order to assess the robustness and accuracy of the proposed algorithm, benchmarking it against an ample spectrum of alternative numerical strategies developed by the authors in recent publications

High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs

Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching

On curl-preserving finite volume discretizations for shallow water equations

The preservation of intrinsic or inherent constraints, like divergence-conditions, has gained increasing interest in numerical simulations of various physical evolution equations. In Torrilhon and Fey, SIAM J. Numer. Anal. (42/4) 2004, a general framework is presented how to incorporate the preservation of a discrete constraint into upwind finite volume methods. This paper applies this framework to the wave equation system and the system of shallow water equations. For the wave equation a curl-preservation for the momentum variable is present and easily identified. The preservation in case of the shallow water system is more involved due to the presence of convection. It leads to the vorticity evolution as generalized curl-constraint. The mechanisms of vorticity generation are discussed. For the numerical discretization special curl-preserving flux distributions are discussed and their incorporation into a finite volume scheme described. This leads to numerical discretizations which are exactly curl-preserving for a specific class of discrete curl-operators. The numerical experiments for the wave equation show a significant improvement of the new method against classical schemes. The extension of the curl-free numerical discretization to the shallow water case is possible after isolating the pressure flux. Simulation examples demonstrate the influence of the modification. The vortex structure is more clearly resolve

A first order hyperbolic framework for large strain computational solid dynamics. Part I: Total Lagrangian isothermal elasticity

This paper introduces a new computational framework for the analysis of large strain fast solid dynamics. The paper builds upon previous work published by the authors (Gil etal., 2014) [48], where a first order system of hyperbolic equations is introduced for the simulation of isothermal elastic materials in terms of the linear momentum, the deformation gradient and its Jacobian as unknown variables. In this work, the formulation is further enhanced with four key novelties. First, the use of a new geometric conservation law for the co-factor of the deformation leads to an enhanced mixed formulation, advantageous in those scenarios where the co-factor plays a dominant role. Second, the use of polyconvex strain energy functionals enables the definition of generalised convex entropy functions and associated entropy fluxes for solid dynamics problems. Moreover, the introduction of suitable conjugate entropy variables enables the derivation of a symmetric system of hyperbolic equations, dual of that expressed in terms of conservation variables. Third, the new use of a tensor cross product [61] greatly facilitates the algebraic manipulations of expressions involving the co-factor of the deformation. Fourth, the development of a stabilised Petrov-Galerkin framework is presented for both systems of hyperbolic equations, that is, when expressed in terms of either conservation or entropy variables. As an example, a polyconvex Mooney-Rivlin material is used and, for completeness, the eigen-structure of the resulting system of equations is studied to guarantee the existence of real wave speeds. Finally, a series of numerical examples is presented in order to assess the robustness and accuracy of the new mixed methodology, benchmarking it against an ample spectrum of alternative numerical strategies, including implicit multi-field Fraeijs de Veubeke-Hu-Washizu variational type approaches and explicit cell and vertex centred Finite Volume schemes

Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics

This article serves as a summary outlining the mathematical entropy analysis of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD equations as they are particularly useful for mathematically modeling a wide variety of magnetized fluids. In order to be self-contained we first motivate the physical properties of a magnetic fluid and how it should behave under the laws of thermodynamics. Next, we introduce a mathematical model built from hyperbolic partial differential equations (PDEs) that translate physical laws into mathematical equations. After an overview of the continuous analysis, we thoroughly describe the derivation of a numerical approximation of the ideal MHD system that remains consistent to the continuous thermodynamic principles. The derivation of the method and the theorems contained within serve as the bulk of the review article. We demonstrate that the derived numerical approximation retains the correct entropic properties of the continuous model and show its applicability to a variety of standard numerical test cases for MHD schemes. We close with our conclusions and a brief discussion on future work in the area of entropy consistent numerical methods and the modeling of plasmas

An upwind cell centred finite volume method for large strain explicit solid dynamics in OpenFOAM

Cotutela Universitat Politècnica de Catalunya i Swansea UniversityIn practical engineering applications involving extremely complex geometries, meshing typically constitutes a large portion of the overall design and analysis time. In the computational mechanics community, the ability to perform calculations on tetrahedral meshes has become increasingly important. For these reasons, automated tetrahedral mesh generation by means of Delaunay and advancing front techniques have recently received increasing attention in a number of applications, namely: crash impact simulations, cardiovascular modelling, blast and fracture modelling. Unfortunately, modern industry codes in solid mechanics typically rely on the use of traditional displacement based Finite Element formulations which possess several distinct disadvantages, namely: (1) reduced order of convergence for strains and stresses in comparison with displacements; (2) high frequency noise in the vicinity of shocks; and (3) numerical instabilities associated with shear locking, volumetric locking and pressure checker-boarding. In order to address the above mentioned shortcomings, a new mixed-based set of equations for solid dynamics formulated in a system of first order hyperbolic conservation laws was introduced. Crucially, the new set of conservation laws has a similar structure to that of the well known Euler equations in the context of Computational Fluid Dynamics (CFD). This enables us to borrow some of the available CFD technologies and to adapt the method in the context of solid dynamics. This thesis builds on the work carried out by Lee et al. 2013 by further developing the upwind cell centred finite volume framework for the numerical analysis of large strain explicit solid dynamics and its tailor-made implementation within the open source code OpenFOAM, extensively used in industrial and academic environments. The object oriented nature of OpenFOAM implementation provides a very efficient platform for future development. In this computational framework, the primary unknown variables are linear momentum and deformation gradient tensor of the system. Moreover, the formulation is further extended for an additional set of geometric strain measures comprising of the co-factor of deformation gradient tensor and the Jacobian of deformation, in order to simulate polyconvex constitutive models ensuring material stability. The domain is spatially discretised using a standard Godunov-type cell centred framework where second order accuracy is achieved by employing a linear reconstruction procedure in conjunction with a slope limiter. This leads to discontinuities in variables at the cell interface which motivate the use of a Riemann solver by introducing an upwind bias into the evaluation of numerical contact fluxes. The acoustic Riemann solver presented is further developed by applying preconditioned dissipation to improve its performance in the near incompressibility regime and extending its range to contact applications. Moreover, two evolutionary frameworks are proposed in this study to satisfy the underlying involutions (or compatibility conditions) of the system. Additionally, the spatial discretisation is alternatively represented through a nodal cell centred finite volume framework for comparison purposes. From a temporal discretisation point of view, a two stage Total Variation Diminishing Runge-Kutta time integrator is employed to ensure second order accuracy. Additionally, inclusion of a global posteriori angular momentum projection procedure enables preservation of angular momenta of the system. Finally, benchmark numerical examples are simulated to demonstrate various aspects of the formulation including mesh convergence, momentum preservation and the locking-free nature of the formulation on complex computational domains.En aplicaciones prácticas de ingeniería que implican geometrías extremadamente complejas, el mallado requiere típicamente una gran parte del tiempo total de diseño y análisis. En la comunidad de mecánica computacional, la capacidad de realizar cálculos sobre mallas tetraédricas está siendo cada vez más importante. Por estas razones, la generación automatizada de mallas tetraédricas por medio de técnicas de Delaunay y frente avanzado han recibido cada vez más atención en ciertas aplicaciones, a saber: simulaciones de impacto, modelado cardiovascular, modelado de explosión y fractura. Por desgracia, los códigos en la industria moderna para mecánica de sólidos se basan normalmente en el uso de formulaciones tradicionales de Elementos Finitos formulados en desplazamientos que poseen varias desventajas: (1) menor orden de convergencia para tensiones y deformaciones; (2) ruido de alta frecuencia cerca de las ondas de choque; y (3) inestabilidades numéricas asociadas con el bloqueo a cortante, el bloqueo volumétrico y oscilaciones de presión. Con el fin de abordar estas deficiencias, se introduce un nuevo conjunto de ecuaciones para mecánica del sólido formulada como un sistema de leyes de conservación de primer orden basada en una formulación mixta. Fundamentalmente, el nuevo sistema de leyes de conservación tiene una estructura similar a la de las famosas ecuaciones de Euler en el contexto de la Dinámica de Fluidos Computacional (CFD). Esto nos permite aprovechar algunas de las tecnologías CFD disponibles y adaptar el método en el contexto de la Mecánica de Sólidos. Esta tesis se basa en el trabajo realizado en Lee et al. 2013 mediante el desarrollo de la estructura de volúmenes finitos centrados en celdas upwind para el análisis numérico de dinámica del sólido explícita en grandes deformaciones y su implementación específicamente diseñada dentro del software de código abierto OpenFOAM, ampliamente utilizado ámbito académico e industrial. Además, la naturaleza orientada a objetos de su implementación proporciona una plataforma muy eficiente para su desarrollo posterior. En este marco computacional, las incógnitas básicas de este sistema son el momento lineal y el tensor gradiente de deformación. Asimismo, la formulación se extiende adicionalmente para un conjunto adicional de medidas de deformación que comprenden el cofactor del tensor gradiente de deformación y el jacobiano de deformación, con el fin de simular modelos constitutivos policonvexos que aseguran la estabilidad del material. El dominio se discretiza espacialmente usando un marco centrado en células de tipo Godunov estándar, donde se consigue la precisión de segundo orden empleando un procedimiento de reconstrucción lineal junto con un limitador de pendiente. Esto conduce a discontinuidades en las variables en la interfase de la célula que motivan el uso de un solucionador de Riemann mediante la introducción de un sesgo contra el viento en la evaluación de flujos de contacto numéricos. El presente solucionador acústico de Riemann es posteriormente desarrollado aplicando disipación pre-condicionada para mejorar su rendimiento en el cercano pero incompresibilidad régimen y extender su gama a aplicaciones de contacto. Además, se proponen dos marcos evolutivos en este estudio para satisfacer las involuciones subyacentes (o condiciones de compatibilidad) del sistema. Además, la discretización espacial se representa alternativamente a través de un marco de volumen finito centrado en células nodales para fines de comparación. Desde el punto de vista de la discretización temporal, se emplea un integrador temporal de Runge-Kutta de dos etapas con Disminución de Variación Total para asegurar segundo orden de precision. Finalmente, se simulan ejemplos numéricos de referencia para demostrar varios aspectos de la formulación que incluyen convergencia de malla, conservación de momento y la naturaleza libre de bloqueo de la formulación en dominios computacionales complejos.Postprint (published version

An upwind cell centred finite volume method for large strain explicit solid dynamics in OpenFOAM

In practical engineering applications involving extremely complex geometries, meshing typically constitutes a large portion of the overall design and analysis time. In the computational mechanics community, the ability to perform calculations on tetrahedral meshes has become increasingly important. For these reasons, automated tetrahedral mesh generation by means of Delaunay and advancing front techniques have recently received increasing attention in a number of applications, namely: crash impact simulations, cardiovascular modelling, blast and fracture modelling. Unfortunately, modern industry codes in solid mechanics typically rely on the use of traditional displacement based Finite Element formulations which possess several distinct disadvantages, namely: (1) reduced order of convergence for strains and stresses in comparison with displacements; (2) high frequency noise in the vicinity of shocks; and (3) numerical instabilities associated with shear locking, volumetric locking and pressure checker-boarding. In order to address the above mentioned shortcomings, a new mixed-based set of equations for solid dynamics formulated in a system of first order hyperbolic conservation laws was introduced. Crucially, the new set of conservation laws has a similar structure to that of the well known Euler equations in the context of Computational Fluid Dynamics (CFD). This enables us to borrow some of the available CFD technologies and to adapt the method in the context of solid dynamics. This thesis builds on the work carried out by Lee et al. 2013 by further developing the upwind cell centred finite volume framework for the numerical analysis of large strain explicit solid dynamics and its tailor-made implementation within the open source code OpenFOAM, extensively used in industrial and academic environments. The object oriented nature of OpenFOAM implementation provides a very efficient platform for future development. In this computational framework, the primary unknown variables are linear momentum and deformation gradient tensor of the system. Moreover, the formulation is further extended for an additional set of geometric strain measures comprising of the co-factor of deformation gradient tensor and the Jacobian of deformation, in order to simulate polyconvex constitutive models ensuring material stability. The domain is spatially discretised using a standard Godunov-type cell centred framework where second order accuracy is achieved by employing a linear reconstruction procedure in conjunction with a slope limiter. This leads to discontinuities in variables at the cell interface which motivate the use of a Riemann solver by introducing an upwind bias into the evaluation of numerical contact fluxes. The acoustic Riemann solver presented is further developed by applying preconditioned dissipation to improve its performance in the near incompressibility regime and extending its range to contact applications. Moreover, two evolutionary frameworks are proposed in this study to satisfy the underlying involutions (or compatibility conditions) of the system. Additionally, the spatial discretisation is alternatively represented through a nodal cell centred finite volume framework for comparison purposes. From a temporal discretisation point of view, a two stage Total Variation Diminishing Runge-Kutta time integrator is employed to ensure second order accuracy. Additionally, inclusion of a global posteriori angular momentum projection procedure enables preservation of angular momenta of the system. Finally, benchmark numerical examples are simulated to demonstrate various aspects of the formulation including mesh convergence, momentum preservation and the locking-free nature of the formulation on complex computational domains.En aplicaciones prácticas de ingeniería que implican geometrías extremadamente complejas, el mallado requiere típicamente una gran parte del tiempo total de diseño y análisis. En la comunidad de mecánica computacional, la capacidad de realizar cálculos sobre mallas tetraédricas está siendo cada vez más importante. Por estas razones, la generación automatizada de mallas tetraédricas por medio de técnicas de Delaunay y frente avanzado han recibido cada vez más atención en ciertas aplicaciones, a saber: simulaciones de impacto, modelado cardiovascular, modelado de explosión y fractura. Por desgracia, los códigos en la industria moderna para mecánica de sólidos se basan normalmente en el uso de formulaciones tradicionales de Elementos Finitos formulados en desplazamientos que poseen varias desventajas: (1) menor orden de convergencia para tensiones y deformaciones; (2) ruido de alta frecuencia cerca de las ondas de choque; y (3) inestabilidades numéricas asociadas con el bloqueo a cortante, el bloqueo volumétrico y oscilaciones de presión. Con el fin de abordar estas deficiencias, se introduce un nuevo conjunto de ecuaciones para mecánica del sólido formulada como un sistema de leyes de conservación de primer orden basada en una formulación mixta. Fundamentalmente, el nuevo sistema de leyes de conservación tiene una estructura similar a la de las famosas ecuaciones de Euler en el contexto de la Dinámica de Fluidos Computacional (CFD). Esto nos permite aprovechar algunas de las tecnologías CFD disponibles y adaptar el método en el contexto de la Mecánica de Sólidos. Esta tesis se basa en el trabajo realizado en Lee et al. 2013 mediante el desarrollo de la estructura de volúmenes finitos centrados en celdas upwind para el análisis numérico de dinámica del sólido explícita en grandes deformaciones y su implementación específicamente diseñada dentro del software de código abierto OpenFOAM, ampliamente utilizado ámbito académico e industrial. Además, la naturaleza orientada a objetos de su implementación proporciona una plataforma muy eficiente para su desarrollo posterior. En este marco computacional, las incógnitas básicas de este sistema son el momento lineal y el tensor gradiente de deformación. Asimismo, la formulación se extiende adicionalmente para un conjunto adicional de medidas de deformación que comprenden el cofactor del tensor gradiente de deformación y el jacobiano de deformación, con el fin de simular modelos constitutivos policonvexos que aseguran la estabilidad del material. El dominio se discretiza espacialmente usando un marco centrado en células de tipo Godunov estándar, donde se consigue la precisión de segundo orden empleando un procedimiento de reconstrucción lineal junto con un limitador de pendiente. Esto conduce a discontinuidades en las variables en la interfase de la célula que motivan el uso de un solucionador de Riemann mediante la introducción de un sesgo contra el viento en la evaluación de flujos de contacto numéricos. El presente solucionador acústico de Riemann es posteriormente desarrollado aplicando disipación pre-condicionada para mejorar su rendimiento en el cercano pero incompresibilidad régimen y extender su gama a aplicaciones de contacto. Además, se proponen dos marcos evolutivos en este estudio para satisfacer las involuciones subyacentes (o condiciones de compatibilidad) del sistema. Además, la discretización espacial se representa alternativamente a través de un marco de volumen finito centrado en células nodales para fines de comparación. Desde el punto de vista de la discretización temporal, se emplea un integrador temporal de Runge-Kutta de dos etapas con Disminución de Variación Total para asegurar segundo orden de precision. Finalmente, se simulan ejemplos numéricos de referencia para demostrar varios aspectos de la formulación que incluyen convergencia de malla, conservación de momento y la naturaleza libre de bloqueo de la formulación en dominios computacionales complejos