518 research outputs found
Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers
In this paper we use the genuinely multidimensional HLL Riemann solvers
recently developed by Balsara et al. to construct a new class of
computationally efficient high order Lagrangian ADER-WENO one-step ALE finite
volume schemes on unstructured triangular meshes. A nonlinear WENO
reconstruction operator allows the algorithm to achieve high order of accuracy
in space, while high order of accuracy in time is obtained by the use of an
ADER time-stepping technique based on a local space-time Galerkin predictor.
The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the
grid, considering the entire Voronoi neighborhood of each node and allows for
larger time steps than conventional one-dimensional Riemann solvers. The
results produced by the multidimensional Riemann solver are then used twice in
our one-step ALE algorithm: first, as a node solver that assigns a unique
velocity vector to each vertex, in order to preserve the continuity of the
computational mesh; second, as a building block for genuinely multidimensional
numerical flux evaluation that allows the scheme to run with larger time steps
compared to conventional finite volume schemes that use classical
one-dimensional Riemann solvers in normal direction. A rezoning step may be
necessary in order to overcome element overlapping or crossing-over. We apply
the method presented in this article to two systems of hyperbolic conservation
laws, namely the Euler equations of compressible gas dynamics and the equations
of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to
fourth order of accuracy in space and time have been carried out. Several
numerical test problems have been solved to validate the new approach
Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes
We present a new family of high order accurate fully discrete one-step
Discontinuous Galerkin (DG) finite element schemes on moving unstructured
meshes for the solution of nonlinear hyperbolic PDE in multiple space
dimensions, which may also include parabolic terms in order to model
dissipative transport processes. High order piecewise polynomials are adopted
to represent the discrete solution at each time level and within each spatial
control volume of the computational grid, while high order of accuracy in time
is achieved by the ADER approach. In our algorithm the spatial mesh
configuration can be defined in two different ways: either by an isoparametric
approach that generates curved control volumes, or by a piecewise linear
decomposition of each spatial control volume into simplex sub-elements. Our
numerical method belongs to the category of direct
Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation
formulation of the governing PDE system is considered and which already takes
into account the new grid geometry directly during the computation of the
numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a
posteriori sub-cell finite volume limiter method, in which the validity of the
candidate solution produced in each cell by an unlimited ADER-DG scheme is
verified against a set of physical and numerical detection criteria. Those
cells which do not satisfy all of the above criteria are flagged as troubled
cells and are recomputed with a second order TVD finite volume scheme. The
numerical convergence rates of the new ALE ADER-DG schemes are studied up to
fourth order in space and time and several test problems are simulated.
Finally, an application inspired by Inertial Confinement Fusion (ICF) type
flows is considered by solving the Euler equations and the PDE of viscous and
resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure
High Order Cell-Centered Lagrangian-Type Finite Volume Schemes with Time-Accurate Local Time Stepping on Unstructured Triangular Meshes
We present a novel cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE)
finite volume scheme on unstructured triangular meshes that is high order
accurate in space and time and that also allows for time-accurate local time
stepping (LTS). The new scheme uses the following basic ingredients: a high
order WENO reconstruction in space on unstructured meshes, an element-local
high-order accurate space-time Galerkin predictor that performs the time
evolution of the reconstructed polynomials within each element, the computation
of numerical ALE fluxes at the moving element interfaces through approximate
Riemann solvers, and a one-step finite volume scheme for the time update which
is directly based on the integral form of the conservation equations in
space-time. The inclusion of the LTS algorithm requires a number of crucial
extensions, such as a proper scheduling criterion for the time update of each
element and for each node; a virtual projection of the elements contained in
the reconstruction stencils of the element that has to perform the WENO
reconstruction; and the proper computation of the fluxes through the space-time
boundary surfaces that will inevitably contain hanging nodes in time due to the
LTS algorithm. We have validated our new unstructured Lagrangian LTS approach
over a wide sample of test cases solving the Euler equations of compressible
gasdynamics in two space dimensions, including shock tube problems, cylindrical
explosion problems, as well as specific tests typically adopted in Lagrangian
calculations, such as the Kidder and the Saltzman problem. When compared to the
traditional global time stepping (GTS) method, the newly proposed LTS algorithm
allows to reduce the number of element updates in a given simulation by a
factor that may depend on the complexity of the dynamics, but which can be as
large as 4.7.Comment: 31 pages, 13 figure
High-Order Unstructured Lagrangian One-Step WENO Finite Volume Schemes for Non-Conservative Hyperbolic Systems: Applications to Compressible Multi-Phase Flows
In this article we present the first better than second order accurate
unstructured Lagrangian-type one-step WENO finite volume scheme for the
solution of hyperbolic partial differential equations with non-conservative
products. The method achieves high order of accuracy in space together with
essentially non-oscillatory behavior using a nonlinear WENO reconstruction
operator on unstructured triangular meshes. High order accuracy in time is
obtained via a local Lagrangian space-time Galerkin predictor method that
evolves the spatial reconstruction polynomials in time within each element. The
final one-step finite volume scheme is derived by integration over a moving
space-time control volume, where the non-conservative products are treated by a
path-conservative approach that defines the jump terms on the element
boundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian
(ALE) method, where the mesh velocity can be chosen independently of the fluid
velocity.
The new scheme is applied to the full seven-equation Baer-Nunziato model of
compressible multi-phase flows in two space dimensions. The use of a Lagrangian
approach allows an excellent resolution of the solid contact and the resolution
of jumps in the volume fraction. The high order of accuracy of the scheme in
space and time is confirmed via a numerical convergence study. Finally, the
proposed method is also applied to a reduced version of the compressible
Baer-Nunziato model for the simulation of free surface water waves in moving
domains. In particular, the phenomenon of sloshing is studied in a moving water
tank and comparisons with experimental data are provided
Unification of Lagrangian staggered-grid hydrodynamics and cell-centered hydrodynamics in one dimension
This paper focuses on the novel scheme to unify both Lagrangian
staggered-grid and cell-centered hydrodynamic methods in one dimension. The
scheme neither contains empirical parameters nor solves the Riemann problem. It
includes two key points: one is the relationship between pressure and velocity,
and the other is Newton's second law. The two methods that make use of this
scheme satisfy the entropy condition and are conservative in total mass,
momentum, and energy. Numerical results show the robustness and accuracy of
both methods.Comment: 22 page
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
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