319 research outputs found
Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling
In the present article we describe a few simple and efficient finite volume
type schemes on moving grids in one spatial dimension combined with appropriate
predictor-corrector method to achieve higher resolution. The underlying finite
volume scheme is conservative and it is accurate up to the second order in
space. The main novelty consists in the motion of the grid. This new dynamic
aspect can be used to resolve better the areas with large solution gradients or
any other special features. No interpolation procedure is employed, thus
unnecessary solution smearing is avoided, and therefore, our method enjoys
excellent conservation properties. The resulting grid is completely
redistributed according the choice of the so-called monitor function. Several
more or less universal choices of the monitor function are provided. Finally,
the performance of the proposed algorithm is illustrated on several examples
stemming from the simple linear advection to the simulation of complex shallow
water waves. The exact well-balanced property is proven. We believe that the
techniques described in our paper can be beneficially used to model tsunami
wave propagation and run-up.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to
Geosciences. Other author's papers can be downloaded at
http://www.denys-dutykh.com
Comparing approaches for numerical modelling of tsunami generation by deformable submarine slides
Tsunami generated by submarine slides are arguably an under-considered risk in comparison to earthquake-generated tsunami. Numerical simulations of submarine slide-generated waves can be used to identify the important factors in determining wave characteristics. Here we use Fluidity, an open source finite element code, to simulate waves generated by deformable submarine slides. Fluidity uses flexible unstructured meshes combined with adaptivity which alters the mesh topology and resolution based on the simulation state, focussing or reducing resolution, when and where it is required. Fluidity also allows a number of different numerical approaches to be taken to simulate submarine slide deformation, free-surface representation, and wave generation within the same numerical framework. In this work we use a multi-material approach, considering either two materials (slide and water with a free surface) or three materials (slide, water and air), as well as a sediment model (sediment, water and free surface) approach. In all cases the slide is treated as a viscous fluid. Our results are shown to be consistent with laboratory experiments using a deformable submarine slide, and demonstrate good agreement when compared with other numerical models. The three different approaches for simulating submarine slide dynamics and tsunami wave generation produce similar waveforms and slide deformation geometries. However, each has its own merits depending on the application. Mesh adaptivity is shown to be able to reduce the computational cost without compromising the accuracy of results
Development and applications of the finite point method to compressible aerodynamics problems
This work deals with the development and application of the Finite Point
Method (FPM) to compressible aerodynamics problems. The research focuses
mainly on investigating the capabilities of the meshless technique to address
practical problems, one of the most outstanding issues in meshless methods.
The FPM spatial approximation is studied firstly, with emphasis on aspects of
the methodology that can be improved to increase its robustness and accuracy.
Suitable ranges for setting the relevant approximation parameters and the
performance likely to be attained in practice are determined. An automatic
procedure to adjust the approximation parameters is also proposed to simplify
the application of the method, reducing problem- and user-dependence
without affecting the flexibility of the meshless technique.
The discretization of the flow equations is carried out following wellestablished
approaches, but drawing on the meshless character of the methodology. In order to meet the requirements of practical applications, the procedures are designed and implemented placing emphasis on robustness and efficiency (a simplification of the basic FPM technique is proposed to this end). The flow solver is based on an upwind spatial discretization of the convective fluxes (using the approximate Riemann solver of Roe) and an explicit time integration scheme. Two additional artificial diffusion schemes are also proposed to suit those cases of study in which computational cost is a major concern. The performance of the flow solver is evaluated in order to determine the potential of the meshless approach. The accuracy, computational cost and parallel scalability of the method are studied in
comparison with a conventional FEM-based technique.
Finally, practical applications and extensions of the flow solution scheme are
presented. The examples provided are intended not only to show the
capabilities of the FPM, but also to exploit meshless advantages. Automatic hadaptive procedures, moving domain and fluid-structure interaction problems,
as well as a preliminary approach to solve high-Reynolds viscous flows, are a
sample of the topics explored.
All in all, the results obtained are satisfactorily accurate and competitive in
terms of computational cost (if compared with a similar mesh-based
implementation). This indicates that meshless advantages can be exploited
with efficiency and constitutes a good starting point towards more challenging
applications
Development and applications of the Finite Point Method to compressible aerodynamics problems
This work deals with the development and application of the Finite Point Method (FPM) to compressible aerodynamics problems. The research focuses mainly on investigating the capabilities of the meshless technique to address practical problems, one of the most outstanding issues in meshless methods.
The FPM spatial approximation is studied firstly, with emphasis on aspects of the methodology that can be improved to increase its robustness and accuracy. Suitable ranges for setting the relevant approximation parameters and the performance likely to be attained in practice are determined. An automatic procedure to adjust the approximation parameters is also proposed to simplify the application of the method, reducing problem- and user-dependence without affecting the flexibility of the meshless technique.
The discretization of the flow equations is carried out following wellestablished approaches, but drawing on the meshless character of the
methodology. In order to meet the requirements of practical applications, the procedures are designed and implemented placing emphasis on robustness and efficiency (a simplification of the basic FPM technique is proposed to this end). The flow solver is based on an upwind spatial discretization of the convective fluxes (using the approximate Riemann solver of Roe) and an explicit time integration scheme. Two additional artificial diffusion schemes are also proposed to suit those cases of study in which computational cost is a major concern. The performance of the flow solver is evaluated in order to determine the potential of the meshless approach. The accuracy, computational cost and parallel scalability of the method are studied in comparison with a conventional FEM-based technique.
Finally, practical applications and extensions of the flow solution scheme are presented. The examples provided are intended not only to show the
capabilities of the FPM, but also to exploit meshless advantages. Automatic hadaptive procedures, moving domain and fluid-structure interaction problems, as well as a preliminary approach to solve high-Reynolds viscous flows, are a sample of the topics explored.
All in all, the results obtained are satisfactorily accurate and competitive in terms of computational cost (if compared with a similar mesh-based
implementation). This indicates that meshless advantages can be exploited with efficiency and constitutes a good starting point towards more challenging applications.En este trabajo se aborda el desarrollo del Método de Puntos Finitos (MPF) y su aplicación a problemas de aerodinámica de flujos compresibles. El objetivo principal es investigar el potencial de la técnica sin malla para la solución de problemas prácticos, lo cual constituye una de las limitaciones más importantes de los métodos sin malla.
En primer lugar se estudia la aproximación espacial en el MPF, haciendo hincapié en aquéllos aspectos que pueden ser mejorados para incrementar la robustez y exactitud de la metodologÃa. Se determinan rangos adecuados para el ajuste de los parámetros de la aproximación y su comportamiento en situaciones prácticas. Se propone además un procedimiento de ajuste automático de estos parámetros a fin de simplificar la aplicación del método y reducir la dependencia de factores como el tipo de problema y la intervención del usuario, sin afectar la flexibilidad de la técnica sin malla.
A continuación se aborda el esquema de solución de las ecuaciones del flujo. La discretización de las mismas se lleva a cabo siguiendo métodos estándar, pero aprovechando las caracterÃsticas de la técnica sin malla. Con el objetivo de abordar problemas prácticos, se pone énfasis en la robustez y eficiencia de la implementación numérica (se propone además una simplificación del procedimiento de solución). El comportamiento del esquema se estudia en detalle para evaluar su potencial y se analiza su exactitud, coste computacional y escalabilidad, todo ello en comparación con un método convencional basado en Elementos Finitos.
Finalmente se presentan distintas aplicaciones y extensiones de la metodologÃa desarrollada. Los ejemplos numéricos pretenden demostrar las
capacidades del método y también aprovechar las ventajas de la metodologÃa sin malla en áreas en que la misma puede ser de especial interés. Los problemas tratados incluyen, entre otras caracterÃsticas, el refinamiento automático de la discretización, la presencia de fronteras móviles e
interacción fluido-estructura, como asà también una aplicación preliminar a flujos compresibles de alto número de Reynolds. Los resultados obtenidos muestran una exactitud satisfactoria. Además, en comparación con una técnica similar basada en Elementos Finitos, demuestran ser competitivos en términos del coste computacional. Esto indica que las ventajas de la metodologÃa sin malla pueden ser explotadas con eficiencia, lo cual constituye un buen punto de partida para el desarrollo de ulteriores aplicaciones.Postprint (published version
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere
We describe a compatible finite element discretisation for the shallow water
equations on the rotating sphere, concentrating on integrating consistent
upwind stabilisation into the framework. Although the prognostic variables are
velocity and layer depth, the discretisation has a diagnostic potential
vorticity that satisfies a stable upwinded advection equation through a
Taylor-Galerkin scheme; this provides a mechanism for dissipating enstrophy at
the gridscale whilst retaining optimal order consistency. We also use upwind
discontinuous Galerkin schemes for the transport of layer depth. These
transport schemes are incorporated into a semi-implicit formulation that is
facilitated by a hybridisation method for solving the resulting mixed Helmholtz
equation. We illustrate our discretisation with some standard rotating sphere
test problems.Comment: accepted versio
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