A cell by cell anisotropic adaptive mesh ALE method

A cell by cell anisotropic adaptive mesh Arbitrary Lagrangian Eulerian (ALE) method for the solution of the Euler equations is described. An efficient approach to equipotential mesh relaxation on anisotropically refined meshes is developed. Results for two test problems are presented.

Adjoint goal-based error norms for adaptive mesh ocean modelling

Flow in the world's oceans occurs at a wide range of spatial scales, from a fraction of a metre up to many thousands of kilometers. In particular, regions of intense flow are often highly localised, for example, western boundary currents, equatorial jets, overflows and convective plumes. Conventional numerical ocean models generally use static meshes. The use of dynamically-adaptive meshes has many potential advantages but needs to be guided by an error measure reflecting the underlying physics. A method of defining an error measure to guide an adaptive meshing algorithm for unstructured tetrahedral finite elements, utilizing an adjoint or goal-based method, is described here. This method is based upon a functional, encompassing important features of the flow structure. The sensitivity of this functional, with respect to the solution variables, is used as the basis from which an error measure is derived. This error measure acts to predict those areas of the domain where resolution should be changed. A barotropic wind driven gyre problem is used to demonstrate the capabilities of the method. The overall objective of this work is to develop robust error measures for use in an oceanographic context which will ensure areas of fine mesh resolution are used only where and when they are required. (c) 2006 Elsevier Ltd. All rights reserved.

Fast three dimensional r-adaptive mesh redistribution

This paper describes a fast and reliable method for redistributing a computational mesh in three dimensions which can generate a complex three dimensional mesh without any problems due to mesh tangling. The method relies on a three dimensional implementation of the parabolic Monge–Ampère (PMA) technique, for finding an optimally transported mesh. The method for implementing PMA is described in detail and applied to both static and dynamic mesh redistribution problems, studying both the convergence and the computational cost of the algorithm. The algorithm is applied to a series of problems of increasing complexity. In particular very regular meshes are generated to resolve real meteorological features (derived from a weather forecasting model covering the UK area) in grids with over 2×107 degrees of freedom. The PMA method computes these grids in times commensurate with those required for operational weather forecasting.

Three-dimensional, fully adaptive simulations of phase-field fluid models

We present an efficient numerical methodology for the 31) computation of incompressible multi-phase flows described by conservative phase-field models We focus here on the case of density matched fluids with different viscosity (Model H) The numerical method employs adaptive mesh refinements (AMR) in concert with an efficient semi-implicit time discretization strategy and a linear, multi-level multigrid to relax high order stability constraints and to capture the flow`s disparate scales at optimal cost. Only five linear solvers are needed per time-step. Moreover, all the adaptive methodology is constructed from scratch to allow a systematic investigation of the key aspects of AMR in a conservative, phase-field setting. We validate the method and demonstrate its capabilities and efficacy with important examples of drop deformation, Kelvin-Helmholtz instability, and flow-induced drop coalescence (C) 2010 Elsevier Inc. All rights reserved

A Robust, Fully Adaptive Hybrid Level-Set/Front-Tracking Method for Two-Phase Flows with an Accurate Surface Tension Computation

We present a variable time step, fully adaptive in space, hybrid method for the accurate simulation of incompressible two-phase flows in the presence of surface tension in two dimensions. The method is based on the hybrid level set/front-tracking approach proposed in [H. D. Ceniceros and A. M. Roma, J. Comput. Phys., 205, 391400, 2005]. Geometric, interfacial quantities are computed from front-tracking via the immersed-boundary setting while the signed distance (level set) function, which is evaluated fast and to machine precision, is used as a fluid indicator. The surface tension force is obtained by employing the mixed Eulerian/Lagrangian representation introduced in [S. Shin, S. I. Abdel-Khalik, V. Daru and D. Juric, J. Comput. Phys., 203, 493-516, 2005] whose success for greatly reducing parasitic currents has been demonstrated. The use of our accurate fluid indicator together with effective Lagrangian marker control enhance this parasitic current reduction by several orders of magnitude. To resolve accurately and efficiently sharp gradients and salient flow features we employ dynamic, adaptive mesh refinements. This spatial adaption is used in concert with a dynamic control of the distribution of the Lagrangian nodes along the fluid interface and a variable time step, linearly implicit time integration scheme. We present numerical examples designed to test the capabilities and performance of the proposed approach as well as three applications: the long-time evolution of a fluid interface undergoing Rayleigh-Taylor instability, an example of bubble ascending dynamics, and a drop impacting on a free interface whose dynamics we compare with both existing numerical and experimental data.

Development of AMREM: a Maxwell's equations solver with Adaptive Mesh Refinement

El principal objetivo de la presente tesis es el de desarrollar y probar un código capaz de resolver las ecuaciones de Maxwell en el dominio del tiempo con Malla Refinada Adaptativa (AMR por sus siglas en inglés). AMR es una técnica de cálculo basada en dividir el dominio físico del problema en distintas mallas rectangulares paralelas a las direcciones cartesianas. Cada una de las mallas tendrá distinta resolución y aquellas con mayor resolución se sitúan allí dónde las ondas electromagnéticas se propagan o interaccionan con los materiales, es decir, dónde mayor precisión es requerida. Como las ondas van desplazándose por todo el dominio, las mayas deberán seguirlas. El principal problema al utilizar esta metodología se puede encontrar en las fronteras internas, dónde las distintas mallas se unen. Ya que el método más corrientemente utilizado para resolver las ecuaciones de Maxwell es el de las diferencias finitas en el dominio del tiempo (FDTD por sus siglas en inglés) , el trabajo comenzó tratando de adaptar AMR a FDTD. Tras descubrirse que esta interacción resultaba en problemas de inestabilidades en las fronteras internas antes citadas, se decidió cambiar a un método basado en volúmenes finitos en el dominio del tiempo (FVTD por sus siglas en inglés). Este se basa en considerar la forma en ecuaciones de conservación de las ecuaciones de Maxwell y aplicar a su resolución un esquema de Godunov. Se ha probado que es clave para el correcto funcionamiento del código la elección de un limitador de flujo que proteja los extremos de la onda de la disipación típica de los métodos de este tipo. Otro problema clásico a la hora de resolver las ecuaciones de Maxwell es el de tratar con las condiciones de frontera física cuando se simulan dominios no acotados, es decir, dónde las ondas deben salir del sistema sin producir ninguna reflexión. Normalmente la solución es la de disponer una banda absorbente en las fronteras físicas. En AMREM se ha desarrollado un nuevo método basado en los campos característicos que con menor requisito de CPU funcina suficientemente bien incluso en los casos más desfaborables. El código ha sido contrastado con soluciones analíticas de diferentes problemas y también su velocidad ha sido comparada con la de Meep, uno de los programas más conocidos del ámbito. También algunas aplicaciones han sido simuladas con el fin de demostrar el amplio espectro de campos en los que AMREM puede funcionar como una útil herramienta.

Error estimation and adaptive mesh refinement for aerodynamic flows

This lecture course covers the theory of so-called duality-based a posteriori error estimation of DG finite element methods. In particular, we formulate consistent and adjoint consistent DG methods for the numerical approximation of both the compressible Euler and Navier-Stokes equations; in the latter case, the viscous terms are discretized based on employing an interior penalty method. By exploiting a duality argument, adjoint-based a posteriori error indicators will be established. Moreover, application of these computable bounds within automatic adaptive finite element algorithms will be developed. Here, a variety of isotropic and anisotropic adaptive strategies, as well as $hp$-mesh refinement will be investigated.

Simulação tridimensional adaptativa da separação das fases de uma mistura bifásica usando a equação de Cahn-Hilliard

Simulamos a separação dos componentes de uma mistura bifásica com a equação de Cahn-Hilliard. Esta equação contém intrincados termos não lineares e derivadas de alta ordem. Além disso, a delgada região de transição entre os componentes da mistura requer muita resolução. Assim, determinar a solução numérica da equação de Cahn-Hilliard não é uma tarefa fácil, principalmente em três dimensões. Conseguimos a resolução exigida no tempo usando uma discretização semi-implícita de segunda ordem. No espaço, obtemos a precisão requerida utilizando malhas refinadas localmente com a estratégia AMR. Essas malhas se adaptam dinamicamente para recobrir a região de transição. O sistema linear proveniente da discretização é solucionado por intermédio de técnicas multinível-multigrid.

Simulation of three-dimensional viscous flow in turbomachinery geometries using a solution-adaptive unstructured mesh methodology

This paper presents a numerical method for the simulation of flow in turbomachinery blade rows using a solution-adaptive mesh methodology. The fully three-dimensional, compressible, Reynolds-averaged Navier-Stokes equations with k-ε turbulence modeling (and low Reynolds number damping terms) are solved on an unstructured mesh formed from tetrahedral finite volumes. At stages in the solution, mesh refinement is carried out based on flagging cell faces with either a fractional variation of a chosen variable (like Mach number) greater than a given threshold or with a mean value of the chosen variable within a given range. Several solutions are presented, including that for the highly three-dimensional flow associated with the corner stall and secondary flow in a transonic compressor cascade, to demonstrate the potential of the new method.

A new type of high-order elements based on the mesh-free interpolations

We propose a new type of high-order elements that incorporates the mesh-free Galerkin formulations into the framework of finite element method. Traditional polynomial interpolation is replaced by mesh-free interpolations in the present high-order elements, and the strain smoothing technique is used for integration of the governing equations based on smoothing cells. The properties of high-order elements, which are influenced by the basis function of mesh-free interpolations and boundary nodes, are discussed through numerical examples. It can be found that the basis function has significant influence on the computational accuracy and upper-lower bounds of energy norm, when the strain smoothing technique retains the softening phenomenon. This new type of high-order elements shows good performance when quadratic basis functions are used in the mesh-free interpolations and present elements prove advantageous in adaptive mesh and nodes refinement schemes. Furthermore, it shows less sensitive to the quality of element because it uses the mesh-free interpolations and obeys the Weakened Weak (W2) formulation as introduced in [3, 5].

A h-Adaptive Algorithm Using Residual Error Estimates for Fluid Flows

Algorithms for adaptive mesh refinement using a residual error estimator are proposed for fluid flow problems in a finite volume framework. The residual error estimator, referred to as the R-parameter is used to derive refinement and coarsening criteria for the adaptive algorithms. An adaptive strategy based on the R-parameter is proposed for continuous flows, while a hybrid adaptive algorithm employing a combination of error indicators and the R-parameter is developed for discontinuous flows. Numerical experiments for inviscid and viscous flows on different grid topologies demonstrate the effectiveness of the proposed algorithms on arbitrary polygonal grids.

Adaptive executions of hyperbolic block-structured AMR applications on GPU systems

A block-structured adaptive mesh refinement (AMR) technique has been used to obtain numerical solutions for many scientific applications. Some block-structured AMR approaches have focused on forming patches of non-uniform sizes where the size of a patch can be tuned to the geometry of a region of interest. In this paper, we develop strategies for adaptive execution of block-structured AMR applications on GPUs, for hyperbolic directionally split solvers. While effective hybrid execution strategies exist for applications with uniform patches, our work considers efficient execution of non-uniform patches with different workloads. Our techniques include bin-packing work units to load balance GPU computations, adaptive asynchronism between CPU and GPU executions using a knapsack formulation, and scheduling communications for multi-GPU executions. Our experiments with synthetic and real data, for single-GPU and multi-GPU executions, on Tesla S1070 and Fermi C2070 clusters, show that our strategies result in up to a 3.23 speedup in performance over existing strategies.

Voronoi, Delaunay, and Block-Structured Mesh Refinement for Solution of the Shallow-Water Equations on the Sphere

Alternative meshes of the sphere and adaptive mesh refinement could be immensely beneficial for weather and climate forecasts, but it is not clear how mesh refinement should be achieved. A finite-volume model that solves the shallow-water equations on any mesh of the surface of the sphere is presented. The accuracy and cost effectiveness of four quasi-uniform meshes of the sphere are compared: a cubed sphere, reduced latitude–longitude, hexagonal–icosahedral, and triangular–icosahedral. On some standard shallow-water tests, the hexagonal–icosahedral mesh performs best and the reduced latitude–longitude mesh performs well only when the flow is aligned with the mesh. The inclusion of a refined mesh over a disc-shaped region is achieved using either gradual Delaunay, gradual Voronoi, or abrupt 2:1 block-structured refinement. These refined regions can actually degrade global accuracy, presumably because of changes in wave dispersion where the mesh is highly nonuniform. However, using gradual refinement to resolve a mountain in an otherwise coarse mesh can improve accuracy for the same cost. The model prognostic variables are height and momentum collocated at cell centers, and (to remove grid-scale oscillations of the A grid) the mass flux between cells is advanced from the old momentum using the momentum equation. Quadratic and upwind biased cubic differencing methods are used as explicit corrections to a fast implicit solution that uses linear differencing.

An adaptive finite element water column model using the Mellor-Yamada level 2.5 turbulence closure scheme

A one-dimensional water column model using the Mellor and Yamada level 2.5 parameterization of vertical turbulent fluxes is presented. The model equations are discretized with a mixed finite element scheme. Details of the finite element discrete equations are given and adaptive mesh refinement strategies are presented. The refinement criterion is an "a posteriori" error estimator based on stratification, shear and distance to surface. The model performances are assessed by studying the stress driven penetration of a turbulent layer into a stratified fluid. This example illustrates the ability of the presented model to follow some internal structures of the flow and paves the way for truly generalized vertical coordinates. (c) 2005 Elsevier Ltd. All rights reserved.