Fig 2 - uploaded by Martin Berzins
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Example triangle used in discretization description replaced by a line integral around the triangular element: Z Ai @f x @x + @f y @y d = I Ci (f x n x + f y n y )ds;
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SPRINT2D is a set of software tools for solving both steady and unsteady partial differential equations in two-space variables. The software consists of a set of coupled modules for mesh generation, spatial discretization, time integration, nonlinear equations, linear algebra, spatial adaptivity, and visualization. The software uses unstructured tr...
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... allow the construction of high order schemes however the centroid of the triangle is deened as the nodal position and the solution value is associated with that point. In Figure 2 for example, the solution at the centroid of triangle i is U i , and the solutions at the centroids of the triangles surrounding triangle i are U j , U k and U l . The coordinates of the mesh point at which a solution value, say U s , is deened are denoted by (x s ; y s ) . ...
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Citations
... For example, Ahmadi et al. [1] have used an adaptive method for solving the Euler equations, but they only solve for steady-state problems and their "mesh adaption" is closer to a partial remeshing than adaptive refinement. Similarly, Berzins et al. [5] have developed a system that incorporates adaptive refinement; they apply "static rezoning" (remeshing in disguise) if their error tolerance is exceeded. Barth and Larson [2] mention an adaptive procedure in passing, but give very little detail as their work is focused on error estimation. ...
An adaptive method based on recursive subdivision of unstructured elements for the solution of conservation laws is presented. The refinement of cells is based on regular subdivision into four children as indicated by a gradient-detector and is carried out "Just-In-Time" before the actual computation on that element takes place. In addition to spatial refinement, temporal refine-ment is carried out in conjunction with "lock-step" time-stepping to guaran-tee the availability of the proper states at the correct times across the mesh on all scales. The approach uses standard slope-limited finite volume meth-ods of MUSCL-Hancock type with slight modifications to cater for different levels of subdivision in adjacent elements. We present some background to AMR and the finite volume framework, the algorithm itself and conclude with numerical examples of linear and non-linear scalar conservation laws in two dimensions.
... Such complexities are further compounded when multiple simulation codes are combined to simulate the interaction of multiple phenomena. Frameworks for PDE simulation are common, including systems such as Diffpack [1], Ellpack [2], Overture [3], POOMA [4], SAMRAI [5], and Sprint 2D [6]. Each of these frameworks have their own strengths and weaknesses, but each are designed to simplify the process of implementing PDE simulations. ...
We describe the Uintah Computational Framework (UCF), a set of software components and libraries that facilitate the simulation
of Partial Differential Equations (PDEs) on Structured AMR (SAMR) grids using hundreds to thousands of processors. The UCF
uses a nontraditional approach to achieving parallelism, employing an abstract taskgraph representation to describe computation
and communication. This representation has a number of advantages that affect the performance of the resulting simulation.
We demonstrate performance of the system on a solid mechanics algorithm, two different Computational Fluid-Dynamics (CFD)
algorithms, and coupled CFD/solids algorithms. We illustrate the performance of the UCF for jobs requiring up to 2000 processors.
... Although in two space dimensional calculations we have used sophisticated space}time error control techniques (Berzins et al., 1998; Tomlin et al., 1997), the computational cost and the need to evaluate spatial mesh adaption has led us to focus here on less costly methods. It is thus primarily for computational e$ciency that, in common with many others, we have used an operator splitting technique. ...
We describe the development of a high-resolution model for air pollution dispersion and chemical transformation. A cell-vertex finite volume scheme based on an adaptive tetrahedral mesh is used to solve the atmospheric diffusion equation. Preliminary studies of dispersion from an accidental release of pollutants into the atmosphere have been carried out. The results show the efficiency and accuracy of this adaptive method compared to fixed methods for mesh refinement. Some comments on the interpolation of input data such as wind fields on unstructured meshes are also made.
... Although in two space dimensional calculations we have used sophisticated space}time error control techniques ( Berzins et al., 1998;Tomlin et al., 1997), the computational cost and the need to evaluate spatial mesh adaption has led us to focus here on less costly methods. It is thus primarily for computational e$ciency that, in common with many others, we have used an operator splitting technique. ...
This paper investigates the solution of a 3D atmospheric dispersion problem using a time-dependent adaptive gridding technique. A cell-vertex finite volume scheme based on tetrahedral elements is used to solve the atmospheric diffusion equation. Preliminary studies of dispersion from a single source in stable, unstable and neutral boundary layers have been carried out. The results show the efficiency of using adaptive grids to represent accurately the structures of plumes in the boundary layer and also the advantage of this method compared to fixed methods for mesh refinement. Some comments about the interpolation of input data such as wind fields onto unstructured meshes are also made.
... Unstructured grids in PDE calculations allow some families of numerical methodssuch as Finite Element Methods (FEM) and Finite Volume Methods (FVM) -to be adopted in problems involving general geometries. As pointed out by Berzins et al. 1998, this issue is presently one of the most important trends in the development of numerical software packages for PDEs. For example, we can mention a (largely incomplete) list of remarkable software products such as DIFFPACK [Bruaset and Langtangen 1996], ELLPACK [Rice and Boisvert 1984], KASKADE [Beck et al. 1995], SPRINT2D-SPRINT3D [Berzins et al. 1998], and UG [Bastian et al. 1997]. ...
... As pointed out by Berzins et al. 1998, this issue is presently one of the most important trends in the development of numerical software packages for PDEs. For example, we can mention a (largely incomplete) list of remarkable software products such as DIFFPACK [Bruaset and Langtangen 1996], ELLPACK [Rice and Boisvert 1984], KASKADE [Beck et al. 1995], SPRINT2D-SPRINT3D [Berzins et al. 1998], and UG [Bastian et al. 1997]. ...
The software interface P2MESH is a collection of C++ class templates suitable for developing prototypes of high-performance PDE solvers on unstructured 2-D meshes. P2MESH supports several discretization methods on triangles and quadrilaterals, such as Finite Volume or Finite Element. The design philosophy of P2MESH does not consider specific model problems or built-in approximation algorithms. The software package is of general purpose and it may also be used as a building block in the implementation of numerical codes both for engineering applications and mathematical problems.
High resolution models of air pollution transport and transformation are necessary in order to test possible abatement strategies based on pollution control and to forecast high pollution episodes. Models are especially relevant for secondary pollutants like ozone and nitrogen dioxide, which are formed in the atmosphere through nonlinear chemical reactions involving primary pollutant species, often far from their sources. Often we are trying to resolve the interactions between plumes from point sources such as power stations and regional pollution tides of ozone formed in other European countries. One method of tackling this problem of different scales is to use different grid sizes, using highly resolved grids in regions where the structure is very fine. Telescopic gridding is currently used in high emission areas or around sensitive receptor points. However, since meteorological conditions vary, this method cannot resolve a priori highly structured regions away from sources, e.g. along plumes. Such refinement can be achieved using adaptive methods which increase resolutions in regions of steep spatial gradients. This article describes the use of 3D adaptive gridding models for pollution transport and reactions using both a layered and a fully adaptive 3D tetrahedral approach and provides examples which show the effect of grid resolution on secondary pollutant formation.
This paper describes a mathematical model and adaptive numerical simulation of the time‐dependent multiphase, multicomponent
flow which occurs when a gas venting process is used to remove a volatile contaminant from a porous medium. The numerical
simulation is adaptive in both space and time and involves the use of a finite element spatial discretization and the SPRINT2D
software [6] for time integration. Results are presented which demonstrate the high quality of the simulation, both in terms
of the length scales of the features that are resolved and the efficiency of the solutions relative to those obtained on fixed
grids.
The software interface P2MESH is a collection of C++ class templates suitable for developing prototypes of high-performance PDE solvers on unstructured 2-D meshes. P2MESH supports several discretization methods on triangles and quadrilaterals, such as finite volume or finite element. The design philosophy of P2MESH does not consider specific model problems or built-in approximation algorithms. The software package is general purpose and it may also be used as a building block in the implementation of numerical codes both for engineering applications and mathematical problems.
This work aims at developing/combining numerical tools adapted to the simulation of the near field of highly underexpanded jets. An overview of the challenging numerical problems related to the complex shock/expansion structure encountered in these flows is given and an efficient and low-cost numerical strategy is proposed to overcome these, even on short computational domains. Based on common upwinding algorithms used on unstructured meshes in a mixed finite-volume/finite-element approach, it relies on an appropriate utilization of zonal anisotropic remeshing algorithms. This methodology is validated for the whole near field of cold air jets issuing from axisymmetric convergent nozzles and yielding various underexpansion ratios. In addition, the most usual corrections of the $k-\epsilon$ model used take into account the compressibility effects on turbulence are precisely assessed.
