A parallel adaptive mesh approach for flowfields with shock waves

A simple and efficient implementation of Adaptive Mesh Refinement (AMR) with distributed-memory approach is presented. Introducing a lookup table including grid connectivity information and simplified algorithms for AMR, the procedures for reconstructing adaptive grids are carried out in parallel, with local data to a large extent. A simple static load-balancing scheme is adopted, and the grids are not repartitioned and no data redistribution is performed. A numerical example on two different parallel computers shows that the proposed implementation of AMR is effective to reduce the computational time for unsteady flows with shock waves. Received 23 October 2000 / Accepted 30 March 2001 Published online 11 June 2002     

The implementation of an aeronautical CFD flow code onto distributed memory parallel systems

The parallelization of an industrially important in‐house computational fluid dynamics (CFD) code for calculating the airflow over complex aircraft configurations using the Euler or Navier–Stokes equations is presented. The code discussed is the flow solver module of the SAUNA CFD suite. This suite uses a novel grid system that may include block‐structured hexahedral or pyramidal grids, unstructured tetrahedral grids or a hybrid combination of both. To assist in the rapid convergence to a solution, a number of convergence acceleration techniques are employed including implicit residual smoothing and a multigrid full approximation storage scheme (FAS). Key features of the parallelization approach are the use of domain decomposition and encapsulated message passing to enable the execution in parallel using a single programme multiple data (SPMD) paradigm. In the case where a hybrid grid is used, a unified grid partitioning scheme is employed to define the decomposition of the mesh. The parallel code has been tested using both structured and hybrid grids on a number of different distributed memory parallel systems and is now routinely used to perform industrial scale aeronautical simulations. Copyright © 2000 John Wiley & Sons, Ltd.     

A numerical algorithm for simulation of the Q-switched fiber laser using the travelling wave model

We develop a finite-difference scheme for approximation of a system of nonlinear PDEs describing the Q-switching process. We construct it by using staggered grids. The transport equations are approximated along characteristics, and quadratic nonlinear functions are linearized using a special selection of staggered grids. The stability analysis proves that a connection between time and space steps arises only due to approximation requirements in order to follow exactly the directions of characteristics. The convergence analysis of this scheme is done in two steps. First, some estimates of the uniform boundedness of the discrete solution are proved. This part of the analysis is done locally, in some neighborhood of the exact solution. Second, on the basis of the obtained estimates, the main stability inequality is proved. The second-order convergence rate with respect to the space and time coordinates follows from this stability estimate. Using the obtained convergence result, we prove that the local stability analysis in the selected neighborhood of the exact solution is sufficient.     

On coordinate transformation and grid stretching for sparse grid pricing of basket options

We evaluate two coordinate transformation techniques in combination with grid stretching for pricing basket options in a sparse grid setting. The sparse grid technique is a basic technique for solving a high-dimensional partial differential equation. By creating a small hypercube sub-grid in the ‘composite’ sparse grid we can also determine hedge parameters accurately. We evaluate these techniques for multi-asset examples with up to five underlying assets in the basket.     

Sparse grids and hybrid methods for the chemical master equation

The direct numerical solution of the chemical master equation (CME) is usually impossible due to the high dimension of the computational domain. The standard method for solution of the equation is to generate realizations of the chemical system by the stochastic simulation algorithm (SSA) by Gillespie and then taking averages over the trajectories. Two alternatives are described here using sparse grids and a hybrid method. Sparse grids, implemented as a combination of aggregated grids are used to address the curse of dimensionality of the CME. The aggregated components are selected using an adaptive procedure. In the hybrid method, some of the chemical species are represented macroscopically while the remaining species are simulated with SSA. The convergence of variants of the method is investigated for a growing number of trajectories. Two signaling cascades in molecular biology are simulated with the methods and compared to SSA results. AMS subject classification (2000)  65C20, 60J25, 92C45     

Evaluation of adaptive mesh refinement and coarsening for the computation of compressible flows on unstructured meshes

The compressible gas flows of interest to aerospace applications often involve situations where shock and expansion waves are present. Decreasing the characteristic dimension of the computational cells in the vicinity of shock waves improves the quality of the computed flows. This reduction in size may be accomplished by the use of mesh adaption procedures. In this paper an analysis is presented of an adaptive mesh scheme developed for an unstructured mesh finite volume upwind computer code. This scheme is tailored to refine or coarsen the computational mesh where gradients of the flow properties are respectively high or low. The refinement and coarsening procedures are applied to the classical gas dynamic problems of the stabilization of shock waves by solid bodies. In particular, situations where oblique shock waves interact with an expansion fan and where bow shocks arise around solid bodies are considered. The effectiveness of the scheme in reducing the computational time, while increasing the solution accuracy, is assessed. It is shown that the refinement procedure alone leads to a number of computational cells which is 20% larger than when alternate passes of refinement and coarsening are used. Accordingly, a reduction of computational time of the same order of magnitude is obtained. Copyright © 2005 John Wiley & Sons, Ltd.     

A multiblock/multilevel mesh refinement procedure for CFD computations

A multiblock/multilevel algorithm with local refinement for general two‐ and three‐dimensional fluid flow is presented. The patched‐based local refinement procedure is presented in detail and algorithmic implementations are also presented. The multiblock implementation is essentially block‐unstructured, i.e. each block having its own local curvilinear co‐ordinate system. Refined grid patches can be put anywhere in the computational domain and can extend across block boundaries. To simplify the implementation, while still maintaining sufficient generality, the refinement is restricted to a refinement of the grid successively halving the grid size within a selected patch. The multiblock approach is implemented within the framework of the well‐known SIMPLE solution strategy. Computational experiments showing the effect of using the multilevel solution procedure are presented for a sample elliptic problem and a few benchmark problems of computational fluid dynamics (CFD). Copyright © 2001 John Wiley & Sons, Ltd.     

Software package for 3D viscous gas flow simulation on multiprocessor computer systems

A parallel software package designed for the numerical simulation of three-dimensional viscous gas flows is presented. The numerical algorithm is based on kinetically consistent difference schemes used on locally refined grids. The software package has been tested in various super-and subsonic flow problems. It provides an opportunity for the direct simulation of turbulent flows. The efficiency of parallelization is analyzed depending on the problem size and the number of processors.     

A numerical technique for laminar swirling flow at the interface between porous and homogenous fluid domains

There have been a few recent numerical implementations of the stress‐jump condition at the interface of conjugate flows, which couple the governing equations for flows in the porous and homogenous fluid domains. These previous demonstration cases were for two‐dimensional, planar flows with simple geometries, for example, flow over a porous layer or flow through a porous plug. The present study implements the interfacial stress‐jump condition for a non‐planar flow with three velocity components, which is more realistic in terms of practical flow applications. The steady, laminar, Newtonian flow in a stirred micro‐bioreactor with a porous scaffold inside was investigated. It is shown how to implement the interfacial jump condition on the radial, axial, and swirling velocity components. To avoid a full three‐dimensional simulation, the flow is assumed to be independent of the azimuthal direction, which makes it an axisymmetric flow with a swirling velocity. The present interface treatment is suitable for non‐flat surfaces, which is achieved by applying the finite volume method based on body‐fitted and multi‐block grids. The numerical simulations show that a vortex breakdown bubble, attached to the free surface, occurs above a certain Reynolds number. The presence of the porous scaffold delays the onset of vortex breakdown and confines it to a region above the scaffold. Copyright © 2008 John Wiley & Sons, Ltd.     

Thin viscous fluid film flows over rotating curvilinear surfaces

Thin viscous fluid film flows over rotating curvilinear surfaces are considered. The steady-state flow is studied for large and finite Ekman numbers. The possibility of obtaining a film of given thickness is considered. The wave formation on the surface of an inhomogeneous film and the development of nonlinear waves are also investigated. The film flow parameters in the cases of flow over a plane disk and a curvilinear surface are compared.