Adaptive mesh refinement and load balancing based on multi-level block-structured Cartesian mesh

We developed a framework for a distributed-memory parallel computer that enables dynamic data management for adaptive mesh refinement and load balancing. We employed simple data structure of the building cube method (BCM) where a computational domain is divided into multi-level cubic domains and each cube has the same number of grid points inside, realising a multi-level block-structured Cartesian mesh. Solution adaptive mesh refinement, which works efficiently with the help of the dynamic load balancing, was implemented by dividing cubes based on mesh refinement criteria. The framework was investigated with the Laplace equation in terms of adaptive mesh refinement, load balancing and the parallel efficiency. It was then applied to the incompressible Navier–Stokes equations to simulate a turbulent flow around a sphere. We considered wall-adaptive cube refinement where a non-dimensional wall distance y⁺ near the sphere is used for a criterion of mesh refinement. The result showed the load imbalance due to y⁺ adaptive mesh refinement was corrected by the present approach. To utilise the BCM framework more effectively, we also tested a cube-wise algorithm switching where an explicit and implicit time integration schemes are switched depending on the local Courant-Friedrichs-Lewy (CFL) condition in each cube.     

自适应网格加密和多重网格求解严重病态方程

本文探讨用自适应网格加密技术结合多重网格法求解由有限元离散导致的病态方程。计算结果表明；自适应多重网格有限元法是求解严重病态方程珠有效方法。     

A gas‐kinetic scheme for the simulation of turbulent flows on unstructured meshes

This paper presents a numerical method for simulating turbulent flows via coupling the Boltzmann BGK equation with Spalart–Allmaras one equation turbulence model. Both the Boltzmann BGK equation and the turbulence model equation are carried out using the finite volume method on unstructured meshes, which is different from previous works on structured grid. The application of the gas‐kinetic scheme is extended to the simulation of turbulent flows with arbitrary geometries. The adaptive mesh refinement technique is also adopted to reduce the computational cost and improve the efficiency of meshes. To organize the unstructured mesh data structure efficiently, a non‐manifold hybrid mesh data structure is extended for polygonal cells. Numerical experiments are performed on incompressible flow over a smooth flat plate and compressible turbulent flows around a NACA 0012 airfoil using unstructured hybrid meshes. These numerical results are found to be in good agreement with experimental data and/or other numerical solutions, demonstrating the applicability of the proposed method to simulate both subsonic and transonic turbulent flows. Copyright © 2016 John Wiley & Sons, Ltd.     

一种新的求解对流占优问题的自适应网格细化方法

在均匀网格上求解对流占优问题时,往往会产生数值震荡现象,因此需要局部加密网格来提高解的精度。针对对流占优问题,设计了一种新的自适应网格细化算法。该方法采用流线迎风SUPG(Petrov-Galerkin)格式求解对流占优问题,定义了网格尺寸并通过后验误差估计子修正来指导自适应网格细化,以泡泡型局部网格生成算法BLMG为网格生成器,通过模拟泡泡在区域中的运动得到了高质量的点集。与其他自适应网格细化方法相比,该方法可在同一框架内实现网格的细化和粗化,同时在所有细化层得到了高质量的网格。数值算例结果表明,该方法在求解对流占优问题时具有更高的数值精度和更好的收敛性。     

An efficient FEM for pressure analysis of oil film in a piston pump

The paper concerns numerical analysis of pressure distribution of an oil film on the valve plate in the variable height gap of an axial piston pump. The analysis employs the finite element method. For determination of oil pressure variations in the gap, the Reynolds equation, commonly applied in the theory of lubrication, is applied. The equation is solved numerically with the use of self-developed program based on the finite element method. In order to obtain high accuracy of the results, an adaptive mesh refinement based on residual estimations of solution errors is applied. The calculation results are represented as dependent on the geometric and working parameters of the pump.     

Continuous adjoint-based error estimation and its application to adaptive discontinuous Galerkin method

An adaptive mesh refinement algorithm based on a continuous adjoint approach is developed. Both the primal equation and the adjoint equation are approximated with the discontinuous Galerkin (DG) method. The proposed adaptive algorithm is used in compressible Euler equations. Numerical tests are made to show the superiority of the proposed adaptive algorithm.     

Numerical simulation of compressible two-phase flow using a diffuse interface method

In this article, a high-resolution diffuse interface method is investigated for simulation of compressible two-phase gas–gas and gas–liquid flows, both in the presence of shock wave and in flows with strong rarefaction waves similar to cavitations. A Godunov method and HLLC Riemann solver is used for discretization of the Kapila five-equation model and a modified Schmidt equation of state (EOS) is used to simulate the cavitation regions. This method is applied successfully to some one- and two-dimensional compressible two-phase flows with interface conditions that contain shock wave and cavitations. The numerical results obtained in this attempt exhibit very good agreement with experimental results, as well as previous numerical results presented by other researchers based on other numerical methods. In particular, the algorithm can capture the complex flow features of transient shocks, such as the material discontinuities and interfacial instabilities, without any oscillation and additional diffusion. Numerical examples show that the results of the method presented here compare well with other sophisticated modeling methods like adaptive mesh refinement (AMR) and local mesh refinement (LMR) for one- and two-dimensional problems.     

A three-dimensional robust volume-of-fluid solver based on the adaptive mesh refinement

The present study provides a three-dimensional volume-of-fluid method based on the adaptive mesh refinement technique. The projection method on the adaptive mesh is introduced for solving the incompressible Navier-Stokes equations. The octree structure mesh is employed to solve the flow velocities and the pressure. The developed solver is applied to simulate the deformation of the cubic droplet driven by the surface tension without the effect of the gravity. The numerical results well predict the shape evolution of the droplet.     

A high-order Runge-Kutta discontinuous Galerkin method with a subcell limiter on adaptive unstructured grids for two-dimensional compressible inviscid flows

A robust, adaptive unstructured mesh refinement strategy for high-order Runge-Kutta discontinuous Galerkin method is proposed. The present work mainly focuses on accurate capturing of sharp gradient flow features like strong shocks in the simulations of two-dimensional inviscid compressible flows. A posteriori finite volume subcell limiter is employed in the shock-affected cells to control numerical spurious oscillations. An efficient cell-by-cell adaptive mesh refinement is implemented to increase the resolution of our simulations. This strategy enables to capture strong shocks without much numerical dissipation. A wide range of challenging test cases is considered to demonstrate the efficiency of the present adaptive numerical strategy for solving inviscid compressible flow problems having strong shocks.     

双曲偏微分方程的局部伪弧长方法研究

重点研究了局部伪弧长方法在处理偏微分方程,尤其是双曲型偏微分方程出现激波间断的奇异性问题,对比分析了全局伪弧长方法空间转化的形式及其网格自适应的性质。为提高求解效率,提出了局部伪弧长方法,利用激波间断的性质,给出了判断奇异点位置以及模板选择的方法,涉及如何处理激波振荡,如何引入弧长参数,以及怎样求解间断等问题。通过数值算例验证了局部伪弧长在激波捕捉和追踪方面的可行性,通过比较局部伪弧长方法与Godunov方法处理不同初值条件的双曲问题,显示出局部伪弧长方法处理双曲偏微分方程的优越性,为伪弧长方法应用到物理问题奠定基础。     

A GPU-accelerated adaptive discontinuous Galerkin method for level set equation

This paper presents a GPU-accelerated nodal discontinuous Galerkin method for the solution of two- and three-dimensional level set (LS) equation on unstructured adaptive meshes. Using adaptive mesh refinement, computations are localised mostly near the interface location to reduce the computational cost. Small global time step size resulting from the local adaptivity is avoided by local time-stepping based on a multi-rate Adams–Bashforth scheme. Platform independence of the solver is achieved with an extensible multi-threading programming API that allows runtime selection of different computing devices (GPU and CPU) and different threading interfaces (CUDA, OpenCL and OpenMP). Overall, a highly scalable, accurate and mass conservative numerical scheme that preserves the simplicity of LS formulation is obtained. Efficiency, performance and local high-order accuracy of the method are demonstrated through distinct numerical test cases.     

Feature‐based adaptive mesh refinement for wingtip vortices

Feature‐based solution‐adaptive mesh refinement is an attractive strategy when it is known a priori that the resolution of certain key features is critical to achieving the objectives of a simulation. In this paper, we apply vortex characterization techniques, which are typically employed to visualize vortices, to identify regions of the computational domain for mesh refinement. We investigate different refinement strategies that are facilitated by these vortex characterization techniques to simulate the flow past a wing in a wind tunnel. Our results, which we compare with experimental data, indicate that it is necessary to refine the region within and near the vortex extent surface to obtain an accurate prediction. Application of the identified mesh refinement strategy also produced observed improvement in the results predicted for a spinning missile with deflected canards. Copyright © 2010 John Wiley & Sons, Ltd.     

A parallel adaptive numerical method with generalized curvilinear coordinate transformation for compressible Navier–Stokes equations

A fourth‐order finite‐volume method for solving the Navier–Stokes equations on a mapped grid with adaptive mesh refinement is proposed, implemented, and demonstrated for the prediction of unsteady compressible viscous flows. The method employs fourth‐order quadrature rules for evaluating face‐averaged fluxes. Our approach is freestream preserving, guaranteed by the way of computing the averages of the metric terms on the faces of cells. The standard Runge–Kutta marching method is used for time discretization. Solutions of a smooth flow are obtained in order to verify that the method is formally fourth‐order accurate when applying the nonlinear viscous operators on mapped grids. Solutions of a shock tube problem are obtained to demonstrate the effectiveness of adaptive mesh refinement in resolving discontinuities. A Mach reflection problem is solved to demonstrate the mapped algorithm on a non‐rectangular physical domain. The simulation is compared against experimental results. Future work will consider mapped multiblock grids for practical engineering geometries. Copyright © 2016 John Wiley & Sons, Ltd.     

Collocated discrete least‐squares (CDLS) meshless method: Error estimate and adaptive refinement

Meshless methods are new approaches for solving partial differential equations. The main characteristic of all these methods is that they do not require the traditional mesh to construct a numerical formulation. They require node generation instead of mesh generation. In other words, there is no pre‐specified connectivity or relationships among the nodes. This characteristic make these methods powerful. For example, an adaptive process which requires high computational effort in mesh‐dependent methods can be very economically solved with meshless methods. In this paper, a posteriori error estimate and adaptive refinement strategy is developed in conjunction with the collocated discrete least‐squares (CDLS) meshless method. For this, an error estimate is first developed for a CDLS meshless method. The proposed error estimator is shown to be naturally related to the least‐squares functional, providing a suitable posterior measure of the error in the solution. A mesh moving strategy is then used to displace the nodal points such that the errors are evenly distributed in the solution domain. Efficiency and effectiveness of the proposed error estimator and adaptive refinement process are tested against two hyperbolic benchmark problems, one with shocked and the other with low gradient smooth solutions. These experiments show that the proposed adaptive process is capable of producing stable and accurate results for the difficult problems considered. Copyright © 2007 John Wiley & Sons, Ltd.     

Solution adaptive grids applied to low Reynolds number flow

A numerical study has been undertaken to investigate the use of a solution adaptive grid for flow around a cylinder in the laminar flow regime. The main purpose of this work is twofold. The first aim is to investigate the suitability of a grid adaptation algorithm and the reduction in mesh size that can be obtained. Secondly, the uniform asymmetric flow structures are ideal to validate the mesh structures due to mesh refinement and consequently the selected refinement criteria. The refinement variable used in this work is a product of the rate of strain and the mesh cell size, and contains two variables C_m and C_str which determine the order of each term. By altering the order of either one of these terms the refinement behaviour can be modified. Copyright © 2003 John Wiley & Sons, Ltd.     

Adaptive resolution refinement for pseudospectral simulations of isotropic homogeneous turbulence

Pseudospectral simulations of homogeneous turbulence provide an important class of benchmark flow problems used for fundamental studies of turbulence and for numerical validation purposes. Depending on the numerical resolution, fully resolved computations of homogeneous turbulence can consume large amounts of central processing unit (CPU) time. Here, we present an approach analogous to adaptive mesh refinement for computations performed in physical space to adaptively refine the spectral resolution for pseudospectral computations of isotropic homogeneous turbulent flows. The method is applied to simulations of two-dimensional and three-dimensional isotropic homogeneous turbulence, and the results are compared with direct numerical simulations (DNS) performed using a fixed fine mesh. Significant savings in computational time are found in each case, with little to no compromise in the accuracy of the solutions.     

High-order schemes for predicting computational aeroacoustic propagation with adaptive mesh refinement

High-order schemes based on block-structured adaptive mesh refinement method are prepared to solve computational aeroacoustic (CAA) problems with an aim at improving computational efficiency. A number of numerical issues associated with high-order schemes on an adaptively refined mesh, such as stability and accuracy are addressed. Several CAA benchmark problems are used to demonstrate the feasibility and efficiency of the approach.     

Numerical investigation of BB-AMR scheme using entropy production as refinement criterion

In this work, a parallel finite volume scheme on unstructured meshes is applied to fluid flow for multidimensional hyperbolic system of conservation laws. It is based on a block-based adaptive mesh refinement strategy which allows quick meshing and easy parallelisation. As a continuation and as an extension of a previous work, the useful numerical density of entropy production is used as mesh refinement criterion combined with a local time-stepping method to preserve the computational time. Then, we numerically investigate its efficiency through several test cases with a confrontation with exact solution or experimental data.     

A computational study of supersonic disk-gap-band parachutes using Large-Eddy Simulation coupled to a structural membrane

Large-scale fluid-structure interaction simulations of compressible flows over flexible supersonic disk-gap-band parachutes are compared with matching experimental results. We utilize adaptive mesh refinement, large-eddy simulation of compressible flow coupled with a thin-shell structural finite-element model. The simulations are carried out in the regime where large canopy-area oscillations are present, around and above Mach 2, where strong nonlinear coupling between the system of bow shocks, turbulent wake and canopy is observed. Comparisons of drag history and its dependence on Mach number are discussed. Furthermore, it is observed that important dynamical features of this coupled system can only be reproduced when sufficient grid resolution is used. Lack of resolution resulted in incorrect flow-physics prediction and, consequently, incorrect fluid-structure interaction coupling.