A Parallel Algorithm for Adaptive Local Refinement of Tetrahedral Meshes Using Bisection

Local mesh refinement is one of the key steps in the implementations of adaptive finite element methods. This paper presents a parallel algorithm for distributed memory parallel computers for adaptive local refinement of tetrahedral meshes using bisection. This algorithm is used in PHG, Parallel Hierarchical Grid Chttp：//lsec. cc. ac. cn/phg/）, a toolbox under active development for parallel adaptive finite element solutions of partial differential equations. The algorithm proposed is characterized by allowing simukaneous refinement of submeshes to arbitrary levels before synchronization between submeshes and without the need of a central coordinator process for managing new vertices. Using the concept of canonical refinement, a simple proof of the independence of the resulting mesh on the mesh partitioning is given, which is useful in better understanding the behaviour of the biseetioning refinement procedure.

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.

Two-Dimensional h-Version Adaptive Remeshing Refinement of FEM

A new method for estimating the posterior error of linear triangular element is presented. According to the specified accuracy, the element size and node spacing function of adaptive h-version mesh refinement are predicted. The main domain is divided by the isolines of node spacing into subregions in which mesh regeneration is realized. This remeshing refinement process implements mesh subdivision and mesh de-refinement in onecycle.

A Numerical Model for Simulating Two-Phase Flow with Adaptive Mesh Refinement

In this study,a numerical model for simulating two-phase flow is developed.The Cartesian grid with Adaptive Mesh Refinement(AMR)is adopted to reduce the computational cost.An explicit projection method is used for the time integration and the Finite Difference Method(FDM)is applied on a staggered grid for the discretization of spatial derivatives.The Volume of Fluid(VOF)method with Piecewise-Linear Interface Calculation(PLIC)is extended to the AMR grid to capture the gas-water interface accurately.A coarse-fine interface treatment method is developed to preserve the flux conservation at the interfaces.Several two-dimensional(2D)and three-dimensional(3D)benchmark cases are carried out for the validation of the model.2D and 3D shear flow tests are conducted to validate the extension of the VOF method to the AMR grid.A 2D linear sloshing case is considered in which the model is proved to have 2nd-order accuracy in space.The efficiency of applying the AMR grid is discussed with a nonlinear sloshing problem.Finally,2D solitary wave past stage and 2D/3D dam break are simulated to demonstrate that the model is able to simulate violent interface problems.

MESH FREE ADAPTIVE ALGORITHM FOR SOLVING EULER EQUATIONS ON STRUCTURED GRID POINTS

A complete mesh free adaptive algorithm （MFAA）, with solution adaptation and geometric adaptation, is developed to improve the resolution of flow features and to replace traditional global refinement techniques in structured grids. Unnecessary redundant points and elements are avoided by using the mesh free local clouds refinement technology in shock influencing regions and regions near large curvature places on the boundary. Inviscid compressible flows over NACA0012 and RAE2822 airfoils are computed. Finally numerical results validate the accuracy of the above method.

Two-Dimensional Euler Adaptive Mesh Method on Detonation

The adaptive mesh refinement （AMR） method is applied in the 2-D Euler multi-component elasticplastic hydrodynamics code （MEPH2Y）. It is applied on detonation. Firstly, the AMR method is described, including a conservative spatial interpolation, the time integration methodology with the adapitve time increment and an adaptive computational region method. The advantage of AMR technique is exhibited by numerical examples, including the 1-D C-J detonation and the 2-D implosion ignited from a single point. Results show that AMR can promote the computational efficiency, keeping the accuracy in interesting regions.

Simulation of shock-induced instability using an essentially conservative adaptive CE/SE method

An essentially conservative adaptive space time conservation element and solution element （CE/SE） method is pro- posed for the effective simulation of shock-induced instability with low computational cost. Its implementation is based on redefined conservation elements （CEs） and solution elements （SEs）, optimized interpolations and a Courant number insensitive CE/SE scheme. This approach is used in two applications, the Woodward double Mach reflection and a two- component Richtmyer-Meshkov instability experiment. This scheme reveals the essential features of the investigated cases, captures small unstable structures, and yields a solution that is consistent with the results from experiments or other high order methods.

Adaptive thermo-fluid moving boundary computations for interfacial dynamics

In this study, we present adaptive moving boundary computation technique with parallel implementation on a distributed memory multi-processor system for large scale thermo-fluid and interfacial flow computations. The solver utilizes Eulerian-Lagrangian method to track moving （Lagrangian） interfaces explicitly on the stationary （Eulerian） Cartesian grid where the flow fields are computed. We address the domain decomposition strategies of Eulerian- Lagrangian method by illustrating its intricate complexity of the computation involved on two different spaces inter- actively and consequently, and then propose a trade-off ap- proach aiming for parallel scalability. Spatial domain decomposition is adopted for both Eulerian and Lagrangian do- main due to easy load balancing and data locality for mini- mum communication between processors. In addition, parallel cell-based unstructured adaptive mesh refinement （AMR） technique is implemented for the flexible local refinement and even-distributed computational workload among processors. Selected cases are presented to highlight the computa- tional capabilities, including Faraday type interfacial waves with capillary and gravitational forcing, flows around varied geometric configurations and induced by boundary conditions and/or body forces, and thermo-fluid dynamics with phase change. With the aid of the present techniques, large scale challenging moving boundary problems can be effectively addressed.

Cavitation Evolution Around a Twist Hydrofoil by Large Eddy Simulation(LES)with Mesh Adaption

The cavitating flow around a Delft Twist-11 hydrofoil is simulated using the large eddy simulation approach.The volume-of-fluid method incorporated with the Schnerr-Sauer cavitation model is utilized to track the water-vapor interface.Adaptive mesh refinement(AMR)is also applied to improve the simulation accuracy automatically.Two refinement levels are conducted to verify the dominance of AMR in predicting cavitating flows.Results show that cavitation features,including the U-type structure of shedding clouds,are consistent with experimental observations.Even a coarse mesh can precisely capture the phase field without increasing the total cell number significantly using mesh adaption.The predicted shedding frequency agrees fairly well with the experimental data under refinement level 2.This study illustrates that AMR is a promising approach to achieve accurate simulations for multiscale cavitating flows within limited computational costs.Finally,the force element method is currently adopted to investigate the lift and drag fluctuations during the evolution of cavitation structure.The mechanisms of lift and drag fluctuations due to cavitation and the interaction between vorticity forces and cavitation are explicitly revealed.

An Adaptive Nonhydrostatic Atmospheric Dynamical Core Using a Multi-Moment Constrained Finite Volume Method

An adaptive 2 D nonhydrostatic dynamical core is proposed by using the multi-moment constrained finite-volume(MCV) scheme and the Berger-Oliger adaptive mesh refinement(AMR) algorithm. The MCV scheme takes several pointwise values within each computational cell as the predicted variables to build high-order schemes based on single-cell reconstruction. Two types of moments, such as the volume-integrated average(VIA) and point value(PV), are defined as constraint conditions to derive the updating formulations of the unknowns, and the constraint condition on VIA guarantees the rigorous conservation of the proposed model. In this study, the MCV scheme is implemented on a height-based, terrainfollowing grid with variable resolution to solve the nonhydrostatic governing equations of atmospheric dynamics. The AMR grid of Berger-Oliger consists of several groups of blocks with different resolutions, where the MCV model developed on a fixed structured mesh can be used directly. Numerical formulations are designed to implement the coarsefine interpolation and the flux correction for properly exchanging the solution information among different blocks. Widely used benchmark tests are carried out to evaluate the proposed model. The numerical experiments on uniform and AMR grids indicate that the adaptive model has promising potential for improving computational efficiency without losing accuracy.

Adaptive Finite Element Method for Steady Convection-Diffusion Equation

This paper examines the numerical solution of the convection-diffusion equation in 2-D. The solution of this equation possesses singularities in the form of boundary or interior layers due to non-smooth boundary conditions. To overcome such singularities arising from these critical regions, the adaptive finite element method is employed. This scheme is based on the streamline diffusion method combined with Neumann-type posteriori estimator. The effectiveness of this approach is illustrated by different examples with several numerical experiments.

Projection-Based Dimensional Reduction of Adaptively Refined Nonlinear Models

Adaptive mesh refinement (AMR) is fairly practiced in the context of high-dimensional, mesh-based computational models. However, it is in its infancy in that of low-dimensional, generalized-coordinate-based computational models such as projection-based reduced-order models. This paper presents a complete framework for projection-based model order reduction (PMOR) of nonlinear problems in the presence of AMR that builds on elements from existing methods and augments them with critical new contributions. In particular, it proposes an analytical algorithm for computing a pseudo-meshless inner product between adapted solution snapshots for the purpose of clustering and PMOR. It exploits hyperreduction—specifically, the energy-conserving sampling and weighting hyperreduction method—to deliver for nonlinear and/or parametric problems the desired computational gains. Most importantly, the proposed framework for PMOR in the presence of AMR capitalizes on the concept of state-local reduced-order bases to make the most of the notion of a supermesh, while achieving computational tractability. Its features are illustrated with CFD applications grounded in AMR and its significance is demonstrated by the reported wall-clock speedup factors.

Large eddy simulation of cavitating flows with dynamic adaptive mesh refinement using OpenFOAM

Cavitating flows are dominated by large gradients of physical properties and quantities containing complicated interfacial structures and lots of multi-scale eddies that need to be accurately characterized using a high-resolution mesh.The present work,within OpenFOAM,proposes an effective modeling framework using the large eddy simulation(LES)approach along with the volume of fluid(VOF)method to simulate the two-phase flow system and applies the Schnerr-Sauer model to calculate the mass-transfer rate between water and vapor.The adaptive mesh refinement(AMR)which is a powerful tool for allocating high-resolution grids only to the region of the greatest concern is adopted for improving the solution of interfacial structures.The effect of grid size is firstly investigated and the time-averaged quantities are verified against the experimental data,and then simulations of cavitating flows are successfully achieved to precisely characterize the features of cavitation with automatically and dynamically refining the mesh.As the refinement only takes place in the interfacial region,high-precision simulations can be achieved with limited computational resources,and the method shows promising prospects for modeling of the multi-scale,time-critical and computationally intensive cavitating flows.

High-order compact finite volume methods on unstructured grids with adaptive mesh refinement for solving inviscid and viscous flows

In the present paper, high-order finite volume schemes on unstructured grids developed in our previous papers are extended to solve three-dimensional inviscid and viscous flows. The highorder variational reconstruction technique in terms of compact stencil is improved to reduce local condition numbers. To further improve the efficiency of computation, the adaptive mesh refinement technique is implemented in the framework of high-order finite volume methods. Mesh refinement and coarsening criteria are chosen to be the indicators for certain flow structures. One important challenge of the adaptive mesh refinement technique on unstructured grids is the dynamic load balancing in parallel computation. To solve this problem, the open-source library p4 est based on the forest of octrees is adopted. Several two-and three-dimensional test cases are computed to verify the accuracy and robustness of the proposed numerical schemes.

Adaptive mesh refinement immersed boundary method for simulations of laminar flows past a moving thin elastic structure

One of the critical issues in numerical simulation of fluid-structure interaction problems is inaccuracy of the solutions,especially for flows past a stationary thin elastic structure where large deformations occur.High resolution is required to capture the flow characteristics near the fluid-structure interface to enhance accuracy of the solutions within proximity of the thin deformable body.Hence,in this work,an algorithm is developed to simulate fluid-structure interactions of moving deformable structures with very thin thicknesses.In this algorithm,adaptive mesh refinement(AMR)is integrated with immersed boundary finite element method(IBFEM)with two-stage pressure-velocity corrections.Despite successive interpolation of the flow field by IBM,the governing equations were solved using a fixed structured mesh,which significantly reduces the computational time associated with mesh reconstruction.The cut-cell IBM is used to predict the body forces while FEM is used to predict deformation of the thin elastic structure in order to integrate the motions of the fluid and solid at the interface.AMR is used to discretize the governing equations and obtain solutions that efficiently capture the thin boundary layer at the fluid-solid interface.The AMR-IBFEM algorithm is first verified by comparing the drag coefficient,lift coefficient,and Strouhal number for a benchmark case(laminar flow past a circular cylinder at Re=100)and the results showed good agreement with those of other researchers.The algorithm is then used to simulate 2-D laminar flows past stationary and moving thin structures positioned perpendicular to the freestream direction.The results also showed good agreement with those obtained from the arbitrary Lagrangian-Eulerian(ALE)algorithm for elastic thin boundaries.It is concluded that the AMR-IBFEM algorithm is capable of predicting the characteristics of laminar flow past an elastic structure with acceptable accuracy(error of-0.02%)with only-1%of the computational time for simulations with full mesh refinement.

界面捕捉Level Set方法的(AMR)数值模拟

在流体力学方程的计算中采用高精度WENO格式,用AMR(adaptive mesh refinement)方法提高流场局部分辨率,在采用Level Set函数标定物质界面的计算中用GFM(ghost fluid method)方法进行界面处理,尝试将AMR技术与界面追踪技术相互融合并应用于数值模拟,对不同的模拟结果进行了比较.

SAMR网格上扩散方程有限体格式的逼近性与两层网格算法

针对结构自适应加密网格(SAMR)上扩散方程的求解,分析几种有限体格式的逼近性,同时设计和分析一种两层网格算法.首先,讨论一种常见的守恒型有限体格式,并给出网格加密区域和细化/粗化插值算子的条件;接着,通过在粗细界面附近引入辅助三角形单元,消除粗细界面处的非协调单元,设计了一种保对称有限体元(SFVE)格式,分析表明,该格式具有更好的逼近性,且对网格加密区域和插值算子的限制更弱;最后,为SFVE格式构造一种两层网格(TL)算法,理论分析和数值实验表明该算法的一致收敛性.

An Adaptive Mesh Refinement Strategy for Immersed Boundary/Interface Methods

An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems involving singular sources.The interface is represented by the zero level set of a Lipschitz functionϕ(x,y).Our adaptive mesh refinement is done within a small tube of|ϕ(x,y)|δwith finer Cartesian meshes.The discrete linear system of equations is solved by a multigrid solver.The AMR methods could obtain solutions with accuracy that is similar to those on a uniform fine grid by distributing the mesh more economically,therefore,reduce the size of the linear system of the equations.Numerical examples presented show the efficiency of the grid refinement strategy.

使用AMR型背景网格的并行SPH方法

在分布式并行SPH方法中根据网格范围内粒子最大光滑长度调整网格尺度的AMR(Adaptive Mesh Refinement)背景网格,确保网格内粒子的相邻粒子位于同一个网格或者相邻的同尺寸网格范围内.与统一尺寸的背景网格相比,在光滑长度空间分布不均匀和随时间变化的情况下能够应用并减小搜索量,提高计算效率.数值结果表明,基于AMR型背景网格的并行SPH方法应用到粒子光滑长度变化很大的数值计算时能有效地降低搜索复杂度.

Multi-Material ALE with AMR for Modeling Hot Plasmas and Cold Fragmenting Materials

We have developed a new 3D multi-physics multi-material code, ALE-AMR, which combines Arbitrary Lagrangian Eulerian （ALE） hydrodynamics with Adaptive Mesh Refinement （AMR） to connect the continuum to the microstructural regimes. The code is unique in its ability to model hot radiating plasmas and cold fragmenting solids. New numerical techniques were developed for many of the physics packages to work efficiently on a dynamically moving and adapting mesh. We use interface reconstruction based on volume fractions of the material components within mixed zones and reconstruct interfaces as needed. This interface reconstruction model is also used for void coalescence and fragmentation. A flexible strength/failure framework allows for pluggable material models, which may require material history arrays to determine the level of accumulated damage or the evolving yield stress in J2 plasticity models. For some applications laser rays are propagating through a virtual composite mesh consisting of the finest resolution representation of the modeled space. A new 2nd order accurate diffusion solver has been implemented for the thermal conduction and radiation transport packages. One application area is the modeling of laser/target effects including debris/shrapnel generation. Other application areas include warm dense matter, EUV lithography, and material wall interactions for fusion devices.