并行有限元网格生成方法及其应用

大型工程数值仿真中,在前处理阶段需要生成千万甚至亿量级的网格,传统的串行网格生成方法由于内存和时间的限制,难以处理如此规模的网格。针对此问题,本文提出了一种大规模网格并行生成方法。首先基于推进波前法对几何模型进行初始体网格划分,接着利用图论理论进行区域分解,并通过表面单元恢复保持其几何精度,然后通过分裂法进行网格的并行生成。将所述方法应用到实际大型工程数值仿真前处理阶段,结果表明所述方法可以获得较好的并行效率,同时所产生的网格质量可以满足后续计算需要。     

Self-adaptive strategy for one-dimensional finite element method based on EEP method with optimal super-convergence order

Based on the newly-developed element energy projection （EEP） method with optimal super-convergence order for computation of super-convergent results, an improved self-adaptive strategy for one-dimensional finite element method （FEM） is proposed. In the strategy, a posteriori errors are estimated by comparing FEM solutions to EEP super-convergent solutions with optimal order of super-convergence, meshes are refined by using the error-averaging method. Quasi-FEM solutions are used to replace the true FEM solutions in the adaptive process. This strategy has been found to be simple, clear, efficient and reliable. For most problems, only one adaptive step is needed to produce the required FEM solutions which pointwise satisfy the user specified error tolerances in the max-norm. Taking the elliptical ordinary differential equation of the second order as the model problem, this paper describes the fundamental idea, implementation strategy and computational algorithm and representative numerical examples are given to show the effectiveness and reliability of the proposed approach.     

A multiscale control volume finite element method for advection–diffusion equations

We present a new stabilized method for advection–diffusion equations, which combines a control volume FEM formulation of the governing equations with a novel multiscale approximation of the total flux. The latter incorporates information about the exact solution that cannot be represented on the mesh. To define this flux, we solve the governing equations along suitable mesh segments under the assumption that the flux varies linearly along these segments. This procedure yields second‐order accurate fluxes on the edges of the mesh. Then, we use curl‐conforming elements of the same order to lift these edge fluxes into the mesh elements. In so doing, we obtain a stabilized control volume FEM formulation that is second‐order accurate and does not require mesh‐dependent stabilization parameters. Numerical convergence studies on uniform and nonuniform grids along with several standard advection tests illustrate the computational properties of the new method. Published 2015. This article is a U.S. Government work and is in the public domain in the USA.     

广义节点有限元法

应用流形方法思想,通过引入广义节点的概念,对传统有限元方法进行改进,建立了可具有任意高阶多项式托值函数的广义节点有限元方法,计算结果表明,广义节点有限元方法较之传统有限元方法有较高的精度。     

A Hermite finite element method for incompressible fluid flow

We describe some Hermite stream function and velocity finite elements and a divergence‐free finite element method for the computation of incompressible flow. Divergence‐free velocity bases defined on (but not limited to) rectangles are presented, which produce pointwise divergence‐free flow fields (∇· u _h≡0). The discrete velocity satisfies a flow equation that does not involve pressure. The pressure can be recovered as a function of the velocity if needed. The method is formulated in primitive variables and applied to the stationary lid‐driven cavity and backward‐facing step test problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Some improvements on free surface simulation by the particle finite element method

The Lagrangian method has become increasingly popular in numerical simulation of free surface problems. In this paper, after a brief review of a recent Lagrangian method, namely the particle finite element method, some issues are discussed and some improvements are made. The least‐square finite element method is adopted to simplify the solving of the Navier–Stokes equations. An adaptive time method is derived to obtain suitable time steps. A mass correction procedure is imported to improve the mass conservation in long time calculations and time discretization scheme is adopted to decrease the pressure oscillations during the calculations. Finally, the method is used to simulate a series of examples and the results are compared with the commercial FLOW3D code. Copyright © 2008 John Wiley & Sons, Ltd.     

Numeric simulation of pollutant dispersion by a control‐volume based on finite element method

The dispersion of pollutants in the environment is an issue of great interest as it directly affects air quality, mainly in large cities. Experimental and numerical tools have been used to predict the behavior of pollutant species dispersion in the atmosphere. A software has been developed based on the control‐volume based on the finite element method in order to obtain two‐dimensional simulations of Navier–Stokes equations and heat or mass transportation in regions with obstacles, varying position of the pollutant source. Numeric results of some applications were obtained and, whenever possible, compared with literature results showing satisfactory accordance. Copyright © 2010 John Wiley & Sons, Ltd.     

基于区域分解的并行有限元程序及其在水下爆炸中的应用

基于MPI标准定义的消息传递接口实现了显式动力学有限元程序的并行计算.通过基于Hilbert空间填充曲线的区域分解算法,实现各区域的独立运算和区域之间共享数据的相互通信.利用程序对水下爆炸自由场中的冲击波传播规律进行了数值模拟,并通过对不同进程数下的并行加速比进行测试,验证了程序的并行效率.     

A control volume finite element method for three-dimensional three-phase flows

A novel control volume finite element method with adaptive anisotropic unstructured meshes is presented for three-dimensional three-phase flows with interfacial tension. The numerical framework consists of a mixed control volume and finite element formulation with a new P₁DG-P₂ elements (linear discontinuous velocity between elements and quadratic continuous pressure between elements). A “volume of fluid” type method is used for the interface capturing, which is based on compressive control volume advection and second-order finite element methods. A force-balanced continuum surface force model is employed for the interfacial tension on unstructured meshes. The interfacial tension coefficient decomposition method is also used to deal with interfacial tension pairings between different phases. Numerical examples of benchmark tests and the dynamics of three-dimensional three-phase rising bubble, and droplet impact are presented. The results are compared with the analytical solutions and previously published experimental data, demonstrating the capability of the present method.     

基于形体中轴的自适应有限元模糊控制算法

空间任意形体均可由形体中轴和中轴半径重构。本文在任意区域的形体中轴生成的基础上，应用中轴半径函数及其变化，建立了有限元网格生成的h型自适应模糊控制。应用结果表明，本算法可大大提高了自适应有限元计算效率和应用范围。     

The optimal control variational principle and finite elements analysis for viscoplastic dynamics

This paper presents the optimal control variational principle for Perzyna model which is one of the main constitutive relation of viscoplasticity in dynamics. And it could also be transformed to solve the parametric quadratic programming problem. The FEM form of this problem and its implementation have also been discussed in the paper.     

Parallel large eddy simulation of turbulent flow around MIRA model using linear equal‐order finite element method

A parallel large eddy simulation code that adopts domain decomposition method has been developed for large‐scale computation of turbulent flows around an arbitrarily shaped body. For the temporal integration of the unsteady incompressible Navier–Stokes equation, fractional 4‐step splitting algorithm is adopted, and for the modelling of small eddies in turbulent flows, the Smagorinsky model is used. For the parallelization of the code, METIS and Message Passing Interface Libraries are used, respectively, to partition the computational domain and to communicate data between processors. To validate the parallel architecture and to estimate its performance, a three‐dimensional laminar driven cavity flow inside a cubical enclosure has been solved. To validate the turbulence calculation, the turbulent channel flows at Re_τ = 180 and 1050 are simulated and compared with previous results. Then, a backward facing step flow is solved and compared with a DNS result for overall code validation. Finally, the turbulent flow around MIRA model at Re = 2.6 × 10⁶ is simulated by using approximately 6.7 million nodes. Scalability curve obtained from this simulation shows that scalable results are obtained. The calculated drag coefficient agrees better with the experimental result than those previously obtained by using two‐equation turbulence models. Copyright © 2007 John Wiley & Sons, Ltd.     

A parallel two-level finite element method for the Navier-Stokes equations

Based on domain decomposition, a parallel two-level finite element method for the stationary Navier-Stokes equations is proposed and analyzed. The basic idea of the method is first to solve the Navier-Stokes equations on a coarse grid, then to solve the resulted residual equations in parallel on a fine grid. This method has low communication complexity. It can be implemented easily. By local a priori error estimate for finite element discretizations, error bounds of the approximate solution are derived. Numerical results are also given to illustrate the high efficiency of the method.     

Adaptive finite element method for an optimal control problem of Stokes flow with L2‐norm state constraint

An adaptive finite element approximation for an optimal control problem of the Stokes flow with an L²‐norm state constraint is proposed. To produce good adaptive meshes, the a posteriori error estimates are discussed. The equivalent residual‐type a posteriori error estimators of the H ¹‐error of state and L²‐error of control are given, which are suitable to carry out the adaptive multi‐mesh finite element approximation. Some numerical experiments are performed to illustrate the efficiency of the a posteriori estimators. Copyright © 2011 John Wiley & Sons, Ltd.     

无穷域势流问题的有限元／差分线法混合求解

提出了一种将有限元和差分线法相结合求解无穷域势流问题的算法。用两同心圆将求解域划分为存在重叠的有限和无限两个区域,在有限和无限域上分别用有限元和差分线法求解Laplace方程边值问题。用差分线法推导出的关系式修正有限元方程,求解该方程组从而得到原问题的解。本算法将求解无穷域问题转化为代数特征值问题和有限域内线性方程组的...     

基于等几何分析的比例边界有限元方法

提出了一种具有比例边界有限元的半解析特性和等几何分析的几何特性的新方法。该新方法是在比例边界有限元框架中用NURBS曲线或曲面精确描述域边界几何形状,同时域边界位移场采用描述几何形状的NURBS形函数等参构造。这种新方法具有比例边界有限元固有的径向解析特性和NURBS的高阶连续性的优点。数值算例显示,与传统的比例边界有限元相比,基于等几何分析的比例边界有限元方法提高了域边界单元和域内应力场的连续性,减少了计算自由度。应用此方法可以用较少的计算自由度获得更高连续阶和更高精度的位移、应力和应变场。     

Adaptive non‐conformal mesh refinement and extended finite element method for viscous flow inside complex moving geometries

Two different techniques to analyze non‐Newtonian viscous flow in complex geometries with internal moving parts and narrow gaps are compared. The first technique is a non‐conforming mesh refinement approach based on the fictitious domain method (FDM), and the second one is the extended finite element method (XFEM). The refinement technique uses one fixed reference mesh, and to impose continuity across non‐conforming regions, constraints using Lagrangian multipliers are used. The size of elements locally in the high shear rate regions is reduced to increase accuracy. FDM is shown to have limitations; therefore, XFEM is applied to decouple the fluid from the internal moving rigid bodies. In XFEM, the discontinuous field variables are captured by using virtual degrees of freedom that serve as enrichment and by applying special integration over the intersected elements. The accuracy of the two methods is demonstrated by direct comparison with results of a boundary‐fitted mesh applied to a two‐dimensional cross section of a twin‐screw extruder. Compared with non‐conforming FDM, XFEM shows a considerable improvement in accuracy around the rigid body, especially in the narrow gap regions. Copyright © 2011 John Wiley & Sons, Ltd.     

Computer simulation of tensile deformation of aluminium bicrystalline specimen by means of elasto-plastic finite element method of crystal

Based on the experimental data of strain-stress relation of single crystal and the complexity of polycrystal deformation, a calculating model is proposed and the calculating program is designed. Finally, the tensile deformation of the bicrystal aluminium specimen is simulated by means of the crystal elasto-plastic finite element method. The strain-stress curve of tension, the details of the slip deformation and the stress distribution are obtained. So the effect of the region near grain boundary is shown obviously.     

The Runge–Kutta control volume discontinuous finite element method for systems of hyperbolic conservation laws

In this paper, a new high‐order and high‐resolution method called the Runge–Kutta control volume discontinuous finite element method (RKCVDFEM) was proposed to solve 1D and 2D systems of hyperbolic conservation laws. Its main advantage lies in the local conservation, and it is simpler than the Runge–Kutta discontinuous Galerkin finite element method (RKDGM). The theoretical analysis showed that the RKCVDFEM has formally an optimal convergence order for 1D systems. Based on logically rectangular grids of irregular quadrilaterals, a scheme for 2D systems was constructed. Some classical problems were simulated and the validity of the method was presented. Copyright © 2010 John Wiley & Sons, Ltd.