Free vibration and dynamic response analysis by Petrov-Galerkin natural element method

In this paper, a Petrov-Galerkin natural element method (PG-NEM) based upon the natural neighbor concept is presented for the free vibration and dynamic response analyses of twodimensional linear elastic structures. A problem domain is discretized with a finite number of nodes and the trial basis functions are defined with the help of the Voronoi diagram. Meanwhile, the test basis functions are supported by Delaunay triangles for the accurate and easy numerical integration with the conventional Gauss quadrature rule. The numerical accuracy and stability of the proposed method are verified through illustrative numerical tests.     

聚合物全三维非稳态非等温多相分层流动成型过程的理论模型和数值模拟

基于聚合物多组分成型技术的工程背景,建立了全三维非稳态非等温多相分层充模流动的理论模型。在综合分析了该理论模型产生数值解的不稳定及发散的主要原因基础上,提出了求解理论模型的稳定快速收敛的数值算法。采用罚函数法,以及速度场分析、温度场分析和流体体积分数分析相分离等方法来降低对计算机的CPU和存储能力的需求,而通过SUPG法、罚函数法和P1+/P1三维单元实现有限元数值分析的稳定性。并基于罚函数法和SUPG法,推导出求解N-S方程、能量方程和全三维多相分层流动成型移动前沿界面和分层界面追踪方程的有限元数值模型,并探讨了全三维非稳态非等温多相分层流动移动前沿界面和分层界面重构技术。最后揭示了粘性包围和界面不稳定的产生机理。     

基于无单元伽辽金法和Matlab遗传算法工具箱的结构形状优化研究

将无网格Galerkin法(element-free Galerkin method,EFGM)和改进的Matlab遗传算法工具箱(matlab genetic toolbox,MGT)相结合,提出了一种新的连续体结构形状优化设计方法。针对形状优化设计的特点,建立了一种规则网格与有限元网格混合的EFG积分背景网格法。详细介绍了M函数文件的修改方法,解决了现有Matlab遗传算法工具箱不能直接施加的非线性与隐性约束条件的难题。最后,完成了一个工程实例的结构形状优化,优化结果表明该方法是有效可行的。     

Bending analysis of mindlin-reissner plates by the element free galerkin method with penalty technique

In this work, a new penalty formulation is proposed for the analysis of Mindlin-Reissner plates by using the element-free Galerkin method. A penalized weak form for the Mindlin-Reissner Plates is constructed through the exterior penalty method to enforce the essential boundary conditions of rotations as well as transverse displacements. In the numerical examples, some typical problems of Mindlin-Reissner plates are analyzed, and parametric studies on the order of integration and the size of influence domain are also carried out. The effect of the types of background cells on the accuracy of numerical solutions is observed and a proper type of background cell for obtaining optimal accuracy is suggested. Further, optimal order of integration and basis order of Moving Least Squares approximation are suggested to efficiently handle the irregularly distributed nodes through the triangular type of background cells. From the numerical tests, it is identified that unlike the finite element method, the proposed element-free Galerkin method with penalty technique gives highly accurate solution without shear locking in dealing with Mindlin-Reissner plates.     

几何非线性问题的无网格法分析

用局部Petrov-Galerkin方法求解几何非线性问题,这是一种真正的无网格方法。这种方法采用移动最小二乘近似函数作为试函数;只包含巾心在所考虑点处的规则局部区域上以及局部边界上的积分;所得系统矩阵是一个带状稀疏矩阵。该方法可以容易推广到求解非线性问题以及非均匀介质力学问题。在涉及几何非线性问题的数值方法中,通常都采用增量和迭代分析的方法。本文从虚功原理出发,用移动最小二乘近似函数的权函数替代虚位移,并在整个分析过程中所有变量的表达格式都是采用全拉格朗日格式。数值算例表明,无网格局部Petrov-Galerkin方法在求解几何非线性问题时仍具有很好的精度。     

透平机械叶轮叶片三维参数化造型及六面体网格生成方法研究

叶轮是向心透平和离心压缩机中的重要部件,其几何形状比较复杂,采用良好的网格进行有限元分析,以获得精确的强度和振动特性,是其安全性的重要保证.针对叶轮结构特点,采用CAGD (computer aided geometric design) 技术实现叶轮的三维实体参数化造型;然后使用同胚原理和BMSweeping(sweeping via background mesh interpolation)等方法建立叶轮六面体网格生成方法.为了保证有限元分析的精确性,建立叶轮上叶片根部倒角、轮盘气道进口处等特殊结构的网格生成方法.100 kW微型燃气轮机的向心叶轮网格划分表明, 使用上述方法生成的网格质量较高,能很好地满足有限元分析要求.     

Numerical Method Based on Compatible Manifold Element for Thin Plate Bending

The typical quadrangular and triangular elements for thin plate bending based on Kirchhoff assumptions are the nonconforming elements with low computational accuracy and limitative application range in finite element method(FEM). Some compatible elements can be developed by the means of supplementing correction functions, increasing nodes in element or on the boundaries, expanding nodal degrees of freedom(DOF), etc, but these elements are inconvenient to apply in practice for the high calculation complexity. In this paper, in order to overcome the defects of thin plate bending finite element, numerical manifold method(NMM) was introduced to solve thin plate bending deformation problem. Rectangular mesh was adopted as mathematical mesh to form finite element cover system, and then 16-cover manifold element was proposed. Numerical manifold formulas were constructed on the basis of minimum potential energy principle, displacement boundary conditions are implemented by penalty function method, and all the element matrixes were derived in details. The 16-cover element has a simple calculation process for employing only the transverse displacement cover DOFs as the basic unknown variables, and has been proved to meet the requirements of completeness and full compatibility. As an application, the presented 16-cover element has been used to analyze bending deformation of square thin plate under different loads and boundary conditions, and the results show that numerical manifold method with compatible element, compared with finite element method, can improve computational accuracy and convergence greatly.     

局部到整体的六面体网格质量优化算法

网格质量是影响有限元分析结果准确性和可靠性的关键因素之一。为了提高六面体网格的质量,提出一种局部到整体的六面体网格质量优化算法。针对每个六面体单元采用双八面体几何变换进行局部规则化操作,得到高质量的规则六面体网格。在此基础上,通过迭代优化一个二次能量函数,对规则化后的网格进行整体拼接和优化,该能量函数由节点内力和表面几何约束组成,节点内力保证了规则化后六面体单元的连接性,表面几何约束保持了原始模型的几何形状。应用实例表明,该方法稳定可靠,显著提高了六面体网格的质量。     

Stress recovery techniques for natural element method in 2-D solid mechanics

This paper is concerned with the stress recovery for the natural element method in which the problem domain is discretized with Delaunay triangles and the structural behavior is approximated with Laplace interpolation functions. Basically, the global and local patch recovery techniques based on the L ₂-projection method are adopted. For the local patch recovery, the local element patches are defined by the supports of each Laplace interpolation function. For the comparison purpose, the local stress recovery is also performed using Lagrange-type basis functions that are used for 3- and 6-node triangular elements. The stresses that are recovered by the present global and local recovery techniques are compared each other and compared with the available analytic solution, in terms of their spatial distributions and the convergence rates. As well, the dependence of the recovered stress field on the type of test basis functions that are used forbnov-Galerkin (BG) and Petrov-Galerkin (PG) natural element methods is also investigated.

     

Development of a criterion for efficient numerical calculation of structural vibration responses

The finite element method is one of the methods widely applied for predicting vibration in mechanical structures. In this paper, the effect of the mesh size of the finite element model on the accuracy of the numerical solutions of the structural vibration problems is investigated with particular focus on obtaining the optimal mesh size with respect to the solution accuracy and computational cost. The vibration response parameters of the natural frequency, modal density, and driving point mobility are discussed. For accurate driving point mobility calculation, the decay method is employed to experimentally determine the internal damping. A uniform plate simply supported at four corners is examined in detail, in which the response parameters are calculated by constructing finite element models with different mesh sizes. The accuracy of the finite element solutions of these parameters is evaluated by comparing with the analytical results as well as estimations based on the statistical energy analysis, or if not available, by testing the numerical convergence. As the mesh size becomes smaller than one quarter of the wavelength of the highest frequency of interest, the solution accuracy improvement is found to be negligible, while the computational cost rapidly increases. For mechanical structures, the finite element analysis with the mesh size of the order of quarter wavelength, combined with the use of the decay method for obtaining internal damping, is found to provide satisfactory predictions for vibration responses.     

An adaptive mesh-independent numerical integration for meshless local petrov-galerkin method

In this paper, an adaptive numerical integration scheme, which does not need non-overlapping and contiguous integration meshes, is proposed for the MLPG (Meshless Local Petrov-Galerkin) method. In the proposed algorithm, the integration points are located between the neighboring nodes to properly consider the irregular nodal distribution, and the nodal points are also included as integration points. For numerical integration without well-defined meshes, the Shepard shape function is adopted to approximate the integrand in the local symmetric weak form, by the values of the integrand at the integration points. This procedure makes it possible to integrate the local symmetric weak form without any integration meshes (non-overlapping and contiguous integration domains). The convergence tests are performed, to investigate the present scheme and several numerical examples are analyzed by using the proposed scheme.     

OBJECT-ORIENTED NUMERICAL MANIFOLD METHOD

0INTRODUCnONNumericalmanifoldmelhod(NMM)isanewlydevelopedgeneralnumericalmethodbaseduPOnmathematicalmethod.FortheNMM,bothcontinuumanddiscontinuitiesarenatural.SothismethodisageneralnumericalmethodforsolvingtheproblemsthatcanbecalculatedbyaaM,DDAoranalyticmethod.ThismethodhasbeenaPPliedtorockengineeringwidely['~SJ.TOstudyNMMandPOPulallzethetwo-dimensionalproblemstothree-dimensionalproblemsconveniently,weintroducedtheObject-orientedPro~ng(OOP)intotheprogramdesignOftheNMManddetai…     

OBJECTORIENTED NUMERICAL MANIFOLD METHOD

The design and management of the objects about the numerical manifold method are studied by abstracting the finite cover system of numerical manifold method as independent data classes and the theoretical basis for the researching and expanding of numerical manifold method is also put forward. The Hammer integration of triangular area coordinates is used in the integration of the element. The calculation result shows that the program is accuracy and effective.     

A simple nodal force distribution method in refined finite element meshes

In finite element analyses, mesh refinement is frequently performed to obtain accurate stress or strain values or to accurately define the geometry. After mesh refinement, equivalent nodal forces should be calculated at the nodes in the refined mesh. If field variables and material properties are available at the integration points in each element, then the accurate equivalent nodal forces can be calculated using an adequate numerical integration. However, in certain circumstances, equivalent nodal forces cannot be calculated because field variable data are not available. In this study, a very simple nodal force distribution method was proposed. Nodal forces of the original finite element mesh are distributed to the nodes of refined meshes to satisfy the equilibrium conditions. The effect of element size should also be considered in determining the magnitude of the distributing nodal forces. A program was developed based on the proposed method, and several example problems were solved to verify the accuracy and effectiveness of the proposed method. From the results, accurate stress field can be recognized to be obtained from refined meshes using the proposed nodal force distribution method. In example problems, the difference between the obtained maximum stress and target stress value was less than 6 % in models with 8-node hexahedral elements and less than 1 % in models with 20-node hexahedral elements or 10-node tetrahedral elements.     

Three dimension profile extrusion simulation using meshfree method

A meshfree method based on reproducing kernel approximation and point collocation is presented for analysis of 3D profile extrusion processes. The point collocation method is a true meshfree method without the employment of a background mesh. We show that, in a point collocation approach, the implementation of program does not need the background mesh, which is a very time consuming process in a Galerkin method for integration. Among the points of the meshfree model, there is no connecting information. A mesh quality control method with mold interpenetration checking based on the Delaunay Bowyer-Watson algorithm is introduced to produce the topological relations, which enable us to show meshfree simulation results using the same procedure as in the finite element method. A C-shape profile extrusion application example is presented and compared with the finite element method to verify the proposed method.     

基于PG原理的三维注射成形流动数值模拟

采用PSPG法和SUPG法，分别在动量方程、连续性方程和能量方程的权函数中加入摄动因子以消除振荡，建立了速度和压力同次插值的三维流体流动控制方程的稳定有限元求解格式。通过比较该数值模拟方法与现有的商业分析软件Moldflow的计算结果表明，基于PG原理的三维注射成形流动数值模拟软件具有良好的精度和稳定性。     

Combined Streamline Upwind Petrov Galerkin method and segregated finite element algorithm for conjugate heat transfer problems

A combined Streamline Upwind Petrov-Galerkin method (SUPG) and segregated finite element algorithm for solving conjugate heat transfer problems where heat conduction in a solid is coupled with heat convection in viscous fluid flow is presented. The Streamline Upwind Petrov-Galerkin method is used for the analysis of viscous thermal flow in the fluid region, while the analysis of heat conduction in solid region is performed by the Galerkin method. The method uses the three-node triangular element with equal-order interpolation functions for all the variables of the velocity components, the pressure and the temperature. The main advantage of the presented method is to consistently couple heat transfer along the fluid-solid interface. Four test cases, which are the conjugate Couette flow problem in parallel plate channel, the counter-flow in heat exchanger, the conjugate natural convection in a square cavity with a conducting wall, and the conjugate natural convection and conduction from heated cylinder in square cavity, are selected to evaluate efficiency of the presented method.     

A second-order time-accurate finite element method for analysis of conjugate heat transfer between solid and unsteady viscous flow

A fractional four-step finite element method for analyzing conjugate heat transfer between solid and unsteady viscous flow is presented. The second-order semi-implicit Crank-Nicolson scheme is used for time integration and the resulting nonlinear equations are linearized without losing the overall time accuracy. The streamline upwind Petrov-Galerkin method (SUPG) is applied for the weighted formulation of the Navier-Stokes equations. The method uses a three-node triangular element with equal-order interpolation functions for all the variables of the velocity components, the pressure and the temperature. The main advantage of the method presented is to consistently couple heat transfer along the fluid-solid interface. Five test cases, which are the lid-driven cavity flow, natural convection in a square cavity, transient flow over a heated circular cylinder, forced convection cooling across rectangular blocks, and conjugate natural convection in a square cavity with a conducting wall, are selected to evaluate the efficiency of the method presented. This paper was recommended for publication in revised form by Associate Editor Kyung-Soo Yang Atipong Malatip received his B.S. degree in Mechanical Engineering from King Mongkut’s University of Technology North Bangkok, Thailand, in 2002. He then received his M.S. degree in Mechanical Engineering Chulalongkorn University, Thailand, in 2005. He is currently pursuing a Ph.D. degree in Mechanical Engineering at Chulalongkorn University. His research interests include computational fluid dynamics and fluid-thermal-structural interaction. Niphon Wansophark received his B.S., M.S., and Ph.D. degrees in Mechanical Engineering from Chulalongkorn University, Thailand in 1996, 2000, and 2007, respectively. He is an Assistant Professor of Mechanical Engineering at Chulalongkorn University, Bangkok, Thailand. His research interests are numerical methods and finite element method. Pramote Dechaumphai received his B.S. degree in Industrial Engineering from Khon-Kaen University, Thailand, in 1974, M.S. degree in Mechanical Engineering from Youngstown State University, USA in 1977, and Ph.D. in Mechanical Engineering from Old Dominion University, USA in 1982. He is currently a Professor of Mechanical Engineering at Chula-longkorn University, Bangkok, Thailand. His research interests are numerical methods, finite element method for thermal stress and computational fluid dynamics analysis.     

二维声学数值计算的径向插值有限元法

针对声学有限元分析中四节点等参单元计算精度低,对网格质量敏感的问题,将无网格径向插值技术引入到标准有限元中,构造径向插值形函数,推导径向插值有限元法(Radial interpolation finite element method,RIFEM)的二维声学数值计算公式。二维声学RIFEM采用标准有限元法形函数构造系统离散方程的声学刚度矩阵和边界积分矢量,保证了声压梯度和边界条件在区域边界的积分精度;采用径向插值形函数构造系统离散方程的质量矩阵,提高了声压数值近似函数的插值精度。对管道二维声腔模型和某轿车二维声腔模型的数值分析结果表明,与标准有限元法和SFEM相比,RIFEM的计算精度更高,对波数、单元尺寸和网格扭曲程度的灵敏度更低。因此RIFEM可以很好地应用于二维声学数值分析,具有广阔的工程应用前景。     

成形充填过程的任意拉格朗日-欧拉有限元与无网格自适应耦合模拟

在任意拉格朗日—欧拉(Arbitrary Lagrangian-Eulerian,ALE)方法框架中发展用于成形充填过程数值模拟的有限元与无网格自适应耦合方案。该方案自适应地在网格发生扭曲的区域采用无网格法空间离散以保证求解的精度和稳定性,而在网格质量较好的区域以及本质边界上保留使用有限元法空间离散以提高计算效率和便于施加本质边界条件,从而在充分发掘有限元法和无网格法各自优点的同时避开它们各自的缺点,在一定程度上保证该方法同时具有较快的计算速度和较好的健壮性。采用ALE参考构形描写充填过程中的熔体质量运动及守恒,可在不频繁变动网格的前提下准确跟踪自由面。应用压力稳定型分步算法求解控制方程,可方便采用不满足Ladyzhenskaja-Babuska-Brezzi(LBB)条件的速度、压力的同低阶插值。进行两个典型算例的数值模拟,结果表明该方法相对于单一的有限元法和无网格法的优越性,以及对含自由面变质量体系流动问题数值模拟的有效性。