三维金属体积成形过程无网格伽辽金方法及其关键技术研究
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摘要
金属体积成形是一种少、无切削的金属加工方法,在现代生产中占有极其重要的地位。对金属体积成形过程进行准确的数值模拟不但可以节省昂贵的实验费用,而且对合理确定成形工艺、保证模具设计的一次性成功具有重要的理论指导意义和工程实用价值。随着数值计算方法和计算机技术的发展,数值模拟分析方法已经成为解决工程问题的重要方法。其中有限元方法(FEM)经过不断的发展成熟,在金属体积成形过程数值模拟中发挥着重要的作用。但在锻造、挤压等大变形成形过程中,金属材料的流动异常复杂,工件变形程度非常大,FEM在处理此类问题时常常面临网格发生畸变而必须进行频繁网格重划分的问题,网格重划分势必造成计算精度和计算效率的下降,甚至导致分析过程无法进行。
无网格方法是近年来兴起的一种新的数值方法,它基于一系列离散节点进行近似,不需要节点的连接信息,克服了有限元法对网格的依赖,而且提供了连续性好、形式灵活的场函数,为大变形成形过程的数值模拟提供了一种有效途径。众多学者开始致力于无网格方法在金属体积成形过程中的应用研究,推动了金属体积成形过程无网格方法数值模拟技术的发展。但对三维非稳态金属体积成形过程的无网格方法数值模拟的研究还不完善,很多关键应用技术有待于进一步的研究。
本文针对金属体积成形过程的特点和工艺要求,提出刚(粘)塑性无网格伽辽金方法(RVPEFGM),并将其应用于三维非稳态金属体积成形过程的数值模拟分析。在对金属体积成形过程刚(粘)塑性无网格伽辽金方法和数值模拟关键技术深入研究的基础上,开发数值模拟分析程序,以期为金属成形过程分析提供一种科学的无网格方法仿真技术工具。
对三维无网格伽辽金方法(EFGM)开展了相关研究。当采用移动最小二乘(MLS)近似方案构造形函数时,会涉及到A~(-1)(x)矩阵的计算,研究了在三维问题中造成A(x)矩阵不可逆的原因,给出了A(x)矩阵不可逆的解决方法,并采用带权正交函数作为基函数,从而避免了对A(x)矩阵的求逆计算。节点影响域尺寸的大小对EFGM的计算精度和计算效率有很大的影响,影响域尺寸的确定要遵循“足够大又尽量小”的原则,为合理确定影响域尺寸,采用动态节点影响域方法,在保证EFGM的计算精度的同时兼顾计算效率。MLS形函数不具备插值性质,为方便EFGM本质边界条件的处理,采用混合变换法对MLS形函数进行局部修正,实现了EFGM本质边界条件的精确施加。研究了三维无网格伽辽金方法的具体实现过程,给出了规则六面体固定背景网格积分方案的实施方法和EFGM的程序设计步骤。研究了基函数、权函数、高斯积分阶次和节点密度等因素对EFGM的计算精度和效率的影响,得到了这些因素对计算精度和效率的影响规律。
将无网格伽辽金方法与刚(粘)塑性流动理论相结合,建立了刚(粘)塑性无网格伽辽金方法(RVPEFGM),并将其应用于三维金属体积成形工艺过程的模拟。基于速度场函数的MLS近似离散形式,推导建立了三维金属体积成形问题中应变速率向量、等效应变速率、体积应变速率的离散格式。采用罚函数法施加体积不变条件,根据不完全广义变分原理,给出系统能量速率泛函的组成,推导出能量速率泛函各组成部分的变分,并获得其矩阵表示形式。研究模具和工件间的摩擦边界条件的处理方法,采用反正切摩擦模型描述摩擦接触边界条件,实现了任意形状模具的三维金属体积成形过程模拟分析中摩擦边界条件在三维局部坐标系下的正确施加,推导建立了三维金属体积成形问题RVPEFGM的刚度方程,在局部坐标系下总装刚度方程,采用Newton-Raphson迭代法对刚度方程进行求解。完善了刚(粘)塑性无网格伽辽金方法基础理论研究,给出了刚(粘)塑性无网格伽辽金方法分析三维金属体积成形过程的模拟分析步骤和程序流程。
实际生产中,大多数金属体积成形工艺过程属于三维非稳态大变形金属塑性成形问题,其成形过程极其复杂,为把RVPEFGM应用于实际生产,并实现分析程序的通用性和自动化,进而发挥无网格方法处理金属体积成形问题的优势。本文结合金属体积成形过程的特点和工艺要求,对三维金属体积成形过程无网格伽辽金方法数值模拟的关键技术进行了深入研究。解决了诸如模具型腔的几何信息描述、接触节点局部坐标系的建立、初始速度场的确定、速度场的收敛性及控制技术、刚性区的处理、体积闭锁问题的处理、工件边界节点与模具的动态接触与脱离的自动处理、区域及边界积分的实施方案等若干关键技术。
采用有限元网格方法描述模具型腔,建立了读取ASCII码格式的STL数据的接口程序。提出了边界接触节点局部坐标系的建立方法,并给出了整体坐标与局部坐标的转换矩阵。金属成形过程中存在着不产生塑性变形的刚性区,为避免刚性区导致的数值计算问题,给出了工件变形过程中刚性区的处理方法。初始速度场的选取直接决定着Newton-Raphson迭代的收敛性和收敛速度,采用直接迭代法生成初始速度场,能够改善Newton-Raphson迭代的收敛性。采用“急降缓升”方法搜索迭代收敛方向,进而确定衰减因子β的取值,提高了Newton-Raphson迭代的收敛速度。针对三维金属体积成形过程RVPEFGM分析中的体积闭锁和压力震荡问题,提出了体积应变率映射法,通过将体积应变率(?)_v映射到低维空间,建立了体积闭锁现象的缓解算法。重点研究了工件边界节点与模具表面接触和脱离的自动处理技术,形成了动态边界自动识别技术的具体算法。提出了根据接触节点到模具型腔三角形面片距离最小的原则进行接触节点位置修正的方法,有效解决了当采用自接触节点到模具型腔三角形面片做垂线进行修正接触节点位置时遇到的“法矢盲区”问题。针对金属体积成形的特点,将单位分解积分法引入到三维金属体积成形RVPEFGM数值积分计算中,解决了使用固定背景网格时因高斯积分点数量减少而致使模拟分析精度下降的问题。
在上述刚(粘)塑性无网格伽辽金方法理论和关键技术研究的基础上,采用C++语言,自主开发了三维金属体积成形过程刚(粘)塑性无网格伽辽金方法数值模拟分析程序,实现了任意形状模具的三维非稳态大变形金属体积成形过程的RVPEFGM分析,并对锻造、挤压等工艺进行了数值模拟分析,通过与有限元软件分析结果和实验结果的比较,验证了所提出的方法及处理技术的正确性。同时在分析对比、实验验证、算例检验的基础上,不断修正模型和完善、维护程序,确保了建立的刚(粘)塑性无网格伽辽金方法的基本理论以及开发的数值模拟程序的准确性、可靠性。
采用自主开发的三维刚(粘)塑性无网格伽辽金方法分析程序,对圆形挤压件等通道弯角挤压过程和转向节预成形件热模锻过程进行了系统模拟分析,获得了相关力学分析结果,对金属流动规律、工件等效应力和等效应变的分布以及载荷力的变化等进行了研究,为掌握等通道弯角挤压机理以及转向节预成形件成形工艺和模具设计的合理确定提供了科学指导,实现了对实际生产工艺和模具的优化设计。
The bulk metal forming process is a kind of metal working method having little or no cutting. It plays a significant role in modern manufacturing. An accurate simulation of the bulk metal forming process accurately can not only save high costs of the experiment, but also have significant theoretical guidance meaning and realistic application value for determining the reasonable forming process and ensuring die design completed at the first time successfully. With the development of calculation method and computer technology, numerical analysis method becomes a powerful tool to analyze engineering problems. Finite element method (FEM) has played an important role in the numerical simulation of bulk metal forming processes. However, when simulating large and severe deformation processes such as forging and extrusion, FEM often encounters some difficulties that meshes become severely distorted and remeshing is necessary because of complicated metal flowing and large deformation. And then the computation process can not be carried on continually and remeshing often brings excessively time-consuming and causes deterioration of computational precision especially.
The meshless method is a new numerical computational method developed in recent years. Compared with FEM, the main benefit of the method is that the approximation field function is constructed entirely in terms of arbitrarily placed nodes of structures without using explicit mesh. And it shows obvious advantages in treating with large deformation problems because it gets rid of the reliance on the mesh and provides continuous and flexible field function. Many researchers have applied meshless method to solve metal forming problems in recent decade and made some achievements. And that drives the development of meshless numerical simulation technology applied for analyzing the metal forming process. But the researches on the application of meshless method in simulating three-dimensional unsteady bulk metal forming process are imperfect, and there are still many key application technologies to be further studied.
Therefore, according to the characteristics of bulk metal forming process and technological requirements, the paper establishes rigid/visco-plastic element free Galerkin method and applies it for the analysis of three-dimensional unsteady bulk metal forming process. On the basis of further studies of rigid/visco-plastic element free Galerkin method and corresponding key numerical simulation techniques, a meshless method numerical analysis procedure for simulating the bulk metal forming process is developed. And a scientific, accurate, reliable, and new meshless method numerical simulation tool is provided to the actual production.
Researches on the three-dimensional element free Galerkin method are carried out. Element free Galerkin method uses moving least square method to construct shapefunction, and to obtain shape function A~(-1) (x) should be calculated firstly. The cases about matrix A(x) appearing irreversible in three-dimensional problems are discussed,and the approaches to treat with this problem are given. The paper employs the orthogonal function with the weight function as the basis function to avoid the processof calculating the inversion of matrix A(x). The size of node influence domain hasgreat effects on computational accuracy and efficiency of element free Galerkin method. The size of node influence domain should be determined following the rule of "sufficiently large and as small as possible". The dynamic node influence domain method is utilized to obtain reasonable node influence domain size, and this method not only guarantees the computational accuracy but also improves the computational efficiency. Due to the lack of Kronecker delta properties in the element free Galerkin method shape functions, the essential boundary conditions can't be imposed directly. In order to exert the boundary conditions directly, the mixed transformation method is adopted to modify shape function, then the modified shape function has the interpolation property and the essential boundary conditions can be enforced exactly and directly. The realization process for the three-dimensional element free Galerkin is studied. The implementation method of the regular hexahedron fixed background cell quadrature technique is introduced. And programming steps for element free Galerkin method are given. Influences of basis function, weight function, Gauss quadrature order and node density on the computational accuracy and efficiency are researched, and the influence law is obtained.
Combining element free Galerkin method and rigid/visco-plastic flow theory, the paper establishes the three-dimensional rigid/visco-plastic element free Galerkin method, and applies it to analyze three-dimensional bulk metal forming process. The velocity field is approximated by MLS method. Based on the expression of velocity field approximation function, the discretized formats of strain rate vector, effective strain rate and volumetric strain rate are established. By employing penalty function method to constrain incompressibility condition, and according to incomplete generalized variational principle, compositions of total energy rate functional are given. Moreover, the variations of the compositions are derived and expressed in matrix forms. The treatment method for the frictional conditions along the workpiece-die interface is studied. The arctangent frictional model is used to implement the frictional boundary conditions. Boundary condition is imposed correctly in the three-dimensional local coordinate system for simulating three-dimensional bulk metal forming process with arbitrarily geometrical shaped dies. The stiffness matrix equation of RVPEFGM is derived and assembled in the local coordinate system. The New-Raphson iterative procedure is implemented to solve stiffness matrix equation until a converged result is obtained. Basic theory research of rigid/visco-plastic element free Galerkin method is improved. Also simulation analysis steps and program flow are presented.
Most bulk metal forming process in practical production belongs to three-dimensional unsteady metal plastic process with severe and complex deformation. Applying RVPEFGM to guide practical production and improving the generality and automation of the analysis program can take advantages of meshless method in treating with bulk metal forming process. According to the characteristics of bulk metal forming process and technological requirements, the related key simulation techniques for analyzing three-dimensional bulk metal forming process used by element free Galerkin method are studied. The problems, such as the description of dies, establishment of local coordinate system for contact points, the way to get initial velocity field, numerical iteration convergence and control method for velocity field, method for dealing with the rigid region, method to release the volumetric locking problem, node detachment and contact criterion, and implement of regional and boundary numerical integration, are solved.
The finite elements are used to describe the mould cavity, and the interface program for reading STL data in ASCII format is developed. The method to establish the local coordinate system for contact point is proposed, and the transformation matrix between global coordinate system and local coordinate system is given. There are rigid regions during metal forming process. The method for avoiding numerical problem caused by rigid region in metal plastic forming is given. Initial velocity field determines the convergence and convergence rate of Newton-Raphson iteration. The direct iteration method is used to get initial velocity field guess for Newton-Raphson iteration. The method of "rapid drop slow ascend" is employed to search convergencedirection and the selection method of the attenuation factorβis given, thus theconvergence velocity of the Newton-Raphson iteration can be improved. Volumetric strain rate projection method is proposed to solve volumetric locking and pressure oscillation problems. A releasing algorithm is realized by modifying the volumetric strain rate in the governing equation. The volumetric strain rate calculated according to velocity field is mapped onto a lower-order space to reduce the number of independent constraint equations. The automatic technique for handling the contact and detachment states among the deforming material boundary and die surfaces is focused. And concrete algorithms of dynamic adjusting techniques are established. The reposition of the contact node is modified in terms of shortest distance principle from the contact node to the triangle patch, thus the "blind area of normal vector" caused by drawing vertical line to triangle patches to search perpendicular point due to scatted die surface meshes can be solved. The partition of unity method is introduced into regional numerical integration to avoid the integration precision loss because of decreasing Gauss integration points in three-dimensional metal forming process simulated by RVPEFGM.
Based on the studies of rigid/visco-plastic element free Galerkin method and related key technology, an analysis program for simulating three-dimensional bulk metal forming processes is developed. And the program is capable of simulating three-dimensional unsteady bulk metal forming processes with severe deformation and arbitrarily shaped dies. The typical metal forming processes such as forging and extrusion processes are analyzed. The effectiveness and validity of the proposed methods and techniques are demonstrated by comparing with the numerical data obtained by using FEM and experimental data. The accuracy and reliability of the rigid/visco-plastic element free Galerkin method and numerical simulation program are ensured by correcting theoretical model and maintaining programs based on the experimental verification and numerical examples analysis.
An equal channel angle extrusion process and a furcation hot forging process are simulated by using the program developed in this paper. Detail mechanical data are obtained. The deformation rules, distributions of the effective stress and effective strain and the load stroke curves are studied. The deforming mechanism of the equal channel angle process is better understood and reasonable forming processes of furcation hot forging are obtained. The scientific guide for optimizing process and die design for practical metal forming production is realized.
无网格方法是近年来兴起的一种新的数值方法,它基于一系列离散节点进行近似,不需要节点的连接信息,克服了有限元法对网格的依赖,而且提供了连续性好、形式灵活的场函数,为大变形成形过程的数值模拟提供了一种有效途径。众多学者开始致力于无网格方法在金属体积成形过程中的应用研究,推动了金属体积成形过程无网格方法数值模拟技术的发展。但对三维非稳态金属体积成形过程的无网格方法数值模拟的研究还不完善,很多关键应用技术有待于进一步的研究。
本文针对金属体积成形过程的特点和工艺要求,提出刚(粘)塑性无网格伽辽金方法(RVPEFGM),并将其应用于三维非稳态金属体积成形过程的数值模拟分析。在对金属体积成形过程刚(粘)塑性无网格伽辽金方法和数值模拟关键技术深入研究的基础上,开发数值模拟分析程序,以期为金属成形过程分析提供一种科学的无网格方法仿真技术工具。
对三维无网格伽辽金方法(EFGM)开展了相关研究。当采用移动最小二乘(MLS)近似方案构造形函数时,会涉及到A~(-1)(x)矩阵的计算,研究了在三维问题中造成A(x)矩阵不可逆的原因,给出了A(x)矩阵不可逆的解决方法,并采用带权正交函数作为基函数,从而避免了对A(x)矩阵的求逆计算。节点影响域尺寸的大小对EFGM的计算精度和计算效率有很大的影响,影响域尺寸的确定要遵循“足够大又尽量小”的原则,为合理确定影响域尺寸,采用动态节点影响域方法,在保证EFGM的计算精度的同时兼顾计算效率。MLS形函数不具备插值性质,为方便EFGM本质边界条件的处理,采用混合变换法对MLS形函数进行局部修正,实现了EFGM本质边界条件的精确施加。研究了三维无网格伽辽金方法的具体实现过程,给出了规则六面体固定背景网格积分方案的实施方法和EFGM的程序设计步骤。研究了基函数、权函数、高斯积分阶次和节点密度等因素对EFGM的计算精度和效率的影响,得到了这些因素对计算精度和效率的影响规律。
将无网格伽辽金方法与刚(粘)塑性流动理论相结合,建立了刚(粘)塑性无网格伽辽金方法(RVPEFGM),并将其应用于三维金属体积成形工艺过程的模拟。基于速度场函数的MLS近似离散形式,推导建立了三维金属体积成形问题中应变速率向量、等效应变速率、体积应变速率的离散格式。采用罚函数法施加体积不变条件,根据不完全广义变分原理,给出系统能量速率泛函的组成,推导出能量速率泛函各组成部分的变分,并获得其矩阵表示形式。研究模具和工件间的摩擦边界条件的处理方法,采用反正切摩擦模型描述摩擦接触边界条件,实现了任意形状模具的三维金属体积成形过程模拟分析中摩擦边界条件在三维局部坐标系下的正确施加,推导建立了三维金属体积成形问题RVPEFGM的刚度方程,在局部坐标系下总装刚度方程,采用Newton-Raphson迭代法对刚度方程进行求解。完善了刚(粘)塑性无网格伽辽金方法基础理论研究,给出了刚(粘)塑性无网格伽辽金方法分析三维金属体积成形过程的模拟分析步骤和程序流程。
实际生产中,大多数金属体积成形工艺过程属于三维非稳态大变形金属塑性成形问题,其成形过程极其复杂,为把RVPEFGM应用于实际生产,并实现分析程序的通用性和自动化,进而发挥无网格方法处理金属体积成形问题的优势。本文结合金属体积成形过程的特点和工艺要求,对三维金属体积成形过程无网格伽辽金方法数值模拟的关键技术进行了深入研究。解决了诸如模具型腔的几何信息描述、接触节点局部坐标系的建立、初始速度场的确定、速度场的收敛性及控制技术、刚性区的处理、体积闭锁问题的处理、工件边界节点与模具的动态接触与脱离的自动处理、区域及边界积分的实施方案等若干关键技术。
采用有限元网格方法描述模具型腔,建立了读取ASCII码格式的STL数据的接口程序。提出了边界接触节点局部坐标系的建立方法,并给出了整体坐标与局部坐标的转换矩阵。金属成形过程中存在着不产生塑性变形的刚性区,为避免刚性区导致的数值计算问题,给出了工件变形过程中刚性区的处理方法。初始速度场的选取直接决定着Newton-Raphson迭代的收敛性和收敛速度,采用直接迭代法生成初始速度场,能够改善Newton-Raphson迭代的收敛性。采用“急降缓升”方法搜索迭代收敛方向,进而确定衰减因子β的取值,提高了Newton-Raphson迭代的收敛速度。针对三维金属体积成形过程RVPEFGM分析中的体积闭锁和压力震荡问题,提出了体积应变率映射法,通过将体积应变率(?)_v映射到低维空间,建立了体积闭锁现象的缓解算法。重点研究了工件边界节点与模具表面接触和脱离的自动处理技术,形成了动态边界自动识别技术的具体算法。提出了根据接触节点到模具型腔三角形面片距离最小的原则进行接触节点位置修正的方法,有效解决了当采用自接触节点到模具型腔三角形面片做垂线进行修正接触节点位置时遇到的“法矢盲区”问题。针对金属体积成形的特点,将单位分解积分法引入到三维金属体积成形RVPEFGM数值积分计算中,解决了使用固定背景网格时因高斯积分点数量减少而致使模拟分析精度下降的问题。
在上述刚(粘)塑性无网格伽辽金方法理论和关键技术研究的基础上,采用C++语言,自主开发了三维金属体积成形过程刚(粘)塑性无网格伽辽金方法数值模拟分析程序,实现了任意形状模具的三维非稳态大变形金属体积成形过程的RVPEFGM分析,并对锻造、挤压等工艺进行了数值模拟分析,通过与有限元软件分析结果和实验结果的比较,验证了所提出的方法及处理技术的正确性。同时在分析对比、实验验证、算例检验的基础上,不断修正模型和完善、维护程序,确保了建立的刚(粘)塑性无网格伽辽金方法的基本理论以及开发的数值模拟程序的准确性、可靠性。
采用自主开发的三维刚(粘)塑性无网格伽辽金方法分析程序,对圆形挤压件等通道弯角挤压过程和转向节预成形件热模锻过程进行了系统模拟分析,获得了相关力学分析结果,对金属流动规律、工件等效应力和等效应变的分布以及载荷力的变化等进行了研究,为掌握等通道弯角挤压机理以及转向节预成形件成形工艺和模具设计的合理确定提供了科学指导,实现了对实际生产工艺和模具的优化设计。
The bulk metal forming process is a kind of metal working method having little or no cutting. It plays a significant role in modern manufacturing. An accurate simulation of the bulk metal forming process accurately can not only save high costs of the experiment, but also have significant theoretical guidance meaning and realistic application value for determining the reasonable forming process and ensuring die design completed at the first time successfully. With the development of calculation method and computer technology, numerical analysis method becomes a powerful tool to analyze engineering problems. Finite element method (FEM) has played an important role in the numerical simulation of bulk metal forming processes. However, when simulating large and severe deformation processes such as forging and extrusion, FEM often encounters some difficulties that meshes become severely distorted and remeshing is necessary because of complicated metal flowing and large deformation. And then the computation process can not be carried on continually and remeshing often brings excessively time-consuming and causes deterioration of computational precision especially.
The meshless method is a new numerical computational method developed in recent years. Compared with FEM, the main benefit of the method is that the approximation field function is constructed entirely in terms of arbitrarily placed nodes of structures without using explicit mesh. And it shows obvious advantages in treating with large deformation problems because it gets rid of the reliance on the mesh and provides continuous and flexible field function. Many researchers have applied meshless method to solve metal forming problems in recent decade and made some achievements. And that drives the development of meshless numerical simulation technology applied for analyzing the metal forming process. But the researches on the application of meshless method in simulating three-dimensional unsteady bulk metal forming process are imperfect, and there are still many key application technologies to be further studied.
Therefore, according to the characteristics of bulk metal forming process and technological requirements, the paper establishes rigid/visco-plastic element free Galerkin method and applies it for the analysis of three-dimensional unsteady bulk metal forming process. On the basis of further studies of rigid/visco-plastic element free Galerkin method and corresponding key numerical simulation techniques, a meshless method numerical analysis procedure for simulating the bulk metal forming process is developed. And a scientific, accurate, reliable, and new meshless method numerical simulation tool is provided to the actual production.
Researches on the three-dimensional element free Galerkin method are carried out. Element free Galerkin method uses moving least square method to construct shapefunction, and to obtain shape function A~(-1) (x) should be calculated firstly. The cases about matrix A(x) appearing irreversible in three-dimensional problems are discussed,and the approaches to treat with this problem are given. The paper employs the orthogonal function with the weight function as the basis function to avoid the processof calculating the inversion of matrix A(x). The size of node influence domain hasgreat effects on computational accuracy and efficiency of element free Galerkin method. The size of node influence domain should be determined following the rule of "sufficiently large and as small as possible". The dynamic node influence domain method is utilized to obtain reasonable node influence domain size, and this method not only guarantees the computational accuracy but also improves the computational efficiency. Due to the lack of Kronecker delta properties in the element free Galerkin method shape functions, the essential boundary conditions can't be imposed directly. In order to exert the boundary conditions directly, the mixed transformation method is adopted to modify shape function, then the modified shape function has the interpolation property and the essential boundary conditions can be enforced exactly and directly. The realization process for the three-dimensional element free Galerkin is studied. The implementation method of the regular hexahedron fixed background cell quadrature technique is introduced. And programming steps for element free Galerkin method are given. Influences of basis function, weight function, Gauss quadrature order and node density on the computational accuracy and efficiency are researched, and the influence law is obtained.
Combining element free Galerkin method and rigid/visco-plastic flow theory, the paper establishes the three-dimensional rigid/visco-plastic element free Galerkin method, and applies it to analyze three-dimensional bulk metal forming process. The velocity field is approximated by MLS method. Based on the expression of velocity field approximation function, the discretized formats of strain rate vector, effective strain rate and volumetric strain rate are established. By employing penalty function method to constrain incompressibility condition, and according to incomplete generalized variational principle, compositions of total energy rate functional are given. Moreover, the variations of the compositions are derived and expressed in matrix forms. The treatment method for the frictional conditions along the workpiece-die interface is studied. The arctangent frictional model is used to implement the frictional boundary conditions. Boundary condition is imposed correctly in the three-dimensional local coordinate system for simulating three-dimensional bulk metal forming process with arbitrarily geometrical shaped dies. The stiffness matrix equation of RVPEFGM is derived and assembled in the local coordinate system. The New-Raphson iterative procedure is implemented to solve stiffness matrix equation until a converged result is obtained. Basic theory research of rigid/visco-plastic element free Galerkin method is improved. Also simulation analysis steps and program flow are presented.
Most bulk metal forming process in practical production belongs to three-dimensional unsteady metal plastic process with severe and complex deformation. Applying RVPEFGM to guide practical production and improving the generality and automation of the analysis program can take advantages of meshless method in treating with bulk metal forming process. According to the characteristics of bulk metal forming process and technological requirements, the related key simulation techniques for analyzing three-dimensional bulk metal forming process used by element free Galerkin method are studied. The problems, such as the description of dies, establishment of local coordinate system for contact points, the way to get initial velocity field, numerical iteration convergence and control method for velocity field, method for dealing with the rigid region, method to release the volumetric locking problem, node detachment and contact criterion, and implement of regional and boundary numerical integration, are solved.
The finite elements are used to describe the mould cavity, and the interface program for reading STL data in ASCII format is developed. The method to establish the local coordinate system for contact point is proposed, and the transformation matrix between global coordinate system and local coordinate system is given. There are rigid regions during metal forming process. The method for avoiding numerical problem caused by rigid region in metal plastic forming is given. Initial velocity field determines the convergence and convergence rate of Newton-Raphson iteration. The direct iteration method is used to get initial velocity field guess for Newton-Raphson iteration. The method of "rapid drop slow ascend" is employed to search convergencedirection and the selection method of the attenuation factorβis given, thus theconvergence velocity of the Newton-Raphson iteration can be improved. Volumetric strain rate projection method is proposed to solve volumetric locking and pressure oscillation problems. A releasing algorithm is realized by modifying the volumetric strain rate in the governing equation. The volumetric strain rate calculated according to velocity field is mapped onto a lower-order space to reduce the number of independent constraint equations. The automatic technique for handling the contact and detachment states among the deforming material boundary and die surfaces is focused. And concrete algorithms of dynamic adjusting techniques are established. The reposition of the contact node is modified in terms of shortest distance principle from the contact node to the triangle patch, thus the "blind area of normal vector" caused by drawing vertical line to triangle patches to search perpendicular point due to scatted die surface meshes can be solved. The partition of unity method is introduced into regional numerical integration to avoid the integration precision loss because of decreasing Gauss integration points in three-dimensional metal forming process simulated by RVPEFGM.
Based on the studies of rigid/visco-plastic element free Galerkin method and related key technology, an analysis program for simulating three-dimensional bulk metal forming processes is developed. And the program is capable of simulating three-dimensional unsteady bulk metal forming processes with severe deformation and arbitrarily shaped dies. The typical metal forming processes such as forging and extrusion processes are analyzed. The effectiveness and validity of the proposed methods and techniques are demonstrated by comparing with the numerical data obtained by using FEM and experimental data. The accuracy and reliability of the rigid/visco-plastic element free Galerkin method and numerical simulation program are ensured by correcting theoretical model and maintaining programs based on the experimental verification and numerical examples analysis.
An equal channel angle extrusion process and a furcation hot forging process are simulated by using the program developed in this paper. Detail mechanical data are obtained. The deformation rules, distributions of the effective stress and effective strain and the load stroke curves are studied. The deforming mechanism of the equal channel angle process is better understood and reasonable forming processes of furcation hot forging are obtained. The scientific guide for optimizing process and die design for practical metal forming production is realized.
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