基于弧长法的有限元逆算法及其在板料成形中的应用
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摘要
在冲压板料成形加工中,毛坯展开计算非常重要。求得板料的展开毛坯,是分析板料变形程度、设计工艺以及拟定工艺规程的前提。合理的毛坯形状和尺寸,可以明显改善冲压过程中板料变形不均匀的现象,充分发挥金属的成形性能。在板料的展开方法中,基于全量理论的有限元逆算法只考虑初始构形及最终构形两个形态,计算速度快,是一种高效的展开算法。在给定工艺条件下,逆算法能快速计算出板料的毛坯形状,以及最终的应力、应变分布和厚度变化等信息。
本文详细阐述了板料成形基本理论,对有限元逆算法的基本原理和实施过程进行了介绍;使用比例加载条件下的材料厚向异性本构关系和简单有效的三角形膜单元,建立了相应的有限元逆算法方程,并编写了计算机程序。然后以ABAQUS为平台,通过其用户单元UEL模块,利用ABAQUS/Standard求解器对有限元逆算法的非线性方程组进行求解。
初始解的确定是有限元逆算法的关键问题,往往会产生单元拓扑关系的变化。因此,初始解不仅能影响逆算法的收敛速度,更能决定逆算法计算结果是否正确。经过研究,提出了弧长法,通过单元追踪计算出最终构形上每个节点到所选取的弧长起始点之间的弧长,能够将产品的最终构形准确的映射到初始平面,并保持每个单元的拓扑关系,所获得初始解能够使有限元逆算法迅速收敛。
用有限元逆算法对圆盒件和类车门件进行了展开计算,并将计算结果与增量有限元软件DYNAFORM和试验的结果进行了分析比较。分析结果显示:在工程精度范围内,有限元逆算法计算结果与增量有限元计算结果基本吻合,但逆算法的计算速度大大高于增量有限元法,节省了大量时间。实例分析表明有限元逆算法是一种有效的板料展开算法,在产品初期设计阶段具有较大的应用价值。
最后在完成三角形膜单元的基础上,本文又推导了基于四节点四边形等参单元有限元逆算法的部分公式,并用弧长法求出了四边形等参单元的初始解,为四边形有限元逆算法的完成打下了良好的基础。
Blank design is important in sheet metal stamping forming process.It is the premise to obtain the blank shape of forming parts to analyze forming distortion degree,design technology as well as draft process specifications.Reasonable blank shape and size can obviously improve the inhomogeneous deformation of blank and fully exploit the metal forming performance.Among the methods of blank design,the Inverse finite elements Approach(IFEA)based on total strain theory,only considering the initial and final stages of the parts,makes IFEA method very fast and high effective.Under some specific conditions,IFEA method can quickly calculate blank shape、final stress、strain、thickness change rate etc.
In the paper,the basic theory of sheet forming is fully studied.The basic theory of IFEA and how to operate it are briefly introduced.Then the anisotropic constitutive relation under proportional loading condition and flat triangular membrane elements of constant thickness having three nodes are adapted,and then the related formula is put forward and the related computer program is implemented.Then we get the solutions of IFEA non-linear equations by the ABAQUS user define element option (UEL) and it's standard calculator.
The key problem of IFEA is the calculation of initial solutions,because of the difficulty to keep the topology of mesh elements.So the initial solutions not only influence the rate of convergence but also determine the results of IFEA are valid or not.Arc-length method is proposed to resolve the difficulty,which map the final product by the arc length of every node of product to the original node.This method can keep topology of all the mesh elements and improve the convergence of IFEA.
Cylindrical cup and door-like product are simulated by IFEA program.Then the results are compared with those simulated by incremental soft ware DYNAFORM and experimental.The comparisons show that inverse approach,which can be used to evaluate the sheet metal forming process quickly,is an effective approach for blank design.So it is a good choice to use this method in the initial stage of die design.
Finally,the IFEA based on the four-node isoparametric element is discussed after the triangular membrane element and part of the finite element equation is established in this paper.Then the initial solution of four-node element is obtained by arc-length method,which is the basic of IFEA based on the four-node element.
本文详细阐述了板料成形基本理论,对有限元逆算法的基本原理和实施过程进行了介绍;使用比例加载条件下的材料厚向异性本构关系和简单有效的三角形膜单元,建立了相应的有限元逆算法方程,并编写了计算机程序。然后以ABAQUS为平台,通过其用户单元UEL模块,利用ABAQUS/Standard求解器对有限元逆算法的非线性方程组进行求解。
初始解的确定是有限元逆算法的关键问题,往往会产生单元拓扑关系的变化。因此,初始解不仅能影响逆算法的收敛速度,更能决定逆算法计算结果是否正确。经过研究,提出了弧长法,通过单元追踪计算出最终构形上每个节点到所选取的弧长起始点之间的弧长,能够将产品的最终构形准确的映射到初始平面,并保持每个单元的拓扑关系,所获得初始解能够使有限元逆算法迅速收敛。
用有限元逆算法对圆盒件和类车门件进行了展开计算,并将计算结果与增量有限元软件DYNAFORM和试验的结果进行了分析比较。分析结果显示:在工程精度范围内,有限元逆算法计算结果与增量有限元计算结果基本吻合,但逆算法的计算速度大大高于增量有限元法,节省了大量时间。实例分析表明有限元逆算法是一种有效的板料展开算法,在产品初期设计阶段具有较大的应用价值。
最后在完成三角形膜单元的基础上,本文又推导了基于四节点四边形等参单元有限元逆算法的部分公式,并用弧长法求出了四边形等参单元的初始解,为四边形有限元逆算法的完成打下了良好的基础。
Blank design is important in sheet metal stamping forming process.It is the premise to obtain the blank shape of forming parts to analyze forming distortion degree,design technology as well as draft process specifications.Reasonable blank shape and size can obviously improve the inhomogeneous deformation of blank and fully exploit the metal forming performance.Among the methods of blank design,the Inverse finite elements Approach(IFEA)based on total strain theory,only considering the initial and final stages of the parts,makes IFEA method very fast and high effective.Under some specific conditions,IFEA method can quickly calculate blank shape、final stress、strain、thickness change rate etc.
In the paper,the basic theory of sheet forming is fully studied.The basic theory of IFEA and how to operate it are briefly introduced.Then the anisotropic constitutive relation under proportional loading condition and flat triangular membrane elements of constant thickness having three nodes are adapted,and then the related formula is put forward and the related computer program is implemented.Then we get the solutions of IFEA non-linear equations by the ABAQUS user define element option (UEL) and it's standard calculator.
The key problem of IFEA is the calculation of initial solutions,because of the difficulty to keep the topology of mesh elements.So the initial solutions not only influence the rate of convergence but also determine the results of IFEA are valid or not.Arc-length method is proposed to resolve the difficulty,which map the final product by the arc length of every node of product to the original node.This method can keep topology of all the mesh elements and improve the convergence of IFEA.
Cylindrical cup and door-like product are simulated by IFEA program.Then the results are compared with those simulated by incremental soft ware DYNAFORM and experimental.The comparisons show that inverse approach,which can be used to evaluate the sheet metal forming process quickly,is an effective approach for blank design.So it is a good choice to use this method in the initial stage of die design.
Finally,the IFEA based on the four-node isoparametric element is discussed after the triangular membrane element and part of the finite element equation is established in this paper.Then the initial solution of four-node element is obtained by arc-length method,which is the basic of IFEA based on the four-node element.
引文
[1]邓陡,王先进,陈鹤峥.金属薄板成形技术[M].兵器工业出版社,1993.
[2]彭颖红.金属塑性成形仿真技术[M].上海交通大学出版社,1999.
[3]梁炳文,胡世光.板金成形塑性理论[M].北京机械工业出版社,1987.
[4]杨晨.钣金展开有限元逆算法[硕士学位论文].南京:南京航空航天大学,2003.
[5]霍同如,徐秉业,宋军.板料成形有限元数值模拟的研究进展[J].力学进展,1996,26(3):379-380.
[6]鲍益东.汽车车身部件一步逆成形有限元法与碰撞仿真研究[博士学位论文].吉林:吉林大学,2004.
[7]唐于芳.有限元逆算法在钣金展开中的应用[硕士学位论文].南京:南京航空航天大学,2007.
[8]金涛.有限元逆算法及其在工艺参数优化中的应用[硕士学位论文].武汉:华中科技大学,2005.
[9]陈国强.有限元逆算法在汽车覆盖件冲压成形数值模拟中的应用[硕士学位论文].武汉:华中科技大学,2004..
[10]兰箭,董湘怀,李志刚.用有限元逆算法计算板料成形毛坯形状和应变分布[J].塑性工程学报,2001,18(3):23-25.
[11]那景新,胡平,焦健等.初始展开面积对一步成形模拟收敛效率的影响[J].哈尔滨工业大学学报,2005,37(6):779-781.
[12]兰箭,董湘怀,李志刚.有限元逆算法与板料成形工艺的评价[J].中国机械工程,2002,13(10):826-829.
[13]汪大年.金属塑性成形原理[M].北京机械工业出版社,1982.
[14]常志华.盒形拉深毛坯展开尺寸的计算方法[J].锻压技术,1991,17(6):15-20.
[15]李乃周,梁炳文.不规则拉延件变形的研究[J].锻压技术,1990,16(3):16-23.
[16]Gerdeen J C,Chen P.Geometric mapping method of computer modeling of sheet.Numiform89,1989
[17]S Kobayashi,J H Kim.Deformation analysis of axis symmetric sheet metal forming processes by the rigid-plastic finite element.Mechanics of Sheet Metal Forming.1978.
[18]Batoz JL,Duroux P,Guo YQ.An efficient algorithm to estimate the large strains in deep drawing.NUMIFORM 89,1989.
[19]Batoz J L.An inverse finite element procedure to estimate the large plastic strain in sheet metal forming.3rd ICTP.1994.
[20]Guo Y Q et al.On the estimation of thickness strains in thin car panels by the inverse approach.NUM IFORM 92:1403-1408.
[21]Batoz J Let al.The inverse approach including bending effects for the analysis and design of sheet metal forming parts.Simulation of Materials Processing:Theory,Methods and Applications,1995.
[22]Lee C H,Huh H.Blank design and strain estimates for sheet metal forming process by a finite element inverse approach with initial guess of linear deformation.Journal of Material Process Technology,1998,82(3):145-155.
[23]谢兰生,杨晨,童国权。基于几何映射法的板金展开有限元逆算法[J].中国机械工程,2004,15(9):832-835.
[24]匡震邦.非线性连续介质力学基础[M].西安:西安交通大学出版社,1989.
[25]R.Hill.A general theory of stability in elesto-Plastic solids.J.Mech.Phys.Solids,1958,11(6):236-238.
[26]Chung K,Richmond R.Ideal Forming I.Homo-geneous Deformation with Minimum Plastic Work.1992,34(8):475-491.
[27]Chung K,Richmond O.Ideal Forming Ⅱ.Sheet Forming with Optimum Deformation.Int.J.1992,34(8):617-633.
[28]何光渝,高永利.Visual Fortran常用数值算法集[M].科学出版社,2002.
[29]Lan Jian,Dong Xiang huai,Li Zhi gang.Inverse finite element approach and its application in sheet metal forming.Journal of Material Process Technology,2005,170(3):624-631.
[30]Ying Huang,Yi Ping Chen,Ru Xu Du.A new approach to solve key issues in multi-step inverse finite-element method in sheet metal stamping,International Journal of Mechanical Sciences,2006,18(3):591-600.
[31]Lee C H,Huh H.Three dimensional multi-step inverse analysis for the optimum blank design in sheet forming processes.Journal of Material Process Technology,1998,80(8):76-82.
[32]兰箭.板料成形的有限元逆算法研究,[博士学位论文].武汉:华中科技大学,2001.
[33]H.Shim,K.Son,K.Kim.Optimum blank design by sensitivity analysis.Journal of Materials Processing Technology,2000,104:191-199.
[34]易国峰,柳玉起,李明林.基于有限元逆算法的板料成形模拟拉深筋的灵敏度优化[J].中国机械工程,2001,16(8):873-876..
[35]Guo C H,Batoz J L.Recent developments on the analysis and optimum design of sheet metal forming parts using a simplified inverse approach.Computers and Structures,2000,78:133-148.
[36]Daryl L.Logan著,伍义生,吴永礼等译。有限元方法基础教程[M].电子工业出版社,2003.
[37]王焕定,焦兆平.有限单元法基础[M].高等教育出版社,2002.
[38]那景新,陆善彬,闫亚坤.改进四节点等参单元在一步成形模拟中的应用[J].固体力学学报,2004,25(1):80-82.
[39]冷纪桐,赵军,张娅.有限元技术基础[M].化学工业出版社,2007.
[40]龙驭球,岑松.广义协调元理论与四边形面积坐标法[M].中国矿业大学出版社,2000.
[2]彭颖红.金属塑性成形仿真技术[M].上海交通大学出版社,1999.
[3]梁炳文,胡世光.板金成形塑性理论[M].北京机械工业出版社,1987.
[4]杨晨.钣金展开有限元逆算法[硕士学位论文].南京:南京航空航天大学,2003.
[5]霍同如,徐秉业,宋军.板料成形有限元数值模拟的研究进展[J].力学进展,1996,26(3):379-380.
[6]鲍益东.汽车车身部件一步逆成形有限元法与碰撞仿真研究[博士学位论文].吉林:吉林大学,2004.
[7]唐于芳.有限元逆算法在钣金展开中的应用[硕士学位论文].南京:南京航空航天大学,2007.
[8]金涛.有限元逆算法及其在工艺参数优化中的应用[硕士学位论文].武汉:华中科技大学,2005.
[9]陈国强.有限元逆算法在汽车覆盖件冲压成形数值模拟中的应用[硕士学位论文].武汉:华中科技大学,2004..
[10]兰箭,董湘怀,李志刚.用有限元逆算法计算板料成形毛坯形状和应变分布[J].塑性工程学报,2001,18(3):23-25.
[11]那景新,胡平,焦健等.初始展开面积对一步成形模拟收敛效率的影响[J].哈尔滨工业大学学报,2005,37(6):779-781.
[12]兰箭,董湘怀,李志刚.有限元逆算法与板料成形工艺的评价[J].中国机械工程,2002,13(10):826-829.
[13]汪大年.金属塑性成形原理[M].北京机械工业出版社,1982.
[14]常志华.盒形拉深毛坯展开尺寸的计算方法[J].锻压技术,1991,17(6):15-20.
[15]李乃周,梁炳文.不规则拉延件变形的研究[J].锻压技术,1990,16(3):16-23.
[16]Gerdeen J C,Chen P.Geometric mapping method of computer modeling of sheet.Numiform89,1989
[17]S Kobayashi,J H Kim.Deformation analysis of axis symmetric sheet metal forming processes by the rigid-plastic finite element.Mechanics of Sheet Metal Forming.1978.
[18]Batoz JL,Duroux P,Guo YQ.An efficient algorithm to estimate the large strains in deep drawing.NUMIFORM 89,1989.
[19]Batoz J L.An inverse finite element procedure to estimate the large plastic strain in sheet metal forming.3rd ICTP.1994.
[20]Guo Y Q et al.On the estimation of thickness strains in thin car panels by the inverse approach.NUM IFORM 92:1403-1408.
[21]Batoz J Let al.The inverse approach including bending effects for the analysis and design of sheet metal forming parts.Simulation of Materials Processing:Theory,Methods and Applications,1995.
[22]Lee C H,Huh H.Blank design and strain estimates for sheet metal forming process by a finite element inverse approach with initial guess of linear deformation.Journal of Material Process Technology,1998,82(3):145-155.
[23]谢兰生,杨晨,童国权。基于几何映射法的板金展开有限元逆算法[J].中国机械工程,2004,15(9):832-835.
[24]匡震邦.非线性连续介质力学基础[M].西安:西安交通大学出版社,1989.
[25]R.Hill.A general theory of stability in elesto-Plastic solids.J.Mech.Phys.Solids,1958,11(6):236-238.
[26]Chung K,Richmond R.Ideal Forming I.Homo-geneous Deformation with Minimum Plastic Work.1992,34(8):475-491.
[27]Chung K,Richmond O.Ideal Forming Ⅱ.Sheet Forming with Optimum Deformation.Int.J.1992,34(8):617-633.
[28]何光渝,高永利.Visual Fortran常用数值算法集[M].科学出版社,2002.
[29]Lan Jian,Dong Xiang huai,Li Zhi gang.Inverse finite element approach and its application in sheet metal forming.Journal of Material Process Technology,2005,170(3):624-631.
[30]Ying Huang,Yi Ping Chen,Ru Xu Du.A new approach to solve key issues in multi-step inverse finite-element method in sheet metal stamping,International Journal of Mechanical Sciences,2006,18(3):591-600.
[31]Lee C H,Huh H.Three dimensional multi-step inverse analysis for the optimum blank design in sheet forming processes.Journal of Material Process Technology,1998,80(8):76-82.
[32]兰箭.板料成形的有限元逆算法研究,[博士学位论文].武汉:华中科技大学,2001.
[33]H.Shim,K.Son,K.Kim.Optimum blank design by sensitivity analysis.Journal of Materials Processing Technology,2000,104:191-199.
[34]易国峰,柳玉起,李明林.基于有限元逆算法的板料成形模拟拉深筋的灵敏度优化[J].中国机械工程,2001,16(8):873-876..
[35]Guo C H,Batoz J L.Recent developments on the analysis and optimum design of sheet metal forming parts using a simplified inverse approach.Computers and Structures,2000,78:133-148.
[36]Daryl L.Logan著,伍义生,吴永礼等译。有限元方法基础教程[M].电子工业出版社,2003.
[37]王焕定,焦兆平.有限单元法基础[M].高等教育出版社,2002.
[38]那景新,陆善彬,闫亚坤.改进四节点等参单元在一步成形模拟中的应用[J].固体力学学报,2004,25(1):80-82.
[39]冷纪桐,赵军,张娅.有限元技术基础[M].化学工业出版社,2007.
[40]龙驭球,岑松.广义协调元理论与四边形面积坐标法[M].中国矿业大学出版社,2000.
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