板料多点成形回弹补偿方法及其数值模拟与实验研究
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摘要
目前，航空航天、汽车及造船等多个领域对金属三维曲面件都有大量需求。传统的三维曲面件生产主要通过模具来实现，模具特别适用于大批量的零件生产模式。随着工业技术发展及人们对个性化产品的需求，新型产品、试制样品不断被研制开发。每一件产品的外形及尺寸均不相同，若采用模具加工，需要花费大量的模具设计、制造、维护等费用，成本高昂，无法获得效益。因此，亟需一种新型技术满足这种单件、小批量加工方式的要求。多点成形正是针对这种需求创建的快速、高效的成形技术，其柔性可调整的多点模具型面，能够快速响应产品个性化要求，加快产品更新换代速度。
与传统模具相比，多点成形技术能够实现无模成形，具有快速柔性成形的优势，但该技术也存在回弹、起皱等成形缺陷，尤其是回弹问题，会严重影响多点成形件的加工精度，无法得到准确的三维曲面零件。多点模具可以根据需要快速调整模具型面，采用补偿方法控制多点成形的回弹是一种有效地、切实可行的手段。本文给出了补偿回弹的曲率计算方法，构建复杂曲面的回弹补偿面，建立了多点成形回弹补偿方法，并通过数值模拟与多点成形实验对该方法进行了验证。
研究的主要内容如下：
（1）基于三种材料模型，推导了简单的单曲率曲面补偿回弹的曲率计算公式及残余应力公式。给出了复杂单曲率曲面的回弹补偿方法，采用结构离散方式将复杂曲面处理成若干微小的简单曲面，基于简单曲面的回弹补偿计算公式，采用曲线插值法或有限差分法得到连续的补偿回弹的曲面。再采用三次B样条法对得到的补偿曲面进一步拟合，得到光滑的曲率连续的曲面。对插值法与差分法这两种计算回弹修正曲面的方法进行了实例计算，并进行了数值模拟和实验验证，发现插值法的成形精度更高。
（2）推导了简单的双曲度曲面回弹前曲率（即补偿回弹的曲率）计算公式，给出了复杂双曲度曲面的回弹补偿方法，该方法对以简单曲面补偿回弹的曲率计算公式为基础，引入离散化思想，根据每块微板元的曲率信息反算板元的特征点，采用插值法得到补偿回弹的曲面上一系列特征点矩阵。再采用贝赛尔曲面拼接法，得到光滑连续的回弹补偿面。研究了多点模具调形成准确的回弹补偿面的方法，多点模具调形中要考虑基本体球冠及公切点位置，依据回弹补偿面反求基本体冲头球心坐标和切点坐标，确定基本体行程，即可构建修正回弹的多点模具型面。
（3）基于ABAQUS软件建立了多点成形有限元模型，描述了模拟过程中单元选择、接触的定义、摩擦系数、约束、位移等边界条件的处理，以及显-隐式结合算法进行多点成形回弹模拟的分析过程，阐述了回弹模拟过程中约束的处理问题。研究了多点成形中弹性垫的使用以及板料厚度、材料、曲率半径等对回弹的影响。指出选用适当厚度、压缩率和弹性模量的弹性垫可以达到良好的表面成形质量，有效降低回弹量；随着板厚的增加，回弹量下降的趋势逐渐变缓；材料的弹性模量越大，工件卸载后回弹量就越小；另外，随着曲率半径减小，回弹量也减小。
（4）模拟了多种典型的复杂曲面件多点成形回弹过程，根据修正的模具型面进行数值模拟，得到了补偿回弹的曲面件，根据目标形状进行数值模拟，得到了未补偿回弹的曲面件。通过回弹前、后的应力云图及Z向位移云图对比，发现回弹后的曲面件应力和曲率减小，曲面形状发生了改变。取模拟结果沿x,y方向的轮廓和目标形状对比，发现补偿回弹的模拟结果与目标轮廓接近，误差很小；未补偿回弹的模拟结果与目标形状存在偏差，误差较大。
（5）采用多点模具成形了多种典型曲面件，并进行了精度测量。发现补偿回弹的曲面件，其测量结果与目标轮廓吻合良好，未补偿回弹的测量结果误差较大。通过模拟、实验结果的误差对比，可以发现采用补偿回弹的模具型面成形的曲面误差范围小，和目标形状吻合，这也验证了文章中所述的回弹补偿计算方法具有良好的预测效果和应用价值。
At present, Sheet metal forming processes are widely used to produce three-dimensionalsurface in many fields, such as aerospace, automotive, aircraft manufacturing, vehicle body,shipbuilding and pressure vessel forming and so on. Stamping is one of the most commonsheet metal forming methods. For conventional stamping process, which involves a matchedsolid die set, its advantages are a short production time and high productivity. Nevertheless,large initial investments and long setup time make its processes inflexible and onlyprofitable for mass production and economically unsuitable for single or small batchproducts. Multi-point forming (MPF), a novel flexible forming technology develops wellrecent years. In MPF, conventional solid stamping dies are replaced by a pair of opposedreconfigurable dies comprised by punch matrices. Based on this technology, dieless, rapidand digital manufacturing of sheet metal parts can be realized. Comparing with conventionalstamping, MPF technology is more suitable for small-lot and individualized production, andresponding to customers’ requirements quickly, meanwhile, accelerating the productrenovation.
Comparing with conventional stamping, MPF technology is with advantages of rapid andflexible forming. However, whether it is conventional stamping or MPF, springback is aninevitable phenomenon in sheet metal forming, and it greatly affects the geometricalaccuracy of products. Springback caused by elastic recovery and release of residual stressafter forming, and the final shape of part depends on the value of springback. Once the valueof springback exceeds the allowable tolerance, it becomes defects and affects the wholeassembly of other parts. Moreover, it is an accumulated effect of the entire processing history.Hence the prediction of springback is difficult and remains an important problem in themanufacturing industry. The die surface of MPF can be adjusted quickly, realizing thereconfiguration of various curve surfaces; therefore, springback compensation method isvery suitable for MPF. Compensation method is an effective, practical method for controlling MPF springback. An algorithm was established for springback compensation inMPF by combination methods of theoretical analysis, numerical simulation and formingexperiment. Calculation formula for curvature before springback was obtained basing onthree material models, the method describing the shape of die-face after springbackmodification was proposed, Simulation and experimental results shows that thecompensation algorithm can effectively control errors caused by springback in MPF.
The main contents and conclusions are as follows:
(1)Calculation formulas for curvature before springback for single-curvature surface wasobtained basing on three material models, For irregular single curvature surface,interpolation and difference method are used to obtain continuous surface with first-orderderivative, then a cubic B-spline fitting method are used to obtain the smooth and continuoussurface. Comparison of the interpolation and difference method, it is found that theinterpolation method is with excellent accuracy.
(2) For irregular doubly curved surface, the calculation formula for curvature beforespringback was obtained basing on three material models. Then, with discrete method, thewhole surface was divided into tiny pieces, interpolation processing and Bezier surfaceblending methods are used to describe the surface after springback compensation. Due to thedie-face of MPF is composed by a series of discrete punch elements, the center position andthe points of tangency of punch elements are obtained basing on the surface after springbackcompensation, the adjusting heights of punch elements can be achieved for MPF diecompensation surface.
(3)Based on the ABAQUS software, the finite element models were established, and theunite selection, contact definition, friction coefficient, constraint and displacement boundarywere described in simulation. Simulating the process of MPF and springback, and theSpringback process is simulated combining explicit and implicit algorithm. The influencingfactors on the springback, such as, using cushion, material, thickness, radius of curvature,were researched. The results show: the greater the thickness is, the springback decreasesgradually; the greater elastic modulus is, smaller the springback after unloading is; with theradius of curvature decreases, the springback is reduced; in addition, with the increase ofcompensation coefficient, the error value is decreasing.
(4)Simulating the springback process of MPF on single-curvature surface and doubly curvedsurface. Effective stress distribution and Z-displacement distribution are achieved.Comparing the simulation result with the target shapes, it is found that the single-curvature surface is easily to generate large springback; for doubly curved surface, if the bendingdirection is coincident, the springback is smaller, otherwise, the springback is bigger.Comparing the simulation result with and without springback compensation, we found thatthe errors is very small with springback compensation, otherwise, the errors is relativelylarge.
(5) A series of experiments were carried out by means of MPF equipment, comparing theresults with compensation and without compensation, it is observed that simulated andmeasured results with compensation matched well with target shapes, and overall errors arerelatively small and satisfied processing precision. The compensation method presented inthis paper provides an effective calculation method to springback compensation ofsingle-curvature and doubly curved surface, and it is of good springback predition effect andapplication value.
与传统模具相比，多点成形技术能够实现无模成形，具有快速柔性成形的优势，但该技术也存在回弹、起皱等成形缺陷，尤其是回弹问题，会严重影响多点成形件的加工精度，无法得到准确的三维曲面零件。多点模具可以根据需要快速调整模具型面，采用补偿方法控制多点成形的回弹是一种有效地、切实可行的手段。本文给出了补偿回弹的曲率计算方法，构建复杂曲面的回弹补偿面，建立了多点成形回弹补偿方法，并通过数值模拟与多点成形实验对该方法进行了验证。
研究的主要内容如下：
（1）基于三种材料模型，推导了简单的单曲率曲面补偿回弹的曲率计算公式及残余应力公式。给出了复杂单曲率曲面的回弹补偿方法，采用结构离散方式将复杂曲面处理成若干微小的简单曲面，基于简单曲面的回弹补偿计算公式，采用曲线插值法或有限差分法得到连续的补偿回弹的曲面。再采用三次B样条法对得到的补偿曲面进一步拟合，得到光滑的曲率连续的曲面。对插值法与差分法这两种计算回弹修正曲面的方法进行了实例计算，并进行了数值模拟和实验验证，发现插值法的成形精度更高。
（2）推导了简单的双曲度曲面回弹前曲率（即补偿回弹的曲率）计算公式，给出了复杂双曲度曲面的回弹补偿方法，该方法对以简单曲面补偿回弹的曲率计算公式为基础，引入离散化思想，根据每块微板元的曲率信息反算板元的特征点，采用插值法得到补偿回弹的曲面上一系列特征点矩阵。再采用贝赛尔曲面拼接法，得到光滑连续的回弹补偿面。研究了多点模具调形成准确的回弹补偿面的方法，多点模具调形中要考虑基本体球冠及公切点位置，依据回弹补偿面反求基本体冲头球心坐标和切点坐标，确定基本体行程，即可构建修正回弹的多点模具型面。
（3）基于ABAQUS软件建立了多点成形有限元模型，描述了模拟过程中单元选择、接触的定义、摩擦系数、约束、位移等边界条件的处理，以及显-隐式结合算法进行多点成形回弹模拟的分析过程，阐述了回弹模拟过程中约束的处理问题。研究了多点成形中弹性垫的使用以及板料厚度、材料、曲率半径等对回弹的影响。指出选用适当厚度、压缩率和弹性模量的弹性垫可以达到良好的表面成形质量，有效降低回弹量；随着板厚的增加，回弹量下降的趋势逐渐变缓；材料的弹性模量越大，工件卸载后回弹量就越小；另外，随着曲率半径减小，回弹量也减小。
（4）模拟了多种典型的复杂曲面件多点成形回弹过程，根据修正的模具型面进行数值模拟，得到了补偿回弹的曲面件，根据目标形状进行数值模拟，得到了未补偿回弹的曲面件。通过回弹前、后的应力云图及Z向位移云图对比，发现回弹后的曲面件应力和曲率减小，曲面形状发生了改变。取模拟结果沿x,y方向的轮廓和目标形状对比，发现补偿回弹的模拟结果与目标轮廓接近，误差很小；未补偿回弹的模拟结果与目标形状存在偏差，误差较大。
（5）采用多点模具成形了多种典型曲面件，并进行了精度测量。发现补偿回弹的曲面件，其测量结果与目标轮廓吻合良好，未补偿回弹的测量结果误差较大。通过模拟、实验结果的误差对比，可以发现采用补偿回弹的模具型面成形的曲面误差范围小，和目标形状吻合，这也验证了文章中所述的回弹补偿计算方法具有良好的预测效果和应用价值。
At present, Sheet metal forming processes are widely used to produce three-dimensionalsurface in many fields, such as aerospace, automotive, aircraft manufacturing, vehicle body,shipbuilding and pressure vessel forming and so on. Stamping is one of the most commonsheet metal forming methods. For conventional stamping process, which involves a matchedsolid die set, its advantages are a short production time and high productivity. Nevertheless,large initial investments and long setup time make its processes inflexible and onlyprofitable for mass production and economically unsuitable for single or small batchproducts. Multi-point forming (MPF), a novel flexible forming technology develops wellrecent years. In MPF, conventional solid stamping dies are replaced by a pair of opposedreconfigurable dies comprised by punch matrices. Based on this technology, dieless, rapidand digital manufacturing of sheet metal parts can be realized. Comparing with conventionalstamping, MPF technology is more suitable for small-lot and individualized production, andresponding to customers’ requirements quickly, meanwhile, accelerating the productrenovation.
Comparing with conventional stamping, MPF technology is with advantages of rapid andflexible forming. However, whether it is conventional stamping or MPF, springback is aninevitable phenomenon in sheet metal forming, and it greatly affects the geometricalaccuracy of products. Springback caused by elastic recovery and release of residual stressafter forming, and the final shape of part depends on the value of springback. Once the valueof springback exceeds the allowable tolerance, it becomes defects and affects the wholeassembly of other parts. Moreover, it is an accumulated effect of the entire processing history.Hence the prediction of springback is difficult and remains an important problem in themanufacturing industry. The die surface of MPF can be adjusted quickly, realizing thereconfiguration of various curve surfaces; therefore, springback compensation method isvery suitable for MPF. Compensation method is an effective, practical method for controlling MPF springback. An algorithm was established for springback compensation inMPF by combination methods of theoretical analysis, numerical simulation and formingexperiment. Calculation formula for curvature before springback was obtained basing onthree material models, the method describing the shape of die-face after springbackmodification was proposed, Simulation and experimental results shows that thecompensation algorithm can effectively control errors caused by springback in MPF.
The main contents and conclusions are as follows:
(1)Calculation formulas for curvature before springback for single-curvature surface wasobtained basing on three material models, For irregular single curvature surface,interpolation and difference method are used to obtain continuous surface with first-orderderivative, then a cubic B-spline fitting method are used to obtain the smooth and continuoussurface. Comparison of the interpolation and difference method, it is found that theinterpolation method is with excellent accuracy.
(2) For irregular doubly curved surface, the calculation formula for curvature beforespringback was obtained basing on three material models. Then, with discrete method, thewhole surface was divided into tiny pieces, interpolation processing and Bezier surfaceblending methods are used to describe the surface after springback compensation. Due to thedie-face of MPF is composed by a series of discrete punch elements, the center position andthe points of tangency of punch elements are obtained basing on the surface after springbackcompensation, the adjusting heights of punch elements can be achieved for MPF diecompensation surface.
(3)Based on the ABAQUS software, the finite element models were established, and theunite selection, contact definition, friction coefficient, constraint and displacement boundarywere described in simulation. Simulating the process of MPF and springback, and theSpringback process is simulated combining explicit and implicit algorithm. The influencingfactors on the springback, such as, using cushion, material, thickness, radius of curvature,were researched. The results show: the greater the thickness is, the springback decreasesgradually; the greater elastic modulus is, smaller the springback after unloading is; with theradius of curvature decreases, the springback is reduced; in addition, with the increase ofcompensation coefficient, the error value is decreasing.
(4)Simulating the springback process of MPF on single-curvature surface and doubly curvedsurface. Effective stress distribution and Z-displacement distribution are achieved.Comparing the simulation result with the target shapes, it is found that the single-curvature surface is easily to generate large springback; for doubly curved surface, if the bendingdirection is coincident, the springback is smaller, otherwise, the springback is bigger.Comparing the simulation result with and without springback compensation, we found thatthe errors is very small with springback compensation, otherwise, the errors is relativelylarge.
(5) A series of experiments were carried out by means of MPF equipment, comparing theresults with compensation and without compensation, it is observed that simulated andmeasured results with compensation matched well with target shapes, and overall errors arerelatively small and satisfied processing precision. The compensation method presented inthis paper provides an effective calculation method to springback compensation ofsingle-curvature and doubly curved surface, and it is of good springback predition effect andapplication value.
引文
[1]常荣福.飞机钣金零件制造技术[M].北京：国防工业出版社，1992.
[2]崔令江.汽车覆盖件冲压成形技术[M].北京：机械工业出版社，2003
[3]赵军，马瑞.板材成形新技术及其发展趋势[J].金属成形工艺，2006,20（6）：1-9
[4] Li Mingzhe, Liu Yuhong, Su Shizhong et al. Multi-point forming: a flexible manufacturing methodfor a3-d surface sheet, Journal of Materials Processing Technology.1999,87:277~280
[5] Ming-Zhe Li, Zhong-Yi Cai, Zhen Sui et al. Multi-point forming technology for sheet metal,Journal of Materials Processing Technology,2002,129(1-3):333~338
[6] Mingzhe LI, Wenzhi FU, Yongsheng PEI and Zhen SUI.2000KN multi-point forming press and itsapplication to the manufacture of high-speed trains，Advanced Technology of Plasticity2002,Proceedings of the7th ICTP, Yokohama, Japan, Oct.,2002,2:979~984
[7] Li MZ, Cai ZY, Liu CG, Fu WZ. Recent developments in multi-point forming technology, Adv TechPlast2005;707-708
[8] Li MZ, Cai ZY, Sui Z, Li XJ. Principle and applications of multi-point matched-die forming forsheet metal. Proc IMechE B: J Eng Manuf2008;222:581-589
[9] Li Mingzhe, Cai Zhongyi, Liu Chunguo, Fu Wenzhi. Recent developments in multi-point formingtechnology, Advanced Technology of Plasticity2005(Edited by P.F. Bariani), Proceedings of the8thInternational Conference on Technology of Plasticity, Edizioni Progetto Padova, Verona, Italy,October9-13,2005,707~708
[10] Li, Ming-Zhe; Cai, Zhong-Yi; Liu, Chun-Guo，Flexible manufacturing of sheet metal parts based ondigitized-die，Robotics and Computer-Integrated Manufacturing，2007，23（1）：107-115
[11]丛莲莲，苏世忠，李明哲.双曲度覆盖件多点成形中回弹的数值模拟，塑性工程学报，2007，14（3）：12~15
[12] M.Z. Li, Z.Q. Hu, Z.Y. Cai, X.P. Gong.Research on new spinning process based on theprinciple of continuous Multi-point forming. The9th International Conference on technology ofplasticity.(ICTP2008)，2008，Gyeongju,Korea
[13]麻桂艳，付文智，李明哲，张传敏，马顺利.中厚板分段多点成形及其数值模拟，北京科技大学学报，2008，30（1）：67~70，76
[14]朱东波，孙琨，李涤尘，卢秉恒.板料成形回弹问题研究新进展.塑性工程学报，2000,7（1）：11-16
[15] Hill R. The mathematical theory of plasticity. Oxford, London,1950
[16] Gardiner, FJ. The springback of metals, Trans. ASME,1957,79:1-9
[17] Proksa F, Zur Theorie des plastischen Blechbiegens, Diss. TH Hannover,1958
[18] Proksa F, Plasticschen Biegen von Blechen, Stablbau,1959,28(2):29-36
[19] Crafoord R. Plastic sheet bending, G teborg,1970
[20]余同希，章亮炽.塑性弯曲理论及其应用[M].北京：科学出版社，1992
[21] TX Yu, W. Johnson, Influence of axial force on the elastic-plastic bending and springback of a beam,J Mech working tech,1982,6:5-21
[22] W. Johnson, TX Yu. Spingback after the biaxial elastic-plastic pure bending of a rectangularplate-Ⅰ, Int J mech sci,1981,23:619-630
[23] W. Johnson, TX Yu. On the range of applicability of results for the springback of anelastic/perfectly plastic rectangular plate after subjecting it to biaxial pure bending-Ⅱ,1981,23:631-637
[24] W. Johnson, TX Yu. On springback after the pure bending of beams and plates of elasticwork-hardening material-Ⅲ, Int J mech sci,1981,23:687-695
[25] Oliveira M.C., Alves J. L., Chaparro B.M., et al. Study on the influence of work-hardeningmodeling in springback prediction[J]. International journal of plasticity,2007,23(3):516~543
[26] Chu Chin-chan, elastic-plastic springback of sheet metals subjected to complex plane strain bendinghistories. International journal of solids and structures,1986,22(10):1071-1081
[27] Pourboghrat F, Chu E. Prediction of springback of side-wall curl in2-D draw bending. J Mater ProcTech1995;50:361-374
[28] Pourboghrat F, Chu E. Springback in plane strain stretch-draw sheet forming, Int J Mech Sci1995;36(3):327-341
[29] Morestin F, Boivin M, Silva C. Elasto plastic formulation using a kinematic hardening model forspringback analysis in sheet metal forming. J Mater Proc Tech1996;56:619-630
[30] Jenn-Terng Gau. A study of the influence of the Bauschinger effect on springback intwo-dimensional sheet metal forming, The USA: Department of Mechanical Engineering, School ofThe Ohio State University.1999
[31] Farhang Pourboghrat, Michael E. Karabin. Richard C. Becher. A hybrid membrane/shell method forcalculating springback of anisotropic sheet metals undergoing axisymmetric loading. Internationaljournal of plasticity,2000,16:677-700
[32] Z.T. Zhang, D. Lee. Development of a new model for plane strain bending and springback analysis.Journal of materials engineering and performance.1995,4:291-299
[33] Z.T. Zhang, S.J. Hu. Stress and residual stress distributions in plane strain bending. Internationaljournal of mechanical sciences,1998,40(6):533-543
[34] Xue P, Yu TX, Chu E. An energy approach for predicting springback of metal sheets afterdouble-curvature forming, Part I: axisymmetric stamping. Int J Mech Sci2001;43:893-1914
[35] Xue P, Yu TX, Chu E. An energy approach for predicting springback of metal sheets afterdouble-curvature forming, Part II: Unequal double-curvature forming. Int J Mech Sci2001;43:1915-1924
[36] Xue P, Yu TX, Chu E. Theoretical prediction of the springback of metal sheets after adouble-curvature forming operation. J Mater Process Technol1999;89-90:65-71
[37] Chuantao Wang, Gary Kinzel, Taylan Altan. Mathematical modeling of plane-strain bending ofsheet and plate. Journal of materials proceeding technology,1993,39:279-304
[38] Chuantao Wang, Gary Kinzel, Taylan Altan. Process simulation and springback control in planestrain sheet bending SAE Technical paper,1993
[39] DJ Zhang, ZS Cui, ZY Chen, XY Ruan. An analytical model for predicting sheet springback afterV-bending. Journal of Zhejiang University Science A,2007,8(2):237–244
[40] DJ Zhang, ZS Cui, XY Ruan, YQ Li. An analytical model for predicting springback and side wallcurl of sheet after U-bending. Computational Materials Science,2007,38:707–715
[41]刘克进.薄板冲压回弹试验研究及数值模拟对比分析.湖南大学硕士学位论文，2004
[42] Parsa MH, al Ahkami SN, Ettehad M. Experimental and finite element study on the spring back ofdouble curved aluminum/polypropylene/aluminum sandwich sheet. Mater Des2010;31(9):4174-4183
[43] Parsa MH, al ahkami SN, Pishbin H, Kazemi M. Investigating spring back phenomena in doublecurved sheet metals forming. Mater Des2012;41:326-337
[44]张乐乐，谭南林，焦风川. ANSYS辅助分析应用基础教程，清华大学出版社，北京交通大学出版社，2006
[45] Ahmad S, Irons BM, Zienkiewicz OC. Analysis of thick and thin shell structures by curved finiteelements. International Journal for Numerical Methods in Engineering,1970,2:419-451
[46] Buchter N et al. Three dimensional extension of non-linear shell formulation based on the enhancedassumed strain concept. International Journal for Numerical Methods in Engineering,1994,7:2551-2568
[47] Sansour C. A theory and finite element formulation of shells at finite deformation involvingthickness change. Archive of Applied Mechanics,1995,65:194-216
[48] Parisch H. A continuum-based shell theory for non-linear applications. International Journal forNumerical Methods in Engineering,1995,38:1855-1883
[49] Hauptmann R, Schweizerhof K. A systematic development of ‘solid-shell’element formulations forlinear and non-linear analyses employing only displacement degrees of freedom. InternationalJournal for Numerical Methods in Engineering,1998,42:49-69
[50] R. A. Fontes Valente. Developments on sheel and solid-shell finite elements technology in nonlinearcontinuum mechanics. PhD Thesis, Faculty of Engineering, University of Porto, December2004
[51] Montmayeur N, Staub C. Springback prediction with OPTRIS. In: Gelin JC, Picart P, editors.Proceedings of NUMISHEET’99.Besancon, France: University of ranche-Compte,1999:41-46
[52]汪晨，张质良.三维S—Rail板料的成形及回弹分析.模具工业，2000,8:17-19
[53] Lembit MK, Jerrell AN, Patricial O, Allan BP, et al. Non-linear finite element analysis ofspringback. Commun Numer Meth Engng1999;15:33-42
[54] A M Prior. Applications of implicit and explicit finite element techniques to metal forming. Journalof Material Proceesing technology[J],1994,45:649-656
[55] Changqing Du, Li Zhang, Neng-Ming Wang. Springback prediction in sheet forming simulation,SAE Trana.1993,940937:707-717
[56] Chuantao Wang, An industrial outlook for springback predictability’Measurement Reliability’ andcompensation technology[C]. Processing of NUMISHEET’2002,2002:597-604
[57]赵军，苏春建，官英平等. Ｕ形件弯曲影响回弹因素模拟分析.锻压技术，2007,3（66）：136-140
[58]罗云华，王磊.高强钢板冲压回弹影响因素研究.锻压技术，2009,34（1）：23-26
[59]谢晖.基于CAE仿真的冲压回弹影响因素研究.湖南大学学报（自然科学版），2003,30（5）：29-34
[60] Asnafi N. On springback of double-curved autobody panels. Int J Mech Sci2001;43:5-37
[61] Thomson P F, Kim J K. Springback and side-wall curl of Galvanized and Galvalume steel sheet.Jounal of Mechanical Working Technology,1989,19:233-238
[62] Livatyali, H., Wu H. C., Altan T.. Prodiction and elimination of springback in straight flangingusing computer-aided design methods: Part2: FEM predictions and tool design. Journal ofMaterials Processing Technology,2002,120(1-3):348-354
[63] Inamdar M. V., Date P. P., Sabnis S. V.. On the effects of geometric parameters on springback insheets of five materials subjected to air vee bending. Journal of Materials Processing Technology,2002,123(10):459-463
[64] Tan Z, Pesson B, Magnusson. An empiric model for controlling springback in V-die bending ofsheet metals. Journal of Materials Processing Technology,1992,34:449-455
[65]赵国伟，王元勋，陈建桥等.材料成形回弹的数值模拟与影响因素.锻压装备与制造经济数学，2005，（3）：55-58
[66]祝洪川，李荣锋，严龙.拉弯回弹试验参数的影响性研究.第十届全国塑性工程学术年会、第三届国际塑性加工先进技术研讨会论文集.南昌，2007
[67] Liu Gang, Lin Z, Xu W, et al. Variable blankholder force in U-shaped part forming for eliminatingspringback error. Journal of Materials Preocessing Technology,2002,120:259-264
[68] R.Ruffini, J.Cao. Using neural network for springback minimization in a channel forming process.Journal of material&manufacturing,1998,107:65-73
[69]刘迪辉，钟志华.拉延筋对回弹的影响机理研究[J].中国机械工程，2005,16（20）：1876-1879
[70] Ben Ayed L, Delameziere A, Batoz J, et al. Optimization of the blankholder force distribution indeep drawing[J].Proceedings of APOMAT, Morschach, Switzerland,2005
[71] Agus Dwi Anggono, Waluyo Adi Siswanto, Badrul Omar. Finite element simulation for springbackprediction compensation[C]. Proceeding of the international conference on advanced science,Engineering and information technology2011, Hotel Equatorial Bangi-Putrajaya, Malaysia,14-15January2011
[72] Sunseri M, Cao J, Karafillis AP, et al. Accommodation of springback error in channel forming usingactive binder force control numerical simulations and experiments [J]. ASME Journal ofEngineering Materials and Technology,1996,118:426
[73]张冬娟，板料冲压成形回弹理论及有限元数值模拟研究[D].上海：上海交通大学，2006
[74]阳湘安，阮锋.模具几何修正的回弹补偿方向分析，塑性工程学报，2010,17（2）：6-10
[75] Karafillis A P, Boyce MC. Tooling design in sheet metal forming using springback calculations,International journal of mechanics science,1992,34(2):113
[76] Wagoner RH, LI M, Gan W. Sheet springback: prediction and design. HSIMP2007: high speedindustrial manufacturing processes, Cetim,2007:1–7.
[77] RH Wagoner, M. Li Simulation of springback: Through-thickness integration International Journalof Plasticity,2007,23(3):345–360
[78] W Gan, RH Wagoner. Die design method for sheet springback. International Journal of MechanicalSciences,2004,46(7):1097-1113
[79]李明哲，姚建国，蔡中义，李淑慧.利用多点反复成形法减小回弹的研究，塑性工程学报，2000,7（1）：22-25
[80]李广权，李达，李东平，李明哲.板材多点成形时复杂边界条件的研究.哈尔滨工业大学学报，2000,32（5）：75-77
[81]严庆光，李明哲，崔相吉.多点成形中回弹量的测量与控制方法研究，材料科学与工艺，2004,12（4）：364-367
[82] Cai ZY, Li MZ, Lan YW. Three-dimensional sheet metal continuous forming process based onflexible roll bending: Principle and experiments. J Mater Proc Tech2012;212:120-127
[83] Cai, Zhong-Yi， Li, Ming-Zhe，A finite element method to generate digitized-die shape from themeasured data of desired part，International Journal of Advanced Manufacturing Technology，2006，30（1-2）：61-69
[84] Cai, Zhong-Yi; Li, Ming-Zhe; Chen, Xi-Di，Digitized die forming system for sheet metal andspringback minimizing technique，International Journal of Advanced Manufacturing Technology,2006，28（11-12）:1089-1096
[85] Cai, ZY; Li, MZ, Smoothing finite-element method for the reconstruction of surface from3Dscattered data, International e-Conference on Computer Science，2005，（5）：19-31
[86]蔡中义，李明哲，李湘吉.板材成形回弹数值分析的静力隐式方法，中国机械工程，2002,13（17）：1458-1461
[87]蔡中义，梁云赋，王少辉，李明哲.多点成形曲面的生成方法与软件开发，中国机械工程，2009，20（22）：2742~2745
[88]蔡中义，张海明，李光俊，李明哲.多点拉形数值模拟及模具型面补偿方法，吉林大学学报（工学版），2008，38（2）：329~333
[89]李婷.板材多点成形件曲面误差分析方法研究[D].长春：吉林大学材料科学与工程学院，2008
[90]王少辉，蔡中义，李明哲，梁云赋，多点拉形中回弹的影响因素研究，塑性工程学报2009，16（4），7~11
[91]王少辉,蔡中义,李明哲.多点拉形中回弹的数值模拟.塑性工程学报,2009,16(2):57-61
[92]张海明，蔡中义，李明哲.板材多点成形回弹补偿与控制的方法研究.塑性工程学报，2007,14（3）：32-35
[93]陈志红，李明哲，高占民.蒙皮多点拉形过程中成形缺陷的数值模拟.塑性工程学报，2007,14（3）：112-116
[94]陈喜娣.柱面件无压边多点成形中成形面的修正与回弹控制[J].五邑大学学报（自然科学版）,2010,24(3):37～40
[95]陈喜娣.板材多点成形中起皱和回弹的数值分析[D].长春：吉林大学材料科学与工程学院，2004
[96]孙刚，李明哲，崔相吉，邓玉山.不同的多点成形工艺方式对回弹的影响，北京科技大学学报，2006，28（3）：274~277
[97]孙刚.多点成形时工艺方式与变形缺陷的数值模拟研究[D].长春：吉林大学材料科学与工程学院，2006
[98]张庆芳，蔡中义，李明哲，张延.多点成形回弹补偿算法及其验证.吉林大学学报（工学版），2012,42（6）：1448-1452
[99]张庆芳，蔡中义，李明哲.板材成形回弹的理论分析与多点模具调形方法的研究，材料科学与工艺，2010,18（sup.1）：9-13
[100] Qingfang Zhang, Zhongyi Cai, Mingzhe Li. Study on springback compensation in Multi-pointforming, Advanced Materials Research,2011,189-193:2957-2960
[101] Qingfang Zhang, Zhongyi Cai, Yan Zhang, Mingzhe Li. Springback compensation method fordoubly curved plate in multi-point forming, Materials and Design,2013,47:377-385
[102]李淑慧，李明哲，蔡中义，姚建国.板材多点弯曲过程及回弹现象的数值模拟.农业机械学报，2000,31（1）：112-115
[103]程万军，李明哲，陈建军.板材多点模成形的成形极限及回弹研究，农业机械学报，2001,32（2）：96-98
[104]陈喜娣，蔡中义，李明哲.板材无压边多点成形中回弹的数值模拟，塑性工程学报，2003,10（5）：10-13
[105]刘启骞.型材多点弯曲中的成形缺陷及其抑制方法的数值模拟研究[D].长春，吉林大学材料科学与工程学院，2011
[106]孙刚，李明哲，蔡中义，付文智.多点成形时工艺方式与工件压痕的关系研究，材料科学与工艺，2004，12（4）：360~363
[107]宋雪松，蔡中义，李明哲，周朝晖.多点成形中弹性垫变形及其对成形结果影响的数值分析，第九届全国塑性工程学术年会论文摘要集，2005：112-115
[108]宋雪松，蔡中义，李明哲.多点成形中压痕的数值模拟及极限成形力的分析,材料科学与工艺，2004,12(4)：368~371
[109]李雪，李明哲，蔡中义.多点成形过程中弹性介质对成形质量的影响.哈尔滨工业大学学报，2005,37（2）：194-197
[110]李雪，李明哲，蔡中义.使用弹性介质的多点成形过程数值模拟研究，塑性工程学报，2003，5：20~24
[111]周朝晖，蔡中义，李明哲.多点模具拉形中回弹的数值分析，锻压技术，2006,2:94-96
[112]王少辉，蔡中义，李明哲，梁云赋.多点拉形中回弹的影响因素研究，塑性工程学报，2009,16（4）：7-11
[113] Fan SJ. A new extracting formula and a new distinguishing means on the one variable cubicequation. Nat. Sci. J Hainan NU1989;2(2):91-98
[114]刘花丽，郝艳莉.计算机辅助工程以弧长为参数的连续差值曲线拟合的生成，中州大学学报，2010,27（5）:118-120
[115]吴家麒，杨冬英，沈林勇，陈建军.基于曲率数据的曲线拟合方法研究，应用科学学报，2003,21（3）：258-262
[116] Clark J. Some Properties of B-Spline[C]. Second USA-JAPAN Computer Conference Proceedings,1975.
[117] Deboor C. A Practical Guide of Spline[J]. Applied Mathenatial Sciences Series,1978,27.
[118]梅向明，黄敬之.微分几何[M].北京：高等教育出版社，2001
[119] Deboor C. On Calculating with B-Spline[J]. J. Approx. Theory,1972,6:50-62.
[120] Cox M G. The Numerical Evaluation of B-Splines[J]. IMA Journal of Applied Mathematics,1971,2(10):134-149.
[121]黄明游,刘播,徐涛.数值计算方法[M].北京:科学出版社,2005.
[122]方忆湘，刘文学.基于集合特性的三次均匀B样条曲线构造描述[J].工程图学学报，2006(2)：96~102
[123]周士森，戴小娟，蔡玗涛.幂强化板材弯板力矩公式的推导，浙江工学院学报，1989,1:97-102
[124]朱心雄等，自由曲线曲面造型技术，北京：科学出版社，2000
[125]徐家川，雷雨成，王玉林，李旭. Bezier曲面拼接误差影响因素，材料工艺设备，2007：48-52
[126] Imageware中光滑拼接曲面的方法，现代设计与先进制造技术，2007,30（5）：43-47
[127] NURBS曲面重构中的几何连续性问题，大连：大连理工大学计算数学专业，2002
[128] Qin H, Terzopoulos D. Dynamic NURBS swung surfaces for physics-based shape design[J]. CADComputer Aided Design,1995,27(2):111-127.
[129] Shreiner D, Woo M, Neider J, et al. OpenGL Programming Guide[M].北京:机械工业出版社,2006.
[130]宋天霞，邹时智，杨文兵.非线性结构有限元计算[M].武汉：华中理工大学出版社，1996
[131] E. Onate, J. Rojek, C. G. Garino. NUMISTAMP: a research project for assessment of finite elementmodels for stamping processes [J]. Journal of Materials Processing Technology,1995,50(1):17-38
[132]赵海鸥. LS-DYNA动力分析指南[M].北京，2003.
[133] O.C. Zienkeiwicz, R.L. Taylor. The Finite Element Method, Volume2:0Solid Mechanics FifthEditon [M]. Butterworth-Heinemann,2000:217-242
[134] M.H. Dirikolu, E. Akdemir. Computer aided modeling of flexible forming process. J. Mater.Process. Technol,2004，148：376-381
[135] Q. Zhang, T.A. Dean, Z.R. Wang. Numerical simulation of deformation in multi-point sandwichforming. Int. J. Mach. Tools Manuf,2006,46:699-707
[136]石亦平，周玉蓉. ABAQUS有限元分析实例详解[M].北京：机械工业出版社，2006
[2]崔令江.汽车覆盖件冲压成形技术[M].北京：机械工业出版社，2003
[3]赵军，马瑞.板材成形新技术及其发展趋势[J].金属成形工艺，2006,20（6）：1-9
[4] Li Mingzhe, Liu Yuhong, Su Shizhong et al. Multi-point forming: a flexible manufacturing methodfor a3-d surface sheet, Journal of Materials Processing Technology.1999,87:277~280
[5] Ming-Zhe Li, Zhong-Yi Cai, Zhen Sui et al. Multi-point forming technology for sheet metal,Journal of Materials Processing Technology,2002,129(1-3):333~338
[6] Mingzhe LI, Wenzhi FU, Yongsheng PEI and Zhen SUI.2000KN multi-point forming press and itsapplication to the manufacture of high-speed trains，Advanced Technology of Plasticity2002,Proceedings of the7th ICTP, Yokohama, Japan, Oct.,2002,2:979~984
[7] Li MZ, Cai ZY, Liu CG, Fu WZ. Recent developments in multi-point forming technology, Adv TechPlast2005;707-708
[8] Li MZ, Cai ZY, Sui Z, Li XJ. Principle and applications of multi-point matched-die forming forsheet metal. Proc IMechE B: J Eng Manuf2008;222:581-589
[9] Li Mingzhe, Cai Zhongyi, Liu Chunguo, Fu Wenzhi. Recent developments in multi-point formingtechnology, Advanced Technology of Plasticity2005(Edited by P.F. Bariani), Proceedings of the8thInternational Conference on Technology of Plasticity, Edizioni Progetto Padova, Verona, Italy,October9-13,2005,707~708
[10] Li, Ming-Zhe; Cai, Zhong-Yi; Liu, Chun-Guo，Flexible manufacturing of sheet metal parts based ondigitized-die，Robotics and Computer-Integrated Manufacturing，2007，23（1）：107-115
[11]丛莲莲，苏世忠，李明哲.双曲度覆盖件多点成形中回弹的数值模拟，塑性工程学报，2007，14（3）：12~15
[12] M.Z. Li, Z.Q. Hu, Z.Y. Cai, X.P. Gong.Research on new spinning process based on theprinciple of continuous Multi-point forming. The9th International Conference on technology ofplasticity.(ICTP2008)，2008，Gyeongju,Korea
[13]麻桂艳，付文智，李明哲，张传敏，马顺利.中厚板分段多点成形及其数值模拟，北京科技大学学报，2008，30（1）：67~70，76
[14]朱东波，孙琨，李涤尘，卢秉恒.板料成形回弹问题研究新进展.塑性工程学报，2000,7（1）：11-16
[15] Hill R. The mathematical theory of plasticity. Oxford, London,1950
[16] Gardiner, FJ. The springback of metals, Trans. ASME,1957,79:1-9
[17] Proksa F, Zur Theorie des plastischen Blechbiegens, Diss. TH Hannover,1958
[18] Proksa F, Plasticschen Biegen von Blechen, Stablbau,1959,28(2):29-36
[19] Crafoord R. Plastic sheet bending, G teborg,1970
[20]余同希，章亮炽.塑性弯曲理论及其应用[M].北京：科学出版社，1992
[21] TX Yu, W. Johnson, Influence of axial force on the elastic-plastic bending and springback of a beam,J Mech working tech,1982,6:5-21
[22] W. Johnson, TX Yu. Spingback after the biaxial elastic-plastic pure bending of a rectangularplate-Ⅰ, Int J mech sci,1981,23:619-630
[23] W. Johnson, TX Yu. On the range of applicability of results for the springback of anelastic/perfectly plastic rectangular plate after subjecting it to biaxial pure bending-Ⅱ,1981,23:631-637
[24] W. Johnson, TX Yu. On springback after the pure bending of beams and plates of elasticwork-hardening material-Ⅲ, Int J mech sci,1981,23:687-695
[25] Oliveira M.C., Alves J. L., Chaparro B.M., et al. Study on the influence of work-hardeningmodeling in springback prediction[J]. International journal of plasticity,2007,23(3):516~543
[26] Chu Chin-chan, elastic-plastic springback of sheet metals subjected to complex plane strain bendinghistories. International journal of solids and structures,1986,22(10):1071-1081
[27] Pourboghrat F, Chu E. Prediction of springback of side-wall curl in2-D draw bending. J Mater ProcTech1995;50:361-374
[28] Pourboghrat F, Chu E. Springback in plane strain stretch-draw sheet forming, Int J Mech Sci1995;36(3):327-341
[29] Morestin F, Boivin M, Silva C. Elasto plastic formulation using a kinematic hardening model forspringback analysis in sheet metal forming. J Mater Proc Tech1996;56:619-630
[30] Jenn-Terng Gau. A study of the influence of the Bauschinger effect on springback intwo-dimensional sheet metal forming, The USA: Department of Mechanical Engineering, School ofThe Ohio State University.1999
[31] Farhang Pourboghrat, Michael E. Karabin. Richard C. Becher. A hybrid membrane/shell method forcalculating springback of anisotropic sheet metals undergoing axisymmetric loading. Internationaljournal of plasticity,2000,16:677-700
[32] Z.T. Zhang, D. Lee. Development of a new model for plane strain bending and springback analysis.Journal of materials engineering and performance.1995,4:291-299
[33] Z.T. Zhang, S.J. Hu. Stress and residual stress distributions in plane strain bending. Internationaljournal of mechanical sciences,1998,40(6):533-543
[34] Xue P, Yu TX, Chu E. An energy approach for predicting springback of metal sheets afterdouble-curvature forming, Part I: axisymmetric stamping. Int J Mech Sci2001;43:893-1914
[35] Xue P, Yu TX, Chu E. An energy approach for predicting springback of metal sheets afterdouble-curvature forming, Part II: Unequal double-curvature forming. Int J Mech Sci2001;43:1915-1924
[36] Xue P, Yu TX, Chu E. Theoretical prediction of the springback of metal sheets after adouble-curvature forming operation. J Mater Process Technol1999;89-90:65-71
[37] Chuantao Wang, Gary Kinzel, Taylan Altan. Mathematical modeling of plane-strain bending ofsheet and plate. Journal of materials proceeding technology,1993,39:279-304
[38] Chuantao Wang, Gary Kinzel, Taylan Altan. Process simulation and springback control in planestrain sheet bending SAE Technical paper,1993
[39] DJ Zhang, ZS Cui, ZY Chen, XY Ruan. An analytical model for predicting sheet springback afterV-bending. Journal of Zhejiang University Science A,2007,8(2):237–244
[40] DJ Zhang, ZS Cui, XY Ruan, YQ Li. An analytical model for predicting springback and side wallcurl of sheet after U-bending. Computational Materials Science,2007,38:707–715
[41]刘克进.薄板冲压回弹试验研究及数值模拟对比分析.湖南大学硕士学位论文，2004
[42] Parsa MH, al Ahkami SN, Ettehad M. Experimental and finite element study on the spring back ofdouble curved aluminum/polypropylene/aluminum sandwich sheet. Mater Des2010;31(9):4174-4183
[43] Parsa MH, al ahkami SN, Pishbin H, Kazemi M. Investigating spring back phenomena in doublecurved sheet metals forming. Mater Des2012;41:326-337
[44]张乐乐，谭南林，焦风川. ANSYS辅助分析应用基础教程，清华大学出版社，北京交通大学出版社，2006
[45] Ahmad S, Irons BM, Zienkiewicz OC. Analysis of thick and thin shell structures by curved finiteelements. International Journal for Numerical Methods in Engineering,1970,2:419-451
[46] Buchter N et al. Three dimensional extension of non-linear shell formulation based on the enhancedassumed strain concept. International Journal for Numerical Methods in Engineering,1994,7:2551-2568
[47] Sansour C. A theory and finite element formulation of shells at finite deformation involvingthickness change. Archive of Applied Mechanics,1995,65:194-216
[48] Parisch H. A continuum-based shell theory for non-linear applications. International Journal forNumerical Methods in Engineering,1995,38:1855-1883
[49] Hauptmann R, Schweizerhof K. A systematic development of ‘solid-shell’element formulations forlinear and non-linear analyses employing only displacement degrees of freedom. InternationalJournal for Numerical Methods in Engineering,1998,42:49-69
[50] R. A. Fontes Valente. Developments on sheel and solid-shell finite elements technology in nonlinearcontinuum mechanics. PhD Thesis, Faculty of Engineering, University of Porto, December2004
[51] Montmayeur N, Staub C. Springback prediction with OPTRIS. In: Gelin JC, Picart P, editors.Proceedings of NUMISHEET’99.Besancon, France: University of ranche-Compte,1999:41-46
[52]汪晨，张质良.三维S—Rail板料的成形及回弹分析.模具工业，2000,8:17-19
[53] Lembit MK, Jerrell AN, Patricial O, Allan BP, et al. Non-linear finite element analysis ofspringback. Commun Numer Meth Engng1999;15:33-42
[54] A M Prior. Applications of implicit and explicit finite element techniques to metal forming. Journalof Material Proceesing technology[J],1994,45:649-656
[55] Changqing Du, Li Zhang, Neng-Ming Wang. Springback prediction in sheet forming simulation,SAE Trana.1993,940937:707-717
[56] Chuantao Wang, An industrial outlook for springback predictability’Measurement Reliability’ andcompensation technology[C]. Processing of NUMISHEET’2002,2002:597-604
[57]赵军，苏春建，官英平等. Ｕ形件弯曲影响回弹因素模拟分析.锻压技术，2007,3（66）：136-140
[58]罗云华，王磊.高强钢板冲压回弹影响因素研究.锻压技术，2009,34（1）：23-26
[59]谢晖.基于CAE仿真的冲压回弹影响因素研究.湖南大学学报（自然科学版），2003,30（5）：29-34
[60] Asnafi N. On springback of double-curved autobody panels. Int J Mech Sci2001;43:5-37
[61] Thomson P F, Kim J K. Springback and side-wall curl of Galvanized and Galvalume steel sheet.Jounal of Mechanical Working Technology,1989,19:233-238
[62] Livatyali, H., Wu H. C., Altan T.. Prodiction and elimination of springback in straight flangingusing computer-aided design methods: Part2: FEM predictions and tool design. Journal ofMaterials Processing Technology,2002,120(1-3):348-354
[63] Inamdar M. V., Date P. P., Sabnis S. V.. On the effects of geometric parameters on springback insheets of five materials subjected to air vee bending. Journal of Materials Processing Technology,2002,123(10):459-463
[64] Tan Z, Pesson B, Magnusson. An empiric model for controlling springback in V-die bending ofsheet metals. Journal of Materials Processing Technology,1992,34:449-455
[65]赵国伟，王元勋，陈建桥等.材料成形回弹的数值模拟与影响因素.锻压装备与制造经济数学，2005，（3）：55-58
[66]祝洪川，李荣锋，严龙.拉弯回弹试验参数的影响性研究.第十届全国塑性工程学术年会、第三届国际塑性加工先进技术研讨会论文集.南昌，2007
[67] Liu Gang, Lin Z, Xu W, et al. Variable blankholder force in U-shaped part forming for eliminatingspringback error. Journal of Materials Preocessing Technology,2002,120:259-264
[68] R.Ruffini, J.Cao. Using neural network for springback minimization in a channel forming process.Journal of material&manufacturing,1998,107:65-73
[69]刘迪辉，钟志华.拉延筋对回弹的影响机理研究[J].中国机械工程，2005,16（20）：1876-1879
[70] Ben Ayed L, Delameziere A, Batoz J, et al. Optimization of the blankholder force distribution indeep drawing[J].Proceedings of APOMAT, Morschach, Switzerland,2005
[71] Agus Dwi Anggono, Waluyo Adi Siswanto, Badrul Omar. Finite element simulation for springbackprediction compensation[C]. Proceeding of the international conference on advanced science,Engineering and information technology2011, Hotel Equatorial Bangi-Putrajaya, Malaysia,14-15January2011
[72] Sunseri M, Cao J, Karafillis AP, et al. Accommodation of springback error in channel forming usingactive binder force control numerical simulations and experiments [J]. ASME Journal ofEngineering Materials and Technology,1996,118:426
[73]张冬娟，板料冲压成形回弹理论及有限元数值模拟研究[D].上海：上海交通大学，2006
[74]阳湘安，阮锋.模具几何修正的回弹补偿方向分析，塑性工程学报，2010,17（2）：6-10
[75] Karafillis A P, Boyce MC. Tooling design in sheet metal forming using springback calculations,International journal of mechanics science,1992,34(2):113
[76] Wagoner RH, LI M, Gan W. Sheet springback: prediction and design. HSIMP2007: high speedindustrial manufacturing processes, Cetim,2007:1–7.
[77] RH Wagoner, M. Li Simulation of springback: Through-thickness integration International Journalof Plasticity,2007,23(3):345–360
[78] W Gan, RH Wagoner. Die design method for sheet springback. International Journal of MechanicalSciences,2004,46(7):1097-1113
[79]李明哲，姚建国，蔡中义，李淑慧.利用多点反复成形法减小回弹的研究，塑性工程学报，2000,7（1）：22-25
[80]李广权，李达，李东平，李明哲.板材多点成形时复杂边界条件的研究.哈尔滨工业大学学报，2000,32（5）：75-77
[81]严庆光，李明哲，崔相吉.多点成形中回弹量的测量与控制方法研究，材料科学与工艺，2004,12（4）：364-367
[82] Cai ZY, Li MZ, Lan YW. Three-dimensional sheet metal continuous forming process based onflexible roll bending: Principle and experiments. J Mater Proc Tech2012;212:120-127
[83] Cai, Zhong-Yi， Li, Ming-Zhe，A finite element method to generate digitized-die shape from themeasured data of desired part，International Journal of Advanced Manufacturing Technology，2006，30（1-2）：61-69
[84] Cai, Zhong-Yi; Li, Ming-Zhe; Chen, Xi-Di，Digitized die forming system for sheet metal andspringback minimizing technique，International Journal of Advanced Manufacturing Technology,2006，28（11-12）:1089-1096
[85] Cai, ZY; Li, MZ, Smoothing finite-element method for the reconstruction of surface from3Dscattered data, International e-Conference on Computer Science，2005，（5）：19-31
[86]蔡中义，李明哲，李湘吉.板材成形回弹数值分析的静力隐式方法，中国机械工程，2002,13（17）：1458-1461
[87]蔡中义，梁云赋，王少辉，李明哲.多点成形曲面的生成方法与软件开发，中国机械工程，2009，20（22）：2742~2745
[88]蔡中义，张海明，李光俊，李明哲.多点拉形数值模拟及模具型面补偿方法，吉林大学学报（工学版），2008，38（2）：329~333
[89]李婷.板材多点成形件曲面误差分析方法研究[D].长春：吉林大学材料科学与工程学院，2008
[90]王少辉，蔡中义，李明哲，梁云赋，多点拉形中回弹的影响因素研究，塑性工程学报2009，16（4），7~11
[91]王少辉,蔡中义,李明哲.多点拉形中回弹的数值模拟.塑性工程学报,2009,16(2):57-61
[92]张海明，蔡中义，李明哲.板材多点成形回弹补偿与控制的方法研究.塑性工程学报，2007,14（3）：32-35
[93]陈志红，李明哲，高占民.蒙皮多点拉形过程中成形缺陷的数值模拟.塑性工程学报，2007,14（3）：112-116
[94]陈喜娣.柱面件无压边多点成形中成形面的修正与回弹控制[J].五邑大学学报（自然科学版）,2010,24(3):37～40
[95]陈喜娣.板材多点成形中起皱和回弹的数值分析[D].长春：吉林大学材料科学与工程学院，2004
[96]孙刚，李明哲，崔相吉，邓玉山.不同的多点成形工艺方式对回弹的影响，北京科技大学学报，2006，28（3）：274~277
[97]孙刚.多点成形时工艺方式与变形缺陷的数值模拟研究[D].长春：吉林大学材料科学与工程学院，2006
[98]张庆芳，蔡中义，李明哲，张延.多点成形回弹补偿算法及其验证.吉林大学学报（工学版），2012,42（6）：1448-1452
[99]张庆芳，蔡中义，李明哲.板材成形回弹的理论分析与多点模具调形方法的研究，材料科学与工艺，2010,18（sup.1）：9-13
[100] Qingfang Zhang, Zhongyi Cai, Mingzhe Li. Study on springback compensation in Multi-pointforming, Advanced Materials Research,2011,189-193:2957-2960
[101] Qingfang Zhang, Zhongyi Cai, Yan Zhang, Mingzhe Li. Springback compensation method fordoubly curved plate in multi-point forming, Materials and Design,2013,47:377-385
[102]李淑慧，李明哲，蔡中义，姚建国.板材多点弯曲过程及回弹现象的数值模拟.农业机械学报，2000,31（1）：112-115
[103]程万军，李明哲，陈建军.板材多点模成形的成形极限及回弹研究，农业机械学报，2001,32（2）：96-98
[104]陈喜娣，蔡中义，李明哲.板材无压边多点成形中回弹的数值模拟，塑性工程学报，2003,10（5）：10-13
[105]刘启骞.型材多点弯曲中的成形缺陷及其抑制方法的数值模拟研究[D].长春，吉林大学材料科学与工程学院，2011
[106]孙刚，李明哲，蔡中义，付文智.多点成形时工艺方式与工件压痕的关系研究，材料科学与工艺，2004，12（4）：360~363
[107]宋雪松，蔡中义，李明哲，周朝晖.多点成形中弹性垫变形及其对成形结果影响的数值分析，第九届全国塑性工程学术年会论文摘要集，2005：112-115
[108]宋雪松，蔡中义，李明哲.多点成形中压痕的数值模拟及极限成形力的分析,材料科学与工艺，2004,12(4)：368~371
[109]李雪，李明哲，蔡中义.多点成形过程中弹性介质对成形质量的影响.哈尔滨工业大学学报，2005,37（2）：194-197
[110]李雪，李明哲，蔡中义.使用弹性介质的多点成形过程数值模拟研究，塑性工程学报，2003，5：20~24
[111]周朝晖，蔡中义，李明哲.多点模具拉形中回弹的数值分析，锻压技术，2006,2:94-96
[112]王少辉，蔡中义，李明哲，梁云赋.多点拉形中回弹的影响因素研究，塑性工程学报，2009,16（4）：7-11
[113] Fan SJ. A new extracting formula and a new distinguishing means on the one variable cubicequation. Nat. Sci. J Hainan NU1989;2(2):91-98
[114]刘花丽，郝艳莉.计算机辅助工程以弧长为参数的连续差值曲线拟合的生成，中州大学学报，2010,27（5）:118-120
[115]吴家麒，杨冬英，沈林勇，陈建军.基于曲率数据的曲线拟合方法研究，应用科学学报，2003,21（3）：258-262
[116] Clark J. Some Properties of B-Spline[C]. Second USA-JAPAN Computer Conference Proceedings,1975.
[117] Deboor C. A Practical Guide of Spline[J]. Applied Mathenatial Sciences Series,1978,27.
[118]梅向明，黄敬之.微分几何[M].北京：高等教育出版社，2001
[119] Deboor C. On Calculating with B-Spline[J]. J. Approx. Theory,1972,6:50-62.
[120] Cox M G. The Numerical Evaluation of B-Splines[J]. IMA Journal of Applied Mathematics,1971,2(10):134-149.
[121]黄明游,刘播,徐涛.数值计算方法[M].北京:科学出版社,2005.
[122]方忆湘，刘文学.基于集合特性的三次均匀B样条曲线构造描述[J].工程图学学报，2006(2)：96~102
[123]周士森，戴小娟，蔡玗涛.幂强化板材弯板力矩公式的推导，浙江工学院学报，1989,1:97-102
[124]朱心雄等，自由曲线曲面造型技术，北京：科学出版社，2000
[125]徐家川，雷雨成，王玉林，李旭. Bezier曲面拼接误差影响因素，材料工艺设备，2007：48-52
[126] Imageware中光滑拼接曲面的方法，现代设计与先进制造技术，2007,30（5）：43-47
[127] NURBS曲面重构中的几何连续性问题，大连：大连理工大学计算数学专业，2002
[128] Qin H, Terzopoulos D. Dynamic NURBS swung surfaces for physics-based shape design[J]. CADComputer Aided Design,1995,27(2):111-127.
[129] Shreiner D, Woo M, Neider J, et al. OpenGL Programming Guide[M].北京:机械工业出版社,2006.
[130]宋天霞，邹时智，杨文兵.非线性结构有限元计算[M].武汉：华中理工大学出版社，1996
[131] E. Onate, J. Rojek, C. G. Garino. NUMISTAMP: a research project for assessment of finite elementmodels for stamping processes [J]. Journal of Materials Processing Technology,1995,50(1):17-38
[132]赵海鸥. LS-DYNA动力分析指南[M].北京，2003.
[133] O.C. Zienkeiwicz, R.L. Taylor. The Finite Element Method, Volume2:0Solid Mechanics FifthEditon [M]. Butterworth-Heinemann,2000:217-242
[134] M.H. Dirikolu, E. Akdemir. Computer aided modeling of flexible forming process. J. Mater.Process. Technol,2004，148：376-381
[135] Q. Zhang, T.A. Dean, Z.R. Wang. Numerical simulation of deformation in multi-point sandwichforming. Int. J. Mach. Tools Manuf,2006,46:699-707
[136]石亦平，周玉蓉. ABAQUS有限元分析实例详解[M].北京：机械工业出版社，2006