汽车车身部件一步逆成形有限元法与碰撞仿真研究
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摘要
在汽车设计制造的整个周期中，车身模具特别是汽车覆盖件模具的设计制造水平，一直是制约汽车产品开发速度与品质的核心因素。基于形变理论的一步逆成形有限元方法作为一种可成形性的快速评价工具非常适合在零件设计阶段和模具设计阶段应用。本文结合国家自然科学基金重点项目 “冲压成形与模具设计的基础理论、计算方法和关键技术”（批准号：19832020）以及国家杰出科学青年基金项目 “基于KBE的冲压模具CAD专家系统理论以及聚合物板材成形理论与计算方法研究”（批准号：10125208），对汽车车身部件一步逆成形有限元法以及它在翻边成形的坯料尺寸预示，弯曲成形的卸载回弹和引入工艺因素的碰撞仿真等方面的应用进行了深入的研究。
一步逆成形有限元法假定成形过程是比例加载的，而且仅考虑初始的毛坯和变形终了的状态，不考虑变形的中间状态，其基本思想为：从最终构形C出发，将其作为最终构形的中面，通过有限元方法确定在满足一定的边界条件下最终构形中各个节点P在初始构形毛坯C0中的位置P0，比较毛坯和最终构形中节点的位置可得到最终构形中应变，应力和厚度的分布。
基于离散的Kichhoff假设，最终构形中任意一点到q点变形梯度张量的逆[F]-1为：
其中： （1）
从上式可以得到左柯西－格林(Cauchy-Green)张量[B]为：
（2）
通过计算[B]-1的特征值和特征向量就可以求出单元面内两个主伸长λ1、λ2和它们的转换矩阵[M]。根据体积不变的假设，单元厚度方向的主伸长λ3可以通过面内的两个主伸长λ1、λ2的求出。最终单元的对数应变可由下式求出：
（3）
一步逆成形有限元法假设板料的弹塑性变形过程与加载路径无关，所以其基于形变理论的弹塑性本构方程为：
（4）
其中为等效应变，为等效应力，各项异性系数r = (r0 + 2r45 + r90)/4。
由于一步逆成形有限元方法不能根据加载历史处理接触问题，所以工具（凸模，凹模和拉伸筋等）对板料的作用都是通过简单的等效外力替代的。于是就可以得到以下的非线性方程组
（5）
上述非线性方程组一般采用Newton-Raphson迭代方法进行求解，其迭代格式为：
（6）
其中ω为迭代收敛松弛因子，[KT(ui)]为第i迭代步的切线刚度矩阵。
一步逆成形有限元法Newton-Raphson迭代初始解{U0}的求解采用基于线弹性反向变形初始解预示算法，其基本思想是忽略各种外载荷，假设最终构形是通过线弹性反向变形到初始构形的，然后根据线弹性有限元方法在初始板料上建立平衡方程，通过迭代求解可以使初始板料的节点残余力满足事先给定的收敛条件，最后求得一步逆成形的初始解。
对那些包含弯曲压料面成形复杂的覆盖件，一步逆成形有限元法必须考虑弯曲压料面的影响。考虑弯曲压料面的一步逆成形有限元法分两个步骤进行，第一步由最终构形回退到事先给定的弯曲压料面上，即中间构形；第二步由弯曲压料面展开成平坯料即初始构形。弯曲压料面的投影算法就是将节点投影到一个给定
的曲面有限元网格上的过程，投影算法过程分成两个步骤即全局搜索和局部搜索。全局搜索确定需要局部求交单元的范围，局部搜索采用弧长法找出中间构形节点所对应的压料面单元。
利用考虑弯曲压料面的一步逆成形有限元法分别对伸长类平面翻边、收缩类平面翻边、伸长类曲面翻边和收缩类曲面翻边这四种典型情况下的翻边成形进行模拟计算并预示其修边线的位置，最后将其预示出的修边线位置用增量法有限元法进行效核以验证该方法的准确性。计算结果表明利用一步逆成形有限元方法可以事先得到所需翻边高度，因此在生产实践中就不需要反复修正修边模轮廓，不仅可以节约大量材料与人工浪费，也可以缩短整个制模与调模周期，大大减少制模成本。
文中采用一步逆成形有限元法模拟板材弯曲成形过程，然后采用非线性弹性静力隐式有限元法预示卸载回弹过程。模拟卸载回弹过程的非线性有限元法是在弹塑性大变形有限元法的基础上，采用以Mindlin理论为基础的Belytschko-Lin-Tsay壳单元模型。
采用逐级更新Lagrange法，板材卸载回弹有限元离散化的单元平衡方程为：
（7）
式中k为局部坐标系下的单元刚度矩阵：
（8）
d是单元节点位移向量：
（9）
f是单元节点内力向量：
（10）
采用上述方法模拟了Numisheet’2002的标准考题之一“无约束圆柱弯曲成形”的卸载回弹过程，并与实验结果进行了比较；预示了货车纵梁的冲压成形过程和脱模卸载回弹过程，回弹模拟后将纵梁6个关键截面的口宽与实测值进行了比较，比较结果说明该回弹数值模拟方法对弯曲成形卸载回弹预示是有效的。
汽车车身部件碰撞仿真是整个汽车碰撞安全性仿真研究的基础。由拉伸实验测出的材料参数是碰撞仿真有限元模型中的一个重要参数。但是拉伸实验所采用的试样通常都取自毛坯而不是最终的冲压件，而加工过程特别是冲压将引起制件
中厚度分布、残余应变和应力等变化。这些变化都会导致碰撞仿真产生误差，然而以?
During the automotive design and manufacture period, it is the main factor of auto product development speed and quality that the design and manufacture level of the auto body die especially auto panel die. The one step inverse forming finite element method based on deformation theory will became a quickly formability evaluation tool and it is applied suitably in part and die design phase. The research contents of this paper are the one step inverse forming FEM of auto body part and applications in the blank sheet prediction of flange forming, the spring back simulation of bend forming and the crash simulation with forming effects. The research works are supported by the key project of National Natural Science Funds “The basic theory, calculation method and key technique of stamping and die design” (19832020) and the key project of National Outstanding Scientific Youth Funds “The CAD expert system theory of stamping design based on KBE and the research on the polymer forming theory and calculation method” (10125208).
The main assumption adopted by the one step inverse forming FEM is the proportional loading assumption. It only considers the initial and the final state without considering the medium state. The basic idea of this method is that we look for the nodal positions in the initial flat blank from the final part, and then the distribution of strain, stress and thickness in the final configuration can be calculated by comparing the nodal position in the initial flat sheet with the one of the final part. Using a generalized Kirchhoff assumption, the initial and the final position vectors of a material point q can be expressed with respect to point p on the mid-surface of C. The inverse
Cauchy-Green left tensor can be obtained by
with (1)
The inverse Cauchy-Green left tensor can be obtained by
(2)
The eigenvalue calculation of [B]-1 gives two principal plane stretches λ1, λ2 and their direction transformation matrix [M]. Then, the thickness stretch λ3 is calculated by the incompressibility assumption. Finally, the logarithmic strains are obtained
(3)
The elastic-plastic deformation is assumed to be independent on the loading path and a total constitutive law is obtained.
(4)
Where and are the total equivalent strain and the equivalent stress. The mean planar isotropic coefficient r is obtained from the three anisotropic coefficients: r = (r0 + 2r45 + r90)/4.
The inverse forming FEM cannot deal with contact problems depending on the loading history, so the tool actions (punch, die and drawbead) are simply represented by some external forces at the final configuration. Then, a non-linear equilibrium system can be obtained.
(5)
This non-linear equilibrium system can be calculated by the Newton-Raphson iterative method.
(6)
Where ω is the relaxation factor and [KT(ui)] is the tangent stiffness matrix in the i iterative step.
The initial solution of one step inverse forming FEM for Newton-Raphson iteration is calculated by the initial solution predictive algorithm based on linear elastic reverse deformation. The basic idea is that we assume the initial flat blank translate into the final part by linear elastic reverse deformation neglecting all of external load. Then the equilibrium equations are established by the linear elastic FEM in the initial flat blank. The nodal residual force vector in the initial flat blank will satisfy the initial convergence conditions by iterative calculation. Finally we can get the initial solution.
The one step inverse forming FEM must consider the curve binder effect for the complicated auto panel with curve binder. The one step inverse forming FEM with curve binder has two steps. The first step is from the final part to the given curve binder surface, which is named medium configuration. The second step is from the curve binder to the initial flat blank. The projection algorithm of the curve binder is t
一步逆成形有限元法假定成形过程是比例加载的，而且仅考虑初始的毛坯和变形终了的状态，不考虑变形的中间状态，其基本思想为：从最终构形C出发，将其作为最终构形的中面，通过有限元方法确定在满足一定的边界条件下最终构形中各个节点P在初始构形毛坯C0中的位置P0，比较毛坯和最终构形中节点的位置可得到最终构形中应变，应力和厚度的分布。
基于离散的Kichhoff假设，最终构形中任意一点到q点变形梯度张量的逆[F]-1为：
其中： （1）
从上式可以得到左柯西－格林(Cauchy-Green)张量[B]为：
（2）
通过计算[B]-1的特征值和特征向量就可以求出单元面内两个主伸长λ1、λ2和它们的转换矩阵[M]。根据体积不变的假设，单元厚度方向的主伸长λ3可以通过面内的两个主伸长λ1、λ2的求出。最终单元的对数应变可由下式求出：
（3）
一步逆成形有限元法假设板料的弹塑性变形过程与加载路径无关，所以其基于形变理论的弹塑性本构方程为：
（4）
其中为等效应变，为等效应力，各项异性系数r = (r0 + 2r45 + r90)/4。
由于一步逆成形有限元方法不能根据加载历史处理接触问题，所以工具（凸模，凹模和拉伸筋等）对板料的作用都是通过简单的等效外力替代的。于是就可以得到以下的非线性方程组
（5）
上述非线性方程组一般采用Newton-Raphson迭代方法进行求解，其迭代格式为：
（6）
其中ω为迭代收敛松弛因子，[KT(ui)]为第i迭代步的切线刚度矩阵。
一步逆成形有限元法Newton-Raphson迭代初始解{U0}的求解采用基于线弹性反向变形初始解预示算法，其基本思想是忽略各种外载荷，假设最终构形是通过线弹性反向变形到初始构形的，然后根据线弹性有限元方法在初始板料上建立平衡方程，通过迭代求解可以使初始板料的节点残余力满足事先给定的收敛条件，最后求得一步逆成形的初始解。
对那些包含弯曲压料面成形复杂的覆盖件，一步逆成形有限元法必须考虑弯曲压料面的影响。考虑弯曲压料面的一步逆成形有限元法分两个步骤进行，第一步由最终构形回退到事先给定的弯曲压料面上，即中间构形；第二步由弯曲压料面展开成平坯料即初始构形。弯曲压料面的投影算法就是将节点投影到一个给定
的曲面有限元网格上的过程，投影算法过程分成两个步骤即全局搜索和局部搜索。全局搜索确定需要局部求交单元的范围，局部搜索采用弧长法找出中间构形节点所对应的压料面单元。
利用考虑弯曲压料面的一步逆成形有限元法分别对伸长类平面翻边、收缩类平面翻边、伸长类曲面翻边和收缩类曲面翻边这四种典型情况下的翻边成形进行模拟计算并预示其修边线的位置，最后将其预示出的修边线位置用增量法有限元法进行效核以验证该方法的准确性。计算结果表明利用一步逆成形有限元方法可以事先得到所需翻边高度，因此在生产实践中就不需要反复修正修边模轮廓，不仅可以节约大量材料与人工浪费，也可以缩短整个制模与调模周期，大大减少制模成本。
文中采用一步逆成形有限元法模拟板材弯曲成形过程，然后采用非线性弹性静力隐式有限元法预示卸载回弹过程。模拟卸载回弹过程的非线性有限元法是在弹塑性大变形有限元法的基础上，采用以Mindlin理论为基础的Belytschko-Lin-Tsay壳单元模型。
采用逐级更新Lagrange法，板材卸载回弹有限元离散化的单元平衡方程为：
（7）
式中k为局部坐标系下的单元刚度矩阵：
（8）
d是单元节点位移向量：
（9）
f是单元节点内力向量：
（10）
采用上述方法模拟了Numisheet’2002的标准考题之一“无约束圆柱弯曲成形”的卸载回弹过程，并与实验结果进行了比较；预示了货车纵梁的冲压成形过程和脱模卸载回弹过程，回弹模拟后将纵梁6个关键截面的口宽与实测值进行了比较，比较结果说明该回弹数值模拟方法对弯曲成形卸载回弹预示是有效的。
汽车车身部件碰撞仿真是整个汽车碰撞安全性仿真研究的基础。由拉伸实验测出的材料参数是碰撞仿真有限元模型中的一个重要参数。但是拉伸实验所采用的试样通常都取自毛坯而不是最终的冲压件，而加工过程特别是冲压将引起制件
中厚度分布、残余应变和应力等变化。这些变化都会导致碰撞仿真产生误差，然而以?
During the automotive design and manufacture period, it is the main factor of auto product development speed and quality that the design and manufacture level of the auto body die especially auto panel die. The one step inverse forming finite element method based on deformation theory will became a quickly formability evaluation tool and it is applied suitably in part and die design phase. The research contents of this paper are the one step inverse forming FEM of auto body part and applications in the blank sheet prediction of flange forming, the spring back simulation of bend forming and the crash simulation with forming effects. The research works are supported by the key project of National Natural Science Funds “The basic theory, calculation method and key technique of stamping and die design” (19832020) and the key project of National Outstanding Scientific Youth Funds “The CAD expert system theory of stamping design based on KBE and the research on the polymer forming theory and calculation method” (10125208).
The main assumption adopted by the one step inverse forming FEM is the proportional loading assumption. It only considers the initial and the final state without considering the medium state. The basic idea of this method is that we look for the nodal positions in the initial flat blank from the final part, and then the distribution of strain, stress and thickness in the final configuration can be calculated by comparing the nodal position in the initial flat sheet with the one of the final part. Using a generalized Kirchhoff assumption, the initial and the final position vectors of a material point q can be expressed with respect to point p on the mid-surface of C. The inverse
Cauchy-Green left tensor can be obtained by
with (1)
The inverse Cauchy-Green left tensor can be obtained by
(2)
The eigenvalue calculation of [B]-1 gives two principal plane stretches λ1, λ2 and their direction transformation matrix [M]. Then, the thickness stretch λ3 is calculated by the incompressibility assumption. Finally, the logarithmic strains are obtained
(3)
The elastic-plastic deformation is assumed to be independent on the loading path and a total constitutive law is obtained.
(4)
Where and are the total equivalent strain and the equivalent stress. The mean planar isotropic coefficient r is obtained from the three anisotropic coefficients: r = (r0 + 2r45 + r90)/4.
The inverse forming FEM cannot deal with contact problems depending on the loading history, so the tool actions (punch, die and drawbead) are simply represented by some external forces at the final configuration. Then, a non-linear equilibrium system can be obtained.
(5)
This non-linear equilibrium system can be calculated by the Newton-Raphson iterative method.
(6)
Where ω is the relaxation factor and [KT(ui)] is the tangent stiffness matrix in the i iterative step.
The initial solution of one step inverse forming FEM for Newton-Raphson iteration is calculated by the initial solution predictive algorithm based on linear elastic reverse deformation. The basic idea is that we assume the initial flat blank translate into the final part by linear elastic reverse deformation neglecting all of external load. Then the equilibrium equations are established by the linear elastic FEM in the initial flat blank. The nodal residual force vector in the initial flat blank will satisfy the initial convergence conditions by iterative calculation. Finally we can get the initial solution.
The one step inverse forming FEM must consider the curve binder effect for the complicated auto panel with curve binder. The one step inverse forming FEM with curve binder has two steps. The first step is from the final part to the given curve binder surface, which is named medium configuration. The second step is from the curve binder to the initial flat blank. The projection algorithm of the curve binder is t
引文
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