CN104573281B - A kind of complex space curved surfaces sheet forming die face design method for considering springback compensation - Google Patents

A kind of complex space curved surfaces sheet forming die face design method for considering springback compensation Download PDF

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CN104573281B
CN104573281B CN201510045973.1A CN201510045973A CN104573281B CN 104573281 B CN104573281 B CN 104573281B CN 201510045973 A CN201510045973 A CN 201510045973A CN 104573281 B CN104573281 B CN 104573281B
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杨忠炯
袁宏亮
周立强
李洪宾
鲁耀中
姜东升
高雨
王卉
周剑奇
董栋
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Central South University
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Abstract

The present invention relates to a kind of complex space curved surfaces sheet forming die face design method for considering springback compensation, plate shaping process of the Finite Element Method Simulation using ideal product profile as die face is used first, obtain the stress matrix of punching press plate node, then the compensation transposed matrix of curved surface node is obtained with the forward direction rebound method iterative of amendment, rear curved surface finally is compensated with the body deformation method fitting nodes of amendment, constantly repeats said process until the die face can stamp out qualified product.Wherein in order to improve compensation speed, by preceding to rebound method amendment, the amount of calculation of each iteration is reduced；And iteration step midpoint is to complex-curved reconstruct, the amendment body deformation algorithm in the inventive method reduces amount of calculation while maintaining body deformation algorithm structure curved surface slickness advantage, realizes the quick design of springback compensation die face.

Description

A kind of complex space curved surfaces sheet forming die face design method for considering springback compensation
Technical field
The present invention relates to a kind of die face design method for considering springback compensation, more particularly, to a kind of consideration springback compensation The design method of complex space curved surfaces sheet forming die face.
Background technology
A large amount of space curved surface thin plates, are to pass through such as automobile panel, sheet metal part, concrete mixer truck helical blade Mould punching obtains, and complex space curved surfaces thin plate resilience error is fairly obvious after punch forming, directly affects follow-up assembling Difficulty and efficiency, therefore need to correct mould repeatedly, untill stamping out morderate part.Traditional moulds modification method is held The continuous cycle is long, cost is high.And rebound phenomenon generally existing, and a kind of part punching rule is not easy to promote on a large scale；It is multiple in addition Miscellaneous space curved surface reconstructs according to conventional method, then not only time-consuming surface fitting effect but also poor.If can find a kind of low time into This and effective springback compensation solution, this raising to complex space curved surfaces sheet parts quality and production efficiency will be Greatly promote.
The complex space curved surfaces sheet forming die face design for solving the problems, such as to consider springback compensation is asked, it is necessary to solve two Topic：First, springback compensation method problem；2nd, complex space curved surfaces reconstruction.
Springback compensation principle is as follows：Because the elastic strain part of plate after punching press can recover, therefore resilience can occur for plate.If By reserve migration of the die face along rebound direction, using just consistent with design curved surface after the plate material to rebound of the mould punching.Resilience Compensation can use the faster forward direction rebound method of convergence to compensate, but each iteration needs to establish new FEM model to count Internal stress is calculated, calculation procedure is numerous and diverse and computationally intensive, if internal stress can be kept constant, but makes it be multiplied by one by previous iteration The zoom factor that result of calculation derives is walked, then can accelerate calculating speed.
The motion vector of limited node on die face can be obtained after springback compensation.If it is fitted using conventional method complicated Space curved surface can then have the problem of die face slickness difference, cause springback compensation failure and because discrete point is very more, fitting with Reparation can take a substantial amount of time.Body deformation method is a kind of model automatic correcting method, but calculating data volume is huge, and die face is The body that thickness is zero, algorithm can be modified according to this feature, simplify calculating process.
ABAQUS is the very strong commercial project analysis finite element software of a kind of powerful and versatility, and ABAQUS is soft in addition Part has very strong autgmentability, it is allowed to user according to oneself need write subprogram；ABAQUS script interfaces are one and are based on The program library of object, embed script Python, but extend Python scripts, be additionally provided about 500 it is right As model.
The content of the invention
It is an object of the invention to provide a kind of complex space curved surfaces sheet forming die face design side for considering springback compensation Method, accelerate consider springback compensation complex space curved surfaces sheet forming die face design speed, using amendment forward direction rebound method and The method that body deformation method combines, wherein the forward direction rebound method corrected realizes springback compensation, the body deformation method of amendment realizes die face Reconstruct, and the two-way Integrated designs of CAD-CAE-CAD are realized based on ABAQUS secondary development, set so as to accelerate die face in engineering practice Meter speed degree.
The quickening of die face design speed will mainly solve the two subject matters, one, the compensation speed of die face how to add block； 2nd, how surface reconstruction speed improves.The solution of specific explanation the two problems of the invention below.
First problem of the present invention is solved using the forward direction rebound method of amendment, before amendment to rebound method be using one with The compensating factor that the stamping parts and design outline deviation of compensation die face constantly adjust is multiplied by the internal stress conduct of a constant size Prestress application using the thin plate neutral surface shape after the deformation under prestressing force effect as die face, and compares this on thin plate Die face stamping parts and design outline deviation △ h, so constantly repeat, until △ h are in the range of allowable error.Using correction algorithm Avoid the need for establishing new punching press FEM model every time afterwards to calculate new internal stress, and only need to be according to the meter of previous iteration step Result is calculated to calculate the new springback compensation factor.
The Second Problem of this law invention realizes that body deformation algorithm can realize three well using the body deformation algorithm of amendment The entity reconstruct after entity deformation is tieed up, but amount of calculation is still very huge.And it is of the invention according to die face feature, by body deformation algorithm Amendment, makes it be specifically applied to the amendment of die face, so as to evade huge invalid computation data, and then accelerates die face structure speed Degree.Die face has following two features：1st, it is body that thickness is zero；2nd, project, have in die face and only one along pressing direction Point and the parameter coordinate value (u, v) on its projection plane are corresponding.Therefore die face can be represented with Bezier curved surfaces or B-spline surface, It is advantageous that any changed on three-dimension curved surface need to only change two parameters, while compensated in view of many times die face When need to only carry out local modification, and Bezier curved surfaces modification one angle point of modification will influence the shape of whole curved surface, therefore of the invention Die face is represented from local modification B-spline surface can be carried out.
A kind of complex space curved surfaces sheet forming die face design method for considering springback compensation, comprises the following steps：
(1) establish and ideal product shape h is shaped as with die faceLPunching press model, with finite element software simulate punch forming Process, plate node stress matrix is F before obtaining resilienceL；
(2) there is ideal product shape h with Finite Element Method SimulationLSheet model in-αnFL(αnRepresent that nth iteration is returned Play compensating factor) internal stress effect under deformation；
(3) plate neutral surface displacement of joint matrix s after deforming is calculatedn, intended here using the body deformation algorithm of amendment Node is closed, correction algorithm is as follows：
(3.1) it is curved surface h to set (u, v)LOn any node parameter coordinate, in stress-αnFLUnder effect, the node moves To (U, V), wherein U=U (u, v), V=V (u, v), it is assumed that phasor function f (u, v) represents the displacement of curved surface node (u, v), i.e.,：
Function f (u, v) is represented with B-spline method, obtains following expression：
In expression formula, n1、n2Respectively curved surface u to, v to nodal point number, mi,jRepresent deformation rear curved surface dnControl vertex, yi,jRepresent deformation front curve hLControl vertex, L, M are constant, represent the power of corresponding B-spline basic function, Ci,LAnd C (u)j,M (v) it is the basic function of B-spline surface, there is following expression：
In formula：0≤u≤1,0≤v≤1；K=0,1,2 ... L；L=0,1,2 ... M
(3.2) by hLNode coordinate Pi,jIt is converted into control vertex coordinate yi,j, shown in transformation for mula such as formula (4)~(7).
First calculate n2Individual u is to polygon
Make Qi,j=yi,j(i=1,2 ..., n1+ 2, j=1,2 ..., n2)
N is calculated again1+ 2 v are to polygon
N can obtain by formula (6) and formula (7)1+ 2 v respectively have n to polygon2+ 2 summits, polygon is formed by it Grid yi,j(i=1,2 ..., n1+2；J=1,2 ..., n2+2).Can the given n of interpolation by the B-spline surface of the mesh definition1 ×n2Individual node.
(3.3)mi,jTo deform rear curved surface dnControl vertex, in order to deformation rear curved surface curvature mutation at node carry out These nodes are handled by adjustment by means of function (8).
Wherein N represents node quantity known to displacement, and T first half is the motion vector of i-th of nodeWith amendment Front and rear curved surface displacement of joint vector f (ui,vi) difference quadratic sum, and the T of latter halfRIt is thin plate strain energy function, diagonal matrix ||β||《1；
TRExpression is as follows：
Wherein μ is a normal matrix, determines each several part (such as stretch-proof, counter-bending) physical deformation energy in curved surface energy function Proportion,
0 ＜ | | μ | | ＜ 1.
(3.4) control vertex mi,jExtremum conditions is taken by T to obtain
(3.5) all m have been tried to achievei,jAfterwards, new curved surface dnExpression formula is as follows：
(4) with Finite Element Method Simulation with dnFor the Sheet Metal Stamping Process of die face, and should in remnants with plate is simulated without modulus method Deformation under power effect, obtains the displacement of joint matrix u after resiliencen；
(5) it is fitted with the body deformation algorithm of amendment by transposed matrix unThe node of sign obtains resilience rear curved surface hn；
(6) h is comparednWith hLDeviation；
(7) the forward direction resilience factor-alpha of the (n+1)th iteration step is calculated if deviation goes beyond the scopen+1, specific solution procedure is as follows：
(7.1) α known to0、α1、…、αn-2、αn△ h corresponding to compensating factor0=h0-hL、△h1=h1-hL、…△hn=hn- hL, following formula is calculated successively；
……
Wherein
(7.2) above-mentioned formula is added to obtain f (α) expression formula：
R in above formulan(α) is remainder of interpolation, and its expression formula is：
Rn(α)=f [α, α0,…,αn](α-α0)(α-α1)…(α-αn-1)(α-αn)
(7.3) after trying to achieve Newton interpolation Equation f (α), Equation f (α)=0 is made, the solution tried to achieve is before following iteration walks To springback compensation factor-alphan+1；
(7.4) skip to step (2) and continue iteration；
(8) stop iteration if deviation is in the range of allowable error and determine dnFor die face.
The beneficial effects of the invention are as follows：The purpose for considering that the die face of springback compensation quickly designs is realized, has been evaded because using Conventional method, there is the problems such as slow convergence rate, compensation curved surface slickness difference, drastically increase mold design efficiency, mitigate The workload of designer.
Brief description of the drawings
Fig. 1 is complex space curved surfaces sheet forming die face design flow chart；
Fig. 2 finite element punching press models；
Fig. 3 is to design node stress before the plate neutral surface resilience that mould punching of the curved surface as die face obtains；
Fig. 4 is to be preceding to springback compensation algorithm schematic diagram；
Fig. 5 resilience postjunction motion vector figures；
Fig. 6 is to be preceding to resilience factor interpolation schematic diagram；
Fig. 7 is to be contrasted using the die face before and after body deformation algorithm；
Fig. 8 is die face comparison diagram before and after concrete mixer truck helical blade springback compensation.
The present invention is elaborated with reference to the accompanying drawings and examples.
Embodiment
It is a kind of consider springback compensation complex space curved surfaces sheet forming die face design method, basic procedure as shown in figure 1, Illustrate the specific implementation process of this method by taking concrete mixer truck helical blade as an example.The punching press model of foundation as shown in Fig. 2 its Mould in the expression of middle the top curved surface, mould under the expression of two guide pillars of band, middle expression blank.Each parameter setting in ABAQUS As shown in table 1.
(1) simulation is with ideal product profile hLFor the punching course of die face, the thin plate node stress F before resilience is obtainedL, Its Stress Map is as shown in Figure 3；
(2) as schematically shown in Figure 4, with node stress matrix FLMultiply diagonal matrix αnObtained-αnFLIt is carried in as prestressing force On plate node, plate is calculated in prestressing force-αnFLDeformation under effect, obtains displacement of joint matrix sn, as shown in Figure 5；
(3) with the body deformation algorithm fitting surface of amendment
(3.1) n is taken here1=n2=100, then number of network nodes is 100 × 100
100 u are first calculated to polygon：
y1,j=P1,j；y102,j=P100,j, j=1,2 ..., 100 (5)
Make Qi,j=yi,j(i=1,2 ..., 102；J=1,2 ..., 100)
102 v are calculated again to polygon：
yi,1=Qi,1；yi,102=Qi,100, i=1,2 ..., 102 (7)
(3.2) B-spline basic function is calculated, takes L=M=3
(3.3) node deformable B-spline function is established
(3.4) curved surface warping function is established
(3.5) d is solvednControl vertex mi,j
OrderObtain mI, j。
(3.6) surface equation solves
(4) curved surface d is obtainednAfterwards, by curved surface dnAs die face and the punching press model similar to Fig. 2 is established, simulates punching press Journey.With the deformation without modulus method simulation plate under residual stress effect, node position (i.e. after resilience) after residual stress effect is obtained Move un；
(5) equally it is fitted to obtain curved surface h with the method as shown in step (3)n,
(6) h is comparednWith ideal product profile hLWhether deviation is in error range；
(7.1) α known to0、α1、…、αn-2、αn△ h corresponding to compensating factor0=h0-hL、△h1=h1-hL、…△hn=hn- hL, following formula is calculated successively；
……
Wherein
(7.2) above-mentioned formula is added to obtain f (α) expression formula：
R in above formulan(α) is remainder of interpolation, and its expression formula is：
Rn(α)=f [α, α0,…,αn](α-α0)(α-α1)…(α-αn-1)(α-αn)
(7.3) after trying to achieve Newton interpolation Equation f (α), Equation f (α)=0 is made, the solution tried to achieve is before following iteration walks To springback compensation factor-alphan+1；
(7.4) use-αn+1FLThe node prestressing force of n iteration step plate is replaced, and skips to step 2；
(8) by d if deviation is in allowed bandnAs die face, design process is completed.
Final springback compensation result is as shown in Figure 8.
The finite element software parameter setting of table 1
In Fig. 4, hLRepresent ideal product thin plate neutrality facial contour；h0Initial mould punching press plate neutral surface resilience is used in expression Each displacement of joint vector afterwards；h1Each displacement of joint is vectorial after representing first time springback compensation mould punching plate neutral surface resilience； Wherein FLInter-node stress matrix before expression plate material to rebound；F1Represent compensation die face d for the first time1Inter-node before punching press plate material to rebound Stress matrix；α0Zero degree iteration (before compensation) springback compensation factor is represented, therefore its diagonal element is all taken as 0；α1Represent first The secondary iteration springback compensation factor, it is all 1 to take diagonal matrix all elements；A, B, C correspond respectively to initial mould punching press plate time Shape, first time compensate shape after die face punching press plate material to rebound after shape, resilience before bullet.
△ h in Fig. 60Represent after springback compensation front mold punching press plate material to rebound neutral surface shape and design outline deviation to Amount, △ h1Represent the die face d after first time springback compensation1Neutral surface shape and design outline bias vector after punching press plate material to rebound, △h2Represent the die face d after second of springback compensation2Punching press plate material to rebound neutral surface shape and design outline bias vector；α0Table Show the springback compensation factor before compensating；α1Represent the first time iteration springback compensation factor, α2Represent second of iteration springback compensation because Son；Point A, B, C represent zero degree, first time, second of iteration respectively, and D represents the preferable springback compensation factor.
(1) represents that the fitting die face of the body deformation algorithm of amendment is not used in Fig. 7, and (2) represent to deform using the body of amendment Die face after algorithm.
Thin plate neutral surface comparison diagram such as Fig. 8 institutes before and after the springback compensation obtained after carrying out springback compensation with the inventive method Show, wherein 1 represents die face after springback compensation, 2 represent die face before springback compensation.It can be seen that fitting surface is smooth, meet design It is required that.
Computing power parameter for numerical simulation：Processor is Intel (R) Core (TM) i3-2120CPU@ 3.30GHZ, inside save as 4.00GB.The use of the conventional algorithm progress springback compensation calculating time is 131min, uses the inventive method The calculating time is 97min, calculates time more conventional numerical method and reduces 25.9%；It is maximum after blade springback compensation after springback compensation Deviation is reduced to 2.6mm from 17.1mm respectively, and resilience deviation reduces 84.8%, springback compensation positive effect.
The characteristics of present industrial product is, the speed of update is fast and pursues and becomes more meticulous, so often product Part category is more and batch is small；And product design is complicated and changeable.And rebound phenomenon generally existing, carry out resilience benefit with commonsense method Repay, compensation speed is slow and the problems such as curved surface slickness easily occurs in compensation rear mold face, so the Fast design method tool of the present invention There are theoretical research and practical significance.

Claims (1)

  1. A kind of 1. complex space curved surfaces sheet forming die face design method for considering springback compensation, it is characterised in that including following step Suddenly：
    (1) establish and ideal product shape h is shaped as with die faceLPunching press model, with finite element software simulate punch forming process, Plate node stress matrix is F before obtaining resilienceL；
    (2) there is ideal product shape h with Finite Element Method SimulationLSheet model in-αnFLInternal stress effect under deformation, αn Represent the nth iteration springback compensation factor；
    (3) using the body deformation algorithm fitting node of amendment, plate neutral surface displacement of joint matrix s after deformation is calculatedn, repair Normal operation method is as follows：
    (3.1) it is curved surface h to set (u, v)LOn any node parameter coordinate, in stress-αnFLUnder effect, the node move to (U, V), wherein U=U (u, v), V=V (u, v), it is assumed that phasor function f (u, v) represents the displacement of curved surface node (u, v), i.e.,：
    <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>U</mi> <mo>-</mo> <mi>u</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>V</mi> <mo>-</mo> <mi>v</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
    Function f (u, v) is represented with B-spline method, obtains following expression：
    <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> </mrow> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>2</mn> </mrow> </munderover> <mrow> <mo>(</mo> <msub> <mi>n</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>y</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <msub> <mi>C</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>L</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> <msub> <mi>C</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>M</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>,</mo> <mn>0</mn> <mo>&amp;le;</mo> <mi>u</mi> <mo>&amp;le;</mo> <mn>1</mn> <mo>;</mo> <mn>0</mn> <mo>&amp;le;</mo> <mi>v</mi> <mo>&amp;le;</mo> <mn>1</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
    In expression formula, n1、n2Respectively curved surface u to, v to nodal point number, mi,jRepresent deformation rear curved surface dnControl vertex, yi,j Represent deformation front curve hLControl vertex, L, M are constant, represent the power of corresponding B-spline basic function, Ci,LAnd C (u)j,M(v) For the basic function of B-spline surface, there is following expression：
    <mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>L</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>L</mi> <mo>!</mo> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>L</mi> <mo>-</mo> <mi>k</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>r</mi> </msup> <mfrac> <mrow> <mo>(</mo> <mi>L</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> <mo>!</mo> </mrow> <mrow> <mi>r</mi> <mo>!</mo> <mrow> <mo>(</mo> <mi>L</mi> <mo>+</mo> <mn>1</mn> <mo>-</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>!</mo> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <mi>u</mi> <mo>+</mo> <mi>L</mi> <mo>-</mo> <mi>k</mi> <mo>-</mo> <mi>r</mi> <mo>)</mo> </mrow> <mi>L</mi> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>C</mi> <mrow> <mi>l</mi> <mo>,</mo> <mi>M</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>M</mi> <mo>!</mo> </mrow> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>r</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>M</mi> <mo>-</mo> <mi>l</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mi>r</mi> </msup> <mfrac> <mrow> <mo>(</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> <mo>!</mo> </mrow> <mrow> <mi>r</mi> <mo>!</mo> <mrow> <mo>(</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>-</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>!</mo> </mrow> </mfrac> <msup> <mrow> <mo>(</mo> <mi>v</mi> <mo>+</mo> <mi>M</mi> <mo>-</mo> <mi>l</mi> <mo>-</mo> <mi>r</mi> <mo>)</mo> </mrow> <mi>M</mi> </msup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
    In formula：0≤u≤1,0≤v≤1；K=0,1,2 ..., L；L=0,1,2 ..., M
    (3.2) by hLNode coordinate Pi,jIt is converted into control vertex coordinate yi,j, transformation for mula such as formula (4)~(7)；
    First calculate n2Individual u is to polygon
    <mrow> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>y</mi> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    Make Qi,j=yi,j, i=1,2 ..., n1+ 2, j=1,2 ..., n2
    N is calculated again1+ 2 v are to polygon
    <mrow> <msub> <mi>y</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>Q</mi> <mrow> <mi>i</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>;</mo> <msub> <mi>y</mi> <mrow> <mi>i</mi> <mo>,</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>Q</mi> <mrow> <mi>i</mi> <mo>,</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </msub> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
    N can obtain by formula (6) and formula (7)1+ 2 v respectively have n to polygon2+ 2 summits, polygonal mesh is formed by it yi,j, i=1,2 ..., n1+2；J=1,2 ..., n2+2；Can the given n of interpolation by the B-spline surface of the mesh definition1×n2 Individual node；
    (3.3)mi,jTo deform rear curved surface dnControl vertex, in order to deformation rear curved surface curvature mutation at node be adjusted, These nodes are handled by means of function (8)；
    <mrow> <mi>T</mi> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msup> <mrow> <mo>&amp;lsqb;</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>v</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>s</mi> <mi>n</mi> <mi>i</mi> </msubsup> <mo>&amp;rsqb;</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>&amp;beta;T</mi> <mi>R</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
    Wherein N represents node quantity known to displacement, and T first half is the motion vector of i-th of nodeBefore and after amendment Curved surface displacement of joint vector f (ui,vi) difference quadratic sum, and the TR of latter half is thin plate strain energy function, diagonal matrix | | β | |《1；
    TRExpression is as follows：
    <mrow> <msub> <mi>T</mi> <mi>R</mi> </msub> <mo>=</mo> <munder> <mo>&amp;Integral;</mo> <mo>&amp;Sigma;</mo> </munder> <mo>{</mo> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msup> <mo>&amp;part;</mo> <mn>2</mn> </msup> <mi>f</mi> </mrow> <mrow> <mo>&amp;part;</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <msup> <mo>&amp;part;</mo> <mn>2</mn> </msup> <mi>f</mi> </mrow> <mrow> <mo>&amp;part;</mo> <msup> <mi>v</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mn>2</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>&amp;mu;</mi> <mo>)</mo> </mrow> <mo>&amp;lsqb;</mo> <mrow> <mfrac> <mrow> <msup> <mo>&amp;part;</mo> <mn>2</mn> </msup> <mi>f</mi> </mrow> <mrow> <mo>&amp;part;</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mfrac> <mrow> <msup> <mo>&amp;part;</mo> <mn>2</mn> </msup> <mi>f</mi> </mrow> <mrow> <mo>&amp;part;</mo> <msup> <mi>v</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>-</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <msup> <mo>&amp;part;</mo> <mn>2</mn> </msup> <mi>f</mi> </mrow> <mrow> <mo>&amp;part;</mo> <mi>u</mi> <mo>&amp;part;</mo> <mi>v</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>&amp;rsqb;</mo> <mo>}</mo> <mi>d</mi> <mi>u</mi> <mi>d</mi> <mi>v</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
    Wherein μ is a normal matrix, determines the proportion of stretch-proof and counter-bending physical deformation energy in curved surface energy function,
    0 ＜ | | μ | | ＜ 1；
    (3.4) control vertex mi,jExtremum conditions is taken by T to obtain
    <mrow> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>T</mi> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> <mo>;</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>2</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
    (3.5) all m have been tried to achievei,jAfterwards, new curved surface dnExpression formula is as follows：
    <mrow> <msub> <mi>d</mi> <mi>n</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>+</mo> <mn>2</mn> </mrow> </munderover> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>2</mn> </mrow> </munderover> <msub> <mi>m</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>L</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> <msub> <mi>C</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>M</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>
    (4) with Finite Element Method Simulation with dnFor the Sheet Metal Stamping Process of die face, and acted on plate is simulated without modulus method in residual stress Under deformation, obtain the displacement of joint matrix u after resiliencen；
    (5) it is fitted with the body deformation algorithm of amendment by transposed matrix unThe node of sign obtains resilience rear curved surface hn；
    (6) h is comparednWith hLDeviation；
    (7) the forward direction resilience factor-alpha of the (n+1)th iteration step is calculated if deviation goes beyond the scopen+1, specific solution procedure is as follows：
    (7.1) α known to0、α1、…、αn-2、αn△ h corresponding to compensating factor0=h0-hL、△h1=h1-hL、…△hn=hn-hL, Following formula is calculated successively；
    Wherein
    (7.2) above-mentioned formula is added to obtain f (α) expression formula：
    <mrow> <mtable> <mtr> <mtd> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mo>&amp;lsqb;</mo> <msub> <mi>&amp;alpha;</mi> <mn>0</mn> </msub> <mo>&amp;rsqb;</mo> <mo>+</mo> <mi>f</mi> <mo>&amp;lsqb;</mo> <msub> <mi>&amp;alpha;</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mn>1</mn> </msub> <mo>&amp;rsqb;</mo> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>-</mo> <msub> <mi>&amp;alpha;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mo>&amp;lsqb;</mo> <msub> <mi>&amp;alpha;</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mn>2</mn> </msub> <mo>&amp;rsqb;</mo> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>-</mo> <msub> <mi>&amp;alpha;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>-</mo> <msub> <mi>&amp;alpha;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mn>...</mn> <mo>+</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>f</mi> <mo>&amp;lsqb;</mo> <msub> <mi>&amp;alpha;</mi> <mn>0</mn> </msub> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>n</mi> </msub> <mo>&amp;rsqb;</mo> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>-</mo> <msub> <mi>&amp;alpha;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>-</mo> <msub> <mi>&amp;alpha;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mn>...</mn> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>-</mo> <msub> <mi>&amp;alpha;</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>R</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>
    R in above formulan(α) is remainder of interpolation, and its expression formula is：
    Rn(α)=f [α, α0,…,αn](α-α0)(α-α1)…(α-αn-1)(α-αn)
    (7.3) after trying to achieve Newton interpolation Equation f (α), Equation f (α)=0 is made, the solution tried to achieve is that the forward direction of following iteration step returns Play compensating factor αn+1；
    (7.4) skip to step (2) and continue iteration；
    (8) stop iteration if deviation is in the range of allowable error and determine dnFor die face.
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