CN114692455A - Modeling method of composite material laminated plate based on segmented equidistant recombination curve - Google Patents
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Abstract
Description
技术领域technical field
本发明涉及复合材料建模技术领域,尤其涉及一种基于分割等距重组曲线的复合材料层合板的建模方法。The invention relates to the technical field of composite material modeling, in particular to a modeling method for composite material laminates based on split equidistant recombination curves.
背景技术Background technique
目前复合材料层合板的有限元模型包括常刚度有限元模型和变刚度有限元模型,而在生成变刚度有限元模型时,为了节省建模时间往往选用理想型层合板结构来替代,但是理想型层合板结构与真实层合板结构还是存在差异的;选用带丝束宽度的变刚度层合板结构则会使得丝束间出现重叠和间隙等缺陷,这些缺陷会使得建模步骤较为复杂;在定义限位变角度轨迹时,传统方法采用已有曲线来对目标层合板进行初始轨迹定义,然而这种方法在一定程度上限制了设计思路;且在有限元模型分析的过程中,用户需自行建模,建模完毕后自行提交分析,结果出来之后还需用户在结果中逐一检索目标数据,导致工作效率较低。At present, the finite element model of composite laminates includes constant stiffness finite element model and variable stiffness finite element model. When generating a variable stiffness finite element model, in order to save modeling time, an ideal laminate structure is often used instead, but the ideal type There are still differences between the laminate structure and the real laminate structure; the selection of the variable stiffness laminate structure with the tow width will cause defects such as overlap and gap between the tows, which will make the modeling steps more complicated; When the trajectory of the displacement angle is changed, the traditional method uses the existing curve to define the initial trajectory of the target laminate, but this method limits the design idea to a certain extent; and in the process of finite element model analysis, the user needs to model by himself. , after the modeling is completed, the analysis is submitted by oneself, and the user needs to retrieve the target data one by one in the results after the results come out, resulting in low work efficiency.
发明内容SUMMARY OF THE INVENTION
本发明公开的一种基于分割等距重组曲线的复合材料层合板的建模方法,解决了现有复合材料层合板建模方法存在的建模步骤复杂、丝束间易出现重叠和间隙、工作效率较低的问题,用户简单输入参数即可得到复合材料常/变刚度层合板带丝束宽度的有限元模型,建模方法简单且其丝束间不会产生重叠和间隙。The present invention discloses a modeling method for composite material laminates based on split equidistant recombination curves, which solves the problems of complex modeling steps, easy overlap and gaps between tows, and work problems existing in the existing composite material laminates modeling methods. For the problem of low efficiency, the user can obtain the finite element model of the composite constant/variable stiffness laminate with tow width simply by inputting parameters. The modeling method is simple and the tows will not produce overlap and gaps.
为达到上述目的,本发明的技术方案具体是这样实现的:In order to achieve the above object, the technical scheme of the present invention is specifically realized in this way:
本发明公开一种基于分割等距重组曲线的复合材料层合板的建模方法,包括以下步骤:The invention discloses a modeling method for a composite material laminate based on a split equidistant recombination curve, comprising the following steps:
输入层合板的尺寸参数、铺层结构、定义曲线函数、孔径及孔的位置信息;Input the dimensional parameters of the laminate, the layer structure, the definition curve function, the hole diameter and the position information of the holes;
对参考轨迹进行散点化处理;Scatter the reference trajectory;
对轨迹点进行拟合并分别对凹凸函数进行等距重组;Fit the trajectory points and reorganize the bump function equidistantly;
输出等距曲线坐标及函数表达式参数;Output isometric curve coordinates and function expression parameters;
生成变刚度层合板模型。Generate a variable stiffness laminate model.
进一步地,对参考轨迹进行散点化处理的具体步骤包括:Further, the specific steps of performing scatter processing on the reference trajectory include:
定义变角度轨迹;Define the variable angle trajectory;
生成一条变角度参考轨迹;Generate a variable angle reference trajectory;
将生成的变角度轨迹数据输入到程序中;Input the generated variable angle trajectory data into the program;
找到参考轨迹上二阶导数为0的分割点;Find the split point on the reference trajectory where the second derivative is 0;
对分割曲线沿曲率降低的方向进行等距偏置;Offset the split curves equally in the direction of decreasing curvature;
对偏置后的轨迹进行平移重组;Perform translational reorganization of the offset trajectory;
求解程函方程的解。Solve the solution of the functional equation.
进一步地,所述对分割曲线沿曲率降低的方向进行等距偏置的距离等于偏置距离与偏置数量的乘积。Further, the distance at which the division curves are equidistantly offset along the direction of decreasing curvature is equal to the product of the offset distance and the number of offsets.
进一步地,所述程函方程为:Further, the described program function equation is:
其中,参考曲线Γ在整个平面域Ω上传播,曲线上的每一个点都具有与曲线正交的恒定速度,T(x)是曲线Γ以等于1/f的速度到达点x所需的时间,T(x)与x到Γ的距离成正比。where the reference curve Γ propagates over the entire planar domain Ω, each point on the curve has a constant velocity orthogonal to the curve, and T(x) is the time required for the curve Γ to reach point x with a velocity equal to 1/f , T(x) is proportional to the distance from x to Γ.
进一步地,采用机器学习的方法对轨迹点进行拟合。Further, a machine learning method is used to fit the trajectory points.
有益技术效果:Beneficial technical effects:
本发明公开一种基于分割等距重组曲线的复合材料层合板的建模方法,包括以下步骤:输入层合板的尺寸参数、铺层结构、定义曲线函数、孔径及孔的位置信息;对参考轨迹进行散点化处理;对轨迹点进行拟合并分别对凹凸函数进行等距重组;输出等距曲线坐标及函数表达式参数;生成变刚度层合板模型,解决了现有复合材料层合板建模方法存在的建模步骤复杂、丝束间易出现重叠和间隙、工作效率较低的问题,用户简单输入参数即可得到复合材料常/变刚度层合板带丝束宽度的有限元模型,建模方法简单且其丝束间不会产生重叠和间隙。The invention discloses a modeling method for composite material laminates based on split equidistant recombination curves. Scattering processing; fitting the trajectory points and equidistantly recombining the concave and convex functions; outputting the equidistant curve coordinates and function expression parameters; The method has the problems of complex modeling steps, easy overlap and gap between tows, and low work efficiency. The user can obtain the finite element model of composite constant/variable stiffness laminate with tow width by simply inputting parameters. The method is simple and there is no overlap or gap between the tows.
附图说明Description of drawings
为了更清楚地说明本发明的技术方案,下面将对实施例描述中所需要使用的附图作简单地介绍。In order to illustrate the technical solutions of the present invention more clearly, the following briefly introduces the accompanying drawings used in the description of the embodiments.
图1为本发明所述的一种基于分割等距重组曲线的复合材料层合板的建模方法的步骤流程图;1 is a flow chart of the steps of a method for modeling a composite material laminate based on a split equidistant recombination curve according to the present invention;
图2为本发明所述的一种基于分割等距重组曲线的复合材料层合板的建模方法的逻辑流程图;Fig. 2 is a logic flow chart of a method for modeling a composite material laminate based on a split equidistant recombination curve according to the present invention;
图3为本发明所述的一种基于分割等距重组曲线的复合材料层合板的建模方法的轨迹分割点示意图;FIG. 3 is a schematic diagram of a trajectory division point of a method for modeling a composite material laminate based on a split equidistant recombination curve according to the present invention;
图4为本发明所述的一种基于分割等距重组曲线的复合材料层合板的建模方法的轨迹偏置示意图;4 is a schematic diagram of trajectory offset of a method for modeling composite laminates based on split equidistant recombination curves according to the present invention;
图5为本发明所述的一种基于分割等距重组曲线的复合材料层合板的建模方法的轨迹重组示意图;5 is a schematic diagram of the trajectory reorganization of a method for modeling a composite laminate based on a split equidistant reorganization curve according to the present invention;
图6为本发明所述的一种基于分割等距重组曲线的复合材料层合板的建模方法的程函方程传播路径图;Fig. 6 is the propagation path diagram of the function equation of the modeling method of the composite material laminate based on the split equidistant recombination curve according to the present invention;
图7为本发明所述的一种基于分割等距重组曲线的复合材料层合板的建模方法的带宽度的纤维铺放示意图;7 is a schematic diagram of fiber laying with width of a modeling method of a composite material laminate based on a split equidistant recombination curve according to the present invention;
图8为本发明所述的一种基于分割等距重组曲线的复合材料层合板的建模方法的机器学习拟合曲线数据图;Fig. 8 is a machine learning fitting curve data diagram of a method for modeling a composite material laminate based on a split equidistant recombination curve according to the present invention;
图9为本发明所述的一种基于分割等距重组曲线的复合材料层合板的建模方法的生成带宽度的变刚度层合板的示意图。FIG. 9 is a schematic diagram of generating a variable stiffness laminate with a width by a modeling method for a composite laminate based on a split equidistant recombination curve according to the present invention.
图10为本发明所述的一种基于分割等距重组曲线的复合材料层合板的建模方法的某种飞机的翼型曲线;10 is an airfoil curve of a certain aircraft based on a modeling method of a composite material laminate based on a split equidistant recombination curve according to the present invention;
图11为本发明所述的一种基于分割等距重组曲线的复合材料层合板的建模方法的二维坐标到三维坐标转换示意图;11 is a schematic diagram of the conversion from two-dimensional coordinates to three-dimensional coordinates of a method for modeling a composite material laminate based on a split equidistant recombination curve according to the present invention;
图12为本发明所述的一种基于分割等距重组曲线的复合材料层合板的建模方法的翼型面铺放角度示意图;12 is a schematic diagram of the laying angle of the airfoil surface of a modeling method for a composite material laminate based on a split equidistant recombination curve according to the present invention;
图13为本发明所述的一种基于分割等距重组曲线的复合材料层合板的建模方法的翼型面纤维铺放示意图。FIG. 13 is a schematic diagram of laying fibers on the airfoil of a modeling method for a composite material laminate based on a split equidistant recombination curve according to the present invention.
具体实施方式Detailed ways
下面详细描述本发明的实施例,所述实施例的示例在附图中示出,其中自始至终相同或类似的标号表示相同或类似的元件或具有相同或类似功能的元件。下面通过参考附图描述的实施例是示例性的,仅用于解释本发明,而不能理解为对本发明的限制。The following describes in detail the embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein the same or similar reference numerals refer to the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary, only used to explain the present invention, and should not be construed as a limitation of the present invention.
下面结合附图对本发明的实施方式进行详细说明。The embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
本发明公开一种基于分割等距重组曲线的复合材料层合板的建模方法,参见图1,具体包括以下步骤:The present invention discloses a modeling method for a composite material laminate based on a split equidistant recombination curve, referring to FIG. 1 , which specifically includes the following steps:
S1:输入层合板的尺寸参数、铺层结构、定义曲线函数、孔径及孔的位置信息;S1: Input the size parameters of the laminate, the layer structure, the definition curve function, the hole diameter and the position information of the holes;
具体地,复合材料层合板的类型包括常刚度层合板和变刚度层合板,根据输入的层合板的尺寸参数,孔信息等来预生成前模型,以便后续对模型进行修改以及保存。Specifically, the types of composite laminates include constant-stiffness laminates and variable-stiffness laminates, and the pre-model is pre-generated according to the input size parameters of the laminate, hole information, etc., so that the model can be modified and saved later.
S2:对参考轨迹进行散点化处理并分别对凹凸函数进行等距重组;S2: Scattering the reference trajectory and equidistantly reorganizing the concave-convex function;
将轨迹散点化后,分别对凹凸函数进行等距重组,以得到重组曲线,具体包括以下步骤:After the trajectories are scattered, the concave and convex functions are recombined equidistantly to obtain the recombination curve, which specifically includes the following steps:
S21:定义变角度轨迹;S21: Define the variable angle track;
S22:生成一条变角度参考轨迹;S22: Generate a variable-angle reference trajectory;
变角度轨迹定义中线性变角度法为:The linear variable angle method in the variable angle trajectory definition is:
其中,θx(x)表示层合板纤维在坐标x处的角度,T0为纤维起始角度,T1为纤维终止角度,a为特征长度;Among them, θ x (x) represents the angle of the laminate fiber at the coordinate x, T 0 is the starting angle of the fiber, T 1 is the ending angle of the fiber, and a is the characteristic length;
变角度轨迹定义中流场函数法为:The flow field function method in the variable angle trajectory definition is:
其中,ψ为等流线,x,y为流线簇铺层的点坐标;Among them, ψ is the iso-streamline, and x, y are the point coordinates of the streamline cluster layup;
所述变角度轨迹定义中贝塞尔曲线法为:The Bezier curve method in the variable angle trajectory definition is:
其中,β1为控制点P1的位置参数;β2为控制点P2在x轴方向上的位置参数;h为控制点P1、P2间在x轴方向上距离。在β1和β2确定情况下,h能决定控制点P2和P3在y轴方向上的位置,Ni,j为三次NURBS曲线的基函数,w为权因子;Among them, β 1 is the position parameter of the control point P 1 ; β 2 is the position parameter of the control point P 2 in the x-axis direction; h is the distance between the control points P 1 and P 2 in the x-axis direction. When β 1 and β 2 are determined, h can determine the position of the control points P 2 and P 3 in the y-axis direction, Ni ,j is the basis function of the cubic NURBS curve, and w is the weight factor;
S23:将生成的变角度轨迹数据输入到程序中;S23: Input the generated variable-angle trajectory data into the program;
S24:找到参考轨迹上二阶导数为0的分割点;S24: Find the segmentation point where the second derivative is 0 on the reference trajectory;
具体地,找到参考轨迹二阶导数为0处的分割点,即f″(x)=0,参见图3;Specifically, find the division point where the second derivative of the reference trajectory is 0, that is, f″(x)=0, see FIG. 3 ;
S25:对分割曲线沿曲率降低的方向进行等距偏置;S25: Perform equidistant offset on the segmentation curve along the direction of decreasing curvature;
具体地,参见图4,以分割点处的法线为界,将曲线视为若干部分,分别对凹凸曲线部分进行沿曲率降低方向的等距偏置,即凹函数向下偏置,凸函数向上偏置,其偏移的距离等于偏置距离与偏置数量的乘积;Specifically, referring to Fig. 4, taking the normal at the dividing point as a boundary, the curve is regarded as several parts, and the concave and convex curve parts are respectively offset along the curvature decreasing direction equidistantly, that is, the concave function is biased downward, and the convex function is offset downward. Offset upward, the offset distance is equal to the product of the offset distance and the offset number;
S26:对偏置后的轨迹进行平移重组;S26: Perform translational reorganization on the offset trajectory;
将各部分偏置后的轨迹重新组合成新的轨迹,可以理解的是,此时得到的如图5中曲线与给定的参考轨迹并不存在平移或平行上的行位关系,即不存在丝束间的重叠和间隙;Recombining the offset trajectories of each part into a new trajectory, it can be understood that the curve obtained at this time as shown in Figure 5 does not have a translation or parallel line relationship with the given reference trajectory, that is, there is no Overlaps and gaps between tows;
S27:求解程函方程的解。S27: Solve the solution of the function equation.
具体地,程函方程可为: Specifically, the program function equation can be:
其中,参考曲线Γ在整个平面域Ω上传播,曲线上的每一个点都具有与曲线正交的恒定速度,T(x)是曲线Γ以等于1/f的速度到达点x所需的时间,T(x)与x到Γ的距离成正比。where the reference curve Γ propagates over the entire planar domain Ω, each point on the curve has a constant velocity orthogonal to the curve, and T(x) is the time required for the curve Γ to reach point x with a velocity equal to 1/f , T(x) is proportional to the distance from x to Γ.
具体地,参见图6,生成一组等距曲线可以分为以下步骤:1)在网格上定义参考曲线Γ来构造上述方程中的初始条件;2)在网格节点处求解上述方程中T的值;3)从每个网格单元的恒定值方向计算T。为了更好的表述等距曲线的算法,本文也给出了等距曲线的参数方程表达式。设平面内光滑曲线L的直角坐标方程为y=f(x),过曲线L上任意点p(x,y)的法线Lf方程为曲线L距离为常数D的等距曲线Ld的点应满足如下参数方程:Specifically, referring to Fig. 6, generating a set of equidistant curves can be divided into the following steps: 1) defining the reference curve Γ on the grid to construct the initial conditions in the above equation; 2) solving T in the above equation at the grid nodes 3) Calculate T from the constant value direction of each grid cell. In order to better describe the algorithm of the equidistant curve, the parametric equation expression of the equidistant curve is also given in this paper. Let the rectangular coordinate equation of the smooth curve L in the plane be y=f(x), and the equation of the normal L f passing through any point p(x, y) on the curve L is: The points of the equidistant curve L d whose distance from the curve L is a constant D should satisfy the following parametric equation:
或or
S3:对轨迹点进行拟合;S3: Fit the trajectory points;
为了得到轨迹点拟合后的轨迹曲线,采用机器学习的方法来对轨迹点进行拟合,由于轨迹点为离散点的集合,若将轨迹点直接连接会将曲线轨迹变为折线段,所以需要对轨迹点进行拟合,由泰勒公式可知对于任意平滑曲线,可以构造一个多项式函数来近似表达曲线,为方便求出多项式系数,本发明中采用高次多项式来表示,构造出函数为:In order to obtain the trajectory curve after fitting the trajectory points, the machine learning method is used to fit the trajectory points. Since the trajectory points are a collection of discrete points, if the trajectory points are directly connected, the curve trajectory will become a polyline segment, so it is necessary to Fitting the trajectory points, it can be known from Taylor's formula that for any smooth curve, a polynomial function can be constructed to approximate the expression curve. In order to conveniently obtain the polynomial coefficients, a high-order polynomial is used in the present invention to represent, and the constructed function is:
f(x)≈pn(x)=a0+a1(x-x0)+a2(x-x0)2+…+an(x-x0)n+Rn(x);f(x)≈p n (x)=a 0 +a 1 (xx 0 )+a 2 (xx 0 ) 2 +…+a n (xx 0 ) n +R n (x);
其中,Rn(x)为n阶泰勒余项;Among them, R n (x) is the nth-order Taylor remainder;
但是拟合出的曲线与目标曲线之间还是存在误差的,为了评判曲线拟合的好坏,以最小二乘法作为损失函数来表示f(x)与待拟合曲线的误差关系,即:However, there is still an error between the fitted curve and the target curve. In order to judge the quality of the curve fitting, the least squares method is used as the loss function to represent the error relationship between f(x) and the curve to be fitted, namely:
令x0=0得Let x 0 =0 get
为了使目标函数最优,使每个Loss对系数偏导数ak1为0,即:In order to make the objective function optimal, the partial derivative a k1 of each Loss to the coefficient is 0, namely:
……...
根据上述公式,使用梯度下降法计算出所有系数ak,采用迭代法来逼近最优解,可以避免一步求出k个方程的最优解。首先,输入迭代次数和学习率learn_rate,在初始化k个系数值后,开始迭代,更新每一次迭代出的系数ak,对应的Loss值在不断减少,到达迭代次数后停止迭代。According to the above formula, all the coefficients a k are calculated by the gradient descent method, and the iterative method is used to approximate the optimal solution, which can avoid finding the optimal solution of k equations in one step. First, enter the number of iterations and the learning rate learn_rate, after initializing k coefficient values, start the iteration, update the coefficient ak obtained in each iteration, the corresponding Loss value is constantly decreasing, and the iteration is stopped after the number of iterations is reached.
其中系数ak对应的梯度值为 更新系数ak为ak=ak+learn_rate*gradient_ak,最后将得到的训练结果保存在本地以便后续调用。The gradient value corresponding to the coefficient a k is The update coefficient ak is ak = ak +learn_rate*gradient_ak, and finally the obtained training result is saved locally for subsequent calls.
S4:输出等距曲线坐标及函数表达式参数;S4: output isometric curve coordinates and function expression parameters;
具体地,参见图7-8,取图4中一条轨迹点为参考点,设定最高阶次为10,迭代次数为5000,学习率learnrate=1e-3后,通过机器学习得到所示10种拟合曲线。可以看到,当多项式拟合阶数为7时,其均方误差达到了最小值,这也意味着当时,曲线拟合是欠拟合的,而当时,曲线是过拟合的,得到其训练结果均方误差meansquare error=0.101后,将此均方误差作为边界值训练其余轨迹曲线,得到如图6所示带宽度铺放曲线;Specifically, referring to Figures 7-8, a trajectory point in Figure 4 is taken as a reference point, the highest order is set to 10, the number of iterations is 5000, and the learning rate is learn rate = 1e-3. After machine learning, the shown 10 is obtained. a fitting curve. It can be seen that when the polynomial fitting order is 7, its mean square error reaches the minimum value, which also means that when the polynomial fitting order is 7 , the curve fit is underfit, and when When , the curve is over-fitted, and after the mean square error of the training result is obtained as meansquare error=0.101, the mean square error is used as the boundary value to train the rest of the trajectory curves, and the width placement curve shown in Figure 6 is obtained;
S5:生成变刚度层合板模型。S5: Generate a variable stiffness laminate model.
具体地,参见图9,层合板参数给定时,若要使层合板面内铺放路径的纤维角度变化等于参考路径纤维角度变化时,则需要根据参考路径铺放角度来设计层合板铺放路径,层合板内纤维角度为<0|45>,满足了设计需求。Specifically, referring to Fig. 9, when the laminate parameters are given, if the fiber angle change of the laying path in the laminate plane is to be equal to the fiber angle change of the reference path, the laying path of the laminate needs to be designed according to the laying angle of the reference path , the fiber angle in the laminate is <0|45>, which meets the design requirements.
利用本发明公开的方法对飞机机翼的复合材料纤维变角度铺放,以某飞机机翼翼型曲线的上半曲线为原型面生成铺放曲面,翼型曲线如图10所示。The method disclosed in the present invention is used to lay the composite fiber of the aircraft wing at variable angles, and the upper half curve of the airfoil curve of an aircraft wing is used as the prototype surface to generate the laying surface. The airfoil curve is shown in Figure 10.
利用分割等距重组曲线的算法,生成平面内纤维铺放轨迹,当从二维平面转变为三位曲面铺放时,其中曲线坐标不再是(x,y),而是变成了(x,y,z)。若要平面轨迹铺放到曲面时,则需要对其进行坐标变换。设平面坐标系下存在一平面Ω1,存在6个特征点,如下图11所示,Ω1的特征点在空间直角坐标系下则变成了(x,y,0),当Ω1向Ω2转变时,Ω1则从平面转变为曲面,Ω1称为Ω2在xy平面下的投影面,此时Ω2的特征点的x,y坐标不变,Ω2在xz面下的投影特征点坐标为(x,z),z(x)是关于剖面曲线的函数。故当已知平面轨迹线坐标(x,y)和剖面曲线坐标(x,z)时,二维坐标到三维坐标转换为:Using the algorithm of dividing the equidistant recombination curve, the in-plane fiber laying trajectory is generated. When the two-dimensional plane is converted to the three-dimensional surface laying, the curve coordinates are no longer (x, y), but become (x , y, z). When the plane track is laid on the surface, it needs to be coordinate transformed. Assuming that there is a plane Ω1 in the plane coordinate system, there are 6 feature points, as shown in Figure 11 below, the feature point of Ω1 becomes (x, y, 0) in the space rectangular coordinate system, when Ω1 transforms to Ω2 , Ω1 is transformed from a plane to a curved surface, and Ω1 is called the projection surface of Ω2 under the xy plane. At this time, the x and y coordinates of the feature points of Ω2 remain unchanged, and the coordinates of the projected feature points of Ω2 under the xz plane are (x, z ), z(x) is a function of the profile curve. Therefore, when the coordinates of the plane track line (x, y) and the coordinates of the profile curve (x, z) are known, the two-dimensional coordinates are converted to the three-dimensional coordinates as:
设轨迹线坐标为(x1,y1),机翼曲线为(x2,y2),得到三维曲面的轨迹坐标由公式9得轨迹坐标(x,y,z)=(x1,y1,y2),在纤维满足不大于最大测地曲率的情况下,实现了简单的从二维到三位的转变,得到如下图12和图13所示的上机翼面铺放。Assuming that the coordinates of the trajectory line are (x 1 , y 1 ), and the wing curve is (x 2 , y 2 ), the trajectory coordinates of the three-dimensional surface are obtained by
本发明公开的一种基于分割等距重组曲线的复合材料层合板的建模方法,将生成轨迹曲线、等距轨迹曲线、拟合轨迹曲线和生成层合板模型等过程简化为只需输入相关参数即可得到,使得建模步骤简便且有效提高了分析效率。The invention discloses a modeling method for composite material laminates based on splitting and equidistant recombination curves, which simplifies the processes of generating trajectory curves, equidistant trajectory curves, fitting trajectory curves, and generating laminate models to only need to input relevant parameters can be obtained, which makes the modeling steps simple and effectively improves the analysis efficiency.
在本说明书的描述中,参考术语“一个实施例”、“一些实施例”、“示例”、“具体示例”、或“一些示例”等的描述意指结合该实施例或示例描述的具体特征、结构、材料或者特点包含于本发明的至少一个实施例或示例中。在本说明书中,对上述术语的示意性表述不一定指的是相同的实施例或示例。而且,描述的具体特征、结构、材料或者特点可以在任何的一个或多个实施例或示例中以合适的方式结合。In the description of this specification, description with reference to the terms "one embodiment," "some embodiments," "example," "specific example," or "some examples", etc., mean specific features described in connection with the embodiment or example , structure, material or feature is included in at least one embodiment or example of the present invention. In this specification, schematic representations of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
以上的实施例仅是对本发明的优选实施方式进行描述,并非对本发明的范围进行限定,在不脱离本发明设计精神的前提下,本领域普通工程技术人员对本发明的技术方案做出的各种变形和改进,均应落入本发明的权利要求书确定的保护范围内。The above embodiments are only to describe the preferred embodiments of the present invention, and do not limit the scope of the present invention. Variations and improvements should fall within the protection scope determined by the claims of the present invention.
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