JP4232566B2 - Shape grasp method in numerical analysis by finite element - Google Patents

Description

The present invention relates to a shape grasp how to grasp the 3-dimensional curved surface shape of the surface elements required by the like stress analysis using the finite element.

  Numerical analysis using a finite-element method (FEM) is widely used as a stress analysis method for mechanical products and building structural materials. FEM divides an object into finite elements having relatively simple characteristics and models them, and analyzes the performance, behavior, and the like of the entire system (see, for example, Patent Document 1).

In recent years, laminated materials containing composite materials such as FRP (fiber reinforced plastics) formed by filling glass fibers or carbon fibers have been used as structural materials for aircraft, ships, automobiles, and the like. The technique described in Patent Document 1 performs modeling by replacing a laminated material with a uniform material having equivalent characteristics, thereby reducing the number of FEM elements in such a laminated material and speeding up computation. .
JP 2002-082998 A (paragraphs 0013 to 0042, FIGS. 1 to 6)

  By the way, in FEM using such a shell model, calculation is performed by dividing an object into plane elements by nodes. It is conceivable that the curvature of the surface element is necessary for the analysis of the model shape. However, in the conventional methods, in order to obtain the curvature of the surface element accurately, it is necessary to make the element sufficiently small with respect to the curvature change, and reducing the number of elements of the shell model itself is not possible. It was difficult.

Therefore, an object of the present invention is to provide a shape grasping method for grasping a three-dimensional curved surface shape of a surface element that enables accurate calculation of the curvature of the surface element even when the number of elements of the shell model is small. .

In order to solve the above problems, a shape grasping method according to the present invention is a shape grasping method for grasping a three-dimensional curved surface shape of a surface element in numerical analysis by a finite element method in which an analysis object is divided into surface elements by nodes. , The position coordinate vector of the target node is X ₀ , the coordinate vector of the neighboring node of this node is X, the unit vector in the direction perpendicular to the curved surface is P , and the symmetric matrix whose component is the reciprocal of the radius of curvature in each direction at the node When substituting the assumed curvature value into the symmetric matrix Q,

の近似解として単位ベクトルＰを求め、得られた単位ベクトルＰにより上記（１）を満たす近似解として対称行列Ｑの各成分を求めることにより、節点における各方向の曲率を対象節点と近傍の節点を通る２次関数により近似することで面要素の曲面形状を把握することを特徴とする。

A unit vector P is obtained as an approximate solution of, and each component of the symmetric matrix Q is obtained as an approximate solution satisfying the above (1) by the obtained unit vector P, whereby the curvature in each direction at the node is determined from the target node and nearby nodes. The curved surface shape of the surface element is grasped by approximating with a quadratic function passing through .

When determining the unit vector P, preferably Rukoto using 0 as the assumed curvature value. A curvature value of 0 corresponds to a case where the element is a plane .

When obtaining each component of the symmetric matrix Q, when the predetermined weighting coefficient is C, it is more preferable to obtain the value of Q as an approximate value from Equation (1) and C QP = 0. The tensor Q has six independent components, but there are too many degrees of freedom of the curved surface shape to express the curved surface with a quadratic function. This expression has a function of suppressing the degree of freedom in the direction in which the curved surface shape is illegally deformed. Further, the weighting has a function of equalizing the contribution of each expression when analytically solving using the least square method.

  According to the present invention, the three-dimensional curved surface shape of the target element can be grasped from the node data of the shell model. As a result, for example, it is possible to easily analyze the deformation of the curved surface due to stress and the accompanying stress component. In particular, it is not necessary to subdivide the object and increase the number of elements for the purpose of grasping the curved surface shape, so it is possible to perform analysis with a small number of elements without reducing the calculation accuracy. Resources can be reduced.

  DESCRIPTION OF EXEMPLARY EMBODIMENTS Hereinafter, preferred embodiments of the invention will be described in detail with reference to the accompanying drawings. In order to facilitate the understanding of the description, the same reference numerals are given to the same components in the drawings as much as possible, and duplicate descriptions are omitted.

  FIG. 1 is a block diagram showing a numerical analysis technique including a curved surface model shape analysis method according to the present invention. In this numerical analysis, the interlaminar stress when a laminated plate is bent is analyzed using FEM. This numerical analysis is performed on various computers, but each means shown below may be configured as a single program or as software including a program group including a plurality of programs. Also good. Furthermore, each unit may operate on a different computer, or each unit may be configured by a plurality of software programs that operate on different computers. In this case, the program includes not only a program having an input / output function itself but also a so-called macro, script, etc. that operates on a specific program.

  First, the FEM model generation unit 20 generates bulk data representing the shape of the target unit (a target plate is represented by a shell model), and a load applied to the plate and plate constraint data. Bulk data may be generated by directly inputting data of each node, but node data may be generated by setting the number of divisions based on shape data such as CAD data. Alternatively, input data may be generated by setting the shape, the number of divisions, and the like using a GUI (Graphical User Interface).

  The FEM analysis means 30 calculates the in-plane stress acting in the plane direction for each element based on the known bulk data and load / constraint conditions based on the known FEM analysis technique. The obtained in-plane stress data is sent to the interlayer stress analysis means 40 together with the bulk data generated by the FEM model generation means 20. The interlaminar stress analysis means 40 includes a shape analysis means 41 for performing the model shape analysis method according to the present invention using bulk data, a stress analysis means 42 for analyzing the in-plane stress, and an interlaminar stress from the model shape and the in-plane stress. It comprises stress calculating means 43 for obtaining stress.

Next, the model shape analysis method in the shape analysis means 41 will be described in detail. FIG. 2 shows an example of a shell model representing a curved surface. Hereinafter, the position coordinate vector of the target node is X ₀ = (x ₀ , y ₀ , z ₀ ), the position coordinate vector around this node is X = (x, y, z), and the target element is represented by S. And In this shape analyzing means 41, P = (Px, Py, Pz) (that is, | P | = 1) is set as a unit vector perpendicular to the surface, and as a tensor representing the curvature.

を用いる。ＱはＱij=Ｑjiとなる対称行列であり、その独立な成分は、Ｑxx、Ｑxy、Ｑxz、Ｑyy、Ｑyz、Ｑzzの６個である。各成分は曲率半径の逆数を表す。例えば、ＱxxはＸ方向の曲率半径の逆数である。

Is used. Q is a symmetric matrix with Qij = Qji, and there are six independent components Qxx, Qxy, Qxz, Qyy, Qyz, and Qzz. Each component represents the reciprocal of the radius of curvature. For example, Qxx is the reciprocal of the radius of curvature in the X direction.

  FIG. 4 is a flowchart of shape analysis in the shape analysis means 41. This calculation is performed for the entire model shape.

First, the coordinate values of the target node and the peripheral nodes are read (step S1). Next, the vector P is calculated (step S2). This is to obtain P by solving (X−X ₀ ) · P = 0 by setting Q = 0 in equation (1). Specifically, when (X−X ₀ ) · P = 0 is expanded, (x−x ₀ ) Px = 0, (y−y ₀ ) Py = 0, and (Z−Z ₀ ) Pz = 0. . Therefore, in each case of Px = 1, Py = 1, and Pz = 1, the target node and its surrounding node coordinates are substituted into the left side of each expanded formula, and the sum of the squares of the values is calculated. Find the smallest three Ps. Then, the solution with the smallest | P | is selected, and P is obtained by standardizing the solution so that | P |

Note that when P is a node around the target element S is rectangular as shown in FIG. 3 and e ₁ to e _4, which represents the vector directed from the node e _j to e _k in V _jk, for example,

により求めてもよい。

You may ask for.

  Next, the value of each component Qxx, Qxy, Qxz, Qyy, Qyz, Qzz of Q is obtained from the equation (1) (step S3). Here, if equation 1 is transformed and expanded,

となる。これは、節点周辺の曲面を２次関数で表現したものである。例えば、曲面がほぼＸＹ面にあるとすると、ｚ²の項は本来不要である。すなわち、図５に示されるように、本来の曲面がＣactであるのに対し、式（１）にｚ²の項が存在することで、Ｃcal1のような曲面をフィッテングしてしまう可能性が出てくる。そこで、本発明では、ＱＰ＝０となる式を条件として追加することでこのｚ²の項を小さくし、滑らかな曲線で節点間をつなぐようにした。

It becomes. This is a representation of a curved surface around a node by a quadratic function. For example, if the curved surface is substantially in the XY plane, the term z ² is not necessary. That is, as shown in FIG. 5, the original curved surface is Cact, but the presence of the z ² term in equation (1) may cause fitting of a curved surface such as Ccal1. Come. Therefore, in the present invention, the expression of QP = 0 is added as a condition to reduce the term of z ² so that the nodes are connected by a smooth curve.

  Specifically, when QP = 0 is expanded by applying a weight C, the following three expressions are obtained.

ここで、Ｃは、式（１）と次元を揃えるため、以下のように設定した。

Here, C was set as follows in order to align the dimension with the formula (1).

なお、Ｃ₀は定数であり、0.1に設定したときに良好な結果が得られた。

C ₀ is a constant, and good results were obtained when set to 0.1.

  When the target node and surrounding node coordinates are input to these equations, the number of the peripheral nodes + 3 equations are obtained. (In the shape model shown in FIG. 2, for example, if 8 nodes are used as the peripheral nodes, 11 equations are obtained.) Since P has already been obtained, the unknown variables are 6 of each component of Q. Yes, the number of expressions exceeds the number of variables. Therefore, each component of Q is obtained by the method of least squares so that the error is minimized.

Next, it is determined whether or not the calculation has been completed for all the nodes in the model (step S4), the nodes are moved until the calculation is completed, and the processes in steps S1 to S3 are repeated to calculate for all the nodes. When the calculation is completed for all nodes, the P and Q of the surface elements are calculated for each element S by averaging the P and Q of the nodes around the element, thereby obtaining the curvature of the surface element (step S5). . At this time, there are two possible directions of P, that is, when facing the front of the surface and when facing the back. Therefore, it is aligned in the direction of the element coordinate e ₃ .

  Here, an example in which the model shape is analyzed from the shape data input to the FEM analysis has been described. However, in the case where the model shape after deformation accompanying stress application is analyzed, the in-plane stress obtained from the FEM analysis means 30 The analysis result may be used in consideration of the movement of each element position.

  Further, in the present embodiment, the case where the numerical analysis is performed using the finite element method has been described. However, instead of the finite element method, the case where another analysis method for modeling and analyzing the object as a finite element is used. The same effect can be obtained.

  The present invention relates to a method for analyzing a curved surface shape, but can also be used to reproduce a curved surface shape.

It is a block diagram which shows the method of numerical analysis including the analysis method of the curved surface model shape which concerns on this invention. An example of a shell model representing a curved surface in FEM analysis is shown. It is a figure which shows the relationship between an object element and a nodal vector. It is a flowchart of shape analysis. It is a figure which shows the influence of an extra freedom degree.

Explanation of symbols

  20 ... FEM model generation means, 30 ... analysis means, 40 ... interlayer stress analysis means, 41 ... shape analysis means, 42 ... stress analysis means, 43 ... stress calculation means.

Claims (3)

A shape grasping method for grasping a three-dimensional curved surface shape of a surface element by a curved surface direction and a radius of curvature in each direction in a numerical analysis by a computer using a finite element method for dividing an analysis object into surface elements by nodes ,
The position coordinate vector of the target node is X ₀ , the coordinate vector of the neighboring node of this node is X, the unit vector in the direction perpendicular to the curved surface centered on the symmetric node is P , and the reciprocal of the radius of curvature in each direction at the node is the component Let Q be a symmetric matrix
Substituting a predetermined curvature value into the symmetric matrix Q,

Obtaining a unit vector P as an approximate solution of
Analyzing each component of the symmetric matrix Q analytically as an approximate solution satisfying the above equation (1) from the obtained unit vector P, the curvature in each direction at the node is expressed by a quadratic function passing through the target node and nearby nodes. A shape grasping method comprising the step of obtaining by approximation . In determining the unit vector P, the shape grasping method according to claim 1, wherein Rukoto using 0 as the assumed curvature value. 3. The value of Q is obtained as an approximate solution of Equation (1) and C QP = 0 , where C is a predetermined weighting coefficient in the step of obtaining curvature in each direction at the node. How to grasp the shape of

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