JPH03211680A - Multiplexing system for super-element - Google Patents

Abstract

PURPOSE:To realize highly efficient and highly precise large-scale structure analysis by executing repeatedly a processing to generate a super-element by using a usual finite element and further, generate the new super-element by using the super-element and the usual finite element. CONSTITUTION:Node data to constitute the super-element is generated, and the element matrix of the usual finite element such as a rigidity matrix, a mass matrix, and a stress matrix, etc., is generated, and these are superposed on a whole matrix, and an element load vector is generated. Then, a normal mode is calculated, and after a bounded mode is calculated, a super-element matrix is generated by using the normal mode and the bounded mode. By repeating this procedure as many times as the number of the super-elements to be generated and handling the super-element too similarly to the usual finite element, the super-element is generated by multiplexing it. Thus, the large-scale structure analysis can be executed at high speed and highly precisely.

Description

[Detailed Description of the Invention] [Industrial Application Field] The present invention relates to a system for performing structural analysis based on the finite element method, and is an ultra-high-performance system that enables large-scale structural analysis to be performed quickly, efficiently, and with high accuracy. Element Multiplexing Method [Prior Art] Conventionally, when performing large-scale structural analysis, many methods have been used to perform analytical calculations using the substructure method. By using hyper-elements created by the substructure mode synthesis method based on the Craig-Bawpton method as analytical elements, it is possible to compensate for the lack of computer capacity even when performing a fairly large-scale analysis. On the other hand, with the improvement of computer capabilities such as supercomputers that enable high-speed calculations and large-capacity calculations, the need for analysis of large-scale structural systems has increased.

In addition, the development of techniques for dividing effective elements has made it easier to create analytical models for large-scale structures.

However, in structural analysis, as the structure becomes larger and more complex, the computer capacity and time required for calculation tend to increase.

Furthermore, when performing vibration response calculations using superelements, in order to improve the accuracy, there is a method of using generalized coordinates that take external force terms into consideration in the reference mode when creating superelements. This method is described, for example, in JP-A-61-270636.

[Problem to be solved by the invention]

In the above-mentioned conventional technology, when performing complex and large-scale structural analysis, many super-elements are generated, resulting in the solution of a large-scale system. Therefore, there was a limit to the saving of computer capacity and shortening of calculation time.

On the other hand, if superelements are repeatedly created and multiplexed, the accuracy of vibration stress in response calculations may decrease.

The first subject of the present invention is truss elements and beam elements.

A superelement is created using general finite elements such as shell elements and solid elements (hereinafter simply referred to as general finite elements), and a new superelement is created using the created superelement and general finite element. By repeating the process of creating new super-elements, large-scale structural analysis can be performed efficiently and with high accuracy.

The second problem of the present invention is to create a superelement using general finite elements, create a new superelement using two or more of the created superelements, and create this new superelement. It consists in doing the tasks repeatedly.

[Means to solve the problem]

The above problem is solved by using super-elements as analytical elements in structural analysis. This is achieved by providing a procedure for creating a hyperelement, and a procedure for creating a hyperelement from a hyperelement and a general finite element.

[Effect]

Since super-elements have a mass matrix, stiffness matrix, and element load vector like general finite elements, by treating super-elements in the same way as general finite elements, it is possible to create multiple super-elements. Become. Furthermore, by using generalized coordinates that take the external force term into account in the calculation of the reference mode, the calculation accuracy of the vibration stress in the vibration response calculation is improved.

〔Example〕

An embodiment of the present invention will be explained with reference to FIG. First, each procedure for creating hyperelements from general finite elements will be explained.

Step 1) to create the node data that constitutes the superelement.
General finite element stiffness matrix, mass matrix,
Create an element matrix such as a stress matrix (hereinafter referred to as element matrix) (Step 2), superimpose them on the overall matrix (Step 3), create element load vectors (Step 4), and calculate the standard mode ( Step 5), Calculation of constraint mode (Step 6)
), a superelement matrix is created using the standard mode and constraint mode (step 7). Each of steps 1 to 7 is repeated for the number of superelements to be created, and the results are output to file A. Next, each procedure for creating multiplexed super-elements will be explained. Step 8) to create node data composing the super-element, step 9) to create an element matrix, superimpose them on the overall matrix (step 10), create element load vectors (step 11), and calculate the standard mode ( After performing step 12) and calculating the constraint mode (step 13), a superelement is created using the reference mode and the constraint mode (step 14). The steps 8 to 14 are repeated as many times as the number of superelements to be created by multiplexing, and the results are output to file B. Here are the steps (Step 9)
Now, let's make it possible to input the data of the general finite element and the hyperelement which are output to files A and B. The hyperelement thus created is used for the analysis of the entire system (step 15).

The concept of the present invention will be explained using FIGS. 2 to 4.

First, a superelement A.32 is created from the general finite element 31. Next, super-element B-33 is created using three super-elements A-32. As a result, as shown in FIG. 4, the entire system consists of one super element A and three super elements B.

The superelement used in the present invention will be explained with reference to FIGS. 5 to 9.

When the continuum is modeled using finite elements as shown in Figure 5, its equation of motion is [M] (M) + [[] (u) = (f)
・=(1). Here, [M is a mass matrix, [K] is a stiffness matrix, (M) is an acceleration vector, (U) is a displacement vector, and (f) is an external vector. The attenuation term is omitted. Introducing -film displacement (q) to equation (1), (u)=['F](q)...
・Conduct the variable conversion in (2) and multiply [IP]T from the left to get [Vco”[M][vl(M)+[:vl”[Kco[
ψ'ko (q) = [W]" (f ) - (3) Furthermore, [M umbrella] = [Vko" [M] ['ψ゛]
...(4) [K skewer] = ['!
']”[K][Ma]...(5)
(g)=[ma]”(f)・
...If we set (6), equation (3) becomes [M umbrella] = (eyebrow + [K umbrella (q) = (g)
...(7) can be written. Here, [V] is (U) and (
q) - Solve equation 0 (7), which is a square matrix giving the transformation corresponding to the discussion, to find (q>) and obtain equation 1 (2
), the solution (u) to be found can be obtained.

Therefore, in order to obtain an advantage in terms of calculation accuracy and efficiency, [t
What is used as F] will be explained below.

The nodes in the finite element model shown in Figure 5 are divided into two types. One is the point indicated by the O mark, and this is called the boundary point.

The other point is the point indicated by the mark .This is called the internal point.

The entire model is divided into three parts in this example, separated by boundary points. [A v-type as shown in Fig. 6 is used. [ΦC] is the constraint mode, which is the deformation mode when all degrees of freedom of the boundary point other than that degree of freedom are constrained for each degree of freedom of the boundary point, and a unit amount of forced displacement is applied to that degree of freedom. , there are as many degrees of freedom as the boundary points. [Φ
N] is a standard mode, which is a natural vibration mode when all boundary points are constrained, and exists as many as the degrees of freedom of internal points.

[V] is a square matrix. If the boundary points and internal points are grouped together as 51 to 57 in the overall model in Figure 5, and the corresponding constraint modes and reference modes are arranged together, all the components will be included in the matrix as shown in Figure 6. O
A submatrix of is created.

Next, if we divide the equation into an equation for the boundary point and an equation for the interior point, and express it by adding the subscript B to the former and the subscript engineering to the latter, we get Equation (1).
The equation of motion is...(8) The variable transformation of equation (2) is as follows. uB on the right side may be expressed as qB, but since it is the same value as uB on the left side, the same symbol was used. qN is the generalized displacement corresponding to the reference mode.

[V] is, but it is clear from the definitions of the constraint mode and the reference mode that the upper half is Xa and O. However, ■B is a unit matrix.

formula %formula%[ ] [ formula (5) [K umbrella] is [K fistko=shiψ]” [K] [’ψ“] becomes.

However, here the constraint mode [ΦC] is [KI! ][Φc]+[K! B]=[Oko...(13
).

The restraint mode is formula (13) So, [K! It is calculated by giving [I] and [KIB].

The -filming load vector (g) in equation (6) is (q) = [ψ]
”(f) Figure 17 illustrates the form of the matrix of equation (7) using the above results.Since the reference mode is the natural vibration mode when the boundary point is fixed, the eigenmode From orthogonality, [ΦN]”[M”][ΦN] and [ΦN
]"[Kf■][ΦN] becomes a diagonal matrix. Furthermore, [ΦN] is normalized by [MI+] to obtain [ΦN]"[
M I I ] If [ΦN] is made into a unit matrix, the diagonal components of [ΦN]T[K-th] [ΦN] will be the square of the natural angular frequency corresponding to each reference mode. The part is indicated by the symbol [Ω2], and in Fig. 7, the part where each 'component of the matrix does not become zero is shown by hatching. For the stiffness matrix, the achieved parts of the internal point equation and the boundary point equation are zero, as determined by equation (12), but for the mass matrix, they are not zero, as determined by equation (11). If all the equations are considered using this formula, it is the same as solving the one-element formula (1), so the results of both will match, but the efficiency of calculation will not improve. Therefore,
Among the standard modes, only the low-order ones are left and the high-order standard modes are omitted to reduce the degree of freedom. Regarding the stiffness matrix, as shown in FIG. 8, equations corresponding to higher-order normal modes are not coupled with other equations.

On the other hand, regarding the mass matrix, it is thought that the influence of the mass term is small in higher-order modes, so even if the part where the higher-order modes are coupled with the restrained mode is ignored, the influence is considered to be small.

The matrix of the substructure created in this way includes a mass matrix, a stiffness matrix, and an element load vector, similar to general finite elements, and this substructure is called a superelement.

Here, regarding the load term, the part related to the higher-order normal mode of ΦNf■ in equation (14) is omitted, but since this part is not zero, especially when a load is applied inside the superelement, Calculated stress values may be inaccurate. Therefore, in the present invention, the superelement has a lower-order normal mode [Φ
Nm], and instead of the higher-order reference mode, [vN
Apply generalized coordinates consisting of r ] and [vs h ]. - Membrane coordinates refer to n quantities in a system of n degrees of freedom. By using this method, the above problems are solved.

Here, the generalized locus 1 used instead of the higher-order standard mode
11. What [ψsr] and [maNh] are will be explained.

The load vector that should be prepared as the element load vector of the super element we are considering now is (Fet) (i = 112 T・
Q), static analysis is performed on this (Fet) with the boundary points fixed. That is, Mouth CI] (ut) = (Fet) −
・-(xs) Obtained by solving this (utHi=1.2
．． ... Q) is made orthogonal with respect to [ΦNm] and [K I I ]. 'Fs-1 is a matrix whose column vectors are the vectors obtained by orthogonalizing the two with respect to [KIT] and normalizing them with [M'l]. Therefore, [vNr] is C"4'Nr]"[K"][ΦN-]
= [O] − (16) [
'V Nrko' [K ''] ['FNr] = [Q H
rco・(17) [Ωsrl is a diagonal matrix. [Φsa] is the eigenvector when the boundary point is fixed, and [Mako is [K] with respect to [ΦNg]
I 1 ] Since they are orthogonal, [ψNr]”[
M ■■ko [ΦN-1=[O]
...(18). The two column vectors of [vNr] do not necessarily have to be orthogonal with respect to [MII], but in the method actually used, they are made orthogonal. Regarding [Mash], vectors can be prepared that are orthogonal to [ΦNN here[vN r E for [KII] and also orthogonal to each other for [K r I ].

That is, [vNhl is [ψ, h]T[K11 [Φs*] = [0
...(19) [mashl”[K”ko[Φ
Nr] = [O] ”・(2
0) [v*hl”[K”][vNh]=[Ωkihl
”.(21) is satisfied. Here, [Ωsh] is a diagonal matrix.

For ['F N h ] obtained as above, [VNh]" (Fe1) = (0) (i = 1
, 2, -2 ) - (22). Therefore, the eighth
[q,] in the figure becomes [O] For [yl 11 ], [vNh ko” [M ” ko[ON *koni[O]
-(23), but ['F Nhko"[M"ko[ΦNr]≠[0ko
-(24). Coordinate transformation matrix [ψ] = [ΦCΦN, ψN r vN
h ] exists. However, what actually needs to be calculated is ΦC4ΦI, ψN.

Since it is known that the corresponding external force term is (0) for ψNh, no calculation is necessary.

As the generalized coordinate calculation method described above, the calculation method shown in FIG. 9 can be used as is. In the first half, p eigenvectors (normative modes) are determined by eigenvalue analysis using the subspace method. Next, for the two modes obtained, equation (15) element load vector (Fe+) (i = 1,
2, . . . 12) are orthogonalized as shown in FIG. 9 to obtain [Fr]. Multiply [Φ] by [M] to obtain [
A matrix [Y&] is prepared whose column vectors are (p+Q) column vectors, which are a combination of [Yk] and [F,].

And [KII] [Xa] = [Ya]
- Solve (25) and [Xal
get. Using this [Xal, [M'']t[K''] is reduced to a partial quantity. Stiffness matrix [K
,] is [L Koni [Mata Eiko T [K Summer! ][Xal
...(26), but formula (25)
Considering, [K & Co = [Xal] Ding [y al
- It can be calculated by (27). The mass matrix [M&] in the subspace is [M al = [also a]T[M■! ] [also ako
-(28)cKa], [Ma]
Eigenvalue problem in one subspace for EK & co[Qa] =
[Mako [Q, HΩa]”・
Solve (29) to find [Q&]. Finally [ΦNt
massr] is [Os a 'F N r ] = [X
ako [Qal -(
30). Furthermore, the element load vector in the original space [Fe3, the element load vector in the space transformed in the constraint mode and standard mode [Gsl] is [ae]
= [ΦNlψsr]”[F el −(
31), which corresponds to the lower half of the right side of equation (14). The upper half is calculated separately, and the two together correspond to the uniform cylindrical point load of a general finite element.

〔Effect of the invention〕

According to the present invention, superelements can be created by multiplexing, so
- degree of freedom of the system to be solved decreases, and a significant reduction in computer capacity can be realized. Accordingly, since the amount of calculation is reduced, the calculation time can also be shortened. In addition, since the superelement to be multiplexed is created using generalized coordinates that take into account the external force term in the reference mode, the external force term corresponding to the omitted higher order reference mode can be made zero, resulting in high accuracy. It is possible to obtain accurate analysis results.

[Brief explanation of drawings]

Fig. 1 is an explanatory diagram of an embodiment of the present invention, Fig. 2 is an explanatory diagram of creating super elements from general finite elements, and Fig. 3 is an explanatory diagram of creating super elements using super elements. , Figure 4 is an explanatory diagram showing the entire system, Figure 5 is an explanatory diagram showing a continuum modeled using general finite elements, and Figure 6 is a square transformation matrix used in the finite element model shown in Figure 5. Figure 7 is an explanatory diagram showing the transformed equation of motion in the form of a matrix. Figure 8 is an explanatory diagram showing [ψ] and its value expressed using generalized coordinates considering external force terms. Explanatory diagram of the equation of motion converted using [ψ], No. 9
The figure is an explanatory diagram showing an example of a method for calculating generalized coordinates in consideration of outer force terms. 5.12...Calculation of standard mode, 6,13...Calculation of restraint mode, 7.14...Creation of super element matrix, 9...Creation of element matrix, 81...External force term name 11￥1 #2 Review v74 Sword No. 3 n 3 No. 5 No. 1 No. 4 Inpo 7

Claims (1)

[Claims] 1. In structural analysis using a computer, in an analysis method that uses hyper-elements as analysis elements, the analysis system is divided into boundary degrees of freedom and internal degrees of freedom, and a unit forced displacement is calculated for each degree of freedom of the boundary. A method for calculating a static deformation mode when a given amount, a procedure for calculating a natural vibration mode when a boundary point is fixed, and a method for calculating a truss element, a beam, using the static deformation mode and the natural vibration mode. A super-element comprising: a procedure for creating a super-element from a general finite element such as an element, a shell element, a solid element, etc.; and a procedure for creating the super-element from the super-element and the general finite element. multiplexing method.