CN102629291A - Analyzing and designing method for structure with holes and structure with components - Google Patents

Abstract

The invention discloses an analyzing and designing method for a structure with holes and a structure with components, and the method is used for solving the technical problem in a conventional finite element analyzing method and an isogeometric analyzing method that the analyzing and designing efficiency of the structure with holes and the structure with components is low. The invention adopts the technical scheme that the structure with holes or the structure with components is divided into two parts, namely a holeless structure and a hole structure; NURBS patches are used for establishing CAD models of the two parts; NURBS primary functions of the CAD models and control points are used for replacing the shape function and the node in the finite element analyzing method, so that the analyzing models of the holeless structure and the hole structure are the same as the CAD models of the holeless structure and the hole structure; when the structure with holes is analyzed, the same part between the holeless structure and the hole structure is deleted; and when the structure with components is analyzed, the same part between the holeless structure and the hole structure is endowed with the material characteristics of the components, so that the analyzing models of the structure with holes and the structure with components are the same as the CAD model of the structure with holes and the structure with components. Therefore, the analyzing and designing accuracy and the analyzing and designing efficiency of the structures are improved.

Description

The analysis design method of hole structure and band modular construction

Technical field

The present invention relates to a kind of analysis design method, the analysis design method of particularly a kind of hole structure and band modular construction.Be applicable to the geometric accuracy demanding occasion of hole structure with the band modular construction.

Background technology

In order to satisfy requirements such as loss of weight, assembling, reduction consumptive material and exhaust blowdown, in mechanical engineering and construction work, be applied to many structures that have various holes, also be hole structure.If hole is filled to another kind of material, just become band modular construction problem, these fill entity can represent connected instrument and equipment etc. on engineering, be referred to as assembly at this.The introducing of hole and assembly will inevitably destroy the integrality of structure, thus a series of problems such as the stress that causes structure is concentrated, decrease of fatigue strength.The shape of hole and assembly, size and position are generally all very big to the influence of structural mechanics response; Aero-Space etc. to the very high field of geometrical scale accuracy requirement in, the influence that hole and assembly bring to structure all need accurately simulate and analyze.

At present; The most frequently used structure analysis method for designing is exactly known finite element method; Commercial finite element analysis software such as ANSYS, NASTRAN etc. have been widely used in the various engineerings field, analyze design in order to any labyrinth is comprised hole structure and band modular construction.Finite element method is at first being converted to it approximate analytical model by the cad model of analytical structure, the unit that then analytical model is divided into a lot of simple shape is analyzed.The raising of finite element analysis precision mainly realizes through unit subdivision, in the unit subdivision process, will keep mutual between analytical model and the cad model, makes the analytical model behind the unit subdivision can approach the cad model of structure more.

Document 1 " Hughes TJR, Cottrell JA, Bazilevs Y.Isogeometric analysis:CAD; finite elements; NURBS, exact geometry and mesh refinement.Computer Methods in Applied Mechanics and Engineering, 2005; 194:4135-95. " discloses a kind of structure analysis method for designing, geometric analysis method such as is called.Adopt the basis function of the non-uniform rational B-spline (NURBS) that is used for geometric modeling among the CAD and shape function and the node that the reference mark replaces finite element method etc. geometric analysis method; Realized organically blending of cad model and analytical model, when structure analysis, can carry out unit subdivision easily and can guarantee that analytical model is identical all the time with cad model.

Yet document 1 etc. geometric analysis method be difficult to hole structure with the band modular construction analyze design; This be since etc. the cad model of geometric analysis method institute analytical structure be only limited to the nurbs surface sheet and set up; The nurbs surface sheet is to be formed by a complete rectangle parameter region mapping, and the cad model that complete rectangle parameter region is difficult to be mapped to hole structure or is with modular construction.Therefore; Etc. geometric analysis method at analytic band pore structure and band during modular construction; Need it is divided into the many minor structures that do not have hole and assembly artificially; And all to set up its cad model to every minor structure,, seldom be used for the analysis design of hole structure and band modular construction etc. geometric analysis method because human factor is big and modeling is complicated with the nurbs surface sheet.Though finite element method is convenient to hole structure and band modular construction are analyzed design, its analytical model that adopts is the approximate of structure C AD model, and the error of analytical model on how much will cause the error of structure analysis design result.In order to improve analysis precision, finite element method need carry out unit subdivision to analytical model and produce more unit, will cause the increase of calculated amount and the reduction of structure analysis design efficiency like this.

Summary of the invention

For overcome existing finite element method with etc. geometric analysis method analyze the low deficiency of design efficiency at hole structure and band modular construction; Make the analytical model of hole structure and band modular construction identical with its cad model and can be convenient to analyze design, the present invention proposes a kind of hole structure and the analysis design method of being with modular construction.This method is divided into non-porous and two parts of hole shape structure to hole structure or band modular construction; These two parts all use a nurbs surface sheet to set up its cad model; NURBS basis function and reference mark with cad model replace shape function and node in the finite element method; Make the analytical model of non-porous and hole shape structure all identical with its cad model; Just part identical with the hole shape structure in the non-porous is deleted during the analytic band pore structure, the material behavior of just giving assembly with part identical with the hole shape structure in the non-porous during analytic band modular construction can guarantee that hole structure is identical with its cad model with the analytical model of band modular construction.

The technical solution adopted for the present invention to solve the technical problems is: the analysis design method of a kind of hole structure and band modular construction is characterized in may further comprise the steps:

(a) with hole structure or band modular construction separated into two parts, the complete non-porous a when part is not perforate, another part is the hole shape structure the same with hole or element shapes.Non-porous and hole shape structure may be defined as according to the expression-form of nurbs surface:

$S(\mathrm{\ξ},\mathrm{\η})=\frac{\underset{i=0}{\overset{n_S}{\mathrm{\Σ}}}\underset{j=0}{\overset{m_S}{\mathrm{\Σ}}}{N}_{i,p}^S\left(\mathrm{\ξ}\right)N_{j,q}^S\left(\mathrm{\η}\right){\mathrm{\ω}}_{i,j}^S{P}_{i,j}}{\underset{i=0}{\overset{n_S}{\mathrm{\Σ}}}\underset{j=0}{\overset{m_S}{\mathrm{\Σ}}}{N}_{i,p}^S\left(\mathrm{\ξ}\right)N_{j,q}^S\left(\mathrm{\η}\right){\mathrm{\ω}}_{i,j}^S}---\left(1\right)$

$H(\mathrm{\ξ},\mathrm{\η})=\frac{\underset{k=0}{\overset{n_H}{\mathrm{\Σ}}}\underset{l=0}{\overset{m_H}{\mathrm{\Σ}}}{N}_{k,p}^H\left(\mathrm{\ξ}\right)N_{l,q}^H\left(\mathrm{\η}\right){\mathrm{\ω}}_{k,l}^H{Q}_{k,l}}{\underset{k=0}{\overset{n_H}{\mathrm{\Σ}}}\underset{l=0}{\overset{m_H}{\mathrm{\Σ}}}{N}_{k,p}^H\left(\mathrm{\ξ}\right)N_{l,q}^H\left(\mathrm{\η}\right){\mathrm{\ω}}_{k,l}^H}$

In the formula, S (ξ, η) and H (ξ, η) expression is abstracted into the non-porous and the hole shape structure of nurbs surface sheet respectively, the subscript of variable or subscript S and H represent non-porous and hole shape structure respectively, the rectangle parameter region of non-porous and hole shape structure is used Ω respectively _SurfAnd Ω _HoleExpression, parameter region Ω _SurfAnd Ω _Holeξ and η both direction are all arranged, and (ξ η) is any coordinate of any on the parameter region.Non-porous parameter region Ω _SurfKnot vector on ξ and η direction is designated as Ξ _SurfAnd H _Surf, then

With

Be respectively to be defined in knot vector Ξ _SurfAnd H _SurfOn non-reasonable B spline base function, n _S+ 1 and m _S+ 1 is respectively the B spline base function

With

Number,

With Subscript p and q be the number of times of B spline base function.Hole shape structural parameters zone Ω _HoleKnot vector on ξ and η direction is designated as Ξ _HoleAnd H _Hole, then

With

Be respectively to be defined in knot vector Ξ _SurfAnd H _SurfOn non-reasonable B spline base function, n _H+ 1 and m _H+ 1 is respectively the B spline base function

With

Number,

With

Subscript p and q be the number of times of B spline base function.

With

Be respectively the weight factor of non-porous and hole shape structure, { P _{I, j}And { Q _{K, l}Be respectively the reference mark of non-porous and hole shape structure.

(b) non-porous is carried out the grid segmentation, promptly at its parameter region Ω _SurfCarry out the node refinement to obtain appropriate nodes vector Ξ _SurfAnd H _Surf, then according to the shape of hole or assembly, size and the position in hole structure or band modular construction thereof from Ξ _SurfAnd H _SurfIn select a part as hole shape structural parameters zones Ω _HoleKnot vector Ξ _HoleAnd H _HoleAt non-porous knot vector Ξ _SurfAnd H _SurfIn repeat to insert node with corresponding position, hole shape structural parameters zone boundary, to obtain the hole shape structure node vector Ξ that first end-node repeats _HoleAnd H _Hole

Like this, hole shape structure node vector Ξ _HoleAnd H _HoleBe respectively non-porous knot vector Ξ _SurfAnd H _SurfIn a part, that is:

With

Hole shape structural parameters zones Ω then _HoleAlso become non-porous parameter region Ω _SurfIn a part, that is:

Because the B spline base function is only relevant with knot vector, and the non-porous knot vector of this moment is at parameter region Ω _HoleInterior part is the same with hole shape structure node vector, so non-porous and hole shape structure are at parameter region Ω _HoleInterior B spline base function also equates, that is: as (ξ, η) ∈ Ω _HoleThe time

$N_{i,p}^S\left(\mathrm{\ξ}\right)=N_{k,p}^H\left(\mathrm{\ξ}\right)$ $N_{j,q}^S\left(\mathrm{\η}\right)=N_{l,q}^H\left(\mathrm{\η}\right)---\left(2\right)$

In the formula, all differ a constant amount between the subscript i of basis function and the k and between j and the l, this is by hole shape structural parameters zone Ω _HoleAt non-porous parameter region Ω _SurfIn determining positions.

(c) according to knot vector Ξ _HoleAnd H _HoleThe hole shape structure is carried out the grid segmentation, again with the reference mark { Q after the segmentation of hole shape structure _{K, l}And weight factor

Go to substitute the reference mark of non-porous appropriate section And weight factor

By hole shape structural parameters zone Ω _HoleAt non-porous parameter region Ω _SurfIn determining positions.Because knot vector Ξ _HoleAnd H _HoleFirst end-node certain multiplicity is all arranged, can know to have only the reference mark according to the local support property of NURBS rational basis function

Corresponding basis function

With

At Ω _HoleGo up non-vanishingly, then pass through the non-porous reference mark { P after the replacement operation _{I, j}And weight factor With hole shape structure control point { Q _{K, l}And weight factor

Relation following: as (ξ, η) ∈ Ω _HoleAnd

With The time:

P _i，j＝Q _k，l ${\mathrm{\ω}}_{i,j}^S={\mathrm{\ω}}_{k,l}^H---\left(3\right)$

In the formula, the definition of the subscript of reference mark and weight factor is the same.

To

Be replaced by hole shape structure control point { Q _{K, l}Afterwards, in order to guarantee { Q _{K, l}With

The coordination at periphery reference mark is merged, and it is right to need

The periphery reference mark is regulated.Control method is following, with { Q _{K, l}Outermost certain reference mark is designated as P _Inner, corresponding with it

Outside reference mark on the L layer grid be designated as P _Outer, obtain outside the hole the 1st layer of new reference mark to the L-1 layer grid according to certain proportionality factors lambda is counter.If L=3, then P _InnerAnd P _OuterBetween need instead ask two new reference mark P ₁And P ₂, on behalf of its place grid, the subscript 1 and 2 at new reference mark be positioned at

Outside the number of plies, suppose P _InnerTo P ₁Distance be d ₁, P ₁To P ₂Distance be d ₂, P ₂To P _OuterDistance be d ₃, d then ₁, d ₂And d ₃Should satisfy:

d ₁/d ₂＝d ₂/d ₃＝λ (4)

In the formula, proportionality factors lambda is generally got the positive number less than 1, with guarantee the closer to the hole or the grid of assembly close more.

Formula (2) and formula (3) substitution formula (1) can be got, as (ξ, η) ∈ Ω _HoleThe time:

S(ξ，η)＝H(ξ，η) (5)

According to following formula, with the non-porous parameter region Ω of this moment _SurfIn hole shape structural parameters zones Ω _HoleDelete, that is: Ω _Surf-Ω _Hole, then corresponding non-porous S (ξ, η) will delete hole shape structure H (ξ η), thereby just becomes the hole structure of being analyzed by inside.So hole structure is by parameter region Ω _Surf-Ω _HoleMapping forms, parameter region Ω _Surf-Ω _HoleIt is a parameter region after rectangular wire frame is pruned.

(d) during the analytic band pore structure just at parameter region Ω _Surf-Ω _HoleGo up material behavior computing unit stiffness matrix, then only need during the analytic band modular construction in addition at parameter region Ω according to structure _HoleGo up material behavior computing unit stiffness matrix, will be assembled into the global stiffness matrix by element stiffness matrix at last, calculate the displacement and the stress of structure again according to boundary condition according to assembly.

The computing method of element stiffness matrix are following: for unit k, relevant with it (p+1) (q+1) individual NURBS rational basis function and reference mark regarded as shape function and node in the finite element method, can calculate the Jacobian matrix J of unit k ^kWith geometric matrix B ^k, the material behavior according to unit k obtains elastic matrix D again ^k, element stiffness matrix k then _e ^kCan be by computes:

${k_e}^k={\∫}_{{\mathrm{\Ω}}^k}{\left(B^k\right)}^T{D}^k{B}^k\det \left[J^k\right]\mathrm{d\Ω}---\left(6\right)$

In the formula, limit of integration Ω ^kBe the rectangular node zone of unit k in the parameter region, be convenient to carry out Gauss integration.

The beneficial effect that the present invention compares prior art is: when adopting the inventive method that hole structure is analyzed design with the band modular construction; Can guarantee all the time that analytical model is identical with the cad model of structure; Improved the precision of structure analysis design; And only need just can satisfy higher analysis precision with less unit, the structure analysis efficiency of design that greatly improves.In the analysis design of embodiment 1 square perforated flat plate structure, the inventive method adopts the quantity of unit to be merely 39.93% of finite element method, just can obtain and the essentially identical stress result of finite element method; In the analysis design of embodiment 2 square band assembly slab constructions, the inventive method adopts the quantity of unit to be merely 39.63% of finite element method, just can obtain and the essentially identical stress result of finite element method.

Below in conjunction with accompanying drawing and embodiment the present invention is elaborated.

Description of drawings

Fig. 1 structural representation that to be the present invention analyzed in first embodiment and second embodiment.

Fig. 2 is the parameter region grid synoptic diagram of Fig. 1 structure.

If the inside of the circular hole among Fig. 1 does not have material then representes square perforated flat plate structure, if there is material circular hole inside then representes square band assembly slab construction.The grid synoptic diagram that Fig. 2 is the non-porous parameter region after carrying out branches such as 16 on ξ and the η direction respectively, non-porous parameter region inside is that hole shape structural parameters zone is represented in the little rectangular area on border to add heavy line.

Embodiment

Following examples see figures.1.and.2.

Embodiment 1: the analysis design of square perforated flat plate structure.

A circular hole that is positioned at its geometric center is arranged on the square perforated flat plate structure.Suppose the length of side L=8mm of square plate, thickness is 1mm, the circle hole radius R=1mm at dull and stereotyped center, and material behavior is: Young modulus E=200GPa, Poisson ratio μ=0.3.The right and left at flat board applies even distributed force 300N/mm, and both sides apply even distributed force 100N/mm up and down.

(a) with square perforated flat plate structure separated into two parts, the complete square non-porous a when part is not perforate, another part is the hole shape structure the same with hole shape.Non-porous and hole shape structure may be defined as according to the expression-form of nurbs surface:

$S(\mathrm{\ξ},\mathrm{\η})=\frac{\underset{i=0}{\overset{n_S}{\mathrm{\Σ}}}\underset{j=0}{\overset{m_S}{\mathrm{\Σ}}}{N}_{i,p}^S\left(\mathrm{\ξ}\right)N_{j,q}^S\left(\mathrm{\η}\right){\mathrm{\ω}}_{i,j}^S{P}_{i,j}}{\underset{i=0}{\overset{n_S}{\mathrm{\Σ}}}\underset{j=0}{\overset{m_S}{\mathrm{\Σ}}}{N}_{i,p}^S\left(\mathrm{\ξ}\right)N_{j,q}^S\left(\mathrm{\η}\right){\mathrm{\ω}}_{i,j}^S}---\left(1\right)$

$H(\mathrm{\ξ},\mathrm{\η})=\frac{\underset{k=0}{\overset{n_H}{\mathrm{\Σ}}}\underset{l=0}{\overset{m_H}{\mathrm{\Σ}}}{N}_{k,p}^H\left(\mathrm{\ξ}\right)N_{l,q}^H\left(\mathrm{\η}\right){\mathrm{\ω}}_{k,l}^H{Q}_{k,l}}{\underset{k=0}{\overset{n_H}{\mathrm{\Σ}}}\underset{l=0}{\overset{m_H}{\mathrm{\Σ}}}{N}_{k,p}^H\left(\mathrm{\ξ}\right)N_{l,q}^H\left(\mathrm{\η}\right){\mathrm{\ω}}_{k,l}^H}$

In the formula, S (ξ, η) and H (ξ, η) expression is abstracted into the non-porous and the hole shape structure of nurbs surface sheet respectively, the subscript of variable or subscript S and H represent non-porous and hole shape structure respectively, the rectangle parameter region of non-porous and hole shape structure is used Ω respectively _SurfAnd Ω _HoleExpression, parameter region Ω _SurfAnd Ω _Holeξ and η both direction are all arranged, and (ξ η) is any coordinate of any on the parameter region.Non-porous parameter region Ω _SurfKnot vector on ξ and η direction is designated as Ξ _SurfAnd H _Surf, then

With

Be respectively to be defined in knot vector Ξ _SurfAnd H _SurfOn non-reasonable B spline base function, n _S+ 1 and m _S+ 1 is respectively the B spline base function With

Number,

With

Subscript p and q be the number of times of B spline base function.Hole shape structural parameters zone Ω _HoleKnot vector on ξ and η direction is designated as Ξ _HoleAnd H _Hole, then

With

Be respectively to be defined in knot vector Ξ _SurfAnd H _SurfOn non-reasonable B spline base function, n _H+ 1 and m _H+ 1 is respectively the B spline base function

With

Number, With Subscript p and q be the number of times of B spline base function.

With Be respectively the weight factor of non-porous and hole shape structure, { P _{I, j}And { Q _{K, l}Be respectively the reference mark of non-porous and hole shape structure.Will confirm non-reasonable B spline base function, weight factor and the reference mark of non-porous and hole shape structure below.

The nurbs surface parameter of non-porous is: the start node vector

The number of times p=q=2 of B spline base function, reference mark { P _{I, j}And weight factor See table 1.The nurbs surface parameter of hole shape structure is: the start node vector

The number of times p=q=2 of B spline base function, reference mark { Q _{K, l}And weight factor

See table 2.

Table 1

Table 2

The unit of the reference mark coordinate in table 1 and the table 2 is mm; The B spline base function of non-porous

and

calculate according to knot vector and

, the B spline base function of hole shape structure

and

according to knot vector He calculate.

(b) at non-porous parameter region Ω _Surfξ and η direction on carry out 16 five equilibriums, the knot vector of this moment is designated as Ξ _SurfAnd H _SurfNon-porous parameter region Ω _SurfMesh lines after the segmentation can be mapped to the mesh lines on the non-porous, judges the quadrilateral mesh wire frame the most close with the border, hole according to the intersection point of hole and non-porous mesh lines, this grid wire frame corresponding node ξ in parameter region ₁=0.375, ξ ₂=0.625, η ₁=0.375 and η ₂=0.625, at ξ ₁, ξ ₂, η ₁And η ₂The place repeats to insert a minor node.With ξ ₁And ξ ₂Between knot vector as Ξ _Hole, η ₁And η ₂Between knot vector as H _Hole, hole shape structural parameters zones Ω then _HoleThe border be with (ξ ₁, η ₁), (ξ ₂, η ₁), (ξ ₂, η ₁) and (ξ ₂, η ₂) be the rectangular area on summit.

Because the B spline base function is only relevant with knot vector, and the non-porous knot vector of this moment is at parameter region Ω _HoleInterior part is the same with hole shape structure node vector, so non-porous and hole shape structure are at parameter region Ω _HoleInterior B spline base function also equates, that is: as (ξ, η) ∈ Ω _HoleThe time:

$N_{i,p}^S\left(\mathrm{\ξ}\right)=N_{k,p}^H\left(\mathrm{\ξ}\right)$ $N_{j,q}^S\left(\mathrm{\η}\right)=N_{l,q}^H\left(\mathrm{\η}\right)---\left(2\right)$

In the formula, target relation is under the basis function: i-k=j-l=6, and k=0 wherein, 1 ..., 5, l=0,1 ..., 5.

(c) according to knot vector Ξ _HoleAnd H _HoleThe hole shape structure is carried out the grid segmentation, again with the reference mark { Q after the segmentation of hole shape structure _{K, l}And weight factor

Substitute Ω in the non-porous parameter region _HoleCorresponding reference mark

And weight factor Because Ξ _HoleAnd H _HoleFirst end-node certain multiplicity is all arranged, can know to have only the reference mark according to the local support property of NURBS rational basis function

Corresponding basis function Kind

At Ω _HoleGo up non-vanishingly, then pass through the non-porous reference mark { P after the replacement operation _{I, j}And weight factor

With hole shape structure control point { Q _{K, l}And weight factor

Relation following: as (ξ, η) ∈ Ω _HoleAnd

With

The time:

P _i，j＝Q _k，l ${\mathrm{\ω}}_{i,j}^S={\mathrm{\ω}}_{k,l}^H---\left(3\right)$

In the formula, the definition of the subscript of reference mark and weight factor is the same.

To

Be replaced by hole shape structure control point { Q _{K, l}Afterwards, in order to guarantee { Q _{K, l}With

The coordination at periphery reference mark is merged, and it is right to need

The periphery reference mark is regulated.With { Q _{K, l}Outermost certain reference mark is designated as P _Inner, corresponding with it Outside reference mark on the 3rd layer of grid be designated as P _Outer,

Outside reference mark on the layers 1 and 2 grid all give up.Existing according to P _InnerAnd P _OuterCounter two new reference mark P that ask between them ₁And P ₂, on behalf of its place grid, new reference mark subscript 1 and 2 be positioned at

Outside the number of plies, suppose P _InnerTo P ₁Distance be d ₁, P ₁To P ₂Distance be d ₂, P ₂To P _OuterDistance be d ₃, d then ₁, d ₂And d ₃Should satisfy:

d ₁/d ₂＝d ₂/d ₃＝λ (4)

In the formula; Proportionality factors lambda gets 0.3; New reference mark on the layers 1 and 2 grid is all counter outside with

obtains by this method, and is close more the closer to the grid in hole to guarantee.

Owing to is hole structure, with the reference mark by analytic target

And weight factor

Be replaced by hole shape structure control point { Q _{K, l}And weight factor

Afterwards, only need to keep the reference mark and the weight factor of hole shape structure boundary, border internal control point and weight factor are all given up.

Formula (2) and formula (3) substitution formula (1) can be got, as (ξ, η) ∈ Ω _HoleThe time:

S(ξ，η)＝H(ξ，η) (5)

According to following formula, if incite somebody to action non-porous parameter region Ω at this moment _SurfIn hole shape structural parameters zones Ω _HoleDelete, that is: Ω _Surf-Ω _Hole, then corresponding non-porous S (ξ, η) will delete hole shape structure H (ξ η), thereby just becomes the square perforated flat plate structure of being analyzed by inside.So square perforated flat plate structure is by parameter region Ω _Surf-Ω _HoleMapping forms, parameter region Ω _Surf-Ω _HoleIt is a parameter region after rectangular wire frame is pruned.

(d) at parameter region Ω _Surf-Ω _HoleLast computing unit stiffness matrix also is assembled into the global stiffness matrix with it.The computing method of element stiffness matrix are following: for unit k, regard relevant 9 NURBS rational basis functions and reference mark with it as in the finite element method shape function and node, can calculate the Jacobian matrix J of unit k ^kWith geometric matrix B ^k, the material behavior according to unit k obtains elastic matrix D again ^k, element stiffness matrix k then _e ^kCan be by computes:

${k_e}^k={\∫}_{{\mathrm{\Ω}}^k}{\left(B^k\right)}^T{D}^k{B}^k\det \left[J^k\right]\mathrm{d\Ω}---\left(6\right)$

In the formula, limit of integration Ω ^kBe the rectangular node zone of unit k in the parameter region, be convenient to carry out Gauss integration.

According to the even distributed force that applies on the square perforated flat plate structure and its global stiffness matrix, just can calculate the mechanical responses such as stress, strain of square perforated flat plate structure.The element number that present embodiment adopts is 240, and unit maximum Von Mises stress result is about 753MPa.The application of finite element analysis software ANSYS is analyzed the square perforated flat plate structure of present embodiment; Adopt the free grid division methods of ANSYS acquiescence; When unit maximum Von Mises stress that element number obtains more than or equal to 601 time converges on 752MPa between the 755MPa, basic identical with the result of present embodiment.It is thus clear that when the element number that adopts as finite element analysis software ANSYS was 601, the element number that present embodiment adopts is merely it 39.93% just can obtain essentially identical with it stress result.

Embodiment 2: the analysis design of square band assembly slab construction.

On the square band assembly slab construction circular components that is positioned at its geometric center is arranged, the material behavior of size, boundary condition and assembly external structure of supposing structure is with embodiment 1, and the material behavior of assembly is: Young modulus E=71GPa, Poisson ratio μ=0.33.

The analytical procedure of present embodiment and embodiment 1 are basic identical; Only need after the reference mark of embodiment 1 step (c) substitutes with weight factor, to keep hole shape structure boundary inner reference mark and weight factor, in the stiffness matrix calculating of embodiment 1 step (d) with the parameter region Ω that deletes _HoleGive the material behavior of assembly and calculate Ω _HoleOn element stiffness matrix.The element number that present embodiment adopts is 256, and unit maximum Von Mises stress result is about 386MPa.The application of finite element analysis software ANSYS is analyzed the square band assembly slab construction of present embodiment; Adopt the free grid division methods of ANSYS acquiescence; When unit maximum Von Mises stress that element number obtains more than or equal to 646 time converges on 385MPa between the 388MPa, basic identical with the result of present embodiment.It is thus clear that when the element number that adopts as finite element analysis software ANSYS was 646, the element number that present embodiment adopts is merely it 39.63% just can obtain essentially identical with it stress result.

Claims (1)

1. the analysis design method of hole structure and band modular construction is characterized in that may further comprise the steps:

(a) with hole structure or band modular construction separated into two parts, the complete non-porous a when part is not perforate, another part is the hole shape structure the same with hole or element shapes; Non-porous and hole shape structure are defined as according to the expression-form of nurbs surface:

$S(\mathrm{\ξ},\mathrm{\η})=\frac{\underset{i=0}{\overset{n_S}{\mathrm{\Σ}}}\underset{j=0}{\overset{m_S}{\mathrm{\Σ}}}{N}_{i,p}^S\left(\mathrm{\ξ}\right)N_{j,q}^S\left(\mathrm{\η}\right){\mathrm{\ω}}_{i,j}^S{P}_{i,j}}{\underset{i=0}{\overset{n_S}{\mathrm{\Σ}}}\underset{j=0}{\overset{m_S}{\mathrm{\Σ}}}{N}_{i,p}^S\left(\mathrm{\ξ}\right)N_{j,q}^S\left(\mathrm{\η}\right){\mathrm{\ω}}_{i,j}^S}---\left(1\right)$

$H(\mathrm{\ξ},\mathrm{\η})=\frac{\underset{k=0}{\overset{n_H}{\mathrm{\Σ}}}\underset{l=0}{\overset{m_H}{\mathrm{\Σ}}}{N}_{k,p}^H\left(\mathrm{\ξ}\right)N_{l,q}^H\left(\mathrm{\η}\right){\mathrm{\ω}}_{k,l}^H{Q}_{k,l}}{\underset{k=0}{\overset{n_H}{\mathrm{\Σ}}}\underset{l=0}{\overset{m_H}{\mathrm{\Σ}}}{N}_{k,p}^H\left(\mathrm{\ξ}\right)N_{l,q}^H\left(\mathrm{\η}\right){\mathrm{\ω}}_{k,l}^H}$

In the formula, S (ξ, η) and H (ξ, η) expression is abstracted into the non-porous and the hole shape structure of nurbs surface sheet respectively, the subscript of variable or subscript S and H represent non-porous and hole shape structure respectively, the rectangle parameter region of non-porous and hole shape structure is used Ω respectively _SurfAnd Ω _HoleExpression, parameter region Ω _SurfAnd Ω _Holeξ and η both direction are all arranged, and (ξ η) is any coordinate of any on the parameter region; Non-porous parameter region Ω _SurfKnot vector on ξ and η direction is designated as Ξ _SurfAnd H _Surf, then

With

Be respectively to be defined in knot vector Ξ _SurfAnd H _SurfOn non-reasonable B spline base function, n _S+ 1 and m _S+ 1 is respectively the B spline base function

With

Number, With

Subscript p and q be the number of times of B spline base function; Hole shape structural parameters zone Ω _HoleKnot vector on ξ and η direction is designated as Ξ _HoleAnd H _Hole, then

With

Be respectively to be defined in knot vector Ξ _SurfAnd H _SurfOn non-reasonable B spline base function, n _H+ 1 and m _H+ 1 is respectively the B spline base function With

Number,

With

Subscript p and q be the number of times of B spline base function;

With

Be respectively the weight factor of non-porous and hole shape structure, { P _{I, j}And { Q _{K, l}Be respectively the reference mark of non-porous and hole shape structure; (b) non-porous is carried out the grid segmentation, promptly at its parameter region Ω _SurfCarry out the node refinement to obtain appropriate nodes vector Ξ _SurfAnd H _Surf, then according to the shape of hole or assembly, size and the position in hole structure or band modular construction thereof from Ξ _SurfAnd H _SurfIn select a part as hole shape structural parameters zones Ω _HoleKnot vector Ξ _HoleAnd H _HoleAt non-porous knot vector Ξ _SurfAnd H _SurfIn repeat to insert node with corresponding position, hole shape structural parameters zone boundary, to obtain the hole shape structure node vector Ξ that first end-node repeats _HoleAnd H _Hole

Like this, hole shape structure node vector Ξ _HoleAnd H _HoleBe respectively non-porous knot vector Ξ _SurfAnd H _SurfIn a part, that is:

With

Hole shape structural parameters zones Ω then _HoleAlso become non-porous parameter region Ω _SurfIn a part, that is:

Because the B spline base function is only relevant with knot vector, and the non-porous knot vector of this moment is at parameter region Ω _HoleInterior part is the same with hole shape structure node vector, so non-porous and hole shape structure are at parameter region Ω _HoleInterior B spline base function also equates, that is: as (ξ, η) ∈ Ω _HoleThe time

$N_{i,p}^S\left(\mathrm{\ξ}\right)=N_{k,p}^H\left(\mathrm{\ξ}\right)$ $N_{j,q}^S\left(\mathrm{\η}\right)=N_{l,q}^H\left(\mathrm{\η}\right)---\left(2\right)$

In the formula, all differ a constant amount between the subscript i of basis function and the k and between j and the l, this is by hole shape structural parameters zone Ω _HoleAt non-porous parameter region Ω _SurfIn determining positions;

(c) according to knot vector Ξ _HoleAnd H _HoleThe hole shape structure is carried out the grid segmentation, again with the reference mark { Q after the segmentation of hole shape structure _{K, l}And weight factor

Go to substitute the reference mark of non-porous appropriate section

And weight factor

By hole shape structural parameters zone Ω _HoleAt non-porous parameter region Ω _SurfIn determining positions; Because knot vector Ξ _HoleAnd H _HoleFirst end-node certain multiplicity is all arranged, can know to have only the reference mark according to the local support property of NURBS rational basis function Corresponding basis function

With

At Ω _HoleGo up non-vanishingly, then pass through the non-porous reference mark { P after the replacement operation _{I, j}And weight factor

With hole shape structure control point { Q _{K, l}And weight factor Relation following: as (ξ, η) ∈ Ω _HoleAnd

With

The time:

P _i，j＝Q _k，l ${\mathrm{\ω}}_{i,j}^S={\mathrm{\ω}}_{k,l}^H---\left(3\right)$

To

Be replaced by hole shape structure control point { Q _{K, l}Afterwards, in order to guarantee { Q _{K, l}With

The coordination at periphery reference mark is merged, and it is right to need The periphery reference mark is regulated; Control method is following, with { Q _{K, l}Outermost certain reference mark is designated as P _Inner, corresponding with it

Outside reference mark on the L layer grid be designated as P _Outer, obtain outside the hole the 1st layer of new reference mark to the L-1 layer grid according to certain proportionality factors lambda is counter; If L=3, then P _InnerAnd P _OuterBetween need instead ask two new reference mark P ₁And P ₂, on behalf of its place grid, the subscript 1 and 2 at new reference mark be positioned at Outside the number of plies, suppose P _InnerTo P ₁Distance be d ₁, P ₁To P ₂Distance be d ₂, P ₂To P _OuterDistance be d ₃, d then ₁, d ₂And d ₃Should satisfy:

d ₁/d ₂＝d ₂/d ₃＝λ (4)

In the formula, proportionality factors lambda is got the positive number less than 1, with guarantee the closer to the hole or the grid of assembly close more;

Formula (2) and formula (3) substitution formula (1) are got, as (ξ, η) ∈ Ω _HoleThe time:

S(ξ，η)＝H(ξ，η)?(5)

According to following formula, with the non-porous parameter region Ω of this moment _SurfIn hole shape structural parameters zones Ω _HoleDelete, that is: Ω _Surf-Ω _Hole, then corresponding non-porous S (ξ, η) inside will be deleted hole shape structure H (ξ η), thereby is just become the hole structure of being analyzed; So hole structure is by parameter region Ω _Surf-Ω _HoleMapping forms, parameter region Ω _Surf-Ω _HoleIt is a parameter region after rectangular wire frame is pruned;

(d) during the analytic band pore structure just at parameter region Ω _Surf-Ω _HoleGo up material behavior computing unit stiffness matrix, then only need during the analytic band modular construction in addition at parameter region Ω according to structure _HoleGo up material behavior computing unit stiffness matrix, will be assembled into the global stiffness matrix by element stiffness matrix at last, calculate the displacement and the stress of structure again according to boundary condition according to assembly;

The computing method of element stiffness matrix are following: for unit k, relevant with it (p+1) (q+1) individual NURBS rational basis function and reference mark regarded as shape function and node in the finite element method, promptly calculate the Jacobian matrix J of unit k ^kWith geometric matrix B ^k, the material behavior according to unit k obtains elastic matrix D again ^k, element stiffness matrix k then _e ^kBy computes:

${k_e}^k={\∫}_{{\mathrm{\Ω}}^k}{\left(B^k\right)}^T{D}^k{B}^k\det \left[J^k\right]\mathrm{d\Ω}---\left(6\right)$

In the formula, limit of integration Ω ^kBe the rectangular node zone of unit k in the parameter region, be convenient to carry out Gauss integration.

Images