CN107577837A - The structural optimization method of quasi-truss is described using subregion interpolation polynomial - Google Patents
The structural optimization method of quasi-truss is described using subregion interpolation polynomial Download PDFInfo
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Abstract
The invention discloses a kind of structural optimization method that quasi-truss is described using subregion interpolation polynomial, design domain is divided into several subdomains first, polynomial interpolating function is constructed according to domain boundary Control point respectively in each subdomain.Density using the position at control point and material at control point is used as design variable.Inside each subdomain material distribution density and direction are calculated according to interpolation polynomial.Lead to these subdomain universal formulation finite elements.Carry out finite element analysis and optimization is analyzed.Cross the Optimum distribution that the density of the position and material of optimal control point at control point realizes truss-like material.On the basis of this, discrete several parameter cans are selected to simplify to obtain truss, or the parameter in the several zonal cooling sections of selection can be obtained by homogeneous isotropism perforated plate.Realize that truss-like structure optimizes by optimizing the distribution of partition boundaries wire shaped and material on partition boundaries line, for solving the problems, such as the Structural Topology Optimization Design of Inhomogeneous Anisotropic material.
Description
Technical field
The present invention relates to a kind of structural optimization method that quasi-truss is described using subregion interpolation polynomial, belong to structure optimization
Design field, it can apply to truss and homogeneous isotropism non-individual body method of topological optimization design with holes.
Background technology
Be frequently necessary in many fields such as space flight and aviation, material, machinery, building and ship and water conservancy to various materials and
Structure optimizes design.These materials and structure not only need to meet the requirement using function, and the material being desirable for
It is as few as possible.How the various materials of optimization design and structure are the problem of these fields are most important.And solve these problems simultaneously
It is not an easy thing.Due to the development of material science and manufacturing technology so that we can manufacture material scope it is continuous
Extension, but how scientifically and rationally designing material and structure can really not solve very well.This belongs to opening up for material and structure
Flutter optimization design problem.
Current various structural topological optimization methods are all to use isotropic material model, directly obtain homogeneous isotropism
Non-individual body with holes.Topological optimization result represents that this kind of method has sawtooth border, grey color density and checkerboard patterns etc. with unit
Numerical computations instability problem is, it is necessary to further handle.Importantly, should be non-homogeneous in topological optimization structural theory
Anisotropy quasi-truss non-individual body.These methods differ farther out away from theoretic optimal solution.Based on the excellent of truss-like material model
Change method, which can directly obtain, is sufficiently close to theoretical majorization of solutions structure.
Traditional optimization method based on truss-like material model using the density of material of finite element site position and direction as
Optimization design variable.For stress constraint volume minimum problem, density of the material in site position is optimized using fully stressed criterion
And direction, the material density of optional position and direction in unit are obtained by interpolation, so as to form the material distribution field of optimization.
In order to which finite element analysis is accurate enough, more unit and node are generally required.Because design variable is defined on site position, knot
Point excessively means that design variable is excessive, and this causes optimization efficiency to reduce.Particularly when using the Mathematical Planning side based on sensitivity
During method research general structure topology optimization problem, because the precision of basis of sensitivity analysis opposed configuration analysis is low, cause the precision of optimization
Also it is not high.In optimization process, often there is acute variation in the direction of each site position in material, it is difficult to is formed rational continuous
Smooth curve, error are larger.Further, because final optimization pass target is material distribution field, with material in the direction of site position
As optimization design variable, then it is indirect method to synthesize distribution field function, adds workload and error.Direct Optimum distribution letter
Number can be more reasonable.
The content of the invention
The invention provides a kind of structural optimization method that quasi-truss is described using subregion interpolation polynomial, solves non-homogeneous
The optimization design of anisotropy quasi-truss non-individual body, member structure and the homogeneous isotropism for finding topological optimization are with holes continuous
Body.The technical solution adopted for the present invention to solve the technical problems is：
The structural optimization method of quasi-truss is described using subregion interpolation polynomial, including：
The first step：Using load point O as origin, approximate oblique coordinates system O α β are established；Curvilinear coordinate axle is by multistage straight line
Section approximate representation；The 1st section of line segment of O α axles and x-axis angle are α10, length l11；O α axles since the 2nd section of line segment, with it is previous
The angular separation α of section line segmenti0(i=2,3 ..., n) is defined, and the length of every section of line segment is l1i(i=1,2 ..., n-1), form control
Make point Pi0(i=1,2 ..., n), n are the number of O α axles division；When reference axis intersects with border, such as Pn0Point, the length of the line segment
Degree is selected on border not as design variable, line segment terminal；90 degree of line segment rotate counterclockwise of the 1st section of O α axles and β10For O β axles the 1st
The direction of section line segment, length l21；O β axles are since the 2nd section of line segment, with the angular separation β with the last period line segment0i(i=2,
3 ..., m) definition, the length of every section of line segment is l2i(i=1,2,3 ..., m), form control point P0i(i=1,2,3 ..., m), m
For the number of O β axles division；The number n and m of two reference axis divisions are previously set, and the two parameters determine computational accuracy；When
Institute is angled initial when being taken as zero, and O α β are exactly rectangular coordinate system；
Second step：Coordinate grid is established, design domain is divided into several quadrangles；Detailed process is：With P10And P01
For starting point, along P10The mean direction of front and rear two sections of line segments rotates counterclockwise β again11Do straight line；Along P01Front and rear two sections of line segments are averaged
Direction rotates counterclockwise α again11Straight line is done, this two straight lines meet at P11, form quadrangle OP10P11P01；Again with P11And P02To rise
Point forms next quadrangle P01P11P12P02；With P02And P11Next quadrangle P is formed for starting point10P20P21P11；With such
Push away, design domain is divided into intersecting net region, form several quadrangles；When institute is angled is initially taken as zero, the net
Lattice are line orthogonal regular grids, and quadrangle is all rectangle.
3rd step：The design domain formed with above-mentioned quadrangle divides finite elements；To form the basic parameter of grid, αij(i
=1,2 ..., n；J=0,1,2 ..., m), βij(i=0,1,2 ..., n-1；J=0,1,2 ..., m), lij(i=1,2；J=1,
2 ..., m-1) and control point position density of material tij(i=0,1,2 ..., n；J=0,1,2 ..., m) it is used as design variable；
Grid lines and interpolating function therein are exactly the direction of material, actual when calculating, and take the direction of two sections of lines before and after control point to be averaged
Value；There is such a grid, actually just obtained the distribution field of material；The density of material of site position by finite element shape
Function obtains according to the density of material and directional interpolation of control point position,
(x in formulai,yi) represent four angle points of quadrangle coordinate, (ξ, η) can be understood as parameter；
Formula (1) constitutes material distribution field function；It is, working as ξ=c constants, η changes as parameter between [- 1,1]
When, the curve that formula (1) represents illustrates the material direction of the point of curve process；Similarly, when η=c constants, ξ as parameter [-
When changing between 1,1], the curve that formula (1) represents illustrates another material direction.Density of material also uses similar formula (1)
Interpolation method
T in formulabiThe density of material in four control points of quadrangle position is represented, it is possible thereby to determine finite element site position
Density of material；
4th step：Finite element analysis, solve displacement of joint, stress and strain；By calculus of finite differences calculating target function on setting
Count the sensitivity of variable；
5th step：KKT conditions are calculated, meet then to stop calculating；Otherwise enter in next step；
6th step：Object function and constraint function line are deployed, using plan QUADRATIC PROGRAMMING METHOD FOR optimization design variable；Control
Set up meter variable αij,βijExcursion in a less scope, return second step.
Compared with background technology, it has the following advantages that the technical program：
The present invention separates design variable with the node of finite element, and design domain is divided into some quadrangle subdomains.Each
Subdomain includes some finite elements.With the material distribution function field in larger grid division interpolation polynomial approximation subregion.Four
Two groups of side shape is to the edge direction just distribution arrangement as two groups of materials, and quadrangle internal material density and direction are by interpolation polynomial
Formula is calculated.
Realize that truss-like structure optimizes by optimizing the distribution of partition boundaries wire shaped and material on partition boundaries line,
For solving the problems, such as the Structural Topology Optimization Design of Inhomogeneous Anisotropic material.
Brief description of the drawings
The invention will be further described with reference to the accompanying drawings and examples.
Fig. 1 quasi-truss optimized variable and structural topology
Fig. 2 is the flow chart of optimization method of the present invention.
Fig. 3 is the initial designs domain and load of one Rectangular Plate Structure of embodiment.
Fig. 4 is the optimal material distribution of the embodiment obtained using the inventive method.
Embodiment
Fig. 1 to Fig. 4 is refer to, the structural optimization method of quasi-truss is described using subregion interpolation polynomial, carries out cantilever square
The topology optimization design of ellbeam
One long L=1.6 rice, high H=1 rice, thick 0.01 meter rectangular design domain are as shown in Figure 2.Lower-left Angle Position effect
The concentrated forces of one 100kN straight down, the right is fixed.Last optimization of material distribution visualized graphs are shown in Fig. 3.Two
Direction division number is n=m=3, and it is origin of coordinates O to take load point.
Optimization Steps are as follows：
1. initial option control parameter：The initial length of both direction is l1j=L/n, l2j=H/m and direction angle alphaij=0,
βij=0, tij=0.2.
2. form grid according to control parameter.Wherein the right the 3rd row grid lines length not as design variable, but with
Intersect with fixed boundary and be defined.
3. the design domain division unit that all sub-field combinations are formed.In each quadrangle subdomain, shape function is selected to make
For interpolating function.Density of material and the direction of site position are obtained according to the density of material at control point and directional interpolation, form knot
Structure stiffness matrix.
4. carrying out finite element analysis, displacement of joint column vector is obtained.Calculated by the strain and elastic matrix of site position
To the strain of the principal direction of stress and principal direction of stress of site position.
5. using derivative of the difference method calculating target function on design variable.It is excellent if KKT optimal conditions are met
Change terminates, and otherwise enters in next step.
6. object function and constraint function line are deployed, using plan QUADRATIC PROGRAMMING METHOD FOR optimization design variable.Return to step
Rapid 2.
It is described above, only present pre-ferred embodiments, therefore the scope that the present invention is implemented can not be limited according to this, i.e., according to
The equivalent changes and modifications that the scope of the claims of the present invention and description are made, all should still it belong in the range of the present invention covers.
Claims (3)
1. the structural optimization method of quasi-truss is described using subregion interpolation polynomial, it is characterised in that including：
The first step：Using load point O as origin, approximate oblique coordinates system O α β are established；Curvilinear coordinate axle is near by multistage straightway
Like expression；The 1st section of line segment of O α axles and x-axis angle are α10, length l11；O α axles since the 2nd section of line segment, with the last period line
The angular separation α of sectioni0(i=2,3 ..., n) is defined, and the length of every section of line segment is l1i(i=1,2 ..., n-1), form control point
Pi0(i=1,2 ..., n), n are the number of O α axles division；When reference axis intersects with border, such as Pn0Point, the length of the line segment is not
As design variable, line segment terminal is selected on border；90 degree of line segment rotate counterclockwise of the 1st section of O α axles and β10For the 1st section of line of O β axles
The direction of section, length l21；O β axles are since the 2nd section of line segment, with the angular separation β with the last period line segment0i(i=2,3 ...,
M) define, the length of every section of line segment is l2i(i=1,2,3 ..., m), form control point P0i(i=1,2,3 ..., m), m are O β axles
The number of division；The number n and m of two reference axis divisions are previously set, and the two parameters determine computational accuracy；When all angles
When degree is initially taken as zero, O α β are exactly rectangular coordinate system；
Second step：Coordinate grid is established, design domain is divided into several quadrangles；Detailed process is：With P10And P01For starting point,
Along P10The mean direction of front and rear two sections of line segments rotates counterclockwise β again11Do straight line；Along P01The mean direction of front and rear two sections of line segments is again
Rotate counterclockwise α11Straight line is done, this two straight lines meet at P11, form quadrangle OP10P11P01；Again with P11And P02Formed for starting point
Next quadrangle P01P11P12P02；With P02And P11Next quadrangle P is formed for starting point10P20P21P11；By that analogy, will set
Meter domain is divided into intersecting net region, forms several quadrangles；When institute is angled is initially taken as zero, the grid is straight line
Orthogonal regular grid, quadrangle are all rectangles.
3rd step：The design domain formed with above-mentioned quadrangle divides finite elements；To form the basic parameter of grid, αij(i=1,
2,…,n；J=0,1,2 ..., m), βij(i=0,1,2 ..., n-1；J=0,1,2 ..., m), lij(i=1,2；J=1,2 ...,
M-1) and control point position density of material tij(i=0,1,2 ..., n；J=0,1,2 ..., m) it is used as design variable；Grid
Line and interpolating function therein are exactly the direction of material, when reality calculates, take the direction average value of two sections of lines before and after control point；Have
Such a grid, has actually just obtained the distribution field of material；The density of material of site position by finite element shape function
Obtained according to the density of material of control point position and directional interpolation,
(x in formulai,yi) represent four angle points of quadrangle coordinate, (ξ, η) can be understood as parameter；
Formula (1) constitutes material distribution field function；It is, work as ξ=c constants, when η changes as parameter between [- 1,1],
The curve that formula (1) represents illustrates the material direction of the point of curve process；Similarly, when η=c constants, ξ is as parameter in [- 1,1]
Between when changing, the curve that formula (1) represents illustrates another material direction.Density of material also uses the interpolation of similar formula (1)
Mode
T in formulabiThe density of material in four control points of quadrangle position is represented, it is possible thereby to determine the material of finite element site position
Expect density；
4th step：Finite element analysis, solve displacement of joint, stress and strain；Become by calculus of finite differences calculating target function on design
The sensitivity of amount；
5th step：KKT conditions are calculated, meet then to stop calculating；Otherwise enter in next step；
6th step：Object function and constraint function line are deployed, using plan QUADRATIC PROGRAMMING METHOD FOR optimization design variable；Control is set
Count variable αij,βijExcursion in a less scope, return second step.
2. the structural optimization method according to claim 1 that quasi-truss is described using subregion interpolation polynomial, its feature are existed
In：In 6th step, such as | αij|<π/2 。
3. the structural optimization method according to claim 2 that quasi-truss is described using subregion interpolation polynomial, its feature are existed
In：In 6th step, | αij|≤π/8,|βij|≤π/8。
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| CN111104764A (en) * | 2020-01-21 | 2020-05-05 | 湖南科技大学 | Structured grid optimization division method for alternating current motor rotor conducting bar thermal analysis model |
| CN113449454A (en) * | 2021-07-21 | 2021-09-28 | 中交第三公路工程局有限公司 | Topological optimization method of novel steel truss structure |
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Cited By (8)
| Publication number | Priority date | Publication date | Assignee | Title |
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| US11455438B2 (en) | 2018-02-01 | 2022-09-27 | Toyota Motor Engineering & Manufacturing North America, Inc. | Methods for topology optimization using a membership variable |
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| CN111104764A (en) * | 2020-01-21 | 2020-05-05 | 湖南科技大学 | Structured grid optimization division method for alternating current motor rotor conducting bar thermal analysis model |
| CN111104764B (en) * | 2020-01-21 | 2020-08-25 | 湖南科技大学 | Structured grid optimization division method for alternating current motor rotor conducting bar thermal analysis model |
| CN113449454A (en) * | 2021-07-21 | 2021-09-28 | 中交第三公路工程局有限公司 | Topological optimization method of novel steel truss structure |
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