CN110069864A - A kind of improvement Topology optimization based on bi-direction evolutionary structural of combination density variable method - Google Patents
A kind of improvement Topology optimization based on bi-direction evolutionary structural of combination density variable method Download PDFInfo
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Abstract
The invention discloses a kind of improvement Topology optimization based on bi-direction evolutionary structural of combination density variable method.The design method includes that two-way penalty improved in density variable method is applied to bi-direction evolutionary structural topology optimization, the method filter sensitivity filtered using sensitivity, then sensitivity is further homogenized using the discrete method of BESO, so that result is easier to restrain, iteration is updated finally for unit and uses dichotomy searching sensitivity threshold value, until iteration convergence.Method proposed by the invention can effectively reduce the number of iterations, save and calculate the time, improve optimization efficiency.
Description
Technical field
The present invention relates to structure equivalence static load technical fields, and in particular to a kind of improvement of combination density variable method is two-way gradually
Into structural topological optimization method.
Background technique
A kind of Topology Optimization Method based on heuristic rule is proposed in Xie in 1993 et al., i.e. evolutional structure optimizes
Method (Evolutionary Structure Optimization, ESO), this method and other based on continuous design variable
Modeling method is the difference is that it originates from stress design technology, it is believed that in design domain, the inoperative material in structure
Material, i.e., those low stresses or the material of low strain dynamic energy density are inefficient, can be removed.The removal of material can pass through
Change the elasticity modulus as stress or strain energy density function or directly leaves out those low stresses or low strain dynamic energy density
Material space.By removing invalid or inefficient material step by step, remaining structure will gradually tend to optimize.
Evolutional structure optimization method is a kind of evolutional structure optimization method that can delete and increase material simultaneously, that is, is being deleted
High stress areas adjacent material is augmented while except inefficient material, the region of initial designs can be smaller, to improve meter
The efficiency of calculation.However primary ESO method can only delete the defect of unit, Querin proposes AESO (Addition
Evolutionary Structure Optimization) method, it can effectively live again bill of materials in solution procedure
Member, this, which can be effectively prevented from, accidentally deletes caused the case where can not restoring due to material.Further, by AESO and ESO
Method is combined together, and constituting can delete but also the two-way BESO (Bi-directional of restorer unit
Evolutionary Structure Optimization, BESO) method.Liu combines ESO with genetic algorithm, leads to
It the genetic evolutions such as crosses to material cell using binary coded form, and using material cell as individual is intersected, made a variation to grasp
Make, to realize the increase and decrease of material cell.
ESO generally uses the indexs such as the stress of structure, sensitivity as evolutionary criterion, and theoretical formula and thought are all more popular
It is understandable, it is easy to be received and used by engineering staff, is difficult to the structural optimization problems expressed with objective function suitable for those.
Summary of the invention
The purpose of the present invention is be in solution equivalence static load method that iterative process is unstable, is easy to appear multiple convergence
The case where, provide a kind of improvement Topology optimization based on bi-direction evolutionary structural of combination density variable method, this method is by density variable method
In penalty be introduced into evolutionary structural optimization method, the sensitivity after using homogenizing as the initial value of each iteration,
Alternative manner is handled using dichotomy, simplifies the iterative process of equivalence static load method, improves the stability of iteration.
To achieve the goals above, the improved BESO method of the present invention is specific as follows, according to equivalence static load method
Mathematical model minimum compliance obeys volume constraint:
ω is submissive angle value, and F is stress load, and U is motion vector, and K is stiffness matrix, ρiPuppet for i-th of unit is close
Degree, δiFor the real density value (0 or 1) of i-th of unit.
It is equal to compliance to unit puppet density derivation according to sensitivity:
It can be in the hope of:
For grid sensitivity filter method, sensitivity information for modified objective function:
D in formulai=rmin- H (i, k), and i ∈ N | H (i, k)≤rmin, rminFor minimum unit diameter d predeterminedmin
Half, H (i, k) be unit i at a distance from close unit k.rminSmaller, cell node sensitivity more levels off to initial sensitivity；
rminBigger, each unit node sensitivity more levels off to identical.By can solve numerical value sharpening to the suitably modified of sensitivity coefficient
Singular problem.
In order to guarantee that the result after iteration restrains, determine further to be homogenized sensitivity using the discrete method of BESO.Tool
Body formula is as follows:
For the sensitivity after homogenizing, k is the sensitivity of e-th of unit, and p (0 < p < 1) is the homogenizing factor.
Sensitivity threshold value is searched using dichotomy.In the iteration initial stage, the minimum value for finding sensitivity is denoted as min, finds
Sensitivity maximum value is denoted as max.Error range is set as 0.00001, sets cycling condition (max-min)/max > 0.00001,
Threshold value th=(max-min)/2 is taken, is then compared threshold value with the sensitivity value of each unit, the big unit of sensitivity value, it will
Its density is set as 1；Conversely, sensitivity is worth small unit, its density is set as 0.001.Finally judge all cell density values
The sum of whether be greater than V*(f*v, as volume threshold): if it is greater than V*, th is assigned to min, execution is then followed by and initially follows
Ring condition (max-min)/max > 0.00001, until convergence, terminates all circulations；If it is less than V*, th is assigned to max,
It is then followed by and executes initial cycle condition (max-min)/max > 0.00001, until convergence, terminates all circulations.
Detailed description of the invention
Fig. 1 is the flow chart for the improvement Topology optimization based on bi-direction evolutionary structural that the present invention combines density variable method.
Fig. 2 is to improve two-way penalty value with the change curve of cell density.
Fig. 3 is two-dimentional cantilever beam example.
Fig. 4 is example optimum results schematic diagram: 4 (a) be conventional method, and 4 (b) be method proposed by the present invention.
Specific embodiment
The present invention will be described in more detail with reference to the accompanying drawing.
The purpose of the present invention is be in solution equivalence static load method that iterative process is unstable, is easy to appear multiple convergence
The case where, provide a kind of improvement Topology optimization based on bi-direction evolutionary structural of combination density variable method, this method is by density variable method
In penalty be introduced into evolutionary structural optimization method, the sensitivity after using homogenizing as the initial value of each iteration,
Alternative manner is handled using dichotomy, simplifies the iterative process of equivalence static load method, improves the stability of iteration.
The improvement BESO method for combining density variable method is specific as follows,
There are also corresponding material parameters with boundary condition for design domain, the load of Definition Model.It gives preset parameter: evolving
Rate ER, filtering radius rmin, volume ratio f.
Volume constraint is obeyed according to the mathematical model minimum compliance of equivalence static load method:
ω is submissive angle value, and F is stress load, and U is motion vector, and K is stiffness matrix, ρiPuppet for i-th of unit is close
Degree, δiFor the real density value (0 or 1) of i-th of unit.
It is equal to compliance to unit puppet density derivation according to sensitivity:
It can be in the hope of:
Then start finite element analysis, the sensitivity of each unit is solved according to above formula.
Next grid sensitivity is filtered, for the sensitivity information of modified objective function, it is existing prevents gridiron pattern
As, it may be assumed that
D in formulai=rmin- H (i, k), and i ∈ N | H (i, k)≤rmin, rminFor minimum unit diameter d predeterminedmin
Half, H (i, k) be unit i at a distance from close unit k.rminSmaller, cell node sensitivity more levels off to initial sensitivity；
rminBigger, each unit node sensitivity more levels off to identical.By can solve numerical value sharpening to the suitably modified of sensitivity coefficient
Singular problem.
In order to guarantee that the result after iteration has better convergence, spirit is further homogenized using the discrete method of BESO
Sensitivity.Specific formula is as follows:
For the sensitivity after homogenizing, k is the sensitivity of e-th of unit, and s (0 < s < 1) is the homogenizing factor.Using upper
To filtered sensitivity, homogenizing is handled formula again.
According to formula Vk+1=Vk(1 ± ER) is iterated, until V according to the value of evolution rate ERk+1≥fVkUntil.
Before above formula iteration stopping, sensitivity threshold value is searched using dichotomy.When finding sensitivity in the iteration initial stage
Minimum value be denoted as min, find sensitivity maximum value and be denoted as max.Error range is set as 0.00001, sets cycling condition
(max-min)/max > 0.00001 takes threshold value th=(max-min)/2, then by the sensitivity value of threshold value and each unit into
Row compares, its density is set as 1 by the big unit of sensitivity value；Conversely, sensitivity is worth small unit, its density is set as
0.001.Finally judge whether the sum of all cell density values are greater than V*(f*v, as volume threshold): if it is greater than V*, by th
It is assigned to min, is then followed by and executes initial cycle condition (max-min)/max > 0.00001, until convergence, terminates all follow
Ring；If it is less than V*, th is assigned to max, is then followed by and executes initial cycle condition (max-min)/max > 0.00001,
Until convergence, terminates all circulations.
Two examples are calculated below according to original Evolutionary structural optimization and method proposed by the present invention.
As shown in Fig. 2, design domain is the two-dimentional cantilever beam structure of 40mm × 200mm, it is divided into 40 × 200 lists
The finite element model of member, model the right midpoint is by a dead load F=1000N effect vertically downward, the left side of cantilever beam
It is clamped.Example others Optimal Parameters are as shown in table 1.
Table 1
Result after optimization is as shown in Figure 3.
Table 2 is the Comparative result of conventional method and this method.
Table 2
From optimum results Fig. 4 (a) and Fig. 4 (b) comparison it can be seen that coming, for unilateral clamped cantilever beam, this hair
Bright proposed method with the optimum results of traditional Evolutionary structural optimization be it is similar, final goal functional value is very close to this
Illustrate that method proposed by the invention is feasible.For this method in view of the number of iteration, this method the number of iterations is 29
It is secondary, and conventional method the number of iterations is 36 times, the number of iterations of this method is fewer than conventional iterative number, therefore illustrates excellent
Change time reduction, optimization efficiency improves 19.4%.Therefore, method efficiency proposed by the present invention is higher than conventional method.
Claims (5)
1. a kind of improvement Topology optimization based on bi-direction evolutionary structural of combination density variable method, which is characterized in that combine variable density
Penalty method in method, and the material interpolation model with penalty factor has been used, improve Evolutionary structural optimization.Compared to
For conventional elements density only has discrete { 0,1 }, it allows density to be provided with continuity, makes iteration result more accurate.
2. the improvement Topology optimization based on bi-direction evolutionary structural of combination density variable method according to claim 1, feature exist
In, it is theoretical in conjunction with " infinite tendency 0-1 ", new penalty is constructed, is unidirectionally punished compared to traditional penalty and is easy to lead
Cause for entity density unidirectionally approach, it has two-way punitive, by set a threshold value come so that small density value to cavity
Evolution, big density value develop to solid element, and specific formula is as follows:
ρiFor the density value of i-th of unit, q and p are coefficient factor, and above formula is two-way interpolating function.
3. the improvement Topology optimization based on bi-direction evolutionary structural of combination density variable method according to claim 1, feature exist
In using grid sensitivity filter method come the sensitivity information of modified objective function:
D in formulai=rmin- H (i, k), and i ∈ N | H (i, k)≤rmin, rminFor minimum unit diameter d predeterminedminOne
Half, H (i, k) are unit i at a distance from close unit k.rminSmaller, cell node sensitivity more levels off to initial sensitivity；rminMore
Greatly, each unit node sensitivity more levels off to identical.By can solve numerical value and sharpen unusual ask to the suitably modified of sensitivity coefficient
Topic.
4. the improvement Topology optimization based on bi-direction evolutionary structural of combination density variable method according to claim 1, feature exist
In in order to guarantee that the result after iteration restrains, decision is further homogenized sensitivity using the discrete method of BESO.Specific formula
It is as follows:
For the sensitivity after homogenizing, k is the sensitivity of e-th of unit, and s (0 < s < 1) is the homogenizing factor.
5. the improvement Topology optimization based on bi-direction evolutionary structural of combination density variable method according to claim 1, feature exist
In searching sensitivity threshold value using dichotomy.In the iteration initial stage, the minimum value for finding sensitivity is denoted as min, finds sensitivity most
Big value is denoted as max.Error range is set as 0.00001, cycling condition (max-min)/max > 0.00001 is set, takes threshold value th
Threshold value, is then compared by=(max-min)/2 with the sensitivity value of each unit, and the big unit of sensitivity value sets its density
It is set to 1；Conversely, sensitivity is worth small unit, its density is set as 0.001.Whether finally judge the sum of all cell density values
Greater than V*(f*v, as volume threshold): if it is greater than V*, th is assigned to min, is then followed by and executes initial cycle condition
(max-min)/max > 0.00001, until convergence, end loop；If it is less than V*, th is assigned to max, is then followed by and holds
Row initial cycle condition (max-min)/max > 0.00001, until convergence, end loop.
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Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN111339616A (en) * | 2020-03-06 | 2020-06-26 | 北京理工大学 | Topology optimization method for maximizing fundamental frequency of mechanical structure |
| CN112287480A (en) * | 2020-10-27 | 2021-01-29 | 北京理工大学 | Mechanical structure topology optimization method based on multi-population genetic algorithm |
| CN113094945A (en) * | 2021-03-22 | 2021-07-09 | 中山大学 | SA-BESO combined topology optimization method |
| CN113130020A (en) * | 2021-04-22 | 2021-07-16 | 湖南科技大学 | Multi-scale bidirectional evolution structure optimization algorithm for materials |
Citations (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN102789533A (en) * | 2012-07-31 | 2012-11-21 | 西北工业大学 | Structure topology optimization design sensitivity filtering method based on density threshold value |
| CN106372347A (en) * | 2016-09-08 | 2017-02-01 | 厦门大学嘉庚学院 | Dynamic response topological optimization method implemented by application of improved bi-directional evolutionary structural optimization (BESO) to equivalent static load method |
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Patent Citations (2)
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| CN102789533A (en) * | 2012-07-31 | 2012-11-21 | 西北工业大学 | Structure topology optimization design sensitivity filtering method based on density threshold value |
| CN106372347A (en) * | 2016-09-08 | 2017-02-01 | 厦门大学嘉庚学院 | Dynamic response topological optimization method implemented by application of improved bi-directional evolutionary structural optimization (BESO) to equivalent static load method |
Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN111339616A (en) * | 2020-03-06 | 2020-06-26 | 北京理工大学 | Topology optimization method for maximizing fundamental frequency of mechanical structure |
| CN112287480A (en) * | 2020-10-27 | 2021-01-29 | 北京理工大学 | Mechanical structure topology optimization method based on multi-population genetic algorithm |
| CN113094945A (en) * | 2021-03-22 | 2021-07-09 | 中山大学 | SA-BESO combined topology optimization method |
| CN113130020A (en) * | 2021-04-22 | 2021-07-16 | 湖南科技大学 | Multi-scale bidirectional evolution structure optimization algorithm for materials |
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