CN111127491B - Multivariable horizontal segmentation method and equipment for cellular structure topology optimization - Google Patents

Multivariable horizontal segmentation method and equipment for cellular structure topology optimization Download PDF

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CN111127491B
CN111127491B CN201911415301.XA CN201911415301A CN111127491B CN 111127491 B CN111127491 B CN 111127491B CN 201911415301 A CN201911415301 A CN 201911415301A CN 111127491 B CN111127491 B CN 111127491B
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honeycomb structure
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夏奇
刘辉
史铁林
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Huazhong University of Science and Technology
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Abstract

The invention discloses a multivariable horizontal segmentation method and equipment for cellular structure topology optimization, belonging to the field of structure design topology optimization and comprising the following steps: the design reference domain D is first divided into cells D k (k 1.. M), then at each unit D k Using multiple basis set functions within
Figure DDA0002351029400000011
And its division function
Figure DDA0002351029400000012
The microstructure is described and cut by a cutting function to achieve shape and topology changes of the microstructure. After the cutting operation, a plurality of virtual microstructures are obtained in each unit, and the virtual microstructures are further combined together through Boolean operation to generate actual microstructures. The method provided by the invention can effectively enlarge the design freedom degree of the optimization of the honeycomb structure, and simultaneously can ensure that the connection between adjacent microstructures does not need any additional constraint to achieve the optimization effect.

Description

Multivariable horizontal segmentation method and equipment for cellular structure topology optimization
Technical Field
The invention belongs to the field of structural design topological optimization, and particularly relates to a multivariable horizontal segmentation method and equipment for cellular structure topological optimization.
Background
Topology optimization of the honeycomb structure helps designers to properly configure the structure on both macro and micro scales, thereby achieving superior performance that is difficult to achieve with only macro scale configuration considerations. For example, good multifunctional response such as stress resistance, heat exchange and shock absorption can be obtained. Furthermore, with the advent of additive manufacturing, honeycomb structures have gained increasing acceptance in practical engineering applications.
Much work has been done to optimize the topology of microstructures. Homogenization-based methods were originally developed for macro-topology optimization and may also be used for this purpose. Along the same reasoning idea, several methods are proposed. In these methods, a microstructured prototype is specified by the designer (e.g., a solid square with rectangular holes) that is explicitly described by several dimensional parameters to transform the topological optimization problem into a dimensional optimization problem that seeks optimal values for the dimensional parameters. After optimization, the microstructures at different locations in the structure may have different shapes or topologies. However, the microstructures of local areas tend to be highly similar, which means that the design freedom is limited to a certain extent by limited dimensional parameters or user-specified microstructure prototypes.
Another type of approach, often referred to as a hierarchical approach or a parallel approach, does not require the designer to specify the microstructure prototypes prior to optimization. They allow microstructures at different locations in the structure to freely develop their own configuration. This approach provides additional flexibility for tailoring local mechanical properties, thereby successfully expanding design freedom however, another problem with cell structure arises. In the united states, adjacent microstructures may not be connected to each other. This disjointing is caused by the assumption that under the scale-splitting assumption of homogenization, the optimization on the macro scale ignores the connectivity between microstructures on the micro scale. The unconnected microstructures do not bear any load and are therefore impractical to use.
Connectivity between microstructures has become an important issue in the topology optimization of cellular structures. In recent years, several topological optimization methods of the honeycomb structure based on different frames are also proposed, but all the methods have a series of limitations such as single optimization direction, poor freedom degree and the like.
Disclosure of Invention
In response to the above deficiencies of the prior art or needs for improvement, the present invention provides a multivariate level segmentation method and apparatus for cellular topology optimization. The method provides a new parameterized format, and has the innovation point that a multivariable cutting level set method (M-VCUT) is provided to expand the design freedom degree of the original VCUT (variable cutting) level set method. In the proposed method, the reference design domain is subdivided into cells, and in the kth cell N basic level set functions are used to represent N microstructure prototypes, all of which remain unchanged in the optimization. Similarly, in the Kth cell, there are N cutting functions that are used to prune the corresponding level set function and change the shape and topology optimized microstructure. A virtual organization is then formed in the kth cell, while the virtual structure is allowed to be empty. And finally, combining all the virtual microstructures of the Kth unit together through Boolean operation to obtain an actual microstructure. When a cell has N virtual microstructures, there are many possible combinations of the sequences. Thus, the level set method effectively expands the design freedom of the optimization of the honeycomb structure.
To achieve the above object, according to one aspect of the present invention, there is provided a multivariate horizontal segmentation method for cellular structure topology optimization, comprising the steps of:
(1) a fixed reference domain D containing the cell structure omega is defined and then divided evenly into M units, denoted as D k (k 1.. M), at each unit D k In which there are N basic level set functions
Figure BDA0002351029380000021
And N cutting functions
Figure BDA0002351029380000022
Each one of which is
Figure BDA0002351029380000023
Representing a pre-defined micro prototype structure that remains unchanged during the design process;
Figure BDA0002351029380000024
is a corresponding cutting
Figure BDA0002351029380000025
A cutting surface of (a);
(2) use of
Figure BDA0002351029380000026
To pair
Figure BDA0002351029380000027
Cutting was performed, as follows:
Figure BDA0002351029380000028
obtaining N virtual microstructures after cutting
Figure BDA0002351029380000031
Then N virtual microstructures are combined together through Boolean operation to obtain the actual microstructure of the kth unit
Figure BDA0002351029380000032
Microstructure combined into final honeycomb structure
Figure BDA0002351029380000033
(3) The objective function is set as:
min C(u)
Figure BDA0002351029380000034
wherein c (u) is the compliance of the honeycomb structure Ω and u is the displacement field of the honeycomb structure Ω;
v is the volume of the honeycomb structure obtained by optimization,
Figure BDA0002351029380000035
is the structural volume upper limit specified for the honeycomb;
(4) when compliance c (u) is minimal:
Figure BDA0002351029380000036
when volume V is minimal:
Figure BDA0002351029380000037
where the superscript T is the transposed matrix symbol, H i Is the global height vector of the cellular structure,
Figure BDA00023510293800000312
is H i Function with respect to time, S k Is a symbol selection matrix;
Figure BDA0002351029380000038
G＝-Ae(u)e(u)
wherein A is the stiffness tensor, e (u) is the strain tensor;
Figure BDA0002351029380000039
is that
Figure BDA00023510293800000310
As a function of the time t,
Figure BDA00023510293800000311
is laplacian, where | is norm, n (x) is a row vector consisting of shape function values of each node of unit x, ds is the derivative of the traction free boundary;
(5) iteratively solving C (u) in the decomposition process of the honeycomb structure omega by a gradient descent method based on the formula (1) and the formula (2), and recording the C (u) obtained by the q-th iteration as C (q-i+1) The corresponding honeycomb volume V is denoted V (q) Then the iteration termination condition is as follows:
Figure BDA0002351029380000041
Figure BDA0002351029380000042
δ c is judgment of complianceThreshold value, δ V And (5) judging the volume, and when the formula (3) and the formula (4) are satisfied simultaneously, terminating iteration and outputting the honeycomb structure at the moment and the corresponding volume and flexibility thereof, namely the optimal topology optimization result of the honeycomb structure.
Further, in the step (2),
Figure BDA0002351029380000043
for is to
Figure BDA0002351029380000044
Function for cutting result of
Figure BDA0002351029380000045
Expressed as:
Figure BDA0002351029380000046
after a cutting operation, a virtual microstructure of the kth cell is obtained
Figure BDA0002351029380000047
The following were used:
Figure BDA0002351029380000048
Figure BDA0002351029380000049
is a partial operator;
Figure BDA00023510293800000410
when N groups are present
Figure BDA00023510293800000411
And
Figure BDA00023510293800000412
after the cutting operation of (a) is completed,n virtual microstructures can be obtained
Figure BDA00023510293800000413
Then N virtual microstructures are combined together through Boolean operation to obtain the actual microstructure of
Figure BDA00023510293800000414
Is provided with
Figure BDA00023510293800000415
Then there is
Figure BDA00023510293800000416
Figure BDA00023510293800000417
Figure BDA00023510293800000418
The final honeycomb structure formed by combining the microstructures is
Figure BDA00023510293800000419
Further, in step (4), the free boundary F of the structure is pulled H Optimized, since the final honeycomb structure omega is divided by the unit D k Is divided into a bottom microscopic actual structure omega k Thus, the free boundary F of the honeycomb structure H Are also divided into
Figure BDA00023510293800000420
Figure BDA00023510293800000421
Wherein the content of the first and second substances,
Figure BDA00023510293800000422
is the actual structure omega k Due to the actual structure omega k From a plurality of virtual microstructures
Figure BDA0002351029380000051
Are composed of, therefore
Figure BDA0002351029380000052
Can be further divided into several segments:
Figure BDA0002351029380000053
wherein the content of the first and second substances,
Figure BDA0002351029380000054
Figure BDA0002351029380000055
free from cell D k Other microstructures within, so:
Figure BDA0002351029380000056
the shape derivative C' of the traction free boundary compliance is:
Figure BDA0002351029380000057
wherein, V n Is the velocity in the direction of the normal vector outside the boundary, a is the stiffness tensor, e (u) is the strain tensor; ds is the traction free boundary F H J ═ 1,. cndot, N;
the shape derivative of the honeycomb volume V is:
Figure BDA0002351029380000058
when the free boundaries of a structure evolve in a cell, the following equation must be satisfied:
Figure BDA0002351029380000059
wherein the content of the first and second substances,
Figure BDA00023510293800000510
is the velocity in the direction of the outward normal vector, t is the time parameter;
Figure BDA00023510293800000511
is laplacian operator, | | x | | is a norm;
defining a shape function for each node of the cell x, then cutting the function
Figure BDA00023510293800000512
Is calculated as a weighted sum of these shape functions, i.e.:
Figure BDA00023510293800000513
Figure BDA00023510293800000514
wherein N is T (x) Is a column vector consisting of the values of the respective node shape functions at cell x, the superscript T is the transposed matrix sign,
Figure BDA00023510293800000515
a column vector consisting of height variables at point x; s k For the symbol selection matrix, H i Is the global height vector of the honeycomb;
according to the cutting conditions:
Figure BDA0002351029380000061
when compliance c (u) is minimal:
Figure BDA0002351029380000062
when volume V is minimal:
Figure BDA0002351029380000063
to achieve the above object, the present invention provides a computer-readable storage medium having stored thereon a computer program which, when executed by a processor, implements the method as described in any of the preceding claims.
To achieve the above object, the present invention provides a multivariable horizontal split device for cellular structure topology optimization, comprising the computer-readable storage medium as described above and a processor for invoking and processing a computer program stored in the computer-readable storage medium.
In general, compared with the prior art, the above technical solution contemplated by the present invention can obtain the following beneficial effects:
(1) the multivariable horizontal segmentation method for the topological optimization of the honeycomb structure can enlarge the design freedom degree of the original method. The present invention divides the reference design domain into cells, but we no longer describe the microstructure using only one basic level set function and one cutting function in each cell. In each cell, a plurality of microstructure prototypes are represented using a plurality of base layer set functions, which are fixed and invariant during the optimization process. The multiple cutting functions are used in each cell to cut its own base layer set function and optimize it to change the shape and topology of the microstructure. After the cutting operation, virtual microstructures are obtained, which are combined together by boolean operations to generate the actual microstructures in each cell. When a cell has N virtual microstructures, it willIs provided with
Figure BDA0002351029380000064
The level set method provided by the invention effectively expands the design freedom of the optimization of the honeycomb structure. Furthermore, this method ensures that the connection between adjacent microstructures can be naturally ensured without any additional constraint.
(2) The invention adopts a partition function
Figure BDA0002351029380000071
Through
Figure BDA0002351029380000072
Will be fully embedded into the cell D after calculation k To obtain the global cutting function Ψ i Because neighboring cells share the same altitude variable on their common boundary, the global cut function Ψ i In the cell is C 0 1 st due to the global cutting function Ψ i And global level set function phi i Are all continuous according to formula
Figure BDA0002351029380000073
It can be known that
Figure BDA0002351029380000074
Is also C 0 Continuous, the joining operation does not transform the joined virtual microstructure into a non-joined microstructure. Therefore, the invention can ensure the continuity of the microstructure under the condition of no external force.
Drawings
Fig. 1 is a schematic diagram illustrating an example of an optimized design of a cantilever structure according to a preferred embodiment of the present invention.
FIG. 2 is the final microstructure optimization results of the optimization example of FIG. 1, including four different cell arrangements, 1 × 2, 3 × 6, 5 × 10 and 7 × 14, the first column (a) (d) (g) (j) being the initial design, the second column (b) (e) (h) (k) being the optimized structure, and the last column (e) (f) (i) (l) being the union of the virtual tissues.
FIG. 3 is a comparison of the optimization results of the optimization example of FIG. 1 using the proposed M-VCUT method and a conventional VCUT; the compliance and iteration count of the optimized structure shown in fig. 3(a) are 71.21 and 27, respectively, and 74.32 and 156, respectively, in fig. 3 (b);
FIG. 4 is a schematic diagram of cell division and dicing;
FIG. 5 is a schematic diagram of the main process steps of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not limit the invention. In addition, the technical features involved in the respective embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
As shown in fig. 5, a preferred multivariate horizontal segmentation method for cellular structure topology optimization in the present invention comprises the following steps:
(1) a fixed reference domain D containing the cell structure omega is defined and then divided evenly into M units, denoted as D k (k 1.. M), in each unit D k In which there are N basic level set functions
Figure BDA0002351029380000081
And N cutting functions
Figure BDA0002351029380000082
Each one of which is
Figure BDA0002351029380000083
Representing a predetermined microscopic prototype structure which remains unchanged during the design process;
Figure BDA0002351029380000084
is a corresponding cutting
Figure BDA0002351029380000085
Of (2)。
(2) The result of the cutting operation is divided by another function
Figure BDA0002351029380000086
To describe, wherein:
Figure BDA0002351029380000087
after one cutting operation, a virtual microstructure of the kth cell is obtained, which can be described as
Figure BDA0002351029380000088
Figure BDA0002351029380000089
Is a partial derivative operator;
Figure BDA00023510293800000810
when N groups are present
Figure BDA00023510293800000811
And with
Figure BDA00023510293800000812
After the cutting operation is completed, N virtual microstructures can be obtained
Figure BDA00023510293800000813
Then N virtual microstructures are combined together through Boolean operation to obtain the actual microstructure of
Figure BDA00023510293800000814
After the operation of the corresponding function set of the N groups is completed, N microstructures can be obtained, and then the N microstructures are combined together through Boolean operation to obtain the actual microstructureMicrostructure of
Figure BDA00023510293800000815
In a horizontal frame, the following functions can be used:
Figure BDA00023510293800000816
to solve the problem.
Thus we can get:
Figure BDA00023510293800000817
Figure BDA00023510293800000818
Figure BDA00023510293800000819
finally we can get the final structure of microcosmic combination as
Figure BDA00023510293800000820
(3) The optimization problem considered by the present invention is the minimum compliance problem, which is defined as:
min C(u)
Figure BDA0002351029380000091
wherein c (u) is the compliance of the honeycomb structure Ω and u is the displacement field of the honeycomb structure Ω; v is the volume of the honeycomb structure obtained by optimization,
Figure BDA0002351029380000092
is the structural volume upper limit specified for the honeycomb;
(4) free edge pulling for structure in level set based topology optimizationBoundary gamma H Optimization is performed. From the above segmentation, Γ can be obtained by the same method H Is also divided into
Figure BDA0002351029380000093
Wherein:
Figure BDA0002351029380000094
wherein the content of the first and second substances,
Figure BDA0002351029380000095
is the actual structure omega k Due to the actual structure omega k From a plurality of virtual microstructures
Figure BDA0002351029380000096
Are composed of, therefore
Figure BDA0002351029380000097
Can be further divided into several segments:
Figure BDA0002351029380000098
wherein the content of the first and second substances,
Figure BDA0002351029380000099
Figure BDA00023510293800000910
free from cell D k Other microstructures within, so:
Figure BDA00023510293800000911
the shape derivative C' of the traction free boundary compliance is:
Figure BDA00023510293800000912
wherein, V n Is the velocity in the direction of the normal vector outside the boundary, a is the stiffness tensor, e (u) is the strain tensor; ds is the traction free boundary Γ H J ═ 1,. N;
the shape derivative of the honeycomb volume V is:
Figure BDA00023510293800000913
when the free boundary of a structure evolves in a cell, the following equation must be satisfied:
Figure BDA00023510293800000914
wherein the content of the first and second substances,
Figure BDA00023510293800000915
is the velocity in the direction of the outward normal vector, t is the time parameter;
Figure BDA00023510293800000916
is a laplacian operator, | x | is a norm;
meanwhile, the unit D can be known according to the step (1) k Each cutting function of
Figure BDA0002351029380000101
Are constructed by inserting a defined set of height variables at the cell nodes, as shown by the black dots in fig. 4. First, a shape function is defined for each node of the cell. Value of function at any x point in cell
Figure BDA0002351029380000102
Is calculated as a weighted sum of these shape functions, i.e.:
Figure BDA0002351029380000103
Figure BDA0002351029380000104
wherein N is T (x) Is a column vector consisting of the values of the respective node shape functions at cell x, the superscript T is the transposed matrix sign,
Figure BDA0002351029380000105
a column vector consisting of height variables at point x; s k Selecting a matrix for the symbol, H i Is the global height vector of the honeycomb;
according to the cutting conditions:
Figure BDA0002351029380000106
when compliance c (u) is minimal:
Figure BDA0002351029380000107
when volume V is minimal:
Figure BDA0002351029380000108
(5) iteratively solving C (u) in the decomposition process of the honeycomb structure omega by a gradient descent method based on the formula (1) and the formula (2), and recording the C (u) obtained by the q-th iteration as C (q-i+1) The corresponding honeycomb volume V is denoted V (q) Then the iteration termination condition is as follows:
Figure BDA0002351029380000109
Figure BDA00023510293800001010
δ c is a compliance determination threshold, δ V And (4) when the volume judgment threshold value is satisfied, terminating iteration and outputting the honeycomb structure and the corresponding volume and flexibility of the honeycomb structure when the iteration is ended, namely the optimal topological optimization result of the honeycomb structure.
In a specific case, we studied the topological optimization of the cantilever structure as shown in fig. 1, and verified the validity of the proposed M-VCUT level set method, and compared it with the VCUT level set method proposed in our previous study. The height and length of the reference field are 3.5 meters and 7 meters, respectively, and the left side is fixed in both the x and y directions. The concentration force f is 1N acting on the right midpoint. Four different cell arrangements, 1X 2, 3X 6, 5X 10 and 7X 14, were used here. With 40 x 40 finite element elements per cell. The upper limit of the volume ratio is 0.5.
The structure topology optimization process is as follows:
firstly, dividing cells, and fixing the grids by adopting 4-node bilinear square cells; a planar stress state is assumed. Extent of cutting function
Figure BDA0002351029380000111
Is set to [ -3, 3 [)]。
Second step setting basic level set function
Figure BDA0002351029380000112
And a cutting function
Figure BDA0002351029380000113
Performing structural decomposition according to the steps (1) to (4);
thirdly, setting an optimization criterion, and performing iterative optimization according to the step (5):
Figure BDA0002351029380000114
Figure BDA0002351029380000115
wherein q is the current iteration number; delta. for the preparation of a coating c Setting 0.5%; delta V Setting 0.5%; n is an empirical parameter, and in this embodiment, n is 5. Furthermore, if the number of iterations reaches 500, the optimization is terminated.
The initial design and optimized structure of different cell arrangements using the M-VCUT level set method is shown in fig. 2. The first column is the initial design, the second is the optimized structure, and the last column is the union of the virtual organizations. The first row relates to a 1 × 2 cell arrangement; the second row is an arrangement of 3 × 6 cells; the third 5 × 10; fourth 7 × 14. The flexibility of the four optimized structures is respectively C 1×2 ＝78.45,C 3×6 ＝80.99,C 5×10 ＝71.80,C 7×14 71.21. The number of iterations is 38, 48, 28, 27 respectively. The optimization result shows that (1) the honeycomb structure can be obtained by adjusting the cutting surface; (2) all neighboring microstructures are connected perfectly without any mismatch.
For comparison, the present invention was also optimized using the original VCUT method for the 7 × 14 cell arrangement and microstructure prototype shown in fig. 2. The results obtained by the M-VCUT and VCUT methods are shown in FIG. 3(a) and FIG. 3(b), respectively. The compliance and number of iterations of the optimized structure shown in fig. 3(a) are 71.21 and 27, respectively, and in fig. 3(b) are 74.32 and 156.
The multivariable horizontal segmentation method for the topological optimization of the honeycomb structure provided by the invention has the advantages that the design freedom of the original method is expanded, the designed microstructure connectivity is ensured, and the iteration times and the flexibility are reduced compared with the original method.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (5)

1. A multivariable horizontal segmentation method for cellular structure topology optimization is characterized by comprising the following steps:
(1) a fixed reference domain D containing the cell structure omega is defined and then divided evenly into M units, denoted D k K 1 … M, in each unit D k In which there are N basic level set functions
Figure FDA0003717455760000011
And N cutting functions
Figure FDA0003717455760000012
1-1 … N, each
Figure FDA0003717455760000013
Representing a pre-defined micro prototype structure that remains unchanged during the design process;
Figure FDA0003717455760000014
is a corresponding cutting
Figure FDA0003717455760000015
A cutting surface of (a);
(2) use of
Figure FDA0003717455760000016
For is to
Figure FDA0003717455760000017
Cutting was performed, as follows:
Figure FDA0003717455760000018
obtaining N virtual microstructures after cutting
Figure FDA0003717455760000019
Then N virtual microstructures are combined together through Boolean operation to obtain the actual microstructure of the kth unit
Figure FDA00037174557600000110
Microstructure combined into final honeycomb structure
Figure FDA00037174557600000111
(3) The objective function is set to:
minC(u)
Figure FDA00037174557600000112
wherein c (u) is the compliance of the honeycomb structure Ω and u is the displacement field of the honeycomb structure Ω;
v is the volume of the honeycomb structure obtained by optimization,
Figure FDA00037174557600000113
is the structural volume upper limit specified for the honeycomb;
(4) when compliance c (u) is minimal:
Figure FDA00037174557600000114
when volume V is minimal:
Figure FDA00037174557600000115
where the superscript T is the transposed matrix symbol, H i Is the global height vector of the cellular structure,
Figure FDA00037174557600000116
is H i Function with respect to time, S k Is a symbol selection matrix;
Figure FDA0003717455760000021
G＝-Ae(u)e(u)
wherein A is the stiffness tensor, e (u) is the strain tensor;
Figure FDA0003717455760000022
is that
Figure FDA0003717455760000023
As a function of the time t,
Figure FDA0003717455760000024
is laplacian, | is norm, n (x) is a row vector consisting of the shape function values of the individual nodes of the element x, ds is the derivative of the pull free boundary;
(5) based on the formula (1) and the formula (2), C (u) is solved iteratively in the decomposition process of the honeycomb structure omega by a gradient descent method, and the C (u) obtained in the q-th iteration is marked as C (q-i+1) The corresponding honeycomb volume V is denoted V (q) Then the iteration termination condition is as follows:
Figure FDA0003717455760000025
Figure FDA0003717455760000026
δ c is a compliance determination threshold, δ V And (3) a volume judgment threshold value, wherein n is an empirical parameter and is an integer, when the formula (3) and the formula (4) are simultaneously satisfied, iteration is terminated, and the honeycomb structure at the moment and the corresponding volume and flexibility of the honeycomb structure, namely the optimal topology optimization result of the honeycomb structure, are output.
2. The multivariate horizontal segmentation method for cellular structure topology optimization according to claim 1, wherein in step (2),
Figure FDA0003717455760000027
to pair
Figure FDA0003717455760000028
Function for cutting result of
Figure FDA0003717455760000029
Expressed as:
Figure FDA00037174557600000210
after a cutting operation, a virtual microstructure of the kth cell is obtained
Figure FDA00037174557600000211
The following were used:
Figure FDA00037174557600000212
Figure FDA00037174557600000213
Figure FDA00037174557600000214
is a partial derivative operator;
Figure FDA00037174557600000215
when N groups are present
Figure FDA00037174557600000216
And
Figure FDA00037174557600000217
after the cutting operation is completed, N virtual microstructures can be obtained
Figure FDA00037174557600000218
Then N virtual microstructures are combined together through Boolean operation to obtain the actual microstructure of
Figure FDA00037174557600000219
Is provided with
Figure FDA0003717455760000031
k is 1 … M, then
Figure FDA0003717455760000032
Figure FDA0003717455760000033
Figure FDA0003717455760000034
The final honeycomb structure formed by combining the microstructures is
Figure FDA0003717455760000035
3. The multivariate horizontal segmentation method for cellular structure topology optimization as defined in claim 2, wherein in step (4), the traction free boundary Γ of the structure is subjected to H Optimized, since the final honeycomb structure omega is divided by the unit D k Is divided into a bottom microscopic actual structure omega k Thus the free traction boundary F of the honeycomb structure H Are also divided into
Figure FDA0003717455760000036
Figure FDA0003717455760000037
Wherein the content of the first and second substances,
Figure FDA0003717455760000038
is a real structure omega k Due to the actual structure omega k From a plurality of virtual microstructures
Figure FDA0003717455760000039
Is composed of, therefore
Figure FDA00037174557600000316
Can be further divided into several segments:
Figure FDA00037174557600000310
wherein the content of the first and second substances,
Figure FDA00037174557600000311
Figure FDA00037174557600000312
free from cell D k Other microstructure effects, so:
Figure FDA00037174557600000313
the shape derivative C' of the traction free boundary compliance is:
Figure FDA00037174557600000314
wherein, V n Is along the boundary external normal measuring methodThe velocity of the direction, A is the stiffness tensor, e (u) is the strain tensor; ds is the traction free boundary F H J ═ 1,. cndot, N;
the shape derivative of the honeycomb volume V is:
Figure FDA00037174557600000315
when the free boundary of a structure evolves in a cell, the following equation must be satisfied:
Figure FDA0003717455760000041
wherein the content of the first and second substances,
Figure FDA0003717455760000048
is the velocity in the direction of the outward normal vector, t is the time parameter;
Figure FDA0003717455760000049
is a laplacian operator, | x | is a norm;
defining a shape function for each node of the cell x, then cutting the function
Figure FDA00037174557600000410
Is calculated as a weighted sum of these shape functions, i.e.:
Figure FDA0003717455760000042
Figure FDA0003717455760000043
wherein N is T (x) Is a column vector consisting of the values of the respective node shape functions at cell x, the superscript T is the transposed matrix sign,
Figure FDA0003717455760000044
a column vector consisting of height variables at point x; s k For the symbol selection matrix, H i Is the global height vector of the honeycomb;
according to the cutting conditions:
Figure FDA0003717455760000045
when compliance c (u) is minimal:
Figure FDA0003717455760000046
when volume V is minimal:
Figure FDA0003717455760000047
4. a computer-readable storage medium, having stored thereon a computer program which, when executed by a processor, implements the method of any one of claims 1 to 3.
5. A multivariate horizontal segmentation device for cellular topology optimization, comprising the computer readable storage medium of claim 4 and a processor for invoking and processing the computer program stored in the computer readable storage medium.
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