CN109002614B - Improved level set topology optimization method for stable pore forming - Google Patents

Improved level set topology optimization method for stable pore forming Download PDF

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CN109002614B
CN109002614B CN201810797119.4A CN201810797119A CN109002614B CN 109002614 B CN109002614 B CN 109002614B CN 201810797119 A CN201810797119 A CN 201810797119A CN 109002614 B CN109002614 B CN 109002614B
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夏奇
田野
史铁林
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Huazhong University of Science and Technology
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Abstract

The invention belongs to the technical field of structural topology optimization and discloses an improved level set topology optimization method for stably forming holes. The method comprises the following steps: (a) dividing a finite element mesh into a design domain, initializing and dividing the design domain into an entity and a hole region, and initially assigning a level set function; (b) calculating a displacement vector of a design domain; (c) calculating the sensitivity of each grid unit in the speed field and the entity area; (d) performing first optimization on the initialized entity area, the hole area and the level set function by using the sensitivity number; (e) respectively updating the level set function of each grid unit in the entity area and the hole area by using the speed field so as to subdivide the entity area and the hole area, thereby realizing the second optimization; (f) and judging whether the result of the second optimization converges according to the flexibility and the volume error. The invention overcomes the defect that the traditional level set topology optimization method can not form holes in the structure, solves the dependence of the optimization problem on the initial design, and is stable and effective.

Description

Improved level set topology optimization method for stable pore forming
Technical Field
The invention belongs to the technical field of structural topology optimization, and particularly relates to an improved level set topology optimization method for stable pore forming.
Background
The structure topology optimization is a novel digital structure design method, a mathematical model containing design variables, objective functions and constraint conditions is established, finite element analysis is carried out on the structure, and material layout meeting the constraint conditions and reaching a design target is iterated in a design domain according to an optimization criterion or a mathematical programming method. The structural topology optimization has great application value in the industries of automobiles, aerospace, high-end equipment and the like.
Level set topology optimization methods and BESO (bidirectional progressive structure optimization) methods are two methods commonly used in structure topology optimization. The level set topology optimization method represents a structure boundary through a zero level set of a level set function of a high one-dimensional space, a design variable is the structure boundary, a relation between the boundary and an objective function is established, an optimal structure is obtained through an optimization algorithm evolution boundary, a BESO method is developed through an ESO (gradual structure optimization) method, and structural optimization is realized through gradually removing low stress units or adding deleted units.
In the traditional level set topology optimization method, holes cannot be generated inside the structure due to the properties of Hamilton-Jacobian equation. Therefore, the initial design of the structure in the level set topology optimization needs many holes, which results in that the method depends on the initial design, especially for the two-dimensional problem, and in order to overcome the dependency on the initial design, the level set method has many improvements, such as combining topological derivatives or utilizing radial basis functions, but these improvements need to define many holes in the design domain in the initial design, wherein the positions, the number and the sizes of the holes have great influence on the final optimization result; in addition, the concept of the improvement measures is complex, numerical analysis is not easy to perform, the improvement measures are difficult to apply to actual engineering, and an effective method which is feasible in engineering application and aims at the topological optimization and stable pore forming of the level set does not exist at present.
Disclosure of Invention
Aiming at the defects or improvement requirements in the prior art, the invention provides an improved level set topology optimization method for stably forming holes, which is characterized in that holes are stably generated by removing invalid materials from the interior of a structure in a mode of twice optimization and repeated iteration to obtain an initial design of each iteration in the level set topology optimization, and then an optimal structure is iterated, particularly the first optimization is carried out.
To achieve the above object, according to one aspect of the present invention, there is provided an improved level set topology optimization method for stable via formation, characterized in that the method comprises the steps of:
(a) carrying out finite element meshing on a design domain of an object to be processed to obtain a plurality of mesh units, establishing a level set function corresponding to each mesh unit one by one, and giving an initial value to the level set function of each mesh unit so as to divide the design domain into an entity region and a hole region;
(b) respectively setting the elastic modulus and Poisson's ratio of the solid area and the hole area according to the material attribute of an object to be processed, respectively calculating an elastic matrix and a rigidity matrix corresponding to the solid area and the hole area according to the elastic modulus and the Poisson's ratio, and then calculating a displacement vector of the designed area;
(c) respectively calculating corresponding speed fields of the entity and the hole area by using the displacement vector and the elastic matrix of the design domain, calculating the sensitivity number of each grid unit in the entity area by using the displacement vector of the design domain and the rigidity matrix of the entity area, and setting an acceptable threshold of the sensitivity number according to the sensitivity number;
(d) comparing the sensitivity number corresponding to each grid unit of the entity area with the acceptable threshold, and deleting the grid unit with the sensitivity number smaller than the acceptable threshold, namely performing first optimization, so that the entity area and the grid area of the design area are subdivided, and meanwhile, updating the level set function of each grid unit according to the entity area and the hole area obtained after the subdivision;
(e) respectively updating the level set function of each grid unit in the entity and hole areas obtained by the first optimization by using the speed fields of the entity and hole areas obtained in the step (c), namely performing second optimization, wherein the updated level set function corresponds to the new entity and hole areas, namely the entity and hole areas obtained by the second optimization;
(f) and (c) calculating the flexibility and the volume error of the entity region optimized for the second time, comparing the flexibility and the volume error with preset error thresholds respectively, if the flexibility and the volume error are both smaller than the error thresholds, determining the entity region and the hole region optimized for the second time as final required results, and otherwise, returning to the step (b).
Further preferably, in step (b), the calculating of the elasticity matrix and the stiffness matrix is preferably performed using the following expression,
Figure GDA0002450283000000031
Figure GDA0002450283000000032
wherein D is an elastic matrix, E is an elastic modulus, v is a Poisson's ratio, KeIs a stiffness matrix, B is a strain matrix for each grid cell, AeIs the area of each grid cell, and dA is the area infinitesimal.
Further preferably, in step (b), the displacement vector of the calculation design field preferably takes the following expression,
Ku＝f
wherein K is a rigidity matrix of the design domain, f is an external force vector received by the design domain, and u is a displacement vector of the design domain.
Further preferably, in step (c), the calculated velocity field preferably takes the following expression,
Vnormal＝D(u)·(u)-λ
wherein, (u) ═ Bu, VnormalIs the velocity field and λ is the lagrange multiplier.
Further preferably, in step (c), the calculation sensitivity number preferably takes the following expression,
Figure GDA0002450283000000041
wherein, αeIs the sensitivity number, u, of each grid cell in the physical areaeIs the displacement vector for each grid cell in the solid area,
Figure GDA0002450283000000042
is a stiffness matrix of the solid area.
Further preferably, in step (c), the acceptable threshold for the set sensitivity number is preferably determined by the following method:
(c1) arranging the sensitivity numbers of all grid units of the entity area in a descending order, presetting the deletion rate of the entity area, and calculating a first heuristic critical value according to the following expression
Figure GDA0002450283000000043
Figure GDA0002450283000000044
Wherein p is the deletion rate, NSIs the total number of grid cells in the physical area;
(c2) calculating the average value of the sensitivity of the grid cells at the junction with the hole area in the entity area
Figure GDA0002450283000000045
Presetting a proportional parameter mu less than 1, and calculating another tentative critical value according to the following expression
Figure GDA0002450283000000046
Figure GDA0002450283000000047
(c3) Taking the tentative threshold value
Figure GDA0002450283000000048
And
Figure GDA0002450283000000049
the smaller value of (a) serves as an acceptable threshold for the sensitivity number.
Further preferably, in step (e), the updating the level set function of each grid cell in the solid and hole areas obtained by the first optimization is preferably performed by using the following expression,
Figure GDA00024502830000000410
wherein phinew(x) Is the level set function obtained for the first optimization and t is time.
Further preferably, in step (f), the compliance and volume errors are preferably calculated according to the following expressions,
Figure GDA0002450283000000051
Figure GDA0002450283000000052
wherein, Cerr、VerrError in compliance and volume, respectively, Ck-i+1、Ck-5-i+1Represents the compliance, V, of the (k-i +1) th and (k-5-i +1) th iterations, respectivelykThe volume of the solid region for the kth iteration,
Figure GDA0002450283000000053
and (3) for the total volume of the entity region after final optimization, i is a set calculation frequency, any positive integer between 1 and 5 is taken, and k is an iteration frequency.
In general, compared with the prior art, the above technical solution contemplated by the present invention can achieve the following beneficial effects:
1. in the invention, by adopting a bidirectional progressive structuring method (BESO), an acceptable threshold of the sensitivity is set by utilizing the sensitivity number corresponding to each grid unit of the entity unit set in the initial assignment, and then the entity area set in the initial assignment is deleted by utilizing the acceptable threshold, so as to form a new hole and an entity area;
2. the invention provides an improved level set topology optimization method for stably forming holes, which utilizes a BESO method to stably generate holes in a structure, and adopts a twice optimization mode to obtain a finally required divided structure, so that the final optimized structure has high reliability and strong practicability;
3. the method provided by the invention is simple, strong in operability, feasible in engineering application, convenient to understand and implement, suitable for all loaded mechanical structures, wide in application range and instructive in structural design in the middle and later stages of engineering.
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FIG. 1 is a flow diagram of a method constructed in accordance with a preferred embodiment of the present invention;
FIG. 2 is a schematic view of a planar cantilever beam structure to be optimized constructed in accordance with a preferred embodiment of the present invention;
FIG. 3 is an initial design in a planar cantilever level set topology optimization constructed in accordance with a preferred embodiment of the present invention;
FIG. 4 is a result of a planar cantilever level set topology optimization constructed in accordance with a preferred embodiment of the present invention;
FIG. 5 is a convergence curve of a planar cantilever level set topological optimization objective function and volume ratio constructed in accordance with a preferred embodiment of the present invention;
FIG. 6 is an initial design in a prior art mid-level set topology optimization;
FIG. 7 is an optimization result in a prior art level set topology optimization.
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 are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
FIG. 1 is a flow diagram of a method constructed in accordance with a preferred embodiment of the present invention, as shown in FIG. 1, a method for improved level set topology optimization for stable pore formation, the method comprising the steps of:
fig. 2 is a schematic diagram of a planar cantilever beam structure to be optimized, constructed according to a preferred embodiment of the present invention, as shown in fig. 2, the present invention is further explained by taking an optimization problem of minimizing the flexibility of the cantilever beam with concentrated loads as an example, given a 2m × 1m rectangular design domain D, a region is fixed at the left end, and a concentrated load 1n is applied vertically downward at the midpoint of the right end.
S1, selecting a four-node square unit to divide a finite element grid into a rectangular design domain D, wherein the grid number is 160 × 80, the side length of the grid is 6.25mm, the node number is 161 × 81, the unit center point coordinate of each grid is x, a level set function phi (x) is defined as follows, and according to the coordinate of the center point of each grid, a level set function phi (x) corresponding to each grid unit is assigned to each grid unit so as to divide the design domain into a solid region and a hole region, the value range of phi (x) in the embodiment is [ -3,3],
Figure GDA0002450283000000077
s2 setting the elastic modulus E of the solid area according to the property of the material used by the cantilever beamsIs 1Pa and a Poisson ratio vs0.3, the elastic modulus of the hole region should be close to 0 and the Poisson's ratio should be consistent with that of the solid region, and the elastic modulus E of the hole region is setvIs 1 × 10-3Pa and Poisson ratio vvIs 0.3, thereby establishing a stiffness matrix K for each grid celleThe calculation formula is as follows:
Figure GDA0002450283000000076
in the formula, B is a strain matrix of each grid cell, a shape function and a displacement function corresponding to each grid cell can be obtained through coordinates of 4 nodes of each grid cell and the side length of the cell, the strain matrix B of each grid cell is obtained by calculation using the shape function and the displacement function of each grid cell, the calculation method of the strain matrix B is the prior art, and is not described here again, D is an elastic matrix, dA is an area infinitesimal, and a is an area infinitesimaleFor the area of grid cell e, the elastic matrix D is calculated as follows:
Figure GDA0002450283000000071
will EsV and vsSubstituting the above formula to obtain the elastic matrix D of the solid regionsCalculating and obtaining a rigidity matrix corresponding to each grid unit in the entity area by using a calculation method of the rigidity matrix, and combining the rigidity matrixes of all grid units in the entity area to form the rigidity matrix of the entity area
Figure GDA0002450283000000072
Similarly, EvV and vvSubstituting the above formula to obtain the elastic matrix D of the hole regionvAnd then obtain
Figure GDA0002450283000000073
Stiffness matrix for solid and hole areas
Figure GDA0002450283000000074
And
Figure GDA0002450283000000075
the method comprises the steps of obtaining a design domain overall stiffness matrix K after assembling, wherein the assembling method is the prior art and is not repeated herein, and obtaining a design domain overall displacement vector u by calculating according to Ku ═ f, wherein f is an external force vector received by the design domain, and the external force vector of the embodiment is a vertically downward 1N concentrated load applied at the midpoint of the right end of the design domain;
s3, obtaining the displacement vector corresponding to each grid cell in the entity area, the hole area and all the grid cells according to the displacement vector of the design domain; respectively calculating corresponding velocity fields V by using respective elastic matrixes and displacement vectors of the solid areas and the hole areasnormalCalculating the sensitivity number α of each grid cell in the solid area by using the displacement vector of each grid cell in the solid area and the rigidity matrix of the solid areaeSetting an acceptable threshold value of the sensitivity number according to the sensitivity number;
each element in the overall displacement vector u represents a displacement vector of one grid cell, the displacement vector of each grid cell is obtained from the overall displacement vector u obtained in step S2, the displacement vectors of the grid cells included in the solid area form the displacement vector of the solid area, and similarly, the displacement vectors of the hole area are obtained,
calculation of velocity field V by shape derivative analysisnormal，VnormalAlong the normal direction to the hole direction of the boundary line between the solid area and the hole area, the calculation formula is as follows:
Vnormal＝D(u)·(u)-λ
where (u) is a strain tensor, and λ is a lagrange multiplier (initial value is set to 10)-3)；
Using a displacement vector u for each grid cell in a real regioneCalculate sensitivity for each grid cell in the entire physical region αeThe calculation formula is as follows:
Figure GDA0002450283000000081
the sensitivity numbers of all the entity areas are sorted in an ascending mode from small to large, and the sensitivity numbers can be obtained
Figure GDA0002450283000000082
NSSetting the deletion rate of the entity regions to be 0.04 and the 0.04N in the sequence for the number of the entity regions with the current structureSThe sensitivity of the bit is recorded as
Figure GDA0002450283000000083
To the reaction solution of 0.04NSThe physical region before the bit is deleted, calculated αeA heuristic threshold of
Figure GDA0002450283000000084
When phi (x) is 0, the average value of the sensitivity of the entity area at the junction between the entity area and the hole area is calculated
Figure GDA0002450283000000085
α can be obtainedeAnother heuristic threshold of
Figure GDA0002450283000000086
Mu is a set parameter, and needs to satisfy 0 ≤ mu ≤ 1, where μ is set to 0.5; comparing two tentative sensitivity values
Figure GDA0002450283000000091
Figure GDA0002450283000000092
Taking the small value between the two as αeIs acceptable threshold αth；
S4 comparing acuity scores α for all solid areaseAnd its threshold value αthWhen α is satisfiede＜αthAnd then removing the solid area, namely inserting holes, obtaining a new hole and a design domain of the solid area, wherein the size of the newly generated hole is the size of the deleted grid unit, and re-assigning the new hole and the new design domain to the level set function according to the distribution of the new hole and the new solid area in the design domain, thereby obtaining a new level set function phinew(x)；
S5 solving Hamilton-Jacobian equation according to respective velocity fields of the solid and the hole areas
Figure GDA0002450283000000093
I.e. the level set function phi is updated againnew(x) That is, the design domain is reinitialized to make the junction between the entity region and the hole region smoothly transit, so as to avoid the too large or too small gradient of the level set function, and the specific process is consistent with the related steps of the traditional level set topology optimization method, which is not repeated herein, and the design domain is updated again to be divided into new entity and hole regions according to the level set function obtained by updating again;
s6, judging whether the optimization result is converged, wherein the convergence criterion is as follows:
Figure GDA0002450283000000094
C＝fTu
wherein, Cerr、VerrError representing compliance and volume of the solid region, respectively, Ck-i+1、Ck-5-i+1Denotes the compliance (how to calculate) for the (k-i +1) th and (k-5-i +1) th iterations, respectively, with compliance C ═ fTu and f are external force vectors received by the design domain, u is an overall displacement vector, and steps S2 to S5 are recorded as an iteration, VkThe volume of the solid region for the kth iteration,
Figure GDA0002450283000000095
in order to optimize the volume upper limit of the solid region structure, 1/2 of the initial volume of the structure is taken in this embodiment, if the structure converges, the optimization process ends, and if the structure does not converge, S2 to S5 are repeated until the structure converges, i is a set number of times, an arbitrary positive integer between 1 and 5 is taken, and k is an iteration number. .
Fig. 3 is an initial design in the topological optimization of the level set of the planar cantilever constructed according to the preferred embodiment of the present invention, as shown in fig. 3, the upper and lower ends of the design domain are set as the hole regions, i.e. the white portions in the figure, and the middle portion of the design domain is the solid region, see the gray portions in the figure, and fig. 4 is the topological optimization result of the level set of the planar cantilever constructed according to the preferred embodiment of the present invention, as shown in fig. 4, after obtaining the optimal solid and hole regions in the figure, it can be known which regions of the cantilever in the structural design are suitable as the solid regions and which regions are suitable for being designed as the hole regions under the existing load.
Fig. 5 is a convergence curve of a plane cantilever level set topology optimization objective function (compliance function) and a volume ratio constructed according to a preferred embodiment of the present invention, and as shown in fig. 5, a final objective function and a volume ratio tend to be stable, i.e., both converge, i.e., a final optimization result obtained by using the method provided by the present invention is an optimal partition structure.
FIG. 6 is an initial design in a prior art mid-level set topology optimization; fig. 7 is an optimization result in the prior art level set topology optimization, and as shown in fig. 6 and 7, it can be seen that the present invention can overcome the dependency of the conventional level set topology optimization method on the initial design, and obtain substantially the same optimization result.
According to the method for stably forming the hole by utilizing the bidirectional progressive structural optimization in the level set topological optimization, the hole is stably generated by utilizing the characteristic that the BESO method is used for removing invalid materials from the interior of the structure, the initial design of each iteration in the level set topological optimization is obtained, the optimal structure is further iterated, the dependence of the traditional level set topological optimization method on the initial design is overcome, the feasibility of engineering application is realized, and the understanding and the implementation are convenient.
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 (8)

1. An improved level set topology optimization method for stabilizing pore formation, the method comprising the steps of:
(a) carrying out finite element meshing on a design domain of an object to be processed to obtain a plurality of mesh units, establishing a level set function corresponding to each mesh unit one by one, and giving an initial value to the level set function of each mesh unit so as to divide the design domain into an entity region and a hole region;
(b) respectively setting the elastic modulus and Poisson's ratio of the solid area and the hole area according to the material attribute of an object to be processed, respectively calculating an elastic matrix and a rigidity matrix corresponding to the solid area and the hole area according to the elastic modulus and the Poisson's ratio, and then calculating a displacement vector of the designed area;
(c) respectively calculating corresponding speed fields of the entity and the hole area by using the displacement vector and the elastic matrix of the design domain, calculating the sensitivity number of each grid unit in the entity area by using the displacement vector of the design domain and the rigidity matrix of the entity area, and setting an acceptable threshold of the sensitivity number according to the sensitivity number;
(d) comparing the sensitivity number corresponding to each grid unit of the entity area with the acceptable threshold, and deleting the grid unit with the sensitivity number smaller than the acceptable threshold, namely performing first optimization, so that the entity area and the grid area of the design area are subdivided, and meanwhile, updating the level set function of each grid unit according to the entity area and the hole area obtained after the subdivision;
(e) respectively updating the level set function of each grid unit in the entity and hole areas obtained by the first optimization by using the speed fields of the entity and hole areas obtained in the step (c), namely performing second optimization, wherein the updated level set function corresponds to the new entity and hole areas, namely the entity and hole areas obtained by the second optimization;
(f) and (c) calculating the flexibility and the volume error of the entity region after the second optimization, respectively comparing the flexibility and the volume error with a preset error threshold, if the flexibility and the volume error are both smaller than the error threshold, determining that the entity region and the hole region after the second optimization are final required results, and otherwise, returning to the step (b).
2. The method of claim 1, wherein in step (b), the calculating of the elasticity matrix and the stiffness matrix is performed using the following expressions,
Figure FDA0002450282990000021
Figure FDA0002450282990000022
wherein D is an elastic matrix, E is an elastic modulus, v is a Poisson's ratio, KeIs a stiffness matrix, B is a strain matrix for each grid cell, AeIs the area of each grid cell, and dA is the area infinitesimal.
3. The method of claim 1, wherein in step (b), the displacement vector of the calculation design field adopts the following expression,
Ku＝f
wherein K is a rigidity matrix of the design domain, f is an external force vector received by the design domain, and u is a displacement vector of the design domain.
4. The method of claim 1, wherein in step (c), the calculated velocity field takes the following expression,
Vnormal＝D(u)·(u)-λ
wherein, (u) ═ Bu, VnormalIs the velocity field and λ is the lagrange multiplier.
5. The method of claim 1, wherein in step (c), the computational acuity number employs the following expression,
Figure FDA0002450282990000023
wherein, αeIs the sensitivity number, u, of each grid cell in the physical areaeIs the displacement vector for each grid cell in the solid area,
Figure FDA0002450282990000024
is a stiffness matrix of the solid area.
6. The method of claim 1, wherein in step (c), said setting an acceptable threshold for the sensitivity number employs the following:
(c1) arranging the sensitivity numbers of all grid units of the entity area in a descending order, presetting the deletion rate of the entity area, and calculating a first heuristic critical value according to the following expression
Figure FDA0002450282990000031
Figure FDA0002450282990000032
Wherein p is the deletion rate, NSIs the total number of grid cells in the physical area;
(c2) calculating the average value of the sensitivity of the grid cells at the junction with the hole area in the entity area
Figure FDA0002450282990000033
Presetting a proportional parameter mu less than 1, and calculating another tentative critical value according to the following expression
Figure FDA0002450282990000034
Figure FDA0002450282990000035
(c3) Taking the tentative threshold value
Figure FDA0002450282990000036
And
Figure FDA0002450282990000037
the smaller value of (a) serves as an acceptable threshold for the sensitivity number.
7. The method according to claim 1, wherein in step (e), the level set function of each grid cell in the solid and hole areas obtained by the first optimization is updated according to the following expression,
Figure FDA0002450282990000038
wherein phinew(x) Is the level set function obtained for the first optimization and t is time.
8. The method of claim 1, wherein in step (f), the compliance and volume errors are calculated according to the following expressions,
Figure FDA0002450282990000039
Figure FDA00024502829900000310
wherein, Cerr、VerrError in compliance and volume, respectively, Ck-i+1、Ck-5-i+1Represents the compliance, V, of the (k-i +1) th and (k-5-i +1) th iterations, respectivelykThe volume of the solid region for the kth iteration,
Figure FDA00024502829900000311
and (3) for the total volume of the entity region after final optimization, i is a set calculation frequency, any positive integer between 1 and 5 is taken, and k is an iteration frequency.
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