CN110955938B - Topological optimization method for sandwich structure with gradient porous sandwich - Google Patents
Topological optimization method for sandwich structure with gradient porous sandwich Download PDFInfo
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Abstract
The invention belongs to the technical field related to structure optimization, and particularly discloses a topological optimization method for a sandwich structure with a gradient porous sandwich. The method comprises the following steps: the sandwich structure is sliced, the density of each slice layer is optimized by adopting a variable thickness method, the thickness of an upper panel and a lower panel is determined according to the density, the density value of each unit in one slice layer of a middle sandwich is optimized, the density value interval is divided into m equal-difference density intervals to obtain the density values of m representative sandwich units, then the selected m representative sandwich units are optimized one by one to obtain an optimized configuration, the optimized configuration is subjected to shape interpolation to obtain the configurations of all sandwich units of the slice layer, the optimized slice layer is periodically repeated in the height direction of the sandwich structure to obtain the configuration of the sandwich structure, and topology optimization is completed. The invention realizes the optimization of the thickness of the upper panel and the lower panel of the sandwich structure, the configuration of the sandwich unit and the gradient distribution of the sandwich unit in the sandwich layer.
Description
Technical Field
The invention relates to the technical field related to structure optimization, in particular to a topological optimization method for a sandwich structure with gradient porous sandwich.
Background
The sandwich structure has a plurality of excellent performances, such as ultra-light weight, high specific bending rigidity/strength, high-efficiency impact energy absorption, and excellent acoustic and thermal properties, and is widely applied to the fields of aerospace, automobile industry and other related structure designs. The multi-scale topological optimization is an effective sandwich structure topological optimization design method, and is very suitable for the design of sandwich structures with lower quality and higher mechanical property requirements. The topological optimization design method of the sandwich structure with the gradient porous sandwich can fully explore the thickness distribution of the upper panel and the lower panel of the sandwich structure and the design potential of the middle sandwich microstructure, and realize the optimal performance of the sandwich structure with the least material consumption or the lowest cost.
Some studies have been made by those skilled in the art for topology optimization of a sandwich structure, as in document 1: G.D.Xu, J.J.ZHai, Z.Tao, Z.H.Wang, C.Su, D.N.Fang, Response of Composite sandwich beams with a graded lattice core, Composite Structures,119(2015) 666-. The method only considers the topological optimization of the sandwich structure and does not consider the upper and lower panel optimization of the sandwich structure. As in document 2: "E.Dragoni, optical mechanical design of four dimensional Structures for Sandwich configurations, Journal of Sandwich Structures & Materials,15(2013) 464-. However, the core of the method is the sandwich structure design with the same microstructure, and the method does not relate to the research on simultaneously optimizing the thicknesses of the upper panel and the lower panel of the sandwich structure and designing the intermediate sandwich structure with the gradient microstructure.
Therefore, the method designs the sandwich structure with the gradient porous sandwich under the condition of simultaneously considering the thickness optimization of the upper panel and the lower panel of the sandwich structure and the microstructure optimization of the middle sandwich with lower calculation cost so as to fully explore a multi-scale design space and furthest improve the performance of the sandwich structure, and is a research hotspot problem to be solved at present.
Disclosure of Invention
Aiming at the defects or improvement requirements in the prior art, the invention provides a topological optimization method for a sandwich structure with a gradient porous sandwich, wherein the characteristics and characteristics of the gradient porous sandwich are combined, each layer of slice of the sandwich structure is optimized, the multi-scale design space of the sandwich structure is fully excavated with lower calculation cost, meanwhile, the connectivity among a plurality of gradient microstructures contained in the middle sandwich of the sandwich structure is ensured, the material potential is fully exerted, the mechanical property of the sandwich structure is improved, and the topological optimization process is realized.
In order to achieve the aim, the invention provides a topological optimization method of a sandwich structure with gradient porous sandwich, which comprises the following steps:
s1, setting the interlayer structure to be optimized as an initial interlayer structure, slicing the interlayer structure into a plurality of slice layers, setting the initial density of each slice layer, optimizing the density of each slice layer by adopting a variable thickness method, determining the thickness of an upper panel, a middle sandwich and a lower panel of the initial interlayer structure according to the optimized density distribution of each slice layer, and acquiring the density value of the middle sandwich;
s2, taking any slice layer in the middle sandwich as an optimization object, discretizing the slice layer into a plurality of units, and optimizing the density value of each unit in the slice layer by adopting a variable thickness method according to the density value of the middle sandwich so as to obtain the density value distribution intervals of all the units in the slice layer after optimization;
s3, dividing the density value distribution interval obtained in the step S2 into m equal difference density intervals, taking the average density of units contained in one equal difference density interval as the density value of a representative sandwich unit, traversing all equal difference intervals to obtain the density values of m representative sandwich units, and optimizing the selected representative sandwich units one by one according to the sequence by adopting a parameterized level set topology optimization method so as to obtain m representative sandwich unit configurations;
s4, according to the m representative sandwich unit configurations obtained in the step S3, shape interpolation is carried out on each unit of the density interval to which each representative sandwich unit belongs to obtain the microstructure of each unit, and then the microstructure of each unit is assembled and backfilled to the corresponding position of the slicing layer where the unit is located, so that an optimized slicing layer with a gradient microstructure is obtained;
s5, according to the one sliced layer with the gradient microstructure obtained in the step S4, the sliced layer is periodically and repeatedly arranged along the height direction of the intermediate sandwich to obtain the complete structure of the optimized intermediate sandwich so as to obtain the optimized sandwich structure, then, a Kriging prediction model of the optimized intermediate sandwich is constructed according to the gradient microstructure, the equivalent property of the sandwich structure with the gradient porous sandwich is predicted according to the Kriging prediction model, whether the obtained sandwich structure meets the preset performance requirement is judged according to the Kriging prediction model, if not, the step S3 is returned, and if so, the topology optimization is completed.
More preferably, step S1 specifically includes the following steps:
s11, setting the sandwich structure to be optimized as an initial sandwich structure, and slicing the sandwich structure into a plurality of slicing layers, wherein the initial density of each slicing layer
The value of (2) is between (0, 1), the density value of each slice layer is optimized by adopting a variable thickness method, so as to obtain the optimized density value of each slice layer
S12 based on optimized density value of each sliced layer
Setting a threshold value, wherein in the sandwich structure, the set of slice layers with density values larger than the threshold value is regarded as an upper panel and a lower panel of the sandwich structure, the set of slice layers with density values smaller than or equal to the threshold value is regarded as an intermediate sandwich of the sandwich structure, an average value of density values of all slice layers in the intermediate sandwich is taken as the density value of the intermediate sandwich, and the density value of the intermediate sandwich is taken as the material volume fraction constraint optimized by the thickness variation method in the step S2.
Preferably, the model for optimizing the density of each sliced layer by using the variable thickness method is as follows:
Find:
Minimize:
Subjectto:
wherein C is the structural compliance of the sandwich structure, G represents the material volume fraction constraint of the sandwich structure,
is a design variable for the relative density of the cells in the sliced layer, X represents the number of sliced layers comprised by the sandwich structure, Ω represents the total design domain comprised by the sandwich structure, εijAnd εklRepresenting the strain field, i, j, k, l-1, 2, …, d, d is the spatial dimension,
is a density of
U is the macroscopic displacement field and v is the spatial component belonging to the permissible displacement
Macroscopic virtual displacement field of0Is the volume of the cell, VmaxExpressed is the maximum volume allowable for the sandwich structure, f is the volume force acting on Ω, τ is the traction force acting on the boundary Γ, ρmax1 and ρminEach of which is 0.001
The upper and lower bounds of the values.
Preferably, in step S2, the model for optimizing the density value of each cell in the sliced layer by using the variable thickness method is:
Find:
Minimize:
Subjectto:
wherein, CsIs the structural flexibility, G, of the slice layer in the intermediate coresThe material volume ratio constraint of the middle sandwich core is expressed,
for design variables of the relative density of the cells in the slice layer, Y is the number of cells in the slice layer, ΩtAnd ΩsDenotes the upper, intermediate and lower panels and the structural domains contained in the sandwich structure, respectively, omega ═ omega-s∪Ωt，εijAnd εklRepresenting the strain field, i, j, k, l-1, 2, …, d, d is the spatial dimension,
is a density of
U is the macroscopic displacement field and v is the permissibleSpace of displacement
Macroscopic virtual displacement field of0Is the volume of the cell or cells,
is the maximum volume allowable for the sandwich, f is the volume force acting on the total design field Ω, τ is the traction force acting on the boundary Γ, fsIs the minimum cell density, ρ, capable of generating the representative sandwich cell configuration in step S3maxRepresenting design variables
The upper bound of the value.
Further preferably, in step S2, the density interval is adaptively determined according to a value range of the density distribution optimized by each cell in the slice layer.
More preferably, step S3 specifically includes the following steps:
s31 dividing the optimized intermediate sandwich density section obtained in the step S2 into m equal difference density sections
Wherein m is the number of the equal difference density sections, and the density value of each unit in the optimized intermediate sandwich core is
All units in the set are taken as a set, and the average density of the units in the set is taken as the density value of the mth representative sandwich unit
S32 Density values of representative Sandwich elements
As material volume fraction constraints, using parameterized level set topologyThe optimization method optimizes the selected m representative sandwich units one by one in sequence, judges whether the optimization result is convergent or not, if yes, sequentially outputs the configurations of the m representative sandwich units by adopting a progressive optimization strategy, and enters step S4, and if not, returns to step S31 until the optimization result is convergent.
Preferably, in step S32, the calculation model of the parameterized level set topology optimization method is:
Find:αm,n(m＝1,2,...,M；n＝1,2,...,N)
Minimize:
Subject to:
wherein M is the number of all units of the center core, N is the number of control points on the micro level set grid for a representative core unit configuration, αm,nIs a design variable on a micro scale, namely an expansion coefficient of a radial basis function established at an nth node to which a micro level set grid of the mth representative sandwich unit configuration belongs, Cs(α) is the structural compliance, ε, of the representative sandwich unit configurationijAnd εklRepresenting strain field, i, j, k, l-1, 2, …, d, d being the spatial dimension, ΦmIs established in the design domain of the m representative sandwich unit configuration
In a parameterized level setIn the method phimSatisfy the requirement of
Is a radial basis function vector, alpha, belonging to the representative sandwich element configurationm(t) is a vector representing the design variable of the representative sandwich element configuration, u is a macroscopic displacement field, u is obtained by transforming the equivalent property of the representative sandwich element configuration
Substituted into the optimization model to obtain v as belonging to the allowable displacement space
Macroscopic virtual displacement field of0A macroscopic finite element cell volume is represented,
represents the density value of the mth representative sandwich element, H () is the Heaviside function used for representing the characteristic function of the structural form,
representing the material volume fraction constraints for the mth representative sandwich element configuration,
and
is a variable quantity
The upper and lower boundaries of the value are taken,
the method is a regularized design variable, and facilitates subsequent optimization algorithm solution.
As a further preference, in the shape interpolation, in step S4, a microstructure having cells of arbitrary density throughout the density interval may be obtained by interpolating a plurality of representative sandwich cell configurations having similar topological configurations.
As a further preferable mode, in step S5, the equivalent property of each representative sandwich element configuration is obtained by a homogenization theory calculation method.
Preferably, in step S5, when the Kriging prediction model is built, several cells are selected in each density interval as sample cells for building the Kriging prediction model.
Generally, compared with the prior art, the above technical solution conceived by the present invention mainly has the following technical advantages:
1. the invention realizes the optimization of the thickness of the upper panel and the lower panel of the sandwich structure, the configuration of the sandwich unit and the gradient distribution of the sandwich unit, exerts the potential of materials to the maximum extent and fully improves the mechanical property of the sandwich structure.
2. The invention adopts a progressive optimization strategy to optimize the selected representative sandwich units one by one in sequence, thereby ensuring that the representative sandwich units have similar topological configurations, and the topological optimization method of various representative sandwich units with similar topological configurations by interpolation of the shape interpolation technology ensures that the gradient microstructure in the sandwich layer has better connectivity.
3. According to the method, equivalent attributes of all microstructures in the gradient porous sandwich are predicted by adopting a Kriging model, so that the calculation cost of microscopic optimization is reduced, the average value of the density values of all the slice layers in the middle sandwich is taken as the density value of the slice layer contained in the middle sandwich of the sandwich structure, and the optimized slice layers are periodically arranged and assembled in the height direction of the slice layer to obtain the specific structure of the middle sandwich, so that the calculation cost of macroscopic optimization is reduced.
4. The invention provides the optimized density interval of the unit in the slice layer determined by self-adaption in the optimization process
Design variables predefined empirically in the optimization model in the prior art are avoided
Value range of (f)s，1]The design freedom degree is greatly expanded due to the limitation of the design space.
5. Compared with the traditional sandwich structure design, the invention not only greatly reduces the calculation cost, but also greatly expands the multi-scale design space, and can effectively improve the mechanical property of the sandwich structure.
Drawings
FIG. 1 is a flow chart of a method for optimizing the topology of a sandwich structure having a gradient porous core constructed in accordance with a preferred embodiment of the present invention;
FIG. 2 is a schematic illustration of a sandwich design domain, loading and boundary conditions constructed in accordance with a preferred embodiment of the present invention;
FIG. 3 is a schematic view of the initial microstructure of a representative sandwich element constructed in accordance with a preferred embodiment of the invention;
FIG. 4 is a schematic illustration of the sandwich structure of FIG. 2 with optimized macro-material layout of the top and bottom sheets and the center core, constructed in accordance with a preferred embodiment of the present invention;
FIG. 5 is a schematic illustration of a progressive optimization strategy for the representative sandwich element of FIG. 3 constructed in accordance with a preferred embodiment of the present invention;
FIG. 6 is a schematic representation of the configuration of four representative sandwich elements and their corresponding gradient microstructures constructed in accordance with a preferred embodiment of the present invention;
FIG. 7 is a schematic view of an optimized gradient cellular structure of a slice layer included in an intermediate core constructed in accordance with a preferred embodiment of the present invention;
fig. 8 is a schematic diagram of an optimal multi-scale design of the sandwich structure of fig. 2 constructed in accordance with a preferred embodiment of the 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 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.
The design domain, load and boundary conditions of the sandwich structure to be optimized in this embodiment are shown in fig. 2, the sandwich structure is shown in fig. 4, and the sandwich structure includes three parts, i.e., an upper panel, a lower panel and a middle sandwich core, the density in the present invention refers to the volume ratio of the volume of the solid part in the whole structure to the volume of the whole structure, the value range is (0, 1), and the generality is not lost, and all physical quantities used in this embodiment are assumed to be dimensionless.
As shown in FIG. 1, the topological optimization method of the sandwich structure with the gradient porous sandwich comprises the following steps:
the method comprises the steps of firstly, setting an interlayer structure to be optimized as an initial interlayer structure, slicing the interlayer structure into a plurality of slice layers, setting the initial density of each slice layer, optimizing the density of each slice layer by adopting a variable thickness method, determining the thickness of an upper panel, a middle sandwich and a lower panel of the initial interlayer structure according to the optimized density distribution of each slice layer, and simultaneously obtaining the density value of the middle sandwich. Setting one sandwich structure as initial sandwich structure, slicing the sandwich structure into several sliced layers, setting the initial density of each sliced layer and the level set function phi of the initial configuration of the representative sandwich unitm(ii) a The method comprises the following steps of optimizing the density of each sliced layer by adopting a topological optimization method based on a variable thickness method, and determining the thickness of an upper panel and a lower panel according to the density distribution of materials in the optimized sandwich structure, wherein the topological optimization method specifically comprises the following substeps:
(1.1) discretization of the sandwich into 20 × 20 × 10 ═ 4000 hexahedral macroscopic elements, Young's modulus E, using the concept of finite elements0(iii) Poisson's ratio μ is 0.3, the densities of macro-units belonging to the same slice layer are set to the same value, and the junction is formed in a sandwich structureThe minimum structural flexibility is taken as a target, an optimization model of the sandwich structure is constructed based on a variable thickness method topological optimization method by combining the material dosage given in advance, namely a constraint condition, and the optimization model is as follows:
Find:
Minimize:
Subject to:
wherein C is the structural compliance of the sandwich structure, G represents the sandwich structure material volume fraction constraint,
is a design variable representing the relative density of the cells, the density values of all cells within said one slice layer being kept the same, X represents the number of slices contained in the sandwich structure, i.e. there are X slices in the sandwich structure, X being any slice therein, Ω represents the total design domain contained in the sandwich structure, εijAnd εklRepresenting the strain field, i, j, k, l-1, 2, …, d, d is the spatial dimension,
is a density of
The elastic tensor of the cell, said
Satisfy the requirement of
Wherein DsolidIs the elastic tensor of the solid element, u is the macroscopic displacement field, v is the space belonging to the allowable displacement
Macroscopic virtual displacement field of0Is the unit volume, VmaxThe maximum volume allowable for the sandwich structure is shown, f is the volume force acting on the macroscopic design domain Ω, τ is the traction force acting on the boundary Γ, ρmax1 and ρminEach design variable is represented by 0.001
The upper and lower bounds of the values.
(1.2) calculating the objective function and the constraint on the design variables
The sensitivity calculation formula is as follows:
ζ＝-∫Ωεij(u)Dsolidεkl(u)dΩ
where ζ is the design variable for the compliance of the sandwich structure over all macro cell densities
Mean (ζ, L) means ζ along the sliced layerAverage sensitivity in the alignment direction, L represents the alignment direction of the slice layer,
is material volume fraction constraint versus design variable
The sensitivity of (2).
(1.3) filtering the sensitivity information of the current macro unit by using the sensitivity information of the adjacent macro unit when calculating the sensitivity information, avoiding the occurrence of numerical instability phenomena such as checkerboard, grid dependency and the like, simultaneously smoothing the configuration of the current sandwich structure, judging whether the objective function meets a set threshold value according to an optimization result, namely a convergence condition, if the objective function meets the convergence condition, outputting the density information of the macro unit contained in the current sandwich structure, otherwise, continuously executing (1.2), and updating the design variable
(1.4) setting a threshold value f based on the information on the density of the macro-units contained in the sandwich structure obtained in (1.3)tThe density value of the slice layer contained in the sandwich structure is specified to be more than ftThe sets of the slice layers are respectively regarded as an upper panel and a lower panel of the sandwich structure, the set of the slice layers with the density value smaller than the threshold value in the sandwich structure is regarded as a middle sandwich of the sandwich structure, and the average value of the density values of the macro units in each slice layer in the middle sandwich is taken as the density value of the macro units in the slice layers contained in the middle sandwich.
And secondly, taking any one slice layer in the middle sandwich as an optimization object, discretizing the slice layer into a plurality of units, and optimizing the density value of each unit in the slice layer by adopting a variable thickness method according to the density value of the middle sandwich so as to obtain the density value distribution interval of all the units in the slice layer after optimization. Taking the density value of the macro unit in the slice layer contained in the middle sandwich obtained in the step one as a given material dosage, namely a constraint condition, and carrying out topological optimization on the slice layer contained in the middle sandwich; the method for optimizing the distribution of materials in a slicing layer contained in the middle sandwich by adopting a topology optimization method based on a variable thickness method specifically comprises the following substeps:
(2.1) using the macro-units constructed in the first step in the process of topology optimization of the sliced layer, namely, the macro-units in one sliced layer are 20 × 20 × 1 ═ 400 hexahedral macro-units, and the material properties are consistent with those in the first step, namely, the Young modulus E 010, Poisson ratio μ0And (3) aiming at the minimum flexibility of the slice layer structure, taking the density value of a macro unit in the slice layer contained in the intermediate sandwich obtained in the step one as a given material dosage, namely a constraint condition, and constructing an optimization model of the slice layer based on a variable thickness method topological optimization method, wherein the optimization model comprises the following steps:
Find:
Minimize:
Subject to:
wherein, CsIs the structural flexibility, G, of a slice layer comprised by said intermediate coresThe volume ratio constraint of the intermediate sandwich material is expressed,
is a design variable representing the relative density of the cells, the intermediate core comprising cells within the slice layer along the sandwichThe density values of the units in the height direction of the core layer are kept consistent, Y represents the number of the units in the slice layer, wherein Y is any one unit, in the third step and the fourth step, each macro unit is considered to be an independent unit cell, the relative density of the unit cell is the material volume ratio constraint in the unit cell optimization design process, and omega is omegas∪Ωt，ΩtAnd ΩsRespectively represents the domains, epsilon, contained by the upper and lower panels and the middle core of the sandwich structure described in step oneijAnd εklRepresenting the strain field, i, j, k, l-1, 2, …, d, d is the spatial dimension,
is a density of
The elastic tensor of the cell, said
Satisfy the requirement of
Wherein DsolidIs the elastic tensor of the solid element, u is the macroscopic displacement field, v is the space belonging to the allowable displacement
Macroscopic virtual displacement field of0Is the volume of a unit of the unit,
the maximum volume allowable for the sandwich is shown, f is the volume force acting on the macro design field omega, tau is the traction force acting on the boundary gamma, fsIs the minimum unit density in step three that can generate the sandwich unit configuration, for simplicity, in the case of two-and three-dimensions fsIs uniformly set to 0.1, ρmaxRepresenting design variables
The upper bound of the value.
(2.2) calculating the objective function and the constraint on the design variables
The sensitivity calculation formula is as follows:
therein, ζsIs the design variable of the flexibility of the slice layer structure to the density of the macro unit in the slice layer
Sensitivity of (d), mean (. zeta.)sP) represents ζsThe average sensitivity along the direction perpendicular to the slice layer, P denotes the direction perpendicular to the slice layer,
is material volume fraction constraint versus design variable
The sensitivity of (2). Based on the sensitivity information of the step, the materials in the optimized slice layer are distributed in a gradient manner, and in addition, the macro units in the direction vertical to the slice layer in the macro units contained in the middle sandwich have the same unit density.
(2.3) filtering the sensitivity information of the current macro-unit using the sensitivity information of neighboring macro-units when calculating the sensitivity information, avoidingAnd (3) generating numerical instability phenomena such as checkerboard and grid dependency, smoothing the boundaries of different density units in the slice layer, judging whether the objective function meets a set threshold value, namely a convergence condition, according to an optimization result, outputting the density information of the macro unit contained in the current sandwich structure if the objective function meets the convergence condition, and otherwise, continuing to execute (2.2) and updating the design variable
And (2.4) outputting the density information of the macro units contained in the current sandwich structure according to the step (2.3) to obtain the density value distribution interval of all the macro units in the optimized slice layer.
And step three, dividing the density value distribution interval obtained in the step two into 4 equal-difference density intervals, taking the average density of units contained in one equal-difference density interval as the density value of a representative sandwich unit, traversing all equal-difference intervals to obtain the density values of 4 representative sandwich units, and optimizing the selected representative sandwich units one by one to obtain 4 representative sandwich unit configurations. Namely, according to the density distribution of macro units in slice layers contained in the optimized middle sandwich, determining a value section of the density value of the macro units, dividing the section into four equal-difference sections, and selecting a representative sandwich unit for optimization design in each section, wherein the method specifically comprises the following substeps:
(3.1) design variables obtained according to (2.3) in step two
Distribution interval of
Dividing the density interval into 4 equal difference density intervals
Wherein m is 1, 2, 3, 4.
(3.2) obtaining the density value of the macro unit in the optimized slice layer obtained in the second step
All units in the set are taken as a set, and the average density of the units in the set is taken as the density value of the mth representative sandwich unit
Density values of representative sandwich elements
As a given amount of material, i.e. a constraint. And (3) dispersing each representative sandwich element into 20 multiplied by 8000 hexahedral micro-elements by using the finite element idea, wherein the material property is consistent with that in the first step, namely the Young modulus E 010, Poisson ratio μ0And (3) constructing an optimization model of the representative unit based on a parameterized level set topology optimization method by taking the minimum structural flexibility of the representative sandwich unit as a target, wherein the optimization model comprises the following steps:
Find:αm,n(m＝1,2,...,M；n＝1,2,...,N)
Minimize:
Subject to:
wherein M is the number of the representative sandwich units, N is the number of control points of one representative sandwich unit on the micro level set grid, and alpham,nIs a design variable on a micro scale, namely, is established on a micro level of the mth representative sandwich unitExpansion coefficient, C, of the radial basis function constructed at the n-th node to which the set mesh belongss(α) is the structural compliance, Φ, of the representative core-sandwiched UnitmIs established in the design domain of the m representative sandwich unit
Level set function of phi in a parameterized level set optimization methodmSatisfy the requirement of
Is the radial basis function vector, alpha, of the representative sandwich elementm(t) is a vector representing the design variable of the representative sandwich element, u is a macroscopic displacement field, and u can be determined by matching the equivalent properties of the representative sandwich element
Substituted into the optimization model to obtain v as belonging to the allowable displacement space
Macroscopic virtual displacement field of0A macroscopic finite element cell volume is represented,
represents the density value of the mth representative sandwich element, H () is the Heaviside function used for representing the characteristic function of the structural form,
representing the material volume fraction constraints of the mth representative sandwich element,
volume fraction of the receiving
The constraint of (a) to (b),
and
is a variable quantity
The upper and lower boundaries of the value are taken,
is a design variable that is regularized and,
and the subsequent optimization algorithm solution is facilitated.
(3.3) calculating the objective function and the constraint on the design variable alpham,nThe sensitivity calculation formula is as follows:
wherein the content of the first and second substances,
is a representative sandwich unit structure flexibility to expansion coefficient alpham,nSensitivity of, delta (. phi.)m) Is the Heaviside function H (phi)m) Is a constant selected based on numerical example, the value of xi is typically 2 to 4 times the size of the grid, DpqrsIs a bullet at any point in a microscopic design domainA sex coefficient matrix, ij 11, 22, 12, representing the horizontal direction, the vertical direction, the shearing direction, respectively, kl, pq, rs are similar to ij,
refers to the cell test strain field in the pq direction,
refers to the unknown strain field caused by the unit test strain field in the pq direction,
is the Mth representative sandwich unit in the design domain
A microscopic displacement field in the inner ij direction,
is the Mth representative sandwich unit in the design domain
A microscopic displacement field in the inner kl direction,
refers to the cell test strain field in the rs direction,
refers to the unknown strain field caused by the unit test strain field in the r direction,
is a radial basis function vector belonging to the representative sandwich unit,
is the volume ratio constraint pair expansion coefficient alpha of a representative sandwich unit materialm,nThe sensitivity of (2).
(3.4) advantage in calculating sensitivity informationFiltering the sensitivity information of the current macro unit by using the sensitivity information of the adjacent macro unit to avoid the occurrence of numerical instability phenomena such as checkerboard, grid dependency and the like, so that the configuration of the microstructure of the representative sandwich unit becomes smooth, judging whether the objective function meets a set threshold value according to an optimization result, namely a convergence condition, if the configuration meets the convergence condition, adopting a progressive optimization strategy to sequentially output four optimized configurations of the representative sandwich unit in sequence, otherwise, continuously executing (3.3), and updating a design variable alpham,n。
Fourthly, according to the configuration of the 4 representative sandwich units obtained in the third step, shape interpolation is carried out on each unit of the density interval to which each representative sandwich unit belongs to obtain the microstructure of each unit, then the microstructure of each unit is assembled and backfilled to the corresponding position of the sliced layer where the unit is located to obtain an optimized sliced layer with a gradient microstructure, the sliced layer is periodically and repeatedly arranged along the height direction of the intermediate sandwich to obtain the complete structure of the optimized intermediate sandwich so as to obtain the optimized sandwich structure, then a prediction model of the optimized intermediate sandwich is constructed according to the gradient microstructure, the equivalent property of the sandwich structure with the gradient porous sandwich is predicted according to the prediction model, whether the obtained sandwich structure meets the preset performance requirement is judged according to the equivalent property, if not, the third step is returned, and if so, realizing the topology optimization process. Namely, shape interpolation is carried out on the configuration of the four optimized representative sandwich unit microstructures based on the shape interpolation technology, so as to obtain a series of gradient microstructures with similar shapes. The method comprises the following steps of constructing a Kriging model by taking the gradient microstructures as sample points and predicting the equivalent property of the gradient porous sandwich of the sandwich structure, and specifically comprises the following substeps:
(4.1) Density value for Macro Unit
The density corresponding to the density interval is interpolated into the density of
The microstructure of the macro unit is obtained by the configuration of the microstructure of the representative sandwich unit, namely the configuration of the microstructure of the macro unit, and then the microstructure of each unit is assembled and backfilled to the corresponding position of the slice layer where the microstructure is located, so that an optimized slice layer with a gradient microstructure is obtained;
(4.2) periodically and repeatedly arranging the sliced layers with the gradient microstructures along the height direction of the intermediate sandwich to obtain an optimized complete structure of the intermediate sandwich, thereby obtaining an optimized sandwich structure.
(4.3) obtaining the equivalent attribute of each representative sandwich unit configuration by a homogenization theory calculation method, selecting a proper number of macro units as samples to construct a Kriging model, and more specifically, selecting a plurality of units as sample units for constructing the prediction model in each density interval when the prediction model is established.
(4.4) predicting the equivalent property of the intermediate sandwich with the gradient microstructure based on the constructed Kriging model;
referring to fig. 2 to 8, the present invention is further illustrated by the design of a sandwich structure with four corner supports. All physical quantities used in the present embodiment are assumed to be dimensionless without loss of generality. As shown in fig. 2, the dimensions of the sandwich structure design domain of the quadrangle support are length L equal to 20, width W equal to 20, height H equal to 10, 5 concentrated loads F are respectively applied to the center and four vertices of the upper surface of the sandwich structure, the size of the load F is 5, the four vertices of the lower surface of the sandwich structure are fixed, the sandwich structure is divided into grids by using 20 × 20 × 10 equal to 4000 hexahedral macro units, the number of macro units included in the slice layer is 20 × 20 × 1 equal to 400, each macro unit is divided into grids by using 20 × 20 × 20 equal to 8000 hexahedral micro units, the initial configuration of the representative sandwich unit is shown in fig. 3, and the material property of the used material is elastic modulus E0The poisson ratio μ is 0.3, 10. The optimization aims at the minimum flexibility value of the sandwich structure, and the limited material consumption is 50%.
Referring to fig. 4, which is a schematic view of the macroscopic material layout of the upper panel, the lower panel and the middle core of the sandwich structure optimized by the variable thickness method, it can be seen that the thickness of the lower panel of the sandwich structure is greater than that of the upper panel, which is very significant for a three-dimensional support structure, because the lower panel is more dominant in resisting bending deformation than the upper panel, and the solid macroscopic elements with higher rigidity and the macroscopic elements with higher density values are mostly located in the region near the load application point or the support application point, which is beneficial for strengthening the structural rigidity near the force transmission path inside the sandwich structure, and the macroscopic elements with gradient microstructures with lower density values are located in the middle core of the sandwich structure, and their main function is to resist the transverse shear deformation in the X-Y plane.
FIG. 5 is a schematic diagram of the progressive optimization strategy used in sequentially optimizing the configuration of four representative sandwich elements one by one. Fig. 6 is an optimal topological configuration of a representative sandwich unit obtained by optimizing the volume ratio of four representative sandwich units in order by using a parametric level set topological optimization method, and a graph of a change of a gradient microstructure of a macro unit obtained by interpolation based on the four representative sandwich unit configurations by using a shape interpolation technique with respect to a density value of the unit, and it can be seen that all the gradient microstructures have good connectivity.
Fig. 7 is a schematic diagram of a gradient porosity structure of a sliced layer obtained by assembling optimized macro-unit microstructures to corresponding positions of the sliced layer. Fig. 8 is a detailed structural diagram of the sandwich structure after multi-scale optimization according to the method provided by the invention. Compared with the prior art, the topological optimization design method for the sandwich structure with the gradient porous sandwich realizes the joint optimization of the thicknesses of the upper panel and the lower panel of the sandwich structure and the sandwich microstructure, and simultaneously ensures that all the gradient microstructures have better connectivity.
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 (10)
1. A topological optimization method for a sandwich structure with gradient porous sandwich is characterized by comprising the following steps:
s1, setting the interlayer structure to be optimized as an initial interlayer structure, slicing the interlayer structure into a plurality of slice layers, setting the initial density of each slice layer, optimizing the density of each slice layer by adopting a variable thickness method, determining the thickness of an upper panel, a middle sandwich and a lower panel of the initial interlayer structure according to the optimized density distribution of each slice layer, and acquiring the density value of the middle sandwich;
s2, taking any slice layer in the middle sandwich as an optimization object, discretizing the slice layer into a plurality of units, and optimizing the density value of each unit in the slice layer by adopting a variable thickness method according to the density value of the middle sandwich so as to obtain the density value distribution intervals of all the units in the slice layer after optimization;
s3, dividing the density value distribution interval obtained in the step S2 into m equal difference density intervals, taking the average density of units contained in one equal difference density interval as the density value of a representative sandwich unit, traversing all equal difference intervals to obtain the density values of m representative sandwich units, and optimizing the selected representative sandwich units one by one according to the sequence by adopting a parameterized level set topology optimization method so as to obtain m representative sandwich unit configurations;
s4, according to the m representative sandwich unit configurations obtained in the step S3, shape interpolation is carried out on each unit of the density interval to which each representative sandwich unit belongs to obtain the microstructure of each unit, and then the microstructure of each unit is assembled and backfilled to the corresponding position of the slicing layer where the unit is located, so that an optimized slicing layer with a gradient microstructure is obtained;
s5, according to the one sliced layer with the gradient microstructure obtained in the step S4, the sliced layer is periodically and repeatedly arranged along the height direction of the intermediate sandwich to obtain the complete structure of the optimized intermediate sandwich so as to obtain the optimized sandwich structure, then, a Kriging prediction model of the optimized intermediate sandwich is constructed according to the gradient microstructure, the equivalent property of the sandwich structure with the gradient porous sandwich is predicted according to the Kriging prediction model, whether the obtained sandwich structure meets the preset performance requirement is judged according to the Kriging prediction model, if not, the step S3 is returned, and if so, the topology optimization is completed.
2. The topology optimization method according to claim 1, wherein the step S1 specifically includes the following steps:
s11, setting the sandwich structure to be optimized as an initial sandwich structure, and slicing the sandwich structure into a plurality of slicing layers, wherein the initial density of each slicing layer
The value of (2) is between (0, 1), the density value of each slice layer is optimized by adopting a variable thickness method, so as to obtain the optimized density value of each slice layer
S12 based on optimized density value of each sliced layer
Setting a threshold value, wherein in the sandwich structure, the set of slice layers with density values larger than the threshold value is regarded as an upper panel and a lower panel of the sandwich structure, the set of slice layers with density values smaller than or equal to the threshold value is regarded as an intermediate sandwich of the sandwich structure, an average value of density values of all slice layers in the intermediate sandwich is taken as the density value of the intermediate sandwich, and the density value of the intermediate sandwich is taken as the material volume fraction constraint optimized by the thickness variation method in the step S2.
3. The topology optimization method according to claim 1, wherein the model for optimizing the density of each sliced layer by using the variable thickness method is:
wherein C is the structural compliance of the sandwich structure, G represents the material volume fraction constraint of the sandwich structure,
is a design variable for the relative density of the cells in the sliced layer, X represents the number of sliced layers comprised by the sandwich structure, Ω represents the total design domain comprised by the sandwich structure, εijAnd εklRepresenting the strain field, i, j, k, l-1, 2, …, d, d is the spatial dimension,
is a density of
U is the macroscopic displacement field and v is the spatial component belonging to the permissible displacement
Macroscopic virtual displacement field of0Is the volume of the cell, VmaxExpressed is the maximum volume allowable for the sandwich structure, f is the volume force acting on Ω, τ is the traction force acting on the boundary Γ, ρmax1 and ρminEach of which is 0.001
The upper and lower bounds of the values.
4. The topology optimization method of claim 1, wherein in step S2, the model for optimizing the density value of each cell in the sliced layer by using the variable thickness method is:
wherein, CsIs the structural flexibility, G, of the slice layer in the intermediate coresThe material volume ratio constraint of the middle sandwich core is expressed,
for design variables of the relative density of the cells in the slice layer, Y is the number of cells in the slice layer, ΩtAnd ΩsDenotes the upper, intermediate and lower panels and the structural domains contained in the sandwich structure, respectively, omega ═ omega-s∪Ωt，εijAnd εklRepresenting the strain field, i, j, k, l-1, 2, …, d, d is the spatial dimension,
is a density of
U is the macroscopic displacement field and v is the spatial component belonging to the permissible displacement
Macroscopic virtual displacement field of0Is the volume of the cell or cells,
is the maximum volume allowable for the sandwich, f is the volume force acting on the total design field Ω, τ is the traction force acting on the boundary Γ, fsIs the minimum cell density, ρ, capable of generating the representative sandwich cell configuration in step S3maxRepresenting design variables
The upper bound of the value.
5. The topology optimization method according to claim 1, wherein in step S2, the density interval is adaptively determined according to a value range of the optimized density distribution of each cell in the slice layer.
6. The topology optimization method according to claim 1, wherein the step S3 specifically includes the following steps:
s31 dividing the optimized intermediate sandwich density section obtained in the step S2 into m equal difference density sections
Wherein m is the number of the equal difference density sections, and the density value of each unit in the optimized intermediate sandwich core is
All units in the set are taken as a set, and the average density of the units in the set is taken as the density value of the mth representative sandwich unit
S32 Density values of representative Sandwich elements
And as material volume ratio constraint, sequentially optimizing the selected m representative sandwich units one by adopting a parameterized level set topological optimization method, judging whether the optimization result is convergent, if so, sequentially outputting the configurations of the m representative sandwich units by adopting a progressive optimization strategy, and entering step S4, otherwise, returning to step S31 until the optimization result is convergent.
7. The topology optimization method according to claim 6, wherein in step S32, the calculation model of the parameterized level set topology optimization method is:
Find:αm,n(m＝1,2,...,M；n＝1,2,...,N)
wherein M is the number of all units of the center core, N is the number of control points on the micro level set grid for a representative core unit configuration, αm,nIs a design variable on a micro scale, namely an expansion coefficient of a radial basis function established at an nth node to which a micro level set grid of the mth representative sandwich unit configuration belongs, Cs(α) is the structural compliance, ε, of the representative sandwich unit configurationijAnd εklRepresenting strain field, i, j, k, l-1, 2, …, d, d being the spatial dimension, ΦmIs established in the design domain of the m representative sandwich unit configuration
Level set function of phi in a parameterized level set optimization methodmSatisfy the requirement of
Is a radial basis function vector, alpha, belonging to the representative sandwich element configurationm(t) is a vector representing the design variable of the representative sandwich element configuration, u is a macroscopic displacement field, u is obtained by transforming the equivalent property of the representative sandwich element configuration
Substituted into the optimization model to obtain v as belonging to the allowable displacement space
Macroscopic virtual displacement field of0A macroscopic finite element cell volume is represented,
represents the density value of the mth representative sandwich element, H () is the Heaviside function used for representing the characteristic function of the structural form,
representing the material volume fraction constraints for the mth representative sandwich element configuration,
and
is a variable quantity
The upper and lower boundaries of the value are taken,
the method is a regularized design variable, and facilitates subsequent optimization algorithm solution.
8. The topology optimization method according to any one of claims 1 to 7, wherein in the shape interpolation, the microstructure of the unit having an arbitrary density throughout the density interval is obtained by interpolating a plurality of representative sandwich unit configurations having similar topological configurations in step S4.
9. The topology optimization method according to any one of claims 1 to 7, wherein in step S5, the equivalent properties of each representative sandwich element configuration are obtained by a homogenization theory calculation method.
10. The topology optimization method of any one of claims 1 to 7, wherein in step S5, in the construction of the Kriging prediction model, a number of cells are selected in each density interval as sample cells for constructing the Kriging prediction model.
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