CN108763778B - Non-probability reliability topological optimization method for continuum structure - Google Patents

Non-probability reliability topological optimization method for continuum structure Download PDF

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CN108763778B
CN108763778B CN201810544826.2A CN201810544826A CN108763778B CN 108763778 B CN108763778 B CN 108763778B CN 201810544826 A CN201810544826 A CN 201810544826A CN 108763778 B CN108763778 B CN 108763778B
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王磊
梁金雄
刘东亮
夏海军
王晓军
邱志平
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Beihang University
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Abstract

The invention discloses a non-probabilistic reliability topological optimization method for integration of a solid material and a truss-like microstructure material. The establishment of a dual-material interpolation model of a solid material and a truss-like microstructure material is indirectly realized through the elastic constant of the material; obtaining an elastic constant of a micro-level truss microstructure equivalent to the rigidity performance of the material on a macro-level by a strain energy equivalent method; describing uncertainty of the optimized model parameters by using an interval model, and establishing reliability displacement as a non-probability reliability index; solving the sensitivity of the reliability displacement to the design variable by using a adjoint vector method; and updating design variables by using an MMA algorithm, and performing iterative optimization until an optimal design scheme is obtained. According to the invention, the integrated design of the entity and the truss-like microstructure material is effectively realized in the topological optimization process of the continuum structure, the influence of uncertainty parameters described by an interval model on the structure performance is reasonably represented, and the light weight, the safety performance and the economical efficiency of the structure are effectively improved.

Description

Non-probability reliability topological optimization method for continuum structure
Technical Field
The invention relates to the field of topological optimization design of a continuum structure, in particular to a non-probability reliability topological optimization method for the continuum structure, which is designed by integrally designing a solid material and a truss-like microstructure material.
Background
The existing industrial products are almost designed and manufactured based on solid materials, and the industrial products based on multi-materials, particularly based on the integrated design of the solid materials and the truss-like microstructure materials, are quite few. Although industrial products based on solid materials are relatively simple to design and process, the material properties of some parts are not fully utilized, and the weight of the structure is occupied but the functions are not sufficiently played. This is particularly true in the aerospace field. The aerospace field is more demanding on the structure, since the weight of the structure directly affects the range, payload, and economic performance of the aircraft. In such circumstances, it is necessary to replace unimportant parts of the structure, such as non-primary load-bearing parts, with other more efficient materials, and the truss-like microstructure material is a very good material. Compared with the solid material, the density and the rigidity of the material are greatly reduced, the material is particularly suitable for certain unimportant and indispensable parts, and the rigidity performance of the solid material and the microstructure material can be fully exerted, so that a better structural lightweight design is realized. With the rapid development of 3D printing technology in recent years, the two-material design scheme has a better industrial manufacturing foundation.
The design idea of the double materials is integrated into the topological optimization design, so that the configuration and the material layout of the component meeting the working requirements can be provided at the conceptual design stage, the structural design of industrial products has great reference significance, the structural design difficulty can be effectively reduced, and the working efficiency can be improved. The material properties of the structure, such as Young modulus, shear modulus and the like, have unavoidable dispersibility in the production process, and external factors, such as load and the like, of the structure in the actual engineering environment are always changed, so that the factors influencing the safety of the structure are not negligible. From traditional design philosophy consideration, can avoid the structure to appear the safety problem through increaseing factor of safety. However, increasing the safety factor of the structural design would result in a significant increase in the structural weight and a significant reduction in economic efficiency, and is not a good solution for aircraft design. Therefore, with continuous progress of topology optimization methods and theoretical research and great enhancement of computer computing power, uncertain topology optimization research is greatly developed.
Reliability optimization designs based on probabilistic models have been successfully applied in topology optimization designs. But the complete probability distribution information of uncertain parameters required by the probabilistic reliability model is usually difficult to obtain in engineering practice. Probability reliability is very sensitive to probability model parameters, and small errors of probability data can cause larger errors in structural reliability calculation. In many cases, although accurate probability distribution data for uncertain parameters is not available, the magnitude or bounds of the parameter uncertainty are easily determined. The interval model is adapted to handle such uncertain but bounded parameters. And (3) providing a concept of non-probability reliability topological optimization of a continuum structure based on the interval model description of the parameters.
Uncertain parameters are described by an interval model and applied to the actual topology optimization technology of engineering, and the technology is not fully developed and effectively applied. The work of the people not only enriches the research of the topological optimization design of the bi-material to a certain extent, but also has important significance for the application of the non-probability reliability topological optimization considering the uncertainty in the engineering.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the defects of the prior art are overcome, and the integrated non-probabilistic reliability topological optimization method of the solid material and the truss-like microstructure material is provided. The invention fully considers the factors of insufficient performance application of single material structural materials and ubiquitous uncertainty in practical engineering, provides an integrated design scheme of solid materials and truss-like microstructure materials, and takes the provided non-probability reliability measurement index as the constraint condition of an optimization model, so that the obtained optimization design result structure is lighter and more economic and efficient.
The technical scheme adopted by the invention is as follows: a solid material and truss-like microstructure material integrated non-probability reliability topological optimization method comprises the following steps:
the first step is as follows: establishing a dual-material interpolation model of the integrated design of the solid material and the truss-like microstructure material based on the traditional material interpolation model with the penalty function:
Figure GDA0002485820830000021
therein ΨiIs the cell stiffness matrix of the ith cell, Ψ1,iA cell stiffness matrix, Ψ, for the i-th cell associated with the solid material2,iA cell stiffness matrix for the ith cell associated with the truss-like microstructured material; x is the number of1,iAnd x2,iRespectively designing variables of the ith unit on the aspect of a solid material layer and a truss-like microstructure material; p (p > 1) is a penalty factor.
The second step is that: using a total of 9 elastic constants E of the elastic modulus, shear modulus and Poisson's ratio in all directions of an anisotropic material (the same applies to an isotropic material)i,x,Ei,y,Ei,z,νi,xy,νi,yz,νi,zx,Gi,xy,Gi,yz,Gi,zxIndirectly realizing the establishment of a bi-material interpolation model:
Figure GDA0002485820830000022
wherein omegai＝[Ei,x,Ei,y,Ei,z,νi,xy,νi,yz,νi,zx,Gi,xy,Gi,yz,Gi,zx]T，Ai＝[Ei,x1,Ei,y1,Ei,z1,νi,xy1,νi,yz1,νi,zx1,Gi,xy1,Gi,yz1,Gi,zx1]TAnd Bi＝[Ei,x2,Ei,y2,Ei,z2,νi,xy2,νi,yz2,νi,zx2,Gi,xy2,Gi,yz2,Gi,zx2]TThe fusion elastic constant vector of the ith unit, the elastic constant vector of the solid material and the elastic constant vector of the truss-like microstructure material are respectively.
The third step: and obtaining the elastic constant of the microstructure of the micro-level truss equivalent to the rigidity performance of the material on the macro-level by a strain energy equivalent method. Applying different typical loading modes and periodic boundary conditions to the truss-like microstructure unit cell, and solving by a finite element method to obtain the equivalent elastic property of the truss-like microstructure material:
Figure GDA0002485820830000031
wherein E1,E2And E3Is the modulus of elasticity in three directions, G12,G23And G31Shear modulus in three directions
Figure GDA0002485820830000032
(p, q ═ 1,2,3) is poisson's ratio.
The fourth step: taking the minimum mass of the structure as an optimization target, taking the displacement of the concentrated load loading node as a constraint bar, and establishing a topological optimization model as follows:
objective function
Figure GDA0002485820830000033
Constraint function bv,r-bv,c≤0,v＝1,2,…,m
Ψ(Ψ1,i,Ψ2,i)b＝F,i＝1,2,…,N
Figure GDA0002485820830000034
Figure GDA0002485820830000035
Where M represents the mass of the structure, ρ1And ρ2Density, V, of solid and truss-like microstructured materials, respectivelyiIs the volume of the ith cell, N is the total number of cells for the optimal region partition, Ψ ∈ CN×NIs the overall stiffness matrix of the cell, b ∈ CN×1Being the total displaced column vector of the cell, F ∈ CN×1As a total load column vector, bv,rIs the actual displacement value of the i-th displacement constraint point, bv,cIs the allowable displacement value of the ith displacement constraint, m is the number of displacement constraints, x1And x2In order to design the lower threshold value of the variable,
Figure GDA0002485820830000036
and
Figure GDA0002485820830000037
is an upper threshold.
The fifth step: and describing the uncertainty of the parameters by the interval model, and establishing reliability displacement as a non-probability reliability index. The original optimization model can be modified by using the reliability displacement as follows:
Figure GDA0002485820830000041
s.t.Lv≤0,v＝1,2,…,m
Figure GDA0002485820830000042
Figure GDA0002485820830000043
Figure GDA0002485820830000044
wherein L isvFor reliable displacement, when LvWhen > 0, there is bv,r-bv,cIf the reliability is higher than 0, the reliability is not satisfied; when L isvIf < 0, there is bv,r-bv,cLess than or equal to 0, indicating that the reliability meets the requirement.
And a sixth step: and D, according to the expression of the reliability displacement obtained in the fifth step, carrying out derivation on the design variable through a complex function derivation rule, and obtaining the sensitivity of the reliability displacement to the design variable by using an adjoint vector method.
The seventh step: substituting the sensitivity of the reliability displacement obtained in the fifth step and the sensitivity of the reliability displacement difference obtained in the sixth step to the design variable into an MMA algorithm to solve the non-probability reliability optimization model to obtain a new design variable.
Eighth step: judging whether a convergence condition is met, if the convergence condition is not met, increasing the number of finished iterations by 1, and returning to the fourth step; otherwise, the iterative process ends.
Compared with the prior art, the invention has the advantages that:
the invention provides an integrated non-probabilistic reliability topological optimization method for a solid material and a truss-like microstructure material, which combines the solid material and the truss-like microstructure material and simultaneously applies the combination to structural topological optimization design so that the materials at all parts of a structure can fully exert the performance of the materials. Compared with a single-material structure, the composite material has the advantages of lower weight, higher efficiency and better economic performance under the condition that the structural performance is not reduced. In addition, the uncertainty of parameters such as load conditions, material attributes and displacement cognition is described by using an interval model, so that the load conditions and the like are closer to the actual working conditions of the structure, the influence of the dispersibility generated by the structural material attributes on the structure safety can be fully considered in the material processing and manufacturing process, and the established reliability displacement is used as a non-probability reliability index measurement standard, so that the safety performance of the structure can be improved, and the economic cost can be effectively reduced.
Drawings
FIG. 1 is a process diagram of the integrated non-probabilistic reliability topology optimization of a solid material and a truss-like microstructure material;
FIG. 2 is an equivalent schematic diagram of macro-microscopic properties of a truss-like microstructure under different unit cell types;
FIG. 3 is a schematic diagram representing voxel periodic boundary conditions;
FIG. 4 is a schematic diagram of interference between the extreme state plane and the standard space, in which FIG. 4(a) is
Figure GDA0002485820830000045
FIG. 4(b) is
Figure GDA0002485820830000046
FIG. 4(c) is
Figure GDA0002485820830000047
FIG. 4(d) is
Figure GDA0002485820830000051
FIG. 4(e) is
Figure GDA0002485820830000052
FIG. 4(f) is
Figure GDA0002485820830000053
Fig. 5 is a schematic diagram of reliability shift definition and calculation, in which fig. 5(a) is the definition of the reliability shift symbol, and fig. 5(b) is the derivation process of the reliability shift.
Detailed Description
The invention is further described with reference to the following figures and detailed description.
As shown in fig. 1, the invention provides an integrated non-probabilistic reliability topological optimization method for a solid material and a truss-like microstructure material, which comprises the following steps:
(1) establishing a dual-material interpolation model of the integrated design of the solid material and the truss-like microstructure material based on the traditional material interpolation model with the penalty function:
Figure GDA0002485820830000054
therein ΨiIs the cell stiffness matrix of the ith cell, Ψ1,iA cell stiffness matrix, Ψ, for the i-th cell associated with the solid material2,iA cell stiffness matrix for the ith cell associated with the truss-like microstructured material; x is the number of1,iAnd x2,iRespectively designing variables of the ith unit on the aspect of a solid material layer and a truss-like microstructure material; p (p > 1) is a penalty factor.
(2) Using a total of 9 elastic constants E of the elastic modulus, shear modulus and Poisson's ratio in all directions of an anisotropic material (the same applies to an isotropic material)i,x,Ei,y,Ei,z,νi,xy,νi,yz,νi,zx,Gi,xy,Gi,yz,Gi,zxIndirectly realizing the establishment of a bi-material interpolation model:
Figure GDA0002485820830000055
wherein omegai＝[Ei,x,Ei,y,Ei,z,νi,xy,νi,yz,νi,zx,Gi,xy,Gi,yz,Gi,zx]T，Ai＝[Ei,x1,Ei,y1,Ei,z1,νi,xy1,νi,yz1,νi,zx1,Gi,xy1,Gi,yz1,Gi,zx1]TAnd Bi＝[Ei,x2,Ei,y2,Ei,z2,νi,xy2,νi,yz2,νi,zx2,Gi,xy2,Gi,yz2,Gi,zx2]TThe fusion elastic constant vector of the ith unit, the elastic constant vector of the solid material and the elastic constant vector of the truss-like microstructure material are respectively.
(3) And obtaining the elastic constant of the microstructure of the micro-level truss equivalent to the rigidity performance of the material on the macro-level by a strain energy equivalent method. Applying different typical loading modes and periodic boundary conditions to the truss-like microstructure unit cell, and solving by a finite element method to obtain the equivalent elastic property of the truss-like microstructure material:
when only two nodes are arranged on each boundary of the representative voxel (as shown in FIG. 3), the coordination of the boundary node deformation is satisfied, and the equivalent elastic performance meeting the precision requirement can be obtained only by the representative voxel formed by one unit cell. Sequentially considering macroscopic line strain
Figure GDA0002485820830000056
And shear strain
Figure GDA0002485820830000057
The periodic boundary equation is as follows:
Figure GDA0002485820830000061
wherein
Figure GDA0002485820830000062
Is a preset macroscopic strain vector; x, Y and Z represent displacements representing three directions of a voxel; subscripts a1, a2, B1, B2, C1, C2 represent arbitrary points on the voxel surface; s represents a side length of a representative voxel.
In a volume V representing a voxelRVEAverage stress after homogenization
Figure GDA0002485820830000063
And average strain
Figure GDA0002485820830000064
Can be expressed as:
Figure GDA0002485820830000065
the equivalent strain energy of a material is:
Figure GDA0002485820830000066
representative strain energies in voxels are:
Figure GDA0002485820830000067
the volume can be integrated as a surface integral by gaussian integration:
Figure GDA0002485820830000068
wherein SRVETo represent the outer surface of a voxel, n is the unit outer normal vector, on which there are:
Figure GDA0002485820830000069
the formula (8) is introduced into formula (7) to obtain:
Figure GDA0002485820830000071
by introducing generalized Hooke's law, one can further derive:
Figure GDA0002485820830000072
wherein E1,E2And E3Is the modulus of elasticity in three directions, G12,G23And G31The shear modulus is in three directions and,
Figure GDA0002485820830000073
is the poisson ratio. Representing the equivalent elastic modulus E of a voxel (or truss-like microstructure material) by applying a suitable loading method and periodic boundary conditions to the representative voxel1,E2,E3,ν12,ν23,ν31,G12,G23，G31I.e. by finite element analysis.
(4) Taking the minimum mass of the structure as an optimization target, taking the displacement of the concentrated load loading node as a constraint bar, and establishing a topological optimization model as follows:
objective function
Figure GDA0002485820830000074
Constraint function bv,r-bv,c≤0,v＝1,2,…,m
Ψ(Ψ1,i,Ψ2,i)b＝F,i＝1,2,…,N
Figure GDA0002485820830000075
Figure GDA0002485820830000076
Where M represents the mass of the structure, ρ1And ρ2Density, V, of solid and truss-like microstructured materials, respectivelyiFor the volume of the ith cell, N is the cell for optimizing the region divisionTotal number of elements, Ψ ∈ CN×NIs the overall stiffness matrix of the cell, b ∈ CN×1Being the total displaced column vector of the cell, F ∈ CN×1As a total load column vector, bv,rIs the actual displacement value of the i-th displacement constraint point, bv,cIs the allowable displacement value of the ith displacement constraint, m is the number of displacement constraints,x 1andx 2in order to design the lower threshold value of the variable,
Figure GDA0002485820830000077
and
Figure GDA0002485820830000078
is an upper threshold.
(5) The uncertainty of the parameters is described by the interval model, and the reliability displacement is established as a non-probability reliability index:
taking into account uncertainties in material properties, external loads and safety displacements, due to the equations of static equilibrium
Figure GDA0002485820830000081
Is a linear equation, and can obtain the constrained displacement by an interval parameter fixed point method
Figure GDA0002485820830000082
The upper and lower bounds of (c). Will be provided with
Figure GDA0002485820830000083
Written in the form of interval mathematics:
Figure GDA0002485820830000084
whereinb v,rAnd
Figure GDA0002485820830000085
are respectively as
Figure GDA0002485820830000086
Lower and upper bounds. Equation (11) is a linear equation, and can be obtained by the interval parameter fixed point method:
Figure GDA0002485820830000087
wherein
Figure GDA0002485820830000088
t ═ m + k denotes the sum of the total stiffness matrix number k (k ═ 1) and the external load number m; g represents different overall stiffness matrix and external load fixed point combinations; w is av1 or 2, w v1 denotes the lower bound of the parameter, w v2 denotes taking the upper bound of the parameter, i.e.
Figure GDA0002485820830000089
(Ψ-1)2＝Ψ -1,
Figure GDA00024858208300000810
Figure GDA00024858208300000811
Constraining the actual displacement of the node by the displacement bv,rAnd a safety displacement bv,cA limit state equation can be defined:
Figure GDA00024858208300000812
and a limit state plane:
Figure GDA00024858208300000813
from the interval mathematical theory, the following standard space can be obtained:
Figure GDA00024858208300000814
wherein Δ bv,rAnd Δ bv,cRespectively belong to a standard interval
Figure GDA00024858208300000815
And
Figure GDA00024858208300000816
a variable of (d);
Figure GDA00024858208300000817
and
Figure GDA00024858208300000818
are respectively as
Figure GDA00024858208300000819
The nominal value of (a) and the interval radius,
Figure GDA00024858208300000820
and
Figure GDA00024858208300000821
are respectively as
Figure GDA00024858208300000822
Nominal value of (d) and interval radius. When the interference condition is satisfied
Figure GDA00024858208300000823
And
Figure GDA00024858208300000824
the extreme state function P divides the standard space into two parts, namely P (b)v,c,bv,r) The part of more than or equal to 0 is a security domain, P (b)v,c,bv,r) The portion < 0 is the fail field (as shown in FIG. 4). Formula (15) may be substituted for formula (14):
Figure GDA00024858208300000825
and:
Figure GDA0002485820830000091
without loss of generality, let Δ bv,c＝1，Δbv,rIs not equal to-1, canObtaining:
Figure GDA0002485820830000092
the non-probabilistic reliability index defined by equation (20) and the area ratio of the security domain to the standard space can be obtained:
Figure GDA0002485820830000093
along with the difference of interference states between the actual displacement interval and the safe displacement interval, states of a safe domain and a failure domain in the standard space are also different. Accordingly, the expression of the reliability index includes 6 cases in total, which is a piecewise function:
Figure GDA0002485820830000094
because the optimization algorithm based on the gradient requires that the reliability index has good continuity and micromability, complete gradient information exists when the MMA algorithm is called to solve large-scale design variables, but when R is usedv1 and RvPartial derivative when equal to 0
Figure GDA0002485820830000095
And
Figure GDA0002485820830000096
the total is 0. At this point, the MMA algorithm will not be able to complete the optimization. To solve this problem, reliability shifts are proposed as a new non-probabilistic reliability index Lv. Reliability shift LvThe distance between the extreme state plane corresponding to the actual reliability and the extreme state plane corresponding to the target reliability, which is obtained by the area ratio defined by equation (20) in each optimization iteration, is defined, and the two planes are parallel in the standard space, i.e., the expression differs by a constant (as shown in fig. 5). When L isvWhen the ratio is less than or equal to 0, Rv≥RcI.e. the reliability meets the requirements. From the distance calculation formula between the two planes, an expression for the reliability distance can be obtained:
Figure GDA0002485820830000101
the optimization model can thus be modified to:
Figure GDA0002485820830000102
(6) and D, according to the expression of the reliability displacement obtained in the fifth step, carrying out derivation on the design variable through a complex function derivation rule, and obtaining the sensitivity of the reliability displacement to the design variable by using an adjoint vector method.
Reliability index LvFor design variable xe,iThe partial derivatives are obtained by (i 1, 2., N, e 1, 2):
Figure GDA0002485820830000103
from formula (22):
Figure GDA0002485820830000104
and:
Figure GDA0002485820830000105
to avoid direct calculation
Figure GDA0002485820830000106
And
Figure GDA0002485820830000107
the proposed adjoint vector method can solve this problem well. The following lagrange equation is established:
Figure GDA0002485820830000111
wherein ζv(v ═ 1,2, …, m) relates toThe companion vector of the static balance equation. Due to the fact that
Figure GDA0002485820830000112
It is obvious that
Figure GDA0002485820830000113
Therefore, equation (26) can be derived by deviatoderivative of the design variables:
Figure GDA0002485820830000114
to avoid calculations
Figure GDA0002485820830000115
Difficulty of (1) to
Figure GDA0002485820830000116
The following can be obtained:
Figure GDA0002485820830000117
it can be seen that equation (28) is a formal and static equilibrium equation
Figure GDA0002485820830000118
Therefore, to find the accompanying vector ζv(v-1, 2, …, m), a unit load, i.e. a load vector f, may be applied only at the corresponding external load loading nodev＝[0,0,…,1,…,0]TObtained by finite element calculation.
Will ζvThe sensitivity of the upper and lower bounds of the actual displacement to the design variables can be found by substituting equation (27):
Figure GDA0002485820830000119
the external load vector F is not changed in the optimization iteration process, so
Figure GDA00024858208300001110
Equation (29) can be further derived as:
Figure GDA00024858208300001111
substituting formula (1) into
Figure GDA00024858208300001112
The following can be obtained:
Figure GDA00024858208300001113
and substituting formula (31) for formula (29) to obtain:
Figure GDA0002485820830000121
thus, the reliability index L is expressed by the expressions (23) to (25) and the expression (32)vFor design variable xe,iThe sensitivity of the method can be completely and efficiently solved.
(7) Substituting the sensitivity of the reliability displacement obtained in the fifth step and the sensitivity of the reliability displacement difference obtained in the sixth step to the design variable into an MMA algorithm to solve the non-probability reliability optimization model to obtain a new design variable.
(8) Judging whether a convergence condition is met, if the convergence condition is not met, increasing the number of finished iterations by 1, and returning to the fourth step; otherwise, the iterative process ends.
In summary, the invention provides an integrated non-probabilistic reliability topological optimization method for a solid material and a truss-like microstructure material. The establishment of a dual-material interpolation model of a solid material and a truss-like microstructure material is indirectly realized through the elastic constant of the material; obtaining an elastic constant of a micro-level truss microstructure equivalent to the rigidity performance of the material on a macro-level by a strain energy equivalent method; describing uncertainty of the optimized model parameters by using an interval model, and establishing reliability displacement as a non-probability reliability index; solving the sensitivity of the reliability displacement to the design variable by using a adjoint vector method; and updating design variables by using an MMA algorithm, and performing iterative optimization until an optimal design scheme is obtained.
The above are only the specific steps of the present invention, and the protection scope of the present invention is not limited in any way; the method can be expanded and applied to the field of topological optimization design of a continuum structure under the displacement constraint, and all technical schemes formed by adopting equivalent transformation or equivalent replacement fall within the protection scope of the invention.
The invention has not been described in detail and is part of the common general knowledge of a person skilled in the art.

Claims (1)

1. A solid material and truss-like microstructure material integrated non-probability reliability topological optimization method is characterized by comprising the following implementation steps:
the first step is as follows: establishing a dual-material interpolation model of the integrated design of the solid material and the truss-like microstructure material based on the traditional material interpolation model with the penalty function:
Figure FDA0002485820820000011
therein ΨiIs the cell stiffness matrix of the ith cell, Ψ1,iA cell stiffness matrix, Ψ, for the i-th cell associated with the solid material2,iA cell stiffness matrix for the ith cell associated with the truss-like microstructured material; x is the number of1,iAnd x2,iRespectively designing variables of the ith unit on the aspect of a solid material layer and a truss-like microstructure material; p is a penalty factor, and p is more than 1;
the second step is that: the elastic modulus, the shear modulus and the Poisson ratio of the anisotropic material or the isotropic material are utilized to have 9 elastic constants E in all directionsi,x,Ei,y,Ei,z,νi,xy,νi,yz,νi,zx,Gi,xy,Gi,yz,Gi,zxIndirectly realizing the establishment of a bi-material interpolation model:
Figure FDA0002485820820000012
wherein omegai＝[Ei,x,Ei,y,Ei,z,νi,xy,νi,yz,νi,zx,Gi,xy,Gi,yz,Gi,zx]T，Ai＝[Ei,x1,Ei,y1,Ei,z1,νi,xy1,νi,yz1,νi,zx1,Gi,xy1,Gi,yz1,Gi,zx1]TAnd Bi＝[Ei,x2,Ei,y2,Ei,z2,νi,xy2,νi,yz2,νi,zx2,Gi,xy2,Gi,yz2,Gi,zx2]TRespectively are a fusion elastic constant vector of the ith unit, an elastic constant vector of the solid material and an elastic constant vector of the truss-like microstructure material;
the third step: obtaining an elastic constant of a micro-layer truss-like microstructure equivalent to the rigidity performance of the material on a macro-layer by a strain energy equivalent method, applying different typical loading modes and periodic boundary conditions to a truss-like microstructure unit cell, and solving by a finite element method to obtain the equivalent elastic performance of the truss-like microstructure material:
Figure FDA0002485820820000021
wherein E1,E2And E3Is the modulus of elasticity in three directions, G12,G23And G31The shear modulus is in three directions and,
Figure FDA0002485820820000022
is the poisson ratio;
the fourth step: taking the minimum mass of the structure as an optimization target, taking the displacement of the concentrated load loading node as a constraint bar, and establishing a topological optimization model as follows:
objective function
Figure FDA0002485820820000023
Constraint function bv,r-bv,c≤0,v＝1,2,…,m
Ψ(Ψ1,i,Ψ2,i)b＝F,i＝1,2,…,N
Figure FDA0002485820820000024
Figure FDA0002485820820000025
Where M represents the mass of the structure, ρ1And ρ2Density, V, of solid and truss-like microstructured materials, respectivelyiIs the volume of the ith cell, N is the total number of cells for the optimal region partition, Ψ ∈ CN×NIs the overall stiffness matrix of the cell, b ∈ CN×1Being the total displaced column vector of the cell, F ∈ CN×1As a total load column vector, bv,rIs the actual displacement value of the i-th displacement constraint point, bv,cIs the allowable displacement value of the ith displacement constraint, m is the number of displacement constraints, x1And x2In order to design the lower threshold value of the variable,
Figure FDA0002485820820000026
and
Figure FDA0002485820820000027
is an upper threshold;
the fifth step: the uncertainty of the parameters is described by the interval model, the reliability displacement is established as a non-probability reliability index, and the original optimization model can be modified as follows by utilizing the reliability displacement:
Figure FDA0002485820820000031
s.t.Lv≤0,v＝1,2,…,m
Figure FDA0002485820820000032
Figure FDA0002485820820000033
Figure FDA0002485820820000034
wherein L isvFor reliable displacement, when LvWhen > 0, there is bv,r-bv,cIf the reliability is higher than 0, the reliability is not satisfied; when L isvIf < 0, there is bv,r-bv,cLess than or equal to 0, representing that the reliability meets the requirement;
and a sixth step: according to the expression of the reliability displacement obtained in the fifth step, the design variable is derived through a complex function derivation rule, and the sensitivity of the reliability displacement to the design variable is obtained by using an adjoint vector method;
the seventh step: substituting the sensitivity of the reliability displacement obtained in the fifth step and the sensitivity of the reliability displacement difference obtained in the sixth step to the design variable into an MMA algorithm to solve the non-probability reliability optimization model to obtain a new design variable;
eighth step: judging whether a convergence condition is met, if the convergence condition is not met, increasing the number of finished iterations by 1, and returning to the fourth step; otherwise, the iterative process ends.
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