CN116187074A - Multi-scale topological optimization method of anisotropic periodic structure material based on isogeometry - Google Patents

Multi-scale topological optimization method of anisotropic periodic structure material based on isogeometry Download PDF

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CN116187074A
CN116187074A CN202310204169.8A CN202310204169A CN116187074A CN 116187074 A CN116187074 A CN 116187074A CN 202310204169 A CN202310204169 A CN 202310204169A CN 116187074 A CN116187074 A CN 116187074A
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张建平
彭江鹏
黄牛武
龚曙光
刘金刚
邱奕
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Hunan Shaofeng Institute Of Applied Mathematics
Xiangtan University
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Abstract

The invention discloses a multi-scale topological optimization method of an anisotropic periodic structure material based on isogeometry, which comprises the following steps: (1) Modeling according to an isogeometric analysis method to obtain discrete point information of the macroscopical periodic structure and various microstructure design domains, and dividing the macroscopical periodic structure design domains into design subdomains; (2) Constructing a coordinate conversion relation between a natural coordinate system and a material coordinate system in the anisotropic material; (3) Performing periodic material distribution optimization, and determining the positions of various microstructures in the macroscopic periodic structure and the volume fractions of the various microstructures by regularizing the relative density; (4) And (3) performing multi-scale parallel topological optimization on the periodic structure/material, and determining a macroscopic periodic structure, various microstructures and equivalent elastic matrixes thereof. The invention carries out anisotropic periodic structure/material multiscale topological optimization based on isogeometric analysis, has accurate result, stable convergence and high efficiency, and can improve structural performance from macroscopic and microscopic two-scale, structure and material two angles.

Description

Multi-scale topological optimization method of anisotropic periodic structure material based on isogeometry
Technical Field
The invention belongs to the field of optimal design in computer aided engineering, and particularly relates to a multi-scale topological optimization method of an anisotropic periodic structure material based on isogeometry.
Background
The periodic porous structure of the composite material has been widely used in the fields of machinery, automobiles, aviation and the like because of the unique configuration and the multifunctional physical characteristics of sound absorption, heat insulation, vibration reduction, collision prevention, light weight, high strength and the like. However, most composite periodic structures designed empirically are generally not optimal structures, and with the application and popularization of various high-performance heterogeneous anisotropic composite materials, the difficulty in designing composite periodic structures with excellent performance is increasing. In addition, with the continuous expansion of the application field of industrial products, the requirements of various complex working environments (such as high temperature, high pressure, high speed and vacuum) on the structural performance and reliability of the industrial products are increasing, so that the periodic structure of the composite material under the pure macroscopic size cannot meet the requirements of the high-precision industry on the structural weight, the functional specialization and the performance integration of the industrial products. Therefore, the adoption of a proper and efficient method to introduce the microstructure design of the material on the microscopic scale on the basis of the periodic structure design on the macroscopic scale is an effective means for further improving the periodic structure performance of the composite material.
Topology optimization is used as an emerging branch of structure optimization, is a calculation design method for automatically generating material layout with maximized performance under related design specifications, can solve the problem of complex structure design which cannot be effectively solved by the traditional trial-and-error method, and provides a powerful system design strategy for a multi-scale periodic structure of a macro-micro composite material. Meanwhile, the rapidly developed additive manufacturing technology provides a convenient way for manufacturing the complex topological structure. The current mainstream topology optimization method comprises the following steps: homogenization (Homogenizatin Method), variable density (Variable Density Method), level Set (LSM), phase field (Phase Field Method), progressive structure optimization (Evolutionary Structural Optimization, ESO), moving deformation assemblies (Moving Morphable Components, MMC), and the like. Among them, the variable density method represented by the punished solid isotropic microstructure optimization method (Solid Isotropic Microstructures with Penalization, SIMP) is most widely used because of its compact model and clear topology. The currently common performance analysis methods are: finite Element Method (Finite Element Method, FEM), boundary Element Method (Boundary Element Method, BEM), finite difference Method (Finite Difference Method, FDM), finite volume Method (Finite Volume Method, FVM), element-free Method, and isogeometric analysis (Isogeometric Analysis, IGA). The former four methods inevitably simplify the model in the process of converting the geometric model into the analysis model, but the inconsistency between the geometric model and the analysis model can cause a certain loss of precision in the subsequent calculation process, and the conversion of the complex geometric model into the analysis model has great difficulty. The isogeometric analysis realizes the unification of the geometric model and the analysis model by taking the spline function describing the geometric shape as the shape function of the analysis calculation solution domain, and the precision of the analysis calculation is ensured by the high-order continuous spline function and the accurate geometric description. Therefore, the topology optimization method combining the SIMP method and the isogeometric analysis is an effective method for designing anisotropic periodic structures/material multi-scale structures.
The topological optimization design of the microstructure with specific properties starts from the reverse homogenization design model proposed by Sigmuld in 90 th century, and in more than 20 years thereafter, a large number of metamaterial microstructures with specific properties such as maximum bulk modulus, maximum shear modulus, negative Poisson ratio, negative thermal expansion coefficient and the like are designed based on the topological optimization method. During the process, researchers also combine the microstructure topological optimization design with the macrostructure topological optimization design, and a multiscale topological optimization design model considering single-type microstructures and multiscale microstructures is provided, so that the material design has more practical value. However, most of the existing research results are based on isotropic materials with homogenization, and the performance of the structure is improved only from the "structure" point of view by deeply digging the potential of material distribution, while the research on the microstructure design and the material/structure multi-scale design of the high-performance anisotropic composite material with good designability is less. In addition, no research results have been reported to introduce macroscopic periodic structural designs into the structural/material multi-scale design framework.
Under the background, the invention provides an anisotropic periodic structure/material multi-scale topological optimization method based on isogeometric analysis, and the product structural performance can be simultaneously improved from the angles of macroscopic scale, microscopic scale, structure and material while the macroscopic periodicity of the structure is ensured.
Disclosure of Invention
In view of the non-optimality in the periodic porous structure performance of most of the composite materials which are currently designed based on experience and the lack of research on the periodic structure/material multiscale topological optimization design of anisotropic composite materials, the technical problem to be solved by the invention is to provide the anisotropic periodic structure/material multiscale topological optimization method based on the isogeometric analysis, which can ensure the macroscopic periodicity of the structure and simultaneously improve the structural performance from the two angles of macroscopic and microscopic scales, structures and materials.
The technical scheme adopted for solving the technical problems is as follows: the method mainly realizes the periodic structure/material multi-scale topological optimization design of the anisotropic composite material through two stages of periodic material distribution optimization and periodic structure/material multi-scale parallel topological optimization, wherein the purpose of the periodic material distribution optimization stage is to determine the distribution positions of various microstructures in a macroscopic periodic structure and the volume fractions of various microstructures in the periodic structure/material multi-scale parallel topological optimization stage, and the purpose of the periodic structure/material multi-scale parallel topological optimization stage is to determine the macroscopic periodic structure, various microstructures and corresponding equivalent elastic matrixes.
The specific implementation steps of the technical scheme of the invention are as follows:
(1) Determining an isogeometric analysis macroscopic periodic structure design domain according to geometric features of a structure in actual engineering, constructing control points and unit information of the macroscopic periodic structure by using NURBS spline surfaces in an isogeometric analysis method, and calculating Gaussian point information and IGA basis function information of the macroscopic periodic structure; determining the volume constraint of the macro periodic structure and the initial relative density of control points of the macro periodic structure according to the performance requirement of the structure in actual engineering, and inputting the main Poisson's ratio, the auxiliary Poisson's ratio, the elastic modulus, the material direction angle and other material properties of the anisotropic material; dividing the design domain of the macro periodic structure into Ms design subdomains, and classifying control points and unit information of the design domain of the macro periodic structure according to a design subdomain division scheme, wherein the division scheme is as follows:
Ms＝Mx×My (1)
Ns＝Nx×Ny (2)
wherein Mx and My are the number of design subdomains in the x and y directions in the design domain, and Ns, nx and Ny represent the total number of control points in the design subdomain and the number of control points in the x and y directions respectively;
(2) Construction of nature in anisotropic materialsCoordinate system (x, y) and material coordinate system
Figure BDA0004110184700000031
Coordinate conversion relation between two coordinate systems to realize rational conversion of material attribute parameters, and elastic matrix under natural coordinate system >
Figure BDA0004110184700000032
The expression of (2) is: />
Figure BDA0004110184700000033
in the formula ,
Figure BDA0004110184700000034
and />
Figure BDA0004110184700000035
An anisotropic material coordinate transformation matrix and an elastic matrix under a material coordinate system respectively, wherein E 1 、E 2 、ν 12 and ν21 Respectively are +.>
Figure BDA0004110184700000036
and />
Figure BDA0004110184700000037
Tensile and compressive modulus in the direction and poisson ratio and satisfy the relation +.>
Figure BDA0004110184700000038
G 12 For shear modulus, θ is the angle between the natural coordinate system and the material coordinate system, defining the ratio of Poisson's ratio in the direction of ζ and η +.>
Figure BDA0004110184700000039
Is a poisson's ratio factor; (3) performing periodic material distribution optimization:
inputting control points, gaussian points, units, IGA basis functions, initial relative densities of the control points, a periodic design subdomain division scheme, anisotropic material properties and coordinate conversion relations of the macroscopic periodic structure initial design domain determined by the step (1) and the step (2); inputting boundary conditions and iteration termination conditions for periodic material distribution optimization;
(II) calculating a displacement field of the macroscopically periodic structure based on the isogeometric analysis: (a) Solving a structural rigidity matrix of the anisotropic material based on an isogeometric analysis theory and a SIMP material interpolation model, and setting a penalty factor pe=1 in the process; (b) Applying a displacement boundary condition and a force load boundary condition at a control point associated with the external load; (c) Establishing a discrete control equation, and solving displacement parameter values of control points in a macroscopic periodic structure design domain; (d) Outputting a displacement vector U and a total force load vector F of a macroscopic periodic structure design domain control point;
(III) establishing a topological optimization mathematical model taking structural flexibility as an objective function based on isogeometric analysis, wherein the expression is as follows:
Figure BDA0004110184700000041
wherein C is the structural flexibility, ρ is the relative density vector of the control point, ne is the number of units of the design domain of the macroscopically periodic structure, K is the overall stiffness matrix,
Figure BDA0004110184700000042
for the initial area of the cell ρ min =0.001 is the control point relative density minimum, V 0 And V represents the structural volume before and after optimization, respectively, ">
Figure BDA0004110184700000043
For a specified volume fraction ρ i,j The relative density of the jth control point in the ith design sub-domain in the design domain for the macroscopically periodic structure; />
Figure BDA0004110184700000044
An IGA basis function taking a unit center coordinate point as a calculation point;
and (IV) solving the sensitivity of the structural flexibility objective function and the volume constraint function of the structural topological optimization model by adopting a concomitant analysis method:
Figure BDA0004110184700000045
Figure BDA0004110184700000046
/>
in the formula ,
Figure BDA0004110184700000047
is an IGA basis function with Gaussian points as calculation points;
(v) applying a periodicity constraint by re-averaging the relative density and objective function sensitivity of control points having the same number within each design sub-field expressed as:
Figure BDA0004110184700000048
Figure BDA0004110184700000049
in the formula ,
Figure BDA00041101847000000410
representing the relative density of the re-average allocation of control points with the same number in each design sub-field, +.>
Figure BDA00041101847000000411
Representing the sensitivity of the target function of the re-average allocation of control points with the same number in each design sub-domain;
(VI) programming according to an Optimization Criterion (OC) method, and updating design variables: inputting the re-average distributed relative density and the sensitivity of the objective function of the current control point, updating the relative density of the control point according to the OC method, solving the total volume of the updated design domain, setting a new interpolation point according to the total product difference before and after updating to judge whether iteration is ended, adopting the relative density of the updated control point and continuing iteration according to the OC method if not ending, and stopping calculating and outputting the relative density of the updated control point if the iteration is ended;
calculating the relative density difference of each control point in the input and output of the step (VI), solving the maximum relative density change value, comparing the maximum change value with the total loop iteration termination condition set in the step (I), judging whether the termination condition is met, if not, feeding the relative density of the control point in the output of the step (VI) back to the step (II) for re-iteration, and if the iteration termination condition is met, ending the iteration and outputting a final control point relative density vector;
(VIII) interpolating based on the IGA basis function and the relative density vector of the control point to obtain the relative density value at the center point of each unit, and taking the relative density value at the center point of each unit as the relative density value of the unit, and carrying out regularization treatment on the obtained relative density vector of each unit to determine the distribution position of each microstructure in the macroscopic periodic structure and the volume fraction of each microstructure, wherein the regularization treatment calculation formula is as follows:
Figure BDA0004110184700000051
in the formula ,
Figure BDA0004110184700000052
is the relative density of the ith unit in the zeta microstructure area in the macroscopic periodic structure;
Figure BDA0004110184700000053
and />
Figure BDA0004110184700000054
Respectively an upper boundary and a lower boundary of the relative density of the units in the zeta microstructure area; />
Figure BDA0004110184700000055
Is the total number of units in the zeta microstructure area of the macroscopic periodic structure; />
Figure BDA0004110184700000056
Regularized relative density for all units in the zeta microstructure area;
the method comprises the following specific steps: (a) Outputting a continuous unit relative density cloud picture of a macroscopic periodic structure, observing the aggregation condition of the unit relative density in the cloud picture, and dividing the cloud picture into a plurality of sections which are not included each other according to the aggregation condition, wherein each section represents a microstructure; (b) Summing up the cell densities in each interval, then averaging, and taking the average relative density as the relative density value of the cells in the interval and the volume fraction of the microstructure represented by the interval; (c) Outputting the relative density of the segmented units of the averaged macroscopic periodic structure, namely the distribution position of various microstructures in the macroscopic periodic structure;
(4) Performing periodic structure/material multi-scale parallel topological optimization:
inputting control points, gaussian points, units, IGA basis functions, periodic design subdomain division schemes, control point relative density vectors, anisotropic material properties and coordinate conversion relations of the initial design domain of the macroscopic periodic structure, distribution positions of various microstructures in the macroscopic periodic structure, volume fractions of various microstructures, boundary conditions and iteration termination conditions of the periodic structure/material multiscale parallel topological optimization, wherein the control points, gaussian points, units and IGA basis functions of the initial design domain of the macroscopic periodic structure are determined in the steps (1), (2) and (3);
Determining the shape of each type of microstructure initial design domain according to the microstructure type determined in the step (I), constructing control points and unit information of each type of microstructure initial design domain by using NURBS spline surfaces in an isogeometric analysis method, calculating Gaussian point information and IGA basis function information of each type of microstructure initial design domain, and determining the initial relative density of the control points of each type of microstructure design domain;
(III) constructing material attribute interpolation models of various microstructures based on an isogeometric analysis theory and an SIMP material interpolation model; firstly, constructing a relative density field of a macroscopic periodic structure and various microstructure design domains, wherein the relative density field has the expression:
Figure BDA0004110184700000057
Figure BDA0004110184700000061
in the formula ,
Figure BDA0004110184700000062
and />
Figure BDA0004110184700000063
Designing the relative densities of arbitrary calculation points and control points within the domain unit for the macroscopically periodic structure, respectively,/->
Figure BDA0004110184700000064
Designing a domain NURBS basis function for the macroscopically periodic structure; />
Figure BDA0004110184700000065
and />
Figure BDA0004110184700000066
Designing the relative densities of arbitrary calculation points and control points in the domain unit for the zeta-like microstructure respectively,/->
Figure BDA0004110184700000067
Designing a domain NURBS basis function for the zeta type microstructure; m and M are macroscopically periodic structures and microstructure identifiers, respectively; the material property interpolation model of various microstructures based on the isogeometric analysis theory and the SIMP material interpolation model can be expressed as:
Figure BDA0004110184700000068
in the formula ,
Figure BDA0004110184700000069
applying punished elastic matrix for zeta microstructure design domain, pk is microstructure control point relative density punishment factor;
and (IV) calculating equivalent elastic matrixes of various microstructures based on an isogeometric analysis theory, an SIMP material interpolation model and an energy homogenization method, wherein the expression is as follows:
Figure BDA00041101847000000610
in the formula ,
Figure BDA00041101847000000611
is the equivalent elastic matrix of the zeta microstructure,>
Figure BDA00041101847000000612
and />
Figure BDA00041101847000000613
Respectively applying displacement fields before and after test strain in the zeta microstructure, ++>
Figure BDA00041101847000000614
Zeta type microstructure design domain area, theta is microstructure type number; />
Figure BDA00041101847000000615
Testing the strain field for linearly independent units within the zeta-like microstructure, +.>
Figure BDA00041101847000000616
Is an unknown strain field within the zeta type microstructure;
(V) calculating a displacement field of the macroscopic periodic structure based on the isogeometric analysis, wherein the method comprises the following specific steps:
(a) Solving an anisotropic material structural rigidity matrix considering various microstructures based on an isogeometric analysis theory and an SIMP material interpolation model; firstly, constructing a macroscopic anisotropic multi-scale material interpolation model based on a SIMP material interpolation model and considering multi-class microstructures:
Figure BDA00041101847000000617
/>
in the formula ,DM Applying punishment to the overall elastic matrix in the macroscopic periodic structure, wherein pe is punishment factor,
Figure BDA00041101847000000618
the relative density of points is arbitrarily calculated for the region where the zeta microstructure in the macroscopic periodic structure is located; the structural stiffness matrix of the anisotropic material based on isogeometric analysis theory and SIMP material interpolation model:
Figure BDA00041101847000000619
in the formula ,ΩM And
Figure BDA00041101847000000620
physical domains and parent spaces of the macroscopically periodic structure, respectively, are mapped according to FIG. 3, J ξη and />
Figure BDA00041101847000000621
The Jacobian transformation matrix is respectively mapped to a parameter domain by a physical domain and mapped to a mother space by the parameter domain;
(b) Applying a displacement boundary condition and a force load boundary condition at a control point associated with the external load;
(c) Establishing a discrete control equation, and solving displacement parameter values of control points in a macroscopic periodic structure design domain;
(d) Outputting a displacement vector U and a total force load vector F of a macroscopic periodic structure design domain control point;
establishing a periodic structure/material multiscale topological optimization mathematical model which takes structural flexibility as an objective function and takes relative densities of macroscopic periodic structure design domain control points and various microstructure design domain control points as design variables based on isogeometric analysis, wherein the expression is as follows:
Figure BDA0004110184700000071
wherein J is an objective function related to the macro design variables and various kinds of microstructure design variables; ρ M and ρm Representing relative density vectors of the control points of the macroscopical periodic structure design domain and the control points of various microstructure design domains respectively;
Figure BDA0004110184700000072
and />
Figure BDA0004110184700000073
Respectively representing a control point relative density vector and a displacement vector of a region where a zeta type microstructure is located in a macroscopic periodic structure design domain;
Figure BDA0004110184700000074
Representing the zeta microstructure design domain control point relative density vector; u (U) M and FM Representing the overall displacement vector and the load vector of the macroscopically periodic structure; />
Figure BDA0004110184700000075
Designing the grid number of the domain for the zeta microstructure; />
Figure BDA0004110184700000076
Designing the total number of units of the domain for the zeta microstructure in the macroscopic periodic structure; />
Figure BDA0004110184700000077
Designing a unit initial area of a domain for a zeta type microstructure in the macroscopic periodic structure; />
Figure BDA0004110184700000078
The initial area of the unit is the zeta microstructure; />
Figure BDA0004110184700000079
and τζ Macroscopic structure and zeta class respectivelyMicrostructure-specified material volume fraction; v (V) M and />
Figure BDA00041101847000000710
The volumes before and after the macro structure optimization are respectively; />
Figure BDA00041101847000000711
and />
Figure BDA00041101847000000712
Respectively optimizing the volumes before and after the zeta microstructure;
Figure BDA00041101847000000713
representing the relative density of the jth control point in the ith design sub-domain in the macrostructure; />
Figure BDA00041101847000000714
Is the relative density of the ith control point within the zeta type microstructure; ρ min =0.001 is the control point relative density minimum; />
Figure BDA00041101847000000715
The IGA basis function takes a unit center coordinate point as a calculation point in the macroscopic periodic structure; />
Figure BDA00041101847000000716
Designing an IGA basis function taking a unit center coordinate point as a calculation point in the domain for the zeta microstructure; />
Figure BDA00041101847000000717
Designing the relative density of domain unit control points for the zeta-like microstructure in the macrostructure; />
Figure BDA00041101847000000718
Designing domain unit control point relative density for the zeta type microstructure;
(VII) solving the sensitivity of the structural flexibility objective function and the volume constraint function of the periodic structure/material multi-scale topological optimization mathematical model to macroscopic design variables by adopting a concomitant analysis method:
Figure BDA0004110184700000081
Figure BDA0004110184700000082
in the formula ,
Figure BDA0004110184700000083
designing the relative density of control points of the region where the zeta-like microstructure of the domain is located for the macroscopic periodic structure;
(VIII) solving the sensitivity of the structural flexibility objective function and the microstructure volume constraint function of the periodic structure/material multi-scale topological optimization mathematical model to the microstructure design variables by adopting a concomitant analysis method:
Figure BDA0004110184700000084
Figure BDA0004110184700000085
in the formula ,
Figure BDA0004110184700000086
the first partial derivative of the homogenized elastic tensor to the microscopic design variable calculated for the zeta microstructure topology has the expression:
Figure BDA0004110184700000087
in the formula ,
Figure BDA0004110184700000088
calculating an IGA basis function matrix of points in the zeta microstructure design domain;
(ix) imposing a periodicity constraint by re-averaging the relative density and objective function sensitivity of control points having the same number within each design sub-domain in the macroscopically periodic structure expressed as:
Figure BDA0004110184700000089
Figure BDA00041101847000000810
in the formula ,
Figure BDA00041101847000000811
representing the relative density of the re-average allocation of control points with the same number in each design sub-domain in the macroscopically periodic structure,/->
Figure BDA00041101847000000812
Representing sensitivity of the target function of re-average allocation of control points with the same number in each design sub-domain in the macro periodic structure;
(X) programming according to an Optimization Criterion (OC) method, and updating a macro periodic structure and various microstructure design variables, wherein the method comprises the following specific steps of: (a) Inputting the re-average distributed relative density and the sensitivity of the objective function of the control point of the current macro periodic structure, updating the relative density of the control point according to the OC method and solving the total volume of the updated design domain, setting a new interpolation point by the total product difference of the macro periodic structure before and after updating to judge whether iteration is ended, adopting the relative density of the updated control point and continuing iteration according to the OC method if not ending, and stopping calculating and outputting the relative density of the updated control point if the iteration is ended; (b) Inputting the relative density and the objective function sensitivity of the control points of the first microstructure, updating the relative density of the control points according to an OC method, solving the total volume of the microstructure design domain after updating, setting a new interpolation point according to the total product difference before and after updating to judge whether iteration is ended, adopting the relative density of the control points after updating and continuing iteration according to the OC method if not ending, and stopping calculating and outputting the relative density of the control points after updating if the iteration is ended; (c) The updating iteration of the other various microstructure design variables is the same as that of the first type microstructure design variable;
(XI) calculating the relative density difference of each macro periodic structure control point input and output in (X), solving the maximum relative density change value, comparing the maximum change value with the total loop iteration termination condition set in (I), judging whether the termination condition is met, if the termination condition is not met, feeding back the relative density of the macro periodic structure design domain control points output in (X) to (III) for re-iteration, and if the iteration termination condition is met, iteratively terminating and outputting the final relative density vectors of the macro periodic structure design domain and the microstructure design domain control points of various types;
and (XII) outputting an optimal macroscopic periodic structure based on isogeometric analysis, various microstructure topological structures and corresponding equivalent elastic matrixes.
The beneficial effects of the invention are as follows: the isogeometric analysis method adopted by the invention realizes the unification of the geometric model and the analysis model by taking the spline function describing the geometric shape as the shape function of the analysis calculation solution domain, and the precision of the analysis calculation is ensured by the high-order continuous spline function and the accurate geometric description; the periodic structure/material multi-scale topological optimization design of the anisotropic composite material is realized through two stages of periodic material distribution optimization and periodic structure/material multi-scale parallel topological optimization, the product structural performance can be simultaneously improved from the two angles of macroscopic scale, microscopic scale, structure and material while the macroscopic periodicity of the structure is ensured, the defect that most of the current periodic porous structures of the composite material which depend on empirical design are not optimal in performance is overcome, and the method for designing the anisotropic periodic structure/material multi-scale structure with clear thought and simple and convenient method is provided, can be closely combined with engineering practice, and has good theoretical research and engineering application values.
Drawings
The invention is described in further detail below with reference to the drawings and examples.
FIG. 1 is a flow chart of programming and calculation of the isogeometric anisotropic periodic structure material multi-scale topology optimization method of the present invention
FIG. 2 is a mapping relationship among physical domain, parameter domain and mother space in the isogeometric analysis method according to the present invention
FIG. 3 is a schematic diagram of a sub-domain partitioning scheme for the design of a macroscopically periodic structure in accordance with the invention
FIG. 4 is a schematic diagram of an orthotropic material coordinate system and a natural coordinate system according to the present invention
FIG. 5 is a schematic diagram of a macroscopically periodic structure design domain of the embodiment of the invention
Fig. 6 is a density distribution of poisson's ratio factor bt=0.6 according to the embodiment of the present invention
Fig. 7 shows a density distribution of poisson's ratio factor bt=1 according to the embodiment of the present invention
Fig. 8 is a density distribution of poisson's ratio factor bt=1.5 according to the embodiment of the present invention
Fig. 9 is a density distribution of poisson's ratio factor bt=2 according to the embodiment of the present invention
Fig. 10 is a regularized segmented density distribution with poisson's ratio factor bt=0.6 in an embodiment of the invention
Fig. 11 is a regularized segmented density distribution with poisson's ratio factor bt=1 in an embodiment of the invention
Fig. 12 is a regularized segmented density distribution with poisson's ratio factor bt=1.5 in an embodiment of the invention
Fig. 13 is a regularized segmented density distribution with poisson's ratio factor bt=2 in an embodiment of the invention
FIG. 14 is a schematic diagram of various types of initial design domains of microstructures according to the embodiment of the present invention
FIG. 15 shows a block periodic structure, various microstructures and corresponding equivalent elastic matrices when the Poisson's ratio factor is 0.6 according to the embodiment of the present invention
FIG. 16 shows a block periodic structure, various microstructures and corresponding equivalent elastic matrices when the Poisson's ratio factor is 1 in the embodiment of the present invention
FIG. 17 shows a block periodic structure, various microstructures and corresponding equivalent elastic matrices when the Poisson's ratio factor is 1.5 according to the embodiment of the present invention
FIG. 18 shows a block periodic structure, various microstructures and corresponding equivalent elastic matrices when the Poisson's ratio factor is 2 in the embodiment of the present invention
Detailed Description
FIG. 1 is a flow chart of programming and calculation of the isogeometric anisotropic periodic structure material multi-scale topological optimization method, referring to FIG. 1, and the specific flow is as follows:
(1) According to the geometric characteristics of the structure in the actual engineering, determining an isogeometric analysis macroscopic periodic structure design domain, constructing control points and unit information of the macroscopic periodic structure by using NURBS spline surfaces in an isogeometric analysis method, and calculating Gaussian point information and IGA basis function information of the macroscopic periodic structure, wherein the formula of the IGA basis function is as follows:
Figure BDA0004110184700000101
Figure BDA0004110184700000102
Figure BDA0004110184700000103
Where ζ is the coordinate variable of any computation point in the parameter space, ζ i+1 Is the coordinate variable of the (i+1) th node in the parameter space, p and q are the orders of the geometric basis functions, u and v respectively represent the coordinate variable of any calculation point in the directions of xi and eta in the parameter space, and N i,p (u) is the one-dimensional B-spline curve value of the ith control point to u coordinate, N j,q (v) For a one-dimensional B-spline curve value of the jth node on the v-coordinate,
Figure BDA0004110184700000111
for the control point P i,j IGA basis function value, ω i,j At control point P for non-uniform B-spline curve i,j Weights at;
determining the volume constraint of the macro periodic structure and the initial relative density of control points of the macro periodic structure according to the performance requirement of the structure in actual engineering, and inputting the main Poisson's ratio, the auxiliary Poisson's ratio, the elastic modulus, the material direction angle and other material properties of the anisotropic material; referring to fig. 2, a macro periodic structure design domain is divided into Ms design subfields, and control points and unit information of the macro periodic structure design domain are classified according to a design subfield division scheme, wherein the division scheme is as follows:
Ms＝Mx×My (4)
Ns＝Nx×Ny (5)
wherein Mx and My are the number of design subdomains in the x and y directions in the design domain, and Ns, nx and Ny represent the total number of control points in the design subdomain and the number of control points in the x and y directions respectively;
(2) Construction of the natural coordinate System (x, y) and Material coordinate System in anisotropic Material as shown in FIG. 3
Figure BDA0004110184700000112
Coordinate conversion relation between two coordinate systems to realize rational conversion of material attribute parameters, and elastic matrix under natural coordinate system>
Figure BDA0004110184700000113
The expression of (2) is:
Figure BDA0004110184700000114
in the formula ,
Figure BDA0004110184700000115
and />
Figure BDA0004110184700000116
Respectively an anisotropic material coordinate transformation matrix and elasticity under a material coordinate systemMatrix, where E 1 、E 2 、ν 12 and ν21 Respectively are +.>
Figure BDA0004110184700000117
and />
Figure BDA0004110184700000118
Tensile and compressive modulus in the direction and poisson ratio and satisfy the relation +.>
Figure BDA0004110184700000119
G 12 For shear modulus, θ is the angle between the natural coordinate system and the material coordinate system, defining the ratio of Poisson's ratio in the direction of ζ and η +.>
Figure BDA00041101847000001110
Is a poisson's ratio factor; (3) performing periodic material distribution optimization:
(3.1) inputting control points, gaussian points, units, IGA basis functions, initial relative densities of the control points, a periodic design subdomain division scheme, anisotropic material properties and coordinate conversion relations of the macroscopic periodic structure initial design domain determined by the step (1) and the step (2); inputting boundary conditions and iteration termination conditions for periodic material distribution optimization;
(3.2) calculating a displacement field of the macroscopic periodic structure based on the isogeometric analysis, wherein the specific steps are as follows:
(a) Solving a structural rigidity matrix of the anisotropic material based on an isogeometric analysis theory and an SIMP material interpolation model; firstly, taking the relative density of control points as a design variable, adopting a quadratic NURBS spline function with higher-order continuity as a shape function of an analytical calculation solution domain, wherein each unit in a discrete design domain is determined by (p+1) times (q+1) pieces of control point information, and the relative density of any point in any unit can be obtained by determining the relative density of (p+1) times (q+1) pieces of control points of the unit and interpolating corresponding NURBS basis functions, and the expression is as follows:
Figure BDA0004110184700000121
in the formula ,ρg For arbitrarily calculating the relative density of points within a cell ρ I In order to control the relative density of the dots,
Figure BDA0004110184700000122
is NURBS basis function; secondly, according to the SIMP material interpolation model, a fictive material with variable relative density between 0 and 1 is introduced, and then the elastic modulus expression based on the SIMP model is as follows:
Figure BDA0004110184700000123
wherein pe is a penalty factor, E 0 For a given modulus of elasticity, ρ, of a solid material g (x) Calculating the relative density of points for any time; finally, constructing a structural rigidity matrix of the anisotropic material based on the isogeometric analysis theory and the SIMP material interpolation model:
Figure BDA0004110184700000124
wherein omega and
Figure BDA0004110184700000125
physical domain and parent space respectively, the mapping relation of which is shown in figure 3, J ξη and />
Figure BDA0004110184700000126
The Jacobian transformation matrix is respectively mapped to a parameter domain by a physical domain and mapped to a mother space by the parameter domain;
(b) Applying a displacement boundary condition and a force load boundary condition at a control point associated with the external load;
(c) Establishing a discrete control equation, and solving displacement parameter values of control points in a macroscopic periodic structure design domain;
(d) Outputting a displacement vector U and a total force load vector F of a macroscopic periodic structure design domain control point;
(3.3) establishing a topological optimization mathematical model taking structural flexibility as an objective function based on isogeometric analysis, wherein the expression is as follows:
Figure BDA0004110184700000127
wherein C is the structural flexibility, ρ is the relative density vector of the control point, ne is the number of units of the macroscopically periodic structural design domain,
Figure BDA0004110184700000128
for the initial area of the cell ρ min =0.001 is the control point relative density minimum, V 0 And V represents the structural volume before and after optimization, respectively, ">
Figure BDA0004110184700000129
For a specified volume fraction ρ i,j The relative density of the jth control point in the ith design sub-domain in the design domain for the macroscopically periodic structure; />
Figure BDA00041101847000001210
An IGA basis function taking a unit center coordinate point as a calculation point;
(3.4) solving the sensitivity of the structural flexibility objective function and the volume constraint function of the structural topology optimization model by adopting a concomitant analysis method:
Figure BDA00041101847000001211
Figure BDA0004110184700000131
in the formula ,
Figure BDA0004110184700000132
is an IGA basis function with Gaussian points as calculation points;
(3.5) applying a periodicity constraint by re-averaging the relative density of control points with the same number and the sensitivity of the objective function within each design sub-field expressed as:
Figure BDA0004110184700000133
Figure BDA0004110184700000134
in the formula ,
Figure BDA0004110184700000135
representing the relative density of the re-average allocation of control points with the same number in each design sub-field, +.>
Figure BDA0004110184700000136
Representing the sensitivity of the target function of the re-average allocation of control points with the same number in each design sub-domain;
(3.6) programming according to an Optimization Criterion (OC) method, and updating design variables: inputting the re-average distributed relative density and the sensitivity of the objective function of the current control point, updating the relative density of the control point according to the OC method, solving the total volume of the updated design domain, setting a new interpolation point according to the total product difference before and after updating to judge whether iteration is ended, adopting the relative density of the updated control point and continuing iteration according to the OC method if not ending, and stopping calculating and outputting the relative density of the updated control point if the iteration is ended; the optimization criterion method is carried out according to the following relation:
Figure BDA0004110184700000137
/>
wherein k is an iteration step, and kappa and beta are respectively a movement limit and a damping coefficient;
Figure BDA0004110184700000138
For the optimization criteria, the expression is:
Figure BDA0004110184700000139
in the formula ,δ1 Lagrange multipliers determined by a dichotomy method;
(3.7) calculating the relative density difference of each control point in the input and output of (3.6), solving the maximum relative density change value, comparing the maximum change value with the total loop iteration termination condition set in (3.1), judging whether the termination condition is met, if the termination condition is not met, feeding the relative density of the control point in the output of (3.6) back to (3.2) for re-iteration, and if the iteration termination condition is met, ending the iteration and outputting a final control point relative density vector; the convergence condition is performed according to the following relation:
Figure BDA00041101847000001310
in the formula ,
Figure BDA00041101847000001311
and />
Figure BDA00041101847000001312
The maximum relative density change values of the k+1st step and the k th step are respectively, and epsilon is an iteration termination condition preset in the step (3.2);
(3.8) calculating the relative density of the units, and carrying out regularization treatment on the relative density to determine the distribution positions of various microstructures in the macroscopic periodic structure and the volume fractions of the various microstructures; first, interpolating to obtain a relative density value at each unit center point based on an IGA basis function and a control point relative density vector, and taking the relative density value at the unit center point as the relative density value of the unit, wherein the unit relative density is solved according to the following relation:
Figure BDA0004110184700000141
Secondly, regularizing the obtained unit relative density vector to determine the distribution position of various microstructures in the macroscopic periodic structure and the volume fraction of various microstructures, wherein the regularized calculation formula is as follows:
Figure BDA0004110184700000142
in the formula ,
Figure BDA0004110184700000143
is the relative density of the ith unit in the zeta microstructure area in the macroscopic periodic structure;
Figure BDA0004110184700000144
and />
Figure BDA0004110184700000145
Respectively an upper boundary and a lower boundary of the relative density of the units in the zeta microstructure area; />
Figure BDA0004110184700000146
Is the total number of units in the zeta microstructure area of the macroscopic periodic structure; />
Figure BDA0004110184700000147
Regularized relative density for all units in the zeta microstructure area;
the method comprises the following specific steps: (a) Outputting a continuous unit relative density cloud picture of a macroscopic periodic structure, observing the aggregation condition of the unit relative density in the cloud picture, and dividing the cloud picture into a plurality of sections which are not included each other according to the aggregation condition, wherein each section represents a microstructure; (b) Summing up the cell densities in each interval, then averaging, and taking the average relative density as the relative density value of the cells in the interval and the volume fraction of the microstructure represented by the interval; (c) Outputting the relative density of the segmented units of the averaged macroscopic periodic structure and the relative density value of the averaged units, namely the distribution positions of various microstructures in the macroscopic periodic structure and the volume fractions of various microstructures;
(4) Performing periodic structure/material multi-scale parallel topological optimization:
(4.1) inputting control points, gaussian points, units, IGA base functions, periodic design subdomain division schemes, control point relative density vectors, anisotropic material properties and coordinate conversion relations, distribution positions of various microstructures in the macroscopic periodic structure and volume fractions and boundary conditions of the various microstructures of the macroscopic periodic structure, which are determined by the steps (1), (2) and (3); determining iteration termination conditions of the periodic structure/material multi-scale parallel topological optimization;
(4.2) determining the shape of each type of microstructure initial design domain according to the microstructure type determined in (4.1), constructing control points and unit information of each type of microstructure initial design domain by using NURBS spline surfaces in an isogeometric analysis method, calculating Gaussian point information and IGA basis function information of each type of microstructure initial design domain, and determining initial relative density of control points of each type of microstructure design domain; (4.3) constructing material attribute interpolation models of various microstructures based on the isogeometric analysis theory and the SIMP material interpolation model; firstly, constructing a relative density field of a macroscopic periodic structure and various microstructure design domains, wherein the relative density field has the expression:
Figure BDA0004110184700000151
Figure BDA0004110184700000152
in the formula ,
Figure BDA0004110184700000153
and />
Figure BDA0004110184700000154
Designing the relative densities of arbitrary calculation points and control points within the domain unit for the macroscopically periodic structure, respectively,/->
Figure BDA0004110184700000155
Designing a domain NURBS basis function for the macroscopically periodic structure; />
Figure BDA0004110184700000156
and />
Figure BDA0004110184700000157
Designing the relative densities of arbitrary calculation points and control points in the domain unit for the zeta-like microstructure respectively,/->
Figure BDA0004110184700000158
Designing a domain NURBS basis function for the zeta type microstructure; m and M are macroscopically periodic structures and microstructure identifiers, respectively; the material property interpolation model of various microstructures based on the isogeometric analysis theory and the SIMP material interpolation model can be expressed as:
Figure BDA0004110184700000159
in the formula ,
Figure BDA00041101847000001510
applying punished elastic matrix for zeta microstructure design domain, pk is microstructure control point relative density punishment factor;
and (4.4) calculating equivalent elastic matrixes of various microstructures based on an isogeometric analysis theory, a SIMP material interpolation model and an energy homogenization method, wherein the expression is as follows:
Figure BDA00041101847000001511
in the formula ,
Figure BDA00041101847000001512
is the equivalent elastic matrix of the zeta microstructure,>
Figure BDA00041101847000001513
and />
Figure BDA00041101847000001514
Respectively applying displacement fields before and after test strain in the zeta microstructure, ++>
Figure BDA00041101847000001515
Zeta type microstructure design domain area, theta is microstructure type number; />
Figure BDA00041101847000001516
Testing the strain field for linearly independent units within the zeta-like microstructure, +.>
Figure BDA00041101847000001517
Is an unknown strain field within the zeta type microstructure;
(4.5) calculating a displacement field of the macroscopic periodic structure based on the isogeometric analysis, wherein the specific steps are as follows:
(a) Solving an anisotropic material structural rigidity matrix considering various microstructures based on an isogeometric analysis theory and an SIMP material interpolation model; firstly, constructing an anisotropic multi-scale material interpolation model based on a SIMP material interpolation model and considering multi-class microstructures:
Figure BDA00041101847000001518
in the formula ,DM Applying punishment to the overall elastic matrix in the macroscopic periodic structure, wherein pe is punishment factor,
Figure BDA00041101847000001519
the relative density of points is arbitrarily calculated for the region where the zeta microstructure in the macroscopic periodic structure is located; the structural stiffness matrix of the anisotropic material based on isogeometric analysis theory and SIMP material interpolation model:
Figure BDA0004110184700000161
in the formula ,ΩM And
Figure BDA0004110184700000162
physical domains and parent spaces of the macroscopically periodic structure, respectively, are mapped according to FIG. 3, J ξη and />
Figure BDA0004110184700000163
The Jacobian transformation matrix is respectively mapped to a parameter domain by a physical domain and mapped to a mother space by the parameter domain;
(b) Applying a displacement boundary condition and a force load boundary condition at a control point associated with the external load;
(c) Establishing a discrete control equation, and solving displacement parameter values of control points in a macroscopic periodic structure design domain;
(d) Outputting a displacement vector U and a total force load vector F of a macroscopic periodic structure design domain control point;
(4.6) establishing a periodic structure/material multi-scale topological optimization mathematical model which takes structural flexibility as an objective function and takes relative densities of macroscopic periodic structure design domain control points and various microstructure design domain control points as design variables based on isogeometric analysis, wherein the expression is as follows:
Figure BDA0004110184700000164
wherein J is an objective function related to a macroscopic design variable and various microstructure design variables; ρ M and ρm Representing relative density vectors of the control points of the macroscopical periodic structure design domain and the control points of various microstructure design domains respectively;
Figure BDA0004110184700000165
and />
Figure BDA0004110184700000166
Respectively representing a control point relative density vector and a displacement vector of a region where a zeta type microstructure is located in a macroscopic periodic structure design domain; />
Figure BDA0004110184700000167
Representing the zeta microstructure design domain control point relative density vector; u (U) M and FM Representing the macroscopic periodic structure overall displacement vector and the load vector, < ->
Figure BDA0004110184700000168
Designing the grid number of the domain for the zeta microstructure; />
Figure BDA00041101847000001619
Designing a unit initial area of a domain for a zeta type microstructure in the macroscopic periodic structure; />
Figure BDA0004110184700000169
The initial area of the unit is the zeta microstructure; />
Figure BDA00041101847000001610
and τζ The volume fractions of the materials respectively specified for the macrostructure and the zeta microstructure; v (V) M and />
Figure BDA00041101847000001611
The volumes before and after the macro structure optimization are respectively; />
Figure BDA00041101847000001612
and />
Figure BDA00041101847000001613
Respectively optimizing the volumes before and after the zeta microstructure; />
Figure BDA00041101847000001614
Representing the relative density of the jth control point in the ith design sub-domain in the macrostructure; />
Figure BDA00041101847000001615
Is the relative density of the ith control point within the zeta type microstructure; ρ min =0.001 is the control point relative density minimum;
Figure BDA00041101847000001616
the IGA basis function takes a unit center coordinate point as a calculation point in the macroscopic periodic structure; />
Figure BDA00041101847000001617
Designing an IGA basis function taking a unit center coordinate point as a calculation point in the domain for the zeta microstructure; />
Figure BDA00041101847000001618
Designing the relative density of domain unit control points for the zeta-like microstructure in the macrostructure; />
Figure BDA0004110184700000171
Designing domain control point relative densities for zeta-like microstructures;
(4.7) solving the sensitivity of the structural flexibility objective function and the volume constraint function of the periodic structure/material multi-scale topological optimization mathematical model to macroscopic design variables by adopting a concomitant analysis method:
Figure BDA0004110184700000172
Figure BDA0004110184700000173
in the formula ,
Figure BDA0004110184700000174
designing the relative density of control points of the region where the zeta-like microstructure of the domain is located for the macroscopic periodic structure;
(4.8) solving the sensitivity of the structural flexibility objective function and the microstructure volume constraint function of the periodic structure/material multi-scale topological optimization mathematical model to the microstructure design variables by adopting a concomitant analysis method:
Figure BDA0004110184700000175
Figure BDA0004110184700000176
in the formula ,
Figure BDA0004110184700000177
the first partial derivative of the homogenized elastic tensor to the microscopic design variable calculated for the zeta microstructure topology has the expression:
Figure BDA0004110184700000178
in the formula ,
Figure BDA0004110184700000179
calculating an IGA basis function matrix of points in the zeta microstructure design domain;
(4.9) applying a periodicity constraint by re-averaging the relative density and objective function sensitivity of control points having the same number within each design sub-domain in the macroscopically periodic structure by:
Figure BDA00041101847000001710
Figure BDA00041101847000001711
in the formula ,
Figure BDA00041101847000001712
representing the relative density of the re-average allocation of control points with the same number in each design sub-domain in the macroscopically periodic structure,/->
Figure BDA00041101847000001713
Representing sensitivity of the target function of re-average allocation of control points with the same number in each design sub-domain in the macro periodic structure;
(4.10) programming according to an Optimization Criterion (OC) method, and updating the macro periodic structure and various microstructure design variables, wherein the method comprises the following specific steps: (a) Inputting the re-average distributed relative density and the sensitivity of the objective function of the control point of the current macro periodic structure, updating the relative density of the control point according to the OC method and solving the total volume of the updated design domain, setting a new interpolation point by the total product difference of the macro periodic structure before and after updating to judge whether iteration is ended, adopting the relative density of the updated control point and continuing iteration according to the OC method if not ending, and stopping calculating and outputting the relative density of the updated control point if the iteration is ended; the macroscopic periodic structure optimization criterion method is carried out according to the following relation:
Figure BDA0004110184700000181
Wherein k is an iteration step, and kappa and beta are respectively a movement limit and a damping coefficient;
Figure BDA0004110184700000182
for the optimization criteria, the expression is:
Figure BDA0004110184700000183
in the formula ,δ1 Lagrange multipliers determined by a dichotomy method;
(b) Inputting the relative density and the objective function sensitivity of the control points of the first microstructure, updating the relative density of the control points according to an OC method, solving the total volume of the microstructure design domain after updating, setting a new interpolation point according to the total product difference before and after updating to judge whether iteration is ended, adopting the relative density of the control points after updating and continuing iteration according to the OC method if not ending, and stopping calculating and outputting the relative density of the control points after updating if the iteration is ended; the microstructure optimization criterion method is carried out according to the following relation:
Figure BDA0004110184700000184
wherein k is an iteration step, and kappa and beta are respectively a movement limit and a damping coefficient;
Figure BDA0004110184700000185
for the optimization criteria, the expression is:
Figure BDA0004110184700000186
in the formula ,λ1 Lagrange multipliers determined by a dichotomy method;
(c) The updating iteration of the other various microstructure design variables is the same as that of the first type microstructure design variable;
(4.11) calculating the relative density difference of each macro periodic structure control point in the input and output process in (4.9), solving the maximum relative density change value, comparing the maximum change value with the total loop iteration termination condition set in (4.1), judging whether the termination condition is met, if not, feeding back the relative density of the macro periodic structure design domain control points in the output process in (4.9) to (4.3) for re-iteration, and if the iteration termination condition is met, iteratively terminating and outputting the final macro periodic structure design domain and the relative density vectors of the various microstructure design domain control points; the convergence condition is performed according to the following relation:
Figure BDA0004110184700000191
in the formula ,
Figure BDA0004110184700000192
and />
Figure BDA0004110184700000193
The maximum relative density change values of the k+1st step and the k th step are respectively, and epsilon is an iteration termination condition preset in the step (4.1);
and (4.12) outputting an optimal macroscopic periodic structure based on the isogeometric analysis and various microstructure topological structures and corresponding equivalent elastic matrixes.
The following is one example of the application of the method of the present invention to engineering practice:
referring to FIG. 5, the present embodiment is a square design domain with a length and width of 100mm, and the material has a main elastic modulus E 1 ＝2.06×10 11 Pa, poisson ratio v 12 Material direction angle θ=0.3; the left side of the design domain is fully constrained, and the center of the right side is subjected to downward concentrated force action F=1000N; the volume constraint of the macroscopic periodic structure is 50%, the material punishment factor is 3, and the initial relative density of the control point is 0.5; the entire design domain is discretized by 120×120 control points and 118×118 units; the design domain is divided into ms=3×3 design subfields; the optimal anisotropic periodic multi-scale structure under poisson ratio factor bt= 0.6,1,1.5,2, the corresponding variable thickness method optimal topology, regularized segmented density distribution, segmented periodic structure, various microstructures and the corresponding equivalent elastic matrix are calculated respectively.
The specific implementation steps of the invention for this example are as follows:
(1) Determining an isogeometric analysis macroscopic periodic structure design domain according to geometric features of a structure in actual engineering, constructing control points and unit information of the macroscopic periodic structure by using NURBS spline surfaces in an isogeometric analysis method, and calculating Gaussian point information and IGA basis function information of the macroscopic periodic structure; determining the volume constraint of the macro periodic structure and the initial relative density of control points of the macro periodic structure according to the performance requirement of the structure in actual engineering, and inputting the main Poisson's ratio, the auxiliary Poisson's ratio, the elastic modulus, the material direction angle and other material properties of the anisotropic material; dividing a macroscopic periodic structure design domain into Ms=2x2 design subdomains, and classifying control points and unit information of the macroscopic periodic structure design domain according to a design subdomain division scheme;
(2) Construction of a natural coordinate System (x, y) and a Material coordinate System in an Anisotropic Material
Figure BDA0004110184700000194
The coordinate conversion relation between the two components,solving an elastic tensor matrix under a natural coordinate system;
(3) And (3) performing periodic material distribution optimization:
(3.1) inputting control points, gaussian points, units, IGA basis functions, initial relative densities of the control points, a periodic design subdomain division scheme, anisotropic material properties and coordinate conversion relations of the macroscopic periodic structure initial design domain determined by the step (1) and the step (2); inputting boundary conditions; setting an iteration termination condition, namely ending iteration when the maximum change value of the relative density of the discrete points before and after updating is smaller than 0.001;
(3.2) solving a structural rigidity matrix of the anisotropic material based on an isogeometric analysis theory and a SIMP material interpolation model; applying a displacement boundary condition and a force load boundary condition at a control point associated with the external load; establishing a discrete control equation, and solving displacement parameter values of control points in a macroscopic periodic structure design domain; outputting a displacement vector U and a total force load vector F of a macroscopic periodic structure design domain control point;
(3.3) establishing a topological optimization mathematical model based on isogeometric analysis and taking the structural flexibility as an objective function, and substituting the displacement vector U and the total force load vector F of the obtained macroscopic periodic structural design domain control points in (3.2) to calculate the structural flexibility;
(3.4) solving the sensitivity of the structural flexibility objective function and the volume constraint function of the structural topological optimization model by adopting a concomitant analysis method; (3.5) imposing a periodicity constraint by re-averaging the relative densities of control points with the same number and the sensitivity of the objective function within each design sub-field;
(3.6) updating design variables according to an Optimization Criterion (OC), solving the total volume of the updated design domain according to the new relative density of the control points, setting new interpolation points according to the total product difference before and after updating to judge whether iteration is ended, adopting the relative density of the updated control points and continuing iteration according to the OC method if not ending, and stopping calculating and outputting the relative density of the updated control points if the iteration is ended;
(3.7) calculating the relative density difference of each control point in the input and output of (3.6), solving the maximum relative density change value, comparing the maximum change value with the total loop iteration termination condition set in (3.1), judging whether the termination condition is met, if the termination condition is not met, feeding the relative density of the control point in the output of (3.6) back to (3.2) for re-iteration, and if the iteration termination condition is met, ending the iteration and outputting a final control point relative density vector;
(3.8) calculating the relative density of the unit, interpolating the relative density vector based on the IGA basis function and the relative density vector of the control point to obtain the relative density value at the center point of each unit, and taking the relative density value at the center point of the unit as the relative density value of the unit, wherein the optimal topology of the variable thickness method when Bt= 0.6,1,1.5,2 is listed in sequence in fig. 6, 7, 8 and 9; observing the aggregation condition of the relative density of units in the cloud picture, dividing the aggregation condition into 4 sections which are not included in each other and are [ rho ] min 0.5), [0.5,0.7), [0.7, 0.8), and [0.8,1 ]]Each interval represents a microstructure; summing up the cell densities in each interval, then averaging, and taking the average relative density as the relative density value of the cells in the interval and the volume fraction of the microstructure represented by the interval; outputting the relative density of the segmented units of the averaged macroscopic periodic structure and the relative density value of the averaged units, namely the distribution positions of various microstructures in the macroscopic periodic structure and the volume fractions of various microstructures; fig. 10, 11, 12, and 13 sequentially list regularized blocked density distributions when bt= 0.6,1,1.5,2;
(4) Performing periodic structure/material multi-scale parallel topological optimization:
(4.1) inputting control points, gaussian points, units, IGA base functions, periodic design subdomain division schemes, control point relative density vectors, anisotropic material properties and coordinate conversion relations, distribution positions of various microstructures in the macroscopic periodic structure and volume fractions and boundary conditions of the various microstructures of the macroscopic periodic structure, which are determined by the steps (1), (2) and (3); setting iteration termination conditions of the periodic structure/material multi-scale parallel topological optimization, namely ending iteration when the maximum change value of the relative density of discrete points before and after updating is smaller than 0.001;
(4.2) determining the shape of each type of microstructure initial design domain according to the 4 types of microstructures determined in (4.1), wherein the middle white part is a cavity material with the relative density of 0.001, the microstructure overall design domain comprises 60×60 control points, the control points and the unit information of each type of microstructure initial design domain are constructed by using NURBS spline surfaces in an isogeometric analysis method, gaussian point information and IGA basis function information of each type of microstructure initial design domain are calculated, and the initial relative density of the control points of each type of microstructure design domain is determined as respective volume constraint;
(4.3) constructing material attribute interpolation models of various microstructures based on the isogeometric analysis theory and the SIMP material interpolation model;
(4.4) calculating equivalent elastic matrixes of various microstructures based on an isogeometric analysis theory, a SIMP material interpolation model and an energy homogenization method, wherein the input test strain is a 3 multiplied by 3 identity matrix;
(4.5) calculating a displacement field of the macroscopic periodic structure based on isogeometric analysis, and firstly, constructing an anisotropic multi-scale material interpolation model based on the SIMP material interpolation model and considering multi-class microstructures; then, solving a structural rigidity matrix of the anisotropic material based on the isogeometric analysis theory and the SIMP material interpolation model; secondly, applying displacement boundary conditions and force load boundary conditions on control points related to external loads; thirdly, establishing a discrete control equation and solving the displacement parameter value of the control point in the macroscopic periodic structure design domain; finally, outputting a displacement vector U and a total force load vector F of the macroscopic periodic structure design domain control point;
(4.6) establishing a periodic structure/material multi-scale topological optimization mathematical model which takes structural flexibility as an objective function and takes relative densities of macroscopic periodic structure design domain control points and various microstructure design domain control points as design variables based on isogeometric analysis, and calculating the structural flexibility;
(4.7) solving the sensitivity of the structural flexibility objective function and the volume constraint function of the periodic structure/material multi-scale topological optimization mathematical model to the macroscopic design variables by adopting a concomitant analysis method;
(4.8) solving the sensitivity of the structural flexibility objective function and the microstructure volume constraint function of the periodic structure/material multi-scale topological optimization mathematical model to the microstructure design variables by adopting a concomitant analysis method;
(4.9) imposing a periodicity constraint by re-averaging the relative densities of control points having the same number and the sensitivity of the objective function within each design sub-domain in the macroscopically periodic structure;
(4.10) updating the macroscopic periodic structure design variable according to an Optimization Criterion (OC), taking a movement limit kappa=0.02 and a damping coefficient beta=0.5, solving the total volume of the updated macroscopic periodic structure design domain according to the new relative density of the control point, setting a new interpolation point according to the total product difference before and after updating to judge whether iteration is ended, adopting the relative density of the updated control point and continuing iteration according to the OC method if not ended, and stopping calculating and outputting the relative density of the updated control point if the iteration is ended;
(4.11) updating various microstructure design variables according to an optimization criterion OC method, taking a movement limit kappa=0.02 and a damping coefficient beta=0.5, solving the total volume of the updated microstructure design domain according to the new relative density of the control point, setting a new interpolation point according to the total product difference before and after updating to judge whether iteration is ended, adopting the relative density of the updated control point and continuing iteration according to the OC method if the iteration is not ended, and stopping calculating and outputting the relative density of the updated control point if the iteration is ended;
(4.12) calculating the relative density difference of each macro periodic structure control point in the input and output process in (4.9), solving the maximum relative density change value, comparing the maximum change value with the total loop iteration termination condition set in (4.1), judging whether the termination condition is met, if not, feeding back the relative density of the macro periodic structure design domain control points in the output process in (4.9) to (4.3) for re-iteration, and if the iteration termination condition is met, iteratively terminating and outputting the final macro periodic structure design domain and the relative density vectors of the various microstructure design domain control points;
and (4.13) outputting an optimal macroscopic periodic structure based on the isogeometric analysis, various microstructure topological structures and corresponding equivalent elastic matrixes, wherein the partitioned periodic structure, various microstructures and corresponding equivalent elastic matrixes when bt= 0.6,1,1.5,2 are sequentially shown in fig. 15, 16, 17 and 18.
Although the present invention has been described in detail with reference to the present embodiment, the above description is not intended to limit the scope of the present invention, and any modification and improvement based on the concept of the present invention is considered as the scope of the present invention.

Claims (3)

1. The multi-scale topological optimization method of the anisotropic periodic structure material based on the isogeometry is characterized by comprising the following steps of:
(1) According to the geometric characteristics of the structure in the actual engineering, determining an isogeometric analysis macroscopic periodic structure design domain, constructing control points and unit information of the macroscopic periodic structure by using NURBS spline surfaces in an isogeometric analysis method, and calculating Gaussian point information and IGA basis function information of the macroscopic periodic structure, wherein the formula of the IGA basis function is as follows:
Figure FDA0004110184690000011
wherein Ni,p (u) is a one-dimensional B-spline curve value of the ith node of the p-order on the u-coordinate,/->
Figure FDA0004110184690000012
Figure FDA0004110184690000013
Where ζ is the coordinate variable of any computation point in the parameter space, ζ i+1 Is the coordinate variable of the (i+1) th node in the parameter space, p and q are the orders of the geometric basis functions, u and v respectively represent the coordinate variable of any calculation point in the directions of xi and eta in the parameter space, and N i,p (u) is the one-dimensional B-spline curve value of the ith control point to u coordinate, N j,q (v) For the one-dimensional B-spline curve value of the jth node on the v-coordinate,/for the j-th node>
Figure FDA0004110184690000014
For the control point P i,j IGA basis function value, ω i,j For non-uniform B-spline curvesAt control point P i,j Weights at; determining the volume constraint of the macro periodic structure and the initial relative density of control points of the macro periodic structure according to the performance requirement of the structure in actual engineering, and inputting the main Poisson's ratio, the auxiliary Poisson's ratio, the elastic modulus, the material direction angle and other material properties of the anisotropic material; referring to fig. 2, a macro periodic structure design domain is divided into Ms design subfields, and control points and unit information of the macro periodic structure design domain are classified according to a design subfield division scheme, wherein the division scheme is as follows: ms=mx×my, ns=nx×ny, where Mx and My are the number of design subfields in the x and y directions in the design field, respectively, and Ns, nx, and Ny represent the total number of control points in the design subfields and the number of control points in the x and y directions, respectively;
(2) Construction of a natural coordinate System (x, y) and a Material coordinate System in an Anisotropic Material
Figure FDA0004110184690000015
Coordinate conversion relation between two coordinate systems to realize rational conversion of material attribute parameters, and elastic matrix under natural coordinate system>
Figure FDA0004110184690000016
The expression of (2) is:
Figure FDA0004110184690000017
wherein />
Figure FDA0004110184690000018
and />
Figure FDA0004110184690000021
An anisotropic material coordinate transformation matrix and an elastic matrix under a material coordinate system respectively, wherein E 1 、E 2 、ν 12 and ν21 Respectively are +.>
Figure FDA0004110184690000022
and />
Figure FDA0004110184690000023
Tensile and compressive modulus in the direction and poisson ratio and satisfy the relation +.>
Figure FDA0004110184690000024
G 12 For shear modulus, θ is the angle between the natural coordinate system and the material coordinate system, defining the ratio of Poisson's ratio in the direction of ζ and η +.>
Figure FDA0004110184690000025
Is a poisson's ratio factor;
(3) Performing periodic material distribution optimization;
(4) And performing periodic structure/material multi-scale parallel topological optimization.
2. The method for multi-scale topological optimization of anisotropic periodic structure material based on isogeometry according to claim 1, wherein the step (3) comprises the following specific steps:
(a) Inputting control points, gaussian points, units, IGA (interior gas phase) basis functions, initial relative densities of the control points, a periodic design subdomain division scheme, anisotropic material properties and coordinate conversion relations of the macroscopic periodic structure initial design domain determined in the step (1) and the step (2); inputting boundary conditions; setting an iteration termination condition;
(b) Solving a structural rigidity matrix of the anisotropic material based on an isogeometric analysis theory and an SIMP material interpolation model; applying a displacement boundary condition and a force load boundary condition at a control point associated with the external load; establishing a discrete control equation, and solving displacement parameter values of control points in a macroscopic periodic structure design domain; outputting a displacement vector U and a total force load vector F of a macroscopic periodic structure design domain control point;
(c) Establishing a topological optimization mathematical model taking structural flexibility as an objective function based on isogeometric analysis:
Figure FDA0004110184690000026
wherein C is the structural flexibility, ρ is the relative density vector of the control point, ne is the number of units of the macroscopically periodic structural design domain,
Figure FDA0004110184690000027
for the initial area of the cell ρ min =0.001 is the control point relative density minimum, V 0 And V represents the structural volume before and after optimization, respectively, ">
Figure FDA0004110184690000028
For a specified volume fraction ρ i,j The relative density of the jth control point in the ith design sub-domain in the design domain for the macroscopically periodic structure; />
Figure FDA0004110184690000029
An IGA basis function taking a unit center coordinate point as a calculation point; substituting the displacement vector U and the total force load vector F of the macro periodic structure design domain control point obtained in the step (b) to calculate the structure flexibility;
(d) Solving the sensitivity of a structure flexibility objective function and a volume constraint function of the structure topology optimization model by adopting a companion analysis method;
(e) Applying a periodicity constraint by re-averaging the relative density of control points with the same number and the sensitivity of the objective function within each design sub-field;
(f) Programming according to an Optimization Criterion (OC) method, and updating design variables: inputting the re-average distributed relative density and the sensitivity of the objective function of the current control point, updating the relative density of the control point according to the OC method, solving the total volume of the updated design domain, setting a new interpolation point according to the total product difference before and after updating to judge whether iteration is ended, adopting the relative density of the updated control point and continuing iteration according to the OC method if not ending, and stopping calculating and outputting the relative density of the updated control point if the iteration is ended;
(g) Calculating the relative density difference of each control point in the input and output process in (f), solving the maximum relative density change value, comparing the maximum change value with the total loop iteration termination condition set in (a), judging whether the termination condition is met, if not, feeding the relative density of the control point in the output process in (b) back to the iteration for re-iteration, and if the iteration termination condition is met, carrying out iteration termination and outputting a final control point relative density vector;
(h) Calculating the relative density of the units, interpolating the relative density value at the center point of each unit based on the IGA base function and the relative density vector of the control point, and regularizing the calculated relative density vector of the units by taking the relative density value at the center point of the unit as the relative density value of the unit to determine the distribution position of various microstructures in the macroscopic periodic structure and the volume fraction of various microstructures, wherein the regularized calculation formula is as follows:
Figure FDA0004110184690000031
/>
wherein ,
Figure FDA0004110184690000032
is the relative density of the ith unit in the zeta microstructure area in the macroscopic periodic structure; />
Figure FDA0004110184690000033
And
Figure FDA0004110184690000034
respectively an upper boundary and a lower boundary of the relative density of the units in the zeta microstructure area; />
Figure FDA0004110184690000035
Is the total number of units in the zeta microstructure area of the macroscopic periodic structure; />
Figure FDA0004110184690000036
For all units in the zeta microstructure areaRelative density after regularization of the relative density of the elements; outputting the relative density of the segmented units of the averaged macroscopic periodic structure and the relative density value of the averaged units, namely the distribution positions of various microstructures in the macroscopic periodic structure and the volume fractions of various microstructures.
3. The method for multi-scale topological optimization of anisotropic periodic structure material based on isogeometry according to claim 1, wherein the step (4) comprises the following specific steps:
(a) Inputting control points, gaussian points, units, IGA basis functions, periodic design subdomain division schemes, control point relative density vectors, anisotropic material properties and coordinate conversion relations, distribution positions of various microstructures in the macroscopic periodic structure and volume fractions and boundary conditions of various microstructures of the macroscopic periodic structure determined by the steps (1), (2) and (3); setting iteration termination conditions of the periodic structure/material multi-scale parallel topological optimization;
(b) Determining the shape of each type of microstructure initial design domain according to the microstructure type determined in the step (a), constructing control points and unit information of each type of microstructure initial design domain by using NURBS spline surfaces in an isogeometric analysis method, calculating Gaussian point information and IGA basis function information of each type of microstructure initial design domain, and determining the initial relative density of the control points of each type of microstructure design domain;
(c) Constructing material attribute interpolation models of various microstructures based on an isogeometric analysis theory and an SIMP material interpolation model;
(d) Calculating equivalent elastic matrixes of various microstructures based on an isogeometric analysis theory, an SIMP material interpolation model and an energy homogenization method;
(e) Calculating a displacement field of the macroscopic periodic structure based on isogeometric analysis, and constructing an anisotropic multi-scale material interpolation model based on the SIMP material interpolation model and considering multi-class microstructures; solving a structural rigidity matrix of the anisotropic material based on an isogeometric analysis theory and an SIMP material interpolation model; applying a displacement boundary condition and a force load boundary condition at a control point associated with the external load; establishing a discrete control equation, and solving displacement parameter values of control points in a macroscopic periodic structure design domain; outputting a displacement vector U and a total force load vector F of a macroscopic periodic structure design domain control point;
(f) Establishing a periodic structure/material multi-scale topological optimization mathematical model which takes structural flexibility as an objective function and takes macroscopic periodic structure design domain control points and relative densities of various microstructure design domain control points as design variables based on isogeometric analysis:
Figure FDA0004110184690000051
wherein J is an objective function related to the macro design variables and various microstructure design variables; ρ M and ρm Representing relative density vectors of the control points of the macroscopical periodic structure design domain and the control points of various microstructure design domains respectively;
Figure FDA0004110184690000052
and />
Figure FDA0004110184690000053
Respectively representing a control point relative density vector and a displacement vector of a region where a zeta type microstructure is located in a macroscopic periodic structure design domain; / >
Figure FDA0004110184690000054
Representing the zeta microstructure design domain control point relative density vector; u (U) M and FM Representing the macroscopic periodic structure overall displacement vector and the load vector, < ->
Figure FDA0004110184690000055
Designing the grid number of the domain for the zeta microstructure; ms and Ns are respectively the number of the design subdomains of the macroscopic periodic structure and the number of control points in the subdomains; />
Figure FDA0004110184690000056
Designing a unit initial area of a domain for a zeta type microstructure in the macroscopic periodic structure; />
Figure FDA0004110184690000057
The initial area of the unit is the zeta microstructure; />
Figure FDA0004110184690000058
and τζ The volume fractions of the materials respectively specified for the macrostructure and the zeta microstructure; v (V) M and />
Figure FDA0004110184690000059
The volumes before and after the macro structure optimization are respectively; />
Figure FDA00041101846900000510
and />
Figure FDA00041101846900000511
Respectively optimizing the volumes before and after the zeta microstructure;
Figure FDA00041101846900000512
representing the relative density of the jth control point in the ith design sub-domain in the macrostructure; />
Figure FDA00041101846900000513
Is the relative density of the ith control point within the zeta type microstructure; ρ min =0.001 is the control point relative density minimum; />
Figure FDA00041101846900000514
The IGA basis function takes a unit center coordinate point as a calculation point in the macroscopic periodic structure; />
Figure FDA00041101846900000515
Taking unit center coordinate point as in zeta microstructure design domainIGA basis functions for the computation points; />
Figure FDA00041101846900000516
Designing the relative density of domain unit control points for the zeta-like microstructure in the macrostructure;
Figure FDA00041101846900000517
designing domain control point relative densities for zeta-like microstructures;
(g) Solving the sensitivity of a structural flexibility objective function and a volume constraint function of a periodic structure/material multi-scale topological optimization mathematical model to macroscopic design variables by adopting an accompanying analysis method;
(h) Solving the sensitivity of a structural flexibility objective function and a microstructure volume constraint function of a periodic structure/material multi-scale topological optimization mathematical model to microstructure design variables by adopting a concomitant analysis method;
(i) The periodicity constraint is imposed by re-averaging the relative density of control points with the same number and the sensitivity of the objective function within each design sub-domain in the macroscopically periodic structure, expressed as:
Figure FDA0004110184690000061
Figure FDA0004110184690000062
in the formula ,
Figure FDA0004110184690000063
representing the relative density of the re-average allocation of control points with the same number in each design sub-domain in the macroscopically periodic structure,/->
Figure FDA0004110184690000064
Representing control with identical numbering in each design sub-domain in a macroscopically periodic structureThe sensitivity of the target function of the re-average distribution of points;
(j) Updating a macroscopic periodic structure design variable according to an Optimization Criterion (OC), solving the total volume of the updated macroscopic periodic structure design domain according to the new relative density of the control points, setting a new interpolation point according to the total product difference before and after updating to judge whether iteration is ended, adopting the relative density of the updated control points and continuing iteration according to the OC method if not ending, and stopping calculating and outputting the relative density of the updated control points if the iteration is ended;
(k) Updating various microstructure design variables according to an optimization criterion OC method, solving the total volume of the updated microstructure design domain according to the new relative density of the control points, setting a new interpolation point according to the total product difference before and after updating to judge whether iteration is ended, adopting the relative density of the updated control points and continuing iteration according to the OC method if not ending, and stopping calculating and outputting the relative density of the updated control points if the iteration is ended;
(l) Calculating the relative density difference of each macro periodic structure control point in the input and output process in (j), solving the maximum relative density change value, comparing the maximum change value with the total loop iteration termination condition set in (a), judging whether the termination condition is met, if not, feeding back the relative density of the macro periodic structure design domain control points output in (j) to (c) for re-iteration, and if the iteration termination condition is met, iteratively terminating and outputting final macro periodic structure design domain and relative density vectors of various microstructure design domain control points;
and (m) outputting an optimal macroscopic periodic structure based on the isogeometric analysis and various microstructure topological structures and corresponding equivalent elastic matrixes.
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