CN116362079A - Multi-material structure topology optimization method based on novel interpolation model - Google Patents

Multi-material structure topology optimization method based on novel interpolation model Download PDF

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CN116362079A
CN116362079A CN202310252549.9A CN202310252549A CN116362079A CN 116362079 A CN116362079 A CN 116362079A CN 202310252549 A CN202310252549 A CN 202310252549A CN 116362079 A CN116362079 A CN 116362079A
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郭旭
杜宗亮
吕昭燕
刘畅
张维声
王鲁斌
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Ningbo Research Institute of Dalian University of Technology
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Abstract

The invention discloses a multi-material structure topology optimization method based on a novel interpolation model, which comprises the following steps: giving an optimization list of the multi-material structure topology optimization problem, obtaining a preset number of grid units, setting calculated boundary constraint and load conditions according to an actual solution problem, introducing a material attribute distinguishing parameter gamma for distinguishing material attributes in each component, generating topology description functions of all components, designing updating changes of the components in the domain, and outputting a final structure optimization configuration until convergence. The invention adopts the multi-material structure topology optimization method based on the novel interpolation model, if the optimized distribution of K different materials in a design domain is considered, only a group of unknown variables gamma describing the distribution category of the materials in the assembly is needed to be introduced K‑1 And associated with topology description functions of the componentBy the method, the material properties in the finite element grids can be described, analysis and calculation processes are simplified, and analysis and calculation efficiency is improved.

Description

Multi-material structure topology optimization method based on novel interpolation model
Technical Field
The invention relates to the technical field of structural optimization, in particular to a multi-material structural topology optimization method based on a novel interpolation model.
Background
The topology optimization is a design method for searching for optimal distribution of materials in a region under the condition of given boundary constraint and load in a designated design region, and the optimization method realizes structure optimization by changing the size, the number, the position, the shape and the like of holes of a structure so as to achieve the effect of changing the geometric shape and the topology of the structure at the same time.
In practical engineering applications, to meet different design requirements, the structure is generally composed of a plurality of heterogeneous materials, for example, in the industrial fields of aerospace, automobiles, ships, railways, electronic manufacturing and the like, the use of multi-material structures is very wide. The multi-material design is an important issue that accompanies the lightweight design of the structure. Compared with a structure filled with a single material, the multi-material structure has the characteristics of light weight, multifunction, better structural performance and the like. In addition, the rapid development of 3D printing technology has prompted the fabrication of multi-material structures, which has also prompted their further research and application, and therefore how to design more economical, reliable multi-material structures by topology optimization techniques is a popular study in the topology optimization field. At present, the topology optimization design of the multi-material structure mainly adopts an implicit topology optimization method based on a structural background finite element unit or node, such as multi-material combined structure topology optimization based on a variable density method and a level set method.
The multi-material structure topology optimization flow based on the variable density method is generally as follows: firstly, defining a design domain of a structure, further dispersing the structure into a finite element grid, adopting a mode of interpolation of elastic modulus of various materials to represent the material state of any point in the design domain, taking units in the design domain as an optimal design variable, and adopting a variable density method to carry out topological optimization design on the structure, so as to obtain the optimal results of overall optimal distribution of the materials in the design domain and reasonable configuration of various materials with different mechanical properties; then carrying out manual identification of materials on the preliminary optimization result obtained by the topology optimization technology, and manually extracting main force transmission paths of the structure, various material distribution forms and geometric characteristic parameters according to the physical material distribution result (usually less clear, fuzzy boundaries and weak units exist) obtained by optimization; then, reestablishing a multi-material structure model according to the identification geometric characteristic parameters, and carrying out a new round of shape and size parameter optimization to obtain an optimal shape and size optimization result; finally, the final multi-material structure optimization design result is obtained through the main two optimization processes.
The topology optimization method of the multi-material combined structure based on the level set method mainly comprises the following steps: firstly, dividing a design domain into a plurality of subregions filled with corresponding materials, then, representing the subregions occupied by different materials and junctions thereof by utilizing a plurality of level set functions established based on background finite element nodes, and optimizing a more reasonable structural topology through mutual evolution among the subregion junctions, wherein the obtained optimization result has no explicit geometric information of a structural basic structural unit, and also needs to carry out subsequent complex post-processing flows such as manual identification, model reconstruction and the like.
Meanwhile, it can be known that the MMC method is an explicit topology optimization method for solving the optimization problem by adopting a clear geometric mode in the field of structure topology optimization, and the mode for solving the problem of multi-material structure topology optimization by adopting the traditional MMC method is generally to place multiple layers of components respectively having different material properties in a design domain, and realize the topology optimization of the multi-material structure by means of movement, deformation, overlapping, mutual fusion and the like of the components respectively representing different materials. This implementation requires the additional introduction of more components as the material types increase, resulting in redundancy of design domain components, a multiple of design variables, and the problem arises essentially from complex material interpolation models.
In summary, the topology optimization of the conventional multi-material structure has the following problems:
1. conventional multi-material topology optimization methods are all based on an implicit framework in which for each type of material, a corresponding density field or level set function defined over the whole design domain must be introduced to describe its structural distribution, so this will inevitably introduce a large number of design variables, with dimensional disasters being particularly serious for three-dimensional problems; in the conventional method for solving the problem of optimized distribution of multiple materials in a structure by adopting a component method, components with corresponding layers are required to be additionally introduced in a design domain based on each material to distinguish the types of the materials, the more the types of the materials are, the more the number of the components is involved, the more design variables are increased, the calculation amount is large, and the calculation efficiency is greatly hindered.
2. The geometric description of the structure by the traditional multi-material topological optimization method depends on implicit pixels or finite element nodes, no explicit geometric information exists, the obtained optimization model cannot be directly connected with some commercial software (such as CAD systems), a large amount of post-processing work is needed to extract the structure boundary, and inconvenience is brought to subsequent design work.
3. The interpolation model adopted by the traditional multi-material topological optimization algorithm generally needs to introduce more additional functions describing different kinds of material densities or level sets along with the increase of material kinds, and has complex interpolation format and lower calculation efficiency.
Disclosure of Invention
The invention provides a multi-material structure topology optimization method based on a novel interpolation model, which solves the defects of the traditional implicit topology optimization method in the multi-material optimization distribution design problem, and greatly improves the design and calculation efficiency of the structure multi-material topology optimization problem, thereby obtaining the optimal optimization configuration.
In order to achieve the above purpose, the invention provides a multi-material structure topology optimization method based on a novel interpolation model, which comprises the following steps:
s1, determining an optimization target, design variables and constraint conditions of a multi-material structure topology optimization problem, and giving an optimization list of the multi-material structure topology optimization problem;
s2, extracting a design domain of a material from an object to be designed, setting material parameters and calculation factors in the design domain, and further performing finite element discretization on the design domain to obtain a preset number of grid cells;
s3, setting calculated boundary constraint and load conditions according to the actual solving problem;
s4, initializing geometric parameters of the components, introducing material attribute distinguishing parameters gamma for distinguishing material attributes in each component, and generating topology description functions of all components, so that an initial single-layer sandwich-shaped component for simultaneously describing multiple materials is placed in a design domain, the inner part of the component is set to be a strong material, the outer part of the component is set to be a weak material, and the non-component area coverage default is set to be a blank material;
s5, finite element analysis is carried out on the design domain, young modulus of a finite element grid unit is calculated according to the multi-material interpolation model and the generated topological description function of each component, so that a rigidity matrix of each grid unit is obtained, and numerical instability phenomenon in the calculation process is reduced by introducing a regularized Haiweide function;
s6, according to the calculated unit stiffness array, calculating the corresponding sensitivity of the objective function and the constraint function respectively, further carrying out optimization solution by utilizing an MMA optimization solver, judging whether the multi-material structure topology optimization problem is converged, if not, updating design variables of the multi-material structure topology optimization problem, namely the explicit geometric characteristic parameters and the material attribute distinguishing parameters gamma of the components, according to the solved sensitivity, further calculating the topology description function in the design domain again according to the optimized design variables, returning to the step S5 to carry out new optimization, and updating the structural configuration, thereby realizing the updating change of the components in the design domain until the final structural optimization configuration is outputted after convergence.
Preferably, the step S4 specifically includes the following steps:
s41, initializing and setting geometric parameters of the component
S411, adopting a movable deformable component as a basic optimization unit component, under the explicit topological optimization framework of the movable deformable component based on Euler description, only taking an initial single-layer sandwich-shaped single component with uniform width as an optimization basic component into consideration, and obtaining the geometric description of the two-dimensional structure of the single component;
s412, adopting topology description function phi i Explicitly describing each component:
Figure BDA0004128338940000041
wherein x is a material point in a design domain, an upper corner mark i is a positive integer greater than or equal to 1 and less than or equal to n, n is the number of components, D is a design domain of the whole structure, and Ω i A material domain internal to the component;
Figure BDA0004128338940000042
a material domain that is a boundary of the component;
wherein,,
Figure BDA0004128338940000043
wherein, (x ', y') is a coordinate in a local coordinate system perpendicular to the assembly centerline; p is a set positive integer; l (L) i Half the length of the assembly; t is t i Half the width of the assembly;
wherein,,
Figure BDA0004128338940000051
in (x) 0i ，y 0i ) Is the center coordinate of the component, namely the origin of the local coordinate system O ' -x ' -y '; θ i Is the inclination angle of the component, i.e. the rotation angle of the local coordinate system relative to the global coordinate system O-x-y;
s413, the initial setting of each component is a sandwich material component, namely a weak material wraps a strong material, and the specific judgment criteria are as follows:
topology description function phi when all components s Satisfy 0<φ s When gamma is less than or equal to gamma, the area in the component is made of weak materials; when phi is s Satisfy gamma<φ s When less than or equal to 1, the area in the component is made of strong material and 0<γ<1；
S42, introducing a material attribute distinguishing parameter gamma to describe the topology of all components:
Figure BDA0004128338940000052
in phi s (x)＝max(φ 1 (x),…,φ n (x))；Ω S A physical material domain that is a component;
Figure BDA0004128338940000057
a weak material domain that is a component; />
Figure BDA0004128338940000058
A strong material domain for the component;
the K-S function approximation max operation is adopted:
Figure BDA0004128338940000053
where λ is a set positive number and n is the number of components.
Preferably, the step S5 specifically includes the following steps:
carrying out finite element numerical analysis on the structure by using a bilinear four-node rectangular unit division grid, wherein the unit elastic modulus is expressed as follows:
Figure BDA0004128338940000054
wherein E is 1 Representing Young's modulus, E, of weak material 2 Representing the Young's modulus of a strong material;
Figure BDA0004128338940000055
the cell density of a weak material is expressed as follows:
Figure BDA0004128338940000061
Figure BDA0004128338940000062
the cell density, which is a strong material, is expressed as follows:
Figure BDA0004128338940000063
(8)
in the method, in the process of the invention,
Figure BDA0004128338940000064
the topology description function value of the whole structure at four nodes of the unit e is that h is an integer which is more than or equal to 1 and less than or equal to 4;
Figure BDA0004128338940000065
in the regularized form of the sea-wazier function, the expression is as follows:
Figure BDA0004128338940000066
in the formula, E is a regularization parameter which represents the width of the smooth transition of the Hexawegian function; alpha is an extremely small positive parameter with a numerical value approaching 0, and is used for simulating the elastic modulus of a blank material, and the stability of the stiffness matrix is ensured in an initial optimization stage.
Preferably, the multi-material structure topology optimization problem described in step S6 is: how to minimize the structural flexibility of various solid materials under fixed volume constraints, assuming that all the materials involved are linearly isotropic materials; the specific optimization steps are:
s61, under an MMC-based explicit topology optimization framework, the expression of the multi-material structure topology optimization problem is as follows:
Find d＝((d 1 ) T ,…,(d n ) T ) T ,γ
Min
Figure BDA0004128338940000067
S.t.
Figure BDA0004128338940000068
Figure BDA0004128338940000069
Figure BDA00041283389400000610
Figure BDA00041283389400000611
Figure BDA00041283389400000612
wherein d i ＝(x 0i ,y 0i ,L i ,t i ,θ i ) T I is 1 or more and n or less; f is an objective function; g β Beta is an integer greater than or equal to 1 and less than or equal to K, wherein K represents the number of material types; |d| is the volume of the design domain; v (V) β Is the volume of the beta-th material; v β Is the volume fraction corresponding to the beta-th material; f (f) β (x) And t represents the volumetric force and Neumann boundary Γ, respectively t Upper face force load;
Figure BDA0004128338940000071
is the fourth-order isotropic elastic tensor of the beta-th solid material,
Figure BDA0004128338940000072
Figure BDA0004128338940000073
Figure BDA0004128338940000074
and delta is the fourth and second order unit tensor, E β And v β Young's modulus and Poisson's ratio of the corresponding materials, respectively; />
Figure BDA0004128338940000075
Representing the design variable vector d i The feasible region to which the method belongs; />
Figure BDA0004128338940000076
Is Dirichlet boundary Γ u A specified displacement thereon; epsilon represents the second order linear strain tensor, u β Representation->
Figure BDA0004128338940000077
A displacement field above; v= (v) 1 ,…,v K ) T Is defined as +.>
Figure BDA0004128338940000078
Heuristic function on and satisfy->
Figure BDA0004128338940000079
Figure BDA00041283389400000710
And v is continuous at the material interface and at Γ u Upper v=0 }, -j->
Figure BDA00041283389400000711
Is a feasible region of v;
s62, specific design variables for the jth parameter associated with the ith component
Figure BDA00041283389400000712
And the sensitivity of γ is calculated as follows:
Figure BDA00041283389400000713
wherein NE is the total unit number of the design domain;
Figure BDA00041283389400000714
the node displacement vector corresponding to the e-th unit is transposed; u (u) e The node displacement vector corresponding to the e-th unit; />
Figure BDA00041283389400000715
Is a stiffness matrix of units composed of solid materials;
Figure BDA00041283389400000716
in the method, in the process of the invention,
Figure BDA00041283389400000717
is a regularized halweseide function;
Figure BDA00041283389400000718
in the formula g 1 Is a volume constraint function of weak materials;
Figure BDA00041283389400000719
occupy the volume of weak material for the e-th cell in the design domain;
Figure BDA0004128338940000081
Figure BDA0004128338940000082
in the formula g 2 Is a volume constraint function of a strong material;
Figure BDA0004128338940000083
occupy the volume of strong material for the e-th cell in the design domain;
Figure BDA0004128338940000084
and (3) finishing to obtain:
Figure BDA0004128338940000085
wherein h is an integer of 1 or more and 4 or less, representing the numbers of the four nodes of the unit,
Figure BDA0004128338940000086
the function values are described for the topology of the overall structure at the four nodes of element e; />
Figure BDA0004128338940000087
Describing function values for the topology of the individual components at the four nodes of element e; />
Figure BDA0004128338940000088
The topology description function values of the kth component in all the components in the design domain at four nodes of the unit e are shown, k is an integer which is more than or equal to 1 and less than or equal to n, and n is the total number of the components;
Figure BDA0004128338940000089
preferably, the update change in step S6 includes at least one of movement, deformation, telescoping, rotation, overlapping, mutual fusion, and any combination thereof.
The invention has the following beneficial effects:
1. the geometric topology of the structure is described by introducing the component with explicit geometric information, geometric parameters such as the central position coordinate, the length, the width, the rotation angle and the like of the component are used as optimized design variables, the size of the multi-material structure can be explicitly controlled, the movement deformation and the like of the component in the optimization process are not dependent on a background finite element grid, the material category is distinguished only by the design variables and is irrelevant to the density of the background grid, the evolution of a solid material region and a boundary is independently carried out, the number of the design variables is greatly reduced, and the calculation efficiency is improved;
2. only a single sandwich component is used for describing the distribution of various materials in a design domain, so that the number of components in the design domain is reduced in a multiplied manner, and the iterative rate of a computer is increased;
3. the optimized result contains the information of the explicit geometric dimension, shape parameters and the like of the component, and can be directly combined with most of the existing CAD/CAE systems to carry out the post-processing process of geometric reconstruction, thereby facilitating the analysis and the solution of the subsequent engineering problems.
The technical scheme of the invention is further described in detail through the drawings and the embodiments.
Drawings
FIG. 1 is a flow chart of a method of topology optimization of a multi-material structure based on a novel interpolation model;
FIG. 2 is a geometric description of the components of a novel interpolation model-based multi-material structure topology optimization method of the present invention;
FIG. 3 is a material distribution diagram of an initial component of a multi-material structure topology optimization method based on a novel interpolation model of the present invention;
FIG. 4 is a diagram of a geometric topology model of a multi-material structure topology optimization method based on a novel interpolation model of the present invention;
FIG. 5 is a diagram of a multi-material interpolation model of a multi-material structure topology optimization method based on a novel interpolation model of the present invention;
FIG. 6 is a structural configuration diagram of a two-bar structure before optimization according to an embodiment;
FIG. 7 is an initial layout of a multi-material employing the present invention in an embodiment;
FIG. 8 is a graph of the results of an optimization using the present invention in an example;
FIG. 9 is a graph showing convergence of the objective function and the volume fraction obtained using the present invention in the example;
FIG. 10 is an initial layout of a multi-material using a conventional MMC in an embodiment;
FIG. 11 is a graph of the optimized results for using conventional MMC multi-materials in the examples;
FIG. 12 is a graph showing convergence of the objective function and the volume fraction obtained by using the conventional MMC multi-material topology optimization method in the example.
Detailed Description
The present invention will be further described with reference to the accompanying drawings, and it should be noted that, while the present embodiment provides a detailed implementation and a specific operation process on the premise of the present technical solution, the protection scope of the present invention is not limited to the present embodiment.
As shown in fig. 1, a multi-material structure topology optimization method based on a novel interpolation model includes the following steps:
s1, determining an optimization target, design variables and constraint conditions of a multi-material structure topology optimization problem, and giving an optimization list of the multi-material structure topology optimization problem;
s2, extracting a design domain of a material from an object to be designed, setting material parameters and calculation factors in the design domain, and further performing finite element discretization on the design domain to obtain a preset number of grid cells;
s3, setting calculated boundary constraint and load conditions according to the actual solving problem;
s4, initializing geometric parameters of the components, introducing material attribute distinguishing parameters gamma for distinguishing material attributes in each component, and generating topology description functions of all components, so that an initial single-layer sandwich-shaped component for simultaneously describing multiple materials is placed in a design domain, the inner part of the component is set to be a strong material, the outer part of the component is set to be a weak material, and the non-component area coverage default is set to be a blank material;
preferably, the step S4 specifically includes the following steps:
s41, initializing and setting geometric parameters of the component
S411, adopting a movable deformable component as a basic optimization unit component, under the explicit topological optimization framework of the movable deformable component based on Euler description, only taking an initial single-layer sandwich-shaped single component with uniform width as an optimization basic component into consideration, and obtaining the geometric description of the two-dimensional structure of the single component;
s412, adopting topology description function phi i Explicitly describing each component:
Figure BDA0004128338940000101
wherein x is a material point in a design domain, an upper corner mark i is a positive integer greater than or equal to 1 and less than or equal to n, n is the number of components, D is a design domain of the whole structure, and Ω i A material domain internal to the component;
Figure BDA0004128338940000102
a material domain that is a boundary of the component;
wherein,,
Figure BDA0004128338940000111
wherein, (x ', y') is a coordinate in a local coordinate system perpendicular to the assembly centerline; p is a set positive integer, in this embodiment, p=6; l (L) i Half the length of the assembly; t is t i Half the width of the assembly;
wherein,,
Figure BDA0004128338940000112
in (x) 0i ，y 0i ) Is the center coordinate of the component, namely the origin of the local coordinate system O ' -x ' -y '; θ i For inclination of the assembly, i.e. local coordinate system relative to global seatThe rotation angle of the standard system O-x-y;
s413, the initial setting of each component is a sandwich material component, namely a weak material wraps a strong material, and the specific judgment criteria are as follows:
topology description function phi when all components s Satisfy 0<φ s When gamma is less than or equal to gamma, the area in the component is made of weak materials; when phi is s Satisfy gamma<φ s When less than or equal to 1, the area in the component is made of strong material and 0<γ<1；
S42, introducing a material attribute distinguishing parameter gamma to describe the topology of all components:
Figure BDA0004128338940000113
in phi s (x)＝max(φ 1 (x),…,φ n (x) -representing a maximum of n number of component material fields placed within the design field; omega shape S A physical material domain that is a component;
Figure BDA0004128338940000114
a weak material domain that is a component; />
Figure BDA0004128338940000115
A strong material domain for the component;
the K-S function approximation max operation is adopted:
Figure BDA0004128338940000116
where λ is a positive number, and in this embodiment, λ is 80.
S5, finite element analysis is carried out on a design domain, young modulus of a finite element grid unit is calculated according to a multi-material interpolation model and a generated topological description function of each component, so that a rigidity matrix of each grid unit is obtained, the types of materials in the component can be judged according to the multi-material interpolation model, a structural configuration and a material distribution form are drawn, and a numerical value instability phenomenon in a calculation process is reduced by introducing a regularized Haweisuide function;
after finite element discretization of the design domain, the elastic modulus of the cells within the design domain is described by the following form:
preferably, the step S5 specifically includes the following steps:
carrying out finite element numerical analysis on the structure by using a bilinear four-node rectangular unit division grid, wherein the unit elastic modulus is expressed as follows:
Figure BDA0004128338940000121
wherein E is 1 Representing Young's modulus, E, of weak material 2 Representing the Young's modulus of a strong material;
Figure BDA0004128338940000122
the cell density of a weak material is expressed as follows:
Figure BDA0004128338940000123
Figure BDA0004128338940000124
the cell density, which is a strong material, is expressed as follows:
Figure BDA0004128338940000125
in the method, in the process of the invention,
Figure BDA0004128338940000126
the topology description function value of the whole structure at four nodes of the unit e is that h is an integer which is more than or equal to 1 and less than or equal to 4;
Figure BDA0004128338940000127
in the regularized form of the sea-wazier function, the expression is as follows:
Figure BDA0004128338940000128
in the formula, E is a regularization parameter which represents the width of the smooth transition of the Hexawegian function; alpha is an extremely small positive parameter with a numerical value approaching 0, and is used for simulating the elastic modulus of a blank material, and the stability of the stiffness matrix is ensured in an initial optimization stage.
S6, according to the calculated unit stiffness array, calculating the corresponding sensitivity of the objective function and the corresponding sensitivity of the constraint function respectively (in the embodiment, calculating the corresponding sensitivity of the objective function and the corresponding sensitivity of the constraint function by using a chained rule), further optimizing and solving by using an MMA optimization solver, judging whether the multi-material structure topology optimization problem reaches convergence, if not, updating the design variables of the multi-material structure topology optimization problem, namely the explicit geometric characteristic parameters and the material attribute distinguishing parameters gamma of the components according to the solved sensitivity, further calculating the topology description function in the design domain again according to the optimized design variables, returning to the step S5 for new optimization, and updating the structural configuration, thereby realizing the updating change of the components in the design domain until the final structure optimization configuration is converged and output.
As can be seen from fig. 4, the solid material domain Ω of the multi-material structure S The geometric topology model and the material distribution form of the system can be determined by the explicit geometric characteristic parameters and the material attribute distinguishing parameters gamma of all components, and the purposes of optimizing the structural shape and the topology can be achieved only by optimizing the parameters.
Preferably, the multi-material structure topology optimization problem described in step S6 is: how to minimize the structural flexibility of various solid materials under fixed volume constraints, assuming that all the materials involved are linearly isotropic materials; the specific optimization steps are:
s61, under an MMC-based explicit topology optimization framework, the expression of the multi-material structure topology optimization problem is as follows:
Find d＝((d 1 ) T ,…,(d n ) T ) T ,γ
Min
Figure BDA0004128338940000131
S.t.
Figure BDA0004128338940000132
Figure BDA0004128338940000133
Figure BDA0004128338940000134
Figure BDA0004128338940000135
Figure BDA0004128338940000136
wherein d i ＝(x 0i ,y 0i ,L i ,t i ,θ i ) T I is 1 or more and n or less; f is an objective function; g β Beta is an integer greater than or equal to 1 and less than or equal to K, wherein K represents the number of material types; |d| is the volume of the design domain; v (V) β Is the volume of the beta-th material; v β Is the volume fraction corresponding to the beta-th material; f (f) β (x) And t represents the volumetric force and Neumann boundary Γ, respectively t Upper face force load;
Figure BDA0004128338940000141
is the fourth-order isotropic elastic tensor of the beta-th solid material,
Figure BDA0004128338940000142
Figure BDA0004128338940000143
Figure BDA0004128338940000144
and delta is the fourth and second order unit tensor, E β And v β Young's modulus and Poisson's ratio of the corresponding materials, respectively; />
Figure BDA0004128338940000145
Representing the design variable vector d i The feasible region to which the method belongs; />
Figure BDA0004128338940000146
Is Dirichlet boundary Γ u A specified displacement thereon; epsilon represents the second order linear strain tensor, u β Representation->
Figure BDA0004128338940000147
A displacement field above; v= (v) 1 ,…,v K ) T Is defined as +.>
Figure BDA0004128338940000148
Heuristic function on and satisfy->
Figure BDA0004128338940000149
Figure BDA00041283389400001410
And v is continuous at the material interface and at Γ u Upper v=0 }, -j->
Figure BDA00041283389400001411
Is a feasible region of v;
s62, specific design variables for the jth parameter associated with the ith component
Figure BDA00041283389400001412
And the sensitivity of γ is calculated as follows:
Figure BDA00041283389400001413
wherein NE is the total unit number of the design domain;
Figure BDA00041283389400001414
the node displacement vector corresponding to the e-th unit is transposed; u (u) e The node displacement vector corresponding to the e-th unit; />
Figure BDA00041283389400001415
Is a stiffness matrix of units composed of solid materials;
Figure BDA00041283389400001416
in the method, in the process of the invention,
Figure BDA00041283389400001417
is a regularized halweseide function;
Figure BDA00041283389400001418
in the formula g 1 A volume constraint function for a first solid material (weak material);
Figure BDA00041283389400001419
occupy the volume of the 1 st solid material for the e-th cell in the design domain;
Figure BDA00041283389400001420
Figure BDA0004128338940000151
in the formula g 2 Volume constraint function for the 2 nd solid material (strong material);
Figure BDA0004128338940000152
occupy the volume of the 2 nd solid material for the e-th cell in the design domain;
Figure BDA0004128338940000153
and (3) finishing to obtain:
Figure BDA0004128338940000154
wherein h is an integer of 1 or more and 4 or less, representing the numbers of the four nodes of the unit,
Figure BDA0004128338940000155
the function values are described for the topology of the overall structure at the four nodes of element e; />
Figure BDA0004128338940000156
Describing function values for the topology of the individual components at the four nodes of element e; />
Figure BDA0004128338940000157
The topology description function values of the kth component in all the components in the design domain at four nodes of the unit e are shown, k is an integer which is more than or equal to 1 and less than or equal to n, and n is the total number of the components;
Figure BDA0004128338940000158
preferably, the update change in step S6 includes at least one of movement, deformation, telescoping, rotation, overlapping, mutual fusion, and any combination thereof.
As shown in fig. 5, the interpolation format is adopted in combination with the MMC topology optimization theory, and only a group of unknown variables gamma describing the material distribution category in the component need to be introduced K-1 And is associated with the topology description function of the component, so that the finite description can be performedThe material property in the cell grid is compared with the prior material interpolation model, the method does not need to rely on the cell density which is conventionally described whether the cell grid is occupied by solid materials or not, and sandwich components can be designed to describe the distribution of the materials, and a layer of components are not required to be added in a design domain correspondingly for each material introduction, so that the efficiency of calculation analysis and calculation processes is greatly simplified and improved.
Examples
According to the method, the classical two-pole structure shown in fig. 6 is subjected to multi-material topological optimization design, the volume fractions of given strong and weak materials are 0.15, a rectangular area with the structural design domain range of 1m multiplied by 2m is adopted, fixed constraint is applied to the left end of the structure, concentrated load with the size of 1N is applied to the middle part of the right end of the structure, the elastic modulus E of the first solid material is set by taking the minimum flexibility, namely the maximum rigidity of the structure as an optimization target 1 Is 1Pa, poisson ratio v 1 Modulus of elasticity E of 0.3 of the second solid material 2 5Pa, poisson ratio v 2 The elastic modulus alpha of the blank material is set to be 1e-9, and the sea-wampee function regularization factor epsilon is 0.01.
The rectangular design domain is divided into 80X 160 grids, the unit type is a bilinear four-node rectangular unit, the unit side length is 12.5mm, in order to facilitate the interpolation of the elastic modulus of a plurality of subsequent materials, a unit stiffness matrix corresponding to the unit elastic modulus is calculated first, an overall stiffness matrix is synthesized, a unit node number matrix and a node coordinate matrix are generated, fixed constraint is set according to the node number and the node coordinate, and corresponding load is applied to a right loading point.
And arranging 2X 4 groups of cross sandwich assemblies in a rectangular design domain, wherein every two cross sandwich assemblies share an initial center coordinate, 16 assemblies are arranged in total, the geometric parameters of the assemblies are all preset with determined initial values, and the initial values of material property distinguishing parameters gamma in the assemblies are set to generate initial vectors of optimization problem design variables.
Setting upper and lower bounds on design variables, and solving topology description function phi of component according to design variables s The distribution of components in the current structure is drawn, and multi-material is utilizedThe interpolation model calculates the unit density corresponding to each solid material
Figure BDA0004128338940000161
And->
Figure BDA0004128338940000162
And generating an overall stiffness matrix K in the design domain, and solving to obtain an overall displacement vector u in the design domain according to f=Ku, wherein f is an external force vector applied to the design domain, namely a concentrated load with the size of 1N applied to the midpoint of the right end of the structure, so that the flexibility and the volume fraction of the structure are calculated.
And calculating the sensitivity of the objective function of the structure and the sensitivity of the volume constraint function corresponding to the two entity materials, optimizing the design variables by adopting an MMA optimization solver, judging whether convergence conditions are reached, if not, updating the design variables, returning to recalculate the topology description function, and updating the distribution and configuration of components in the design domain until the final convergence to obtain an optimal optimization result.
The initial setting of the method is to place a layer of sandwich components with weak materials wrapping strong materials in a design domain, the objective function f takes 2.521, and the traditional method is to place upper and lower layers of solid components with different materials in the design domain, the objective function f takes 2.625, and the number of components of the method is reduced exponentially as the plurality of materials of the method are only represented by one layer of components, and the number of design variables is reduced correspondingly.
Meanwhile, as shown in fig. 9 and fig. 12, compared with the prior topological optimization, the optimization result has no problems of gray level units, checkerboards and the like, the boundary is smoother, the structural topology and the material distribution are more reasonable, and the objective function reaches a better calculation result.
Therefore, the invention adopts the multi-material structure topology optimization method based on the novel interpolation model, and only needs to introduce a group of description componentsUnknown variable gamma of internal material distribution class K-1 And the material properties in the finite element mesh can be described by being related to the topology description function of the component, so that the analysis and calculation processes are simplified, and the analysis and calculation efficiency is improved.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solution of the present invention and not for limiting it, and although the present invention has been described in detail with reference to the preferred embodiments, it will be understood by those skilled in the art that: the technical scheme of the invention can be modified or replaced by the same, and the modified technical scheme cannot deviate from the spirit and scope of the technical scheme of the invention.

Claims (5)

1. A multi-material structure topology optimization method based on a novel interpolation model is characterized by comprising the following steps of: the method comprises the following steps:
s1, determining an optimization target, design variables and constraint conditions of a multi-material structure topology optimization problem, and giving an optimization list of the multi-material structure topology optimization problem;
s2, extracting a design domain of a material from an object to be designed, setting material parameters and calculation factors in the design domain, and further performing finite element discretization on the design domain to obtain a preset number of grid cells;
s3, setting calculated boundary constraint and load conditions according to the actual solving problem;
s4, initializing geometric parameters of the components, introducing material attribute distinguishing parameters gamma for distinguishing material attributes in each component, and generating topology description functions of all components, so that an initial single-layer sandwich-shaped component for simultaneously describing multiple materials is placed in a design domain, the inner part of the component is set to be a strong material, the outer part of the component is set to be a weak material, and the non-component area coverage default is set to be a blank material;
s5, finite element analysis is carried out on the design domain, young modulus of a finite element grid unit is calculated according to the multi-material interpolation model and the generated topological description function of each component, so that a rigidity matrix of each grid unit is obtained, and numerical instability phenomenon in the calculation process is reduced by introducing a regularized Haiweide function;
s6, according to the calculated unit stiffness array, calculating the corresponding sensitivity of the objective function and the constraint function respectively, further carrying out optimization solution by utilizing an MMA optimization solver, judging whether the multi-material structure topology optimization problem is converged, if not, updating design variables of the multi-material structure topology optimization problem, namely the explicit geometric characteristic parameters and the material attribute distinguishing parameters gamma of the components, according to the solved sensitivity, further calculating the topology description function in the design domain again according to the optimized design variables, returning to the step S5 to carry out new optimization, and updating the structural configuration, thereby realizing the updating change of the components in the design domain until the final structural optimization configuration is outputted after convergence.
2. The multi-material structure topology optimization method based on the novel interpolation model according to claim 1, wherein the method comprises the following steps: the step S4 specifically includes the following steps:
s41, initializing and setting geometric parameters of the component
S411, adopting a movable deformable component as a basic optimization unit component, under the explicit topological optimization framework of the movable deformable component based on Euler description, only taking an initial single-layer sandwich-shaped single component with uniform width as an optimization basic component into consideration, and obtaining the geometric description of the two-dimensional structure of the single component;
s412, adopting topology description function phi i Explicitly describing each component:
Figure FDA0004128338930000021
wherein x is a material point in a design domain, an upper corner mark i is a positive integer greater than or equal to 1 and less than or equal to n, n is the number of components, D is a design domain of the whole structure, and Ω i A material domain internal to the component;
Figure FDA0004128338930000024
a material domain that is a boundary of the component;
wherein,,
Figure FDA0004128338930000022
wherein, (x ', y') is a coordinate in a local coordinate system perpendicular to the assembly centerline; p is a set positive integer; l (L) i Half the length of the assembly; t is t i Half the width of the assembly;
wherein,,
Figure FDA0004128338930000023
in (x) 0i ，y 0i ) Is the center coordinate of the component, namely the origin of the local coordinate system O ' -x ' -y '; θ i Is the inclination angle of the component, i.e. the rotation angle of the local coordinate system relative to the global coordinate system O-x-y;
s413, the initial setting of each component is a sandwich material component, namely a weak material wraps a strong material, and the specific judgment criteria are as follows:
topology description function phi when all components s Satisfy 0<φ s When gamma is less than or equal to gamma, the area in the component is made of weak materials; when phi is s Satisfy gamma<φ s When less than or equal to 1, the area in the component is made of strong material and 0<γ<1；
S42, introducing a material attribute distinguishing parameter gamma to describe the topology of all components:
Figure FDA0004128338930000031
in phi s (x)＝max(φ 1 (x),…,φ n (x))；Ω S A physical material domain that is a component;
Figure FDA00041283389300000311
a weak material domain that is a component; />
Figure FDA00041283389300000312
A strong material domain for the component;
the K-S function approximation max operation is adopted:
Figure FDA0004128338930000032
where λ is a set positive number and n is the number of components.
3. The multi-material structure topology optimization method based on the novel interpolation model according to claim 2, wherein the method comprises the following steps: the step S5 specifically comprises the following steps:
carrying out finite element numerical analysis on the structure by using a bilinear four-node rectangular unit division grid, wherein the unit elastic modulus is expressed as follows:
Figure FDA0004128338930000033
wherein E is 1 Representing Young's modulus, E, of weak material 2 Representing the Young's modulus of a strong material;
Figure FDA0004128338930000034
the cell density of a weak material is expressed as follows:
Figure FDA0004128338930000035
Figure FDA0004128338930000036
the cell density, which is a strong material, is expressed as follows:
Figure FDA0004128338930000037
in the method, in the process of the invention,
Figure FDA0004128338930000038
the topology description function value of the whole structure at four nodes of the unit e is that h is an integer which is more than or equal to 1 and less than or equal to 4;
Figure FDA0004128338930000039
in the regularized form of the sea-wazier function, the expression is as follows:
Figure FDA00041283389300000310
in the formula, E is a regularization parameter which represents the width of the smooth transition of the Hexawegian function; alpha is an extremely small positive parameter with a numerical value approaching 0, and is used for simulating the elastic modulus of a blank material, and the stability of the stiffness matrix is ensured in an initial optimization stage.
4. The multi-material structure topology optimization method based on the novel interpolation model according to claim 1, wherein the method comprises the following steps: the topology optimization problem of the multi-material structure described in step S6 is as follows: how to minimize the structural flexibility of various solid materials under fixed volume constraints, assuming that all the materials involved are linearly isotropic materials; the specific optimization steps are:
s61, under an MMC-based explicit topology optimization framework, the expression of the multi-material structure topology optimization problem is as follows:
Find d＝((d 1 ) T ,…,(d n ) T ) T ,γ
Figure FDA0004128338930000041
Figure FDA0004128338930000042
Figure FDA0004128338930000043
Figure FDA0004128338930000044
Figure FDA0004128338930000045
wherein d i ＝(x 0i ,y 0i ,L i ,t i ,θ i ) T I is 1 or more and n or less; f is an objective function; g β Beta is an integer greater than or equal to 1 and less than or equal to K, wherein K represents the number of material types; |d| is the volume of the design domain; v (V) β Is the volume of the beta-th material; v β Is the volume fraction corresponding to the beta-th material; f (f) β (x) And t represents the volumetric force and Neumann boundary Γ, respectively t Upper face force load;
Figure FDA0004128338930000046
fourth order isotropic elastic tensor, beta-th order solid material->
Figure FDA0004128338930000047
Figure FDA0004128338930000048
Figure FDA0004128338930000049
And delta is respectively four-order and two-orderTensor of order unit, E β And v β Young's modulus and Poisson's ratio of the corresponding materials, respectively; />
Figure FDA00041283389300000410
Representing the design variable vector d i The feasible region to which the method belongs; />
Figure FDA00041283389300000411
Is Dirichlet boundary Γ u A specified displacement thereon; epsilon represents the second order linear strain tensor, u β Representing omega Sβ A displacement field above; v= (v) 1 ,…,v K ) T Is defined as +.>
Figure FDA00041283389300000412
Heuristic function on and satisfy->
Figure FDA00041283389300000413
Figure FDA0004128338930000051
And v is continuous at the material interface and at Γ u Upper v=0 }, -j->
Figure FDA0004128338930000052
Is a feasible region of v;
s62, specific design variables for the jth parameter associated with the ith component
Figure FDA0004128338930000053
And the sensitivity of γ is calculated as follows:
Figure FDA0004128338930000054
wherein NE is the total unit number of the design domain;
Figure FDA0004128338930000055
the node displacement vector corresponding to the e-th unit is transposed; u (u) e The node displacement vector corresponding to the e-th unit; />
Figure FDA0004128338930000056
Is a stiffness matrix of units composed of solid materials;
Figure FDA0004128338930000057
in the method, in the process of the invention,
Figure FDA0004128338930000058
is a regularized halweseide function;
Figure FDA0004128338930000059
in the formula g 1 Is a volume constraint function of weak materials;
Figure FDA00041283389300000510
occupy the volume of weak material for the e-th cell in the design domain;
Figure FDA00041283389300000511
Figure FDA00041283389300000512
in the formula g 2 Is a volume constraint function of a strong material;
Figure FDA00041283389300000513
occupy the volume of strong material for the e-th cell in the design domain;
Figure FDA00041283389300000514
and (3) finishing to obtain:
Figure FDA00041283389300000515
wherein h is an integer of 1 or more and 4 or less, representing the numbers of the four nodes of the unit,
Figure FDA0004128338930000061
the function values are described for the topology of the overall structure at the four nodes of element e; />
Figure FDA0004128338930000062
Describing function values for the topology of the individual components at the four nodes of element e; />
Figure FDA0004128338930000063
The topology description function values of the kth component in all the components in the design domain at four nodes of the unit e are shown, k is an integer which is more than or equal to 1 and less than or equal to n, and n is the total number of the components;
Figure FDA0004128338930000064
5. the multi-material structure topology optimization method based on the novel interpolation model according to claim 1, wherein the method comprises the following steps: the update change in step S6 includes at least one of movement, deformation, telescoping, rotation, overlapping, mutual fusion, and any combination thereof.
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