CN115203997A - Dot matrix-entity composite structure topology optimization method based on multivariate design - Google Patents

Dot matrix-entity composite structure topology optimization method based on multivariate design Download PDF

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CN115203997A
CN115203997A CN202210602022.XA CN202210602022A CN115203997A CN 115203997 A CN115203997 A CN 115203997A CN 202210602022 A CN202210602022 A CN 202210602022A CN 115203997 A CN115203997 A CN 115203997A
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刘继凯
张乘虎
李磊
黄嘉奇
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SUZHOU RESEARCH INSTITUTE SHANDONG UNIVERSITY
Shandong University
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Abstract

The invention discloses a dot matrix-entity composite structure topological optimization method based on multivariate design, which solves the problem that the dot matrix-entity composite structure optimization of multivariate design can not be realized in the prior art, has the beneficial effect of expanding the design space of a dot matrix material filling structure, and has the following specific scheme: a dot matrix-entity composite structure topological optimization method based on multivariate design comprises the steps of establishing a dot matrix material structure configuration containing a plurality of design variables; obtaining macroscopic equivalent physical properties of the lattice materials corresponding to different design variables; setting a design definition domain of multi-design variables for the dot matrix-entity material, and establishing a dot matrix-entity multi-material interpolation model; constructing a multivariate lattice-entity composite structure topological optimization mathematical model; calculating the target function value corresponding to the current design variable and the sensitivity information of the current design variable to the target function and the constraint function; and judging the convergence of the optimization iteration according to the updated design variable.

Description

Dot matrix-entity composite structure topology optimization method based on multivariate design
Technical Field
The invention relates to the field of structure optimization, in particular to a dot matrix-entity composite structure topology optimization method based on multivariate design.
Background
The statements in this section merely provide background information related to the present disclosure and may not constitute prior art.
In recent years, the rapid development of additive manufacturing technology has realized the processing and manufacturing of structural members with complex geometric shapes, and more innovative designs are converted from concepts to actual products, especially lattice material filling designs. The lattice material is a high-performance lightweight material containing a porous microstructure, has multifunctional attributes such as heat dissipation, shock absorption and sound insulation, and has wide application prospects in the fields of daily commodities, architectural decoration, automobile industry, aerospace and the like.
The structural topology optimization is based on given boundary conditions and constraints, and the optimal distribution of materials in a structural design domain is searched, so that the structural target performance is optimal. The design of a high-performance non-uniform lattice filling structure by using a topological optimization method has become a hotspot in the field of structural optimization design. Compared with the complete lattice filling design, the lattice-solid composite structure can realize better mechanical property. In recent years, some progress has been made in designing lattice-entity composite structures based on topological optimization methods. However, the inventor finds that the existing method cannot realize the optimization of the lattice-solid composite structure of the multivariate design and cannot realize the cooperative optimization of the lattice-solid material of the multivariate design under the condition of restricting the variation range of the relative density of the lattice material.
Disclosure of Invention
Aiming at the defects in the prior art, the invention aims to provide a dot matrix-solid composite structure topological optimization method based on multivariate design, which realizes the optimization of the dot matrix-solid composite structure of the multivariate design and realizes the cooperative optimization of the dot matrix-solid material under the condition of restricting the variation range of the relative density of the dot matrix material.
In order to achieve the purpose, the invention is realized by the following technical scheme:
a dot matrix-entity composite structure topological optimization method based on multivariate design comprises the following steps:
establishing a lattice material structure configuration containing a plurality of design variables;
calculating a plurality of sample point data, namely obtaining the macroscopic equivalent physical attributes of the lattice materials corresponding to different design variables;
based on a plurality of sample point data, establishing an interpolation model for mapping a mathematical relationship between design variables in the dot matrix material and physical attributes of the dot matrix material;
setting design definition domains of multiple design variables for the dot matrix-solid material, including a dot matrix material definition domain and a solid material definition domain, and establishing a dot matrix-solid multi-material interpolation model according to an interpolation model for mapping mathematical relations between the design variables in the dot matrix material and physical attributes of the dot matrix material;
constructing a topological optimization mathematical model of a multivariable lattice-entity composite structure;
calculating a target function value corresponding to a current design variable and sensitivity information of the current design variable to a target function and a constraint function based on a dot matrix-entity multi-material interpolation model and a multivariable dot matrix-entity composite structure topological optimization mathematical model, and updating the design variable;
and judging the convergence of the optimization iteration according to the updated design variable, and finishing the optimization if the iteration is converged.
The topology optimization method can effectively improve the mechanical property of the dot matrix-solid composite structure based on the multivariate design, and establishes a dot matrix-solid multi-material interpolation model according to the dot matrix material definition domain and the solid material definition domain to realize the collaborative optimization of the dot matrix-solid composite structure of the multivariate design.
According to the dot matrix-entity composite structure topology optimization method based on multivariate design, the convergence of optimization iteration is judged according to the updated design variables, if the iteration converges, the optimization is completed, otherwise, the next step is carried out;
judging whether the current design variable meets the updating condition of the material definition domain or not according to the updated design variable, if so, updating the material definition domain, otherwise, not updating; and the objective function value corresponding to the current design variable and the sensitivity information of the current design variable to the objective function and the constraint function are calculated again.
The lattice-solid composite structure topology optimization method based on multivariate design is as described above, the multiple design variables refer to definition parameters of the lattice material, and the definition parameters of the lattice material comprise width parameters or length parameters or angle parameters or composition coefficient parameters of the lattice material structure configuration.
In establishing the lattice material structure configuration, the variation range of the design variables is given.
The method for optimizing the topology of the lattice-entity composite structure based on the multivariate design as described above, the macroscopic equivalent physical properties of the lattice material corresponding to the different design variables include the relative density of the lattice material and the equivalent elastic matrix of the lattice material.
In the method for optimizing the topology of the lattice-entity composite structure based on the multivariate design, the fitting relation of the interpolation model is determined according to the fitting constant calculated based on the sample point data and the least square method and the defining parameters of the lattice material.
The lattice material definition domain is that a plurality of design variables are between the minimum value of the definition parameters of the predefined lattice material and the maximum value of the definition parameters of the predefined lattice material.
In the method for optimizing the topology of the lattice-entity composite structure based on the multivariate design, the entity material definition domain is that a plurality of design variables are all larger than the maximum value of the definition parameters of the predefined lattice material, at this time, the plurality of design variables are cooperatively changed, and the cooperative change range of the plurality of design variables is between the maximum value of the definition parameters of the lattice material and the maximum value of the change range of the design variables.
The dot matrix-entity multi-material interpolation model is obtained based on the dot matrix material definition domain, the entity material definition domain, the design variables and the interpolation model for mapping the mathematical relationship between the design variables and the physical properties of the dot matrix material in the dot matrix material.
After the multivariate design-based lattice-entity composite structure topological optimization mathematical model is constructed, the topological optimization structure parameters are defined, and the optimization iteration initial design is set.
The dot matrix-entity composite structure topology optimization method based on the multivariate design is characterized in that based on a dot matrix-entity multi-material interpolation model and a multivariate dot matrix-entity composite structure topology optimization mathematical model, material properties corresponding to design variables at the moment are calculated, finite element analysis is carried out, structure displacement is calculated, and then a target function value corresponding to the current design variables is calculated;
and updating the design variables by using a moving progression method.
The beneficial effects of the invention are as follows:
1) The invention can effectively improve the mechanical property of the lattice material filling structure and greatly expand the design space of the lattice material filling structure by providing a topological optimization method and based on the lattice-entity composite structure design.
2) The invention establishes a dot matrix-entity multi-material interpolation model according to the dot matrix material definition domain and the entity material definition domain based on the multivariate design through the provision of a topological optimization method, realizes the cooperative optimization of the dot matrix-entity composite structure of the multivariate design, and furthest exerts the design potential of the dot matrix-entity composite structure; the method has wide applicability to the structural optimization problem of compounding the dot matrix materials and the entities in various forms.
3) The lattice-entity multi-material interpolation model is set based on the lattice material definition domain and the entity material definition domain, so that the relative density change range of the lattice material can be effectively controlled in the optimization process; and the distribution of the lattice material and the solid material is in accordance with the working condition, so that the lattice material has excellent mechanical properties and excellent multifunctional properties of the lattice material.
4) The method establishes an interpolation model for mapping the mathematical relationship between the design variables and the physical properties of the dot matrix material based on a plurality of sample point data, can efficiently predict the material properties corresponding to different design variables, and greatly saves the calculation time.
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The accompanying drawings, which are incorporated in and constitute a part of this specification, are included to provide a further understanding of the invention, and are incorporated in and constitute a part of this specification, illustrate exemplary embodiments of the invention and together with the description serve to explain the invention and not to limit the invention.
FIG. 1 is a flow chart of the method for optimizing the topology of a lattice-solid composite structure based on multivariate design according to the present invention;
FIG. 2 is a schematic illustration of design variables in a single structure of the lattice material of the present invention;
FIG. 3 is a schematic diagram of a cantilever structure to be optimized and boundary conditions according to the present invention;
FIG. 4 is a diagram of the distribution evolution process of design variables during the cantilever topology optimization process of the present invention;
figure 5 is a structural detail view of the cantilever topology optimization result of the present invention.
In the figure: the spacing or size between each other is exaggerated to show the location of the portions, and the illustration is merely for illustrative purposes.
Detailed Description
It is to be understood that the following detailed description is exemplary and is intended to provide further explanation of the invention as claimed. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of exemplary embodiments according to the invention. As used herein, the singular forms "a", "an", and "the" are intended to include the plural forms as well, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof;
as introduced by the background art, the prior art has the problem that the lattice-entity composite structure optimization of the multivariable design cannot be realized, and in order to solve the technical problem, the invention provides a lattice-entity composite structure topological optimization method based on the multivariable design.
In an exemplary embodiment of the present invention, referring to fig. 1, a method for optimizing a lattice-entity composite topology based on multivariate design includes the following steps:
the method comprises the following steps: designing a lattice material structure configuration containing a plurality of design variables;
the multiple design variables refer to definition parameters of the lattice materials, the definition parameters of the lattice materials can be two, three, four or other quantities, and the definition parameters of the lattice materials comprise width parameters or length parameters or angle parameters or composition coefficient parameters of lattice material structure configurations;
in the process of establishing the lattice material structure configuration, the variation range of the design variable corresponding to the lattice material is given.
Specifically, one lattice material structure configuration, such as that shown in fig. 2, contains four design variables: alpha, beta, gamma and delta respectively correspond to the widths of four rods in the lattice material structure. Wherein the variation ranges of the four design variables are defined as:
l 1 ≤α，β，γ，δ≤l 2 (1)
in the formula I 1 And l 2 Respectively, the minimum value of the definition parameter of the predefined lattice material in the lattice material structure and the maximum value of the definition parameter of the predefined lattice material, namely the predefined lattice materialThe minimum and maximum of the rod width.
Step two: and calculating a plurality of sample point data, namely obtaining the macroscopic equivalent physical properties of the lattice materials corresponding to different design variables.
Specifically, the macroscopically equivalent physical properties of the lattice material include the relative density ρ of the lattice material L And equivalent elastic matrix D of lattice material L . Relative density of lattice material rho L ：
Figure BDA0003670068620000071
In the formula, V lattice Volume of lattice material structure, V domain The volume of the domains is designed for the lattice material structure.
Equivalent elastic matrix D of lattice material L The calculation can be based on the existing numerical homogenization method.
Step three: based on a plurality of sample point data, an interpolation model for mapping the mathematical relationship between the design variables and the macroscopic equivalent physical attributes of the lattice materials is established. The fitting relation of the interpolation model is as follows:
Figure BDA0003670068620000081
in the formula (I), the compound is shown in the specification,
Figure BDA0003670068620000082
is an equivalent elastic matrix D L Coefficient of (1), u i (i =1 to 15) are fitting constants calculated based on the sample points and the least square method.
Step four: setting a design variable definition domain of the multi-design variables for the dot matrix-solid material, namely a dot matrix material definition domain and a solid material definition domain, and establishing a dot matrix-solid multi-material interpolation model according to an interpolation model for mapping mathematical relations between the design variables and physical properties of the dot matrix material in the dot matrix material.
Specifically, the lattice material definition domain is that four design variables are all between the preset valuesBetween the minimum rod width and the minimum rod width, and four variables are relatively independent at this time, i.e./ 1 ≤α，β，γ，δ≤l 2 ；
The solid material definition domain is that four design variables are all larger than the maximum value l of the predefined rod width 2 And at the moment, the four design variables change cooperatively, and the maximum value of the change range of the four design variables is set as l 3 I.e. l 2 ＜α＝β＝γ＝δ≤l 3 . It should be noted that when α = β = γ = δ = l 3 When the method is used, corresponding to complete entity material attributes, the method can promote the entity material corresponding to design variables to be converged to the maximum value l of the variation range of the design variables in the optimization process by applying a material attribute penalty 3 。
The lattice-entity multi-material interpolation model comprises equivalent elastic matrixes and relative density information of lattice materials, and the constructed lattice-entity multi-material interpolation model comprises the following steps:
Figure BDA0003670068620000091
in the formula (I), the compound is shown in the specification,
Figure BDA0003670068620000092
interpolating a material equivalent elastic matrix D in a model for multiple materials E Coefficient of (1), p E For relative density, p is the material property penalty factor,
Figure BDA0003670068620000093
and ω are constants in the interpolation model, respectively, that can be calculated by α = β = γ = δ = l 2 And α = β = γ = δ = l 3 And solving the corresponding material attribute.
Step five: and constructing a topological optimization mathematical model of the multivariate lattice-entity composite structure. Taking a structure flexibility minimization topological optimization model under the constraint of set volume fraction as an example, the multivariate lattice-entity composite structure topological optimization mathematical model is as follows:
Figure BDA0003670068620000094
Figure BDA0003670068620000095
s.t.：F＝KU， (5)
Figure BDA0003670068620000096
l 1 ≤α E ，β E ，γ E ，δ E ≤l 3 ，E＝1，2，...，N.
wherein the content of the first and second substances,
Figure BDA0003670068620000097
representing a design variable total function, including four design variable information in all finite element units, wherein N is the number of the finite element units, C is the structural flexibility, F is the load vector borne by the structure, U is displacement, K is a structural total stiffness matrix, and U is S Is a unit displacement, K E Is a unit stiffness matrix, V is the total volume of the structure in the optimization process, V 0 Domain volume for structural design, f volume fraction constraint, l 1 And l 3 Respectively, the lower and upper limits of the design variable variation.
Step six: defining topological optimization structure parameters and setting optimization iteration initial design;
specifically, the initial design is set as a lattice material uniform filling design with a set relative density, and the variation range of the four design variables is defined as a lattice material definition domain.
Step seven: calculating the material attribute corresponding to the design variable at the moment based on the lattice-entity multi-material interpolation model constructed in the step four, performing finite element analysis, calculating to obtain the structure displacement, calculating the target function value corresponding to the current design variable and the sensitivity information of the current design variable to the target function and the constraint function,
specifically, the sensitivity of the current design variables to the objective function is:
Figure BDA0003670068620000101
in the formula, x E Designing a variable total function for a cell, including information of four design variables in the cell, B is a cell strain matrix, omega E Is a finite element volume of the element,
Figure BDA0003670068620000102
and calculating by the lattice-solid multi-material interpolation model constructed in the step four.
The sensitivity of the design variables to the constraint function is:
Figure BDA0003670068620000103
Figure BDA0003670068620000104
and calculating by the lattice-solid multi-material interpolation model constructed in the step four.
Step eight: from the sensitivity information obtained in step seven, the design variables are updated using the MMA algorithm (moving asymptote method).
Step nine: and judging whether the updated design variables are converged. If not, continuing the step ten; if yes, ending the solution, and outputting a topology optimization result, namely the optimal distribution of the four design variables at each position in the macrostructure.
Specifically, the sign of iteration convergence is that the design variable change value updated by the optimization algorithm is smaller than a set value or the maximum iteration number exceeds a preset number.
Step ten: inputting the updated design variables into the checking mechanism of the multivariate material definition domain, and returning to the seventh step.
Specifically, the inspection mechanism for multivariate material domains is as follows:
when the multivariate material definition domain is defined for the lattice materialThe domain, if the four updated variables of the optimization algorithm reach the upper limit of the lattice material definition domain at the same time, namely, alpha = beta = gamma = delta = l 2 If the material definition domain is changed into the entity material definition domain in the next iteration, otherwise, the material definition domain is not changed; when the material definition domain is the entity material definition domain, if the four updated variables of the optimization algorithm reach the lower limit of the entity material definition domain at the same time, namely, alpha = beta = gamma = delta = l 2 Then in the next iteration the material definition domain is changed to a lattice material definition domain, otherwise the material definition domain is not changed.
The following describes a method for optimizing a lattice-entity composite structure topology based on multivariate design, which is proposed by the present invention, with reference to examples.
All physical quantities used in the present embodiment are assumed to be dimensionless. The dimensions and boundary conditions of the two-dimensional cantilever beam structure are shown in fig. 3, the structure dimensions are defined as 40 × 20, the finite unit grid dimensions are 1 × 1, and the volume constraint is 50% of the original volume constraint. The initial design was to fill uniformly with four lattice materials of equal rod width and relative density 0.5.
FIG. 4 is a progression of evolution of four design variables during the optimization process. Fig. 5 shows a detailed diagram of the lattice-solid composite structure design obtained by the final optimization.
As can be seen from fig. 4 and 5, the lattice-entity composite structure topology optimization method based on multivariate design provided by the invention realizes lattice-entity material cooperative optimization under the condition of constraining the density variation range of the lattice material, and greatly expands the design space of the lattice material filling structure; the solid material and the high-density lattice material are distributed at the position with high strain energy of the structural unit in the optimized structure, and the distribution of the lattice material and the solid material is matched with the working condition, so that the optimized structure has excellent mechanical property and excellent multifunctional property of the lattice material. Meanwhile, the topological optimization method is high in optimization efficiency and good in structural connectivity.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (10)

1. A dot matrix-entity composite structure topological optimization method based on multivariate design is characterized by comprising the following steps:
establishing a lattice material structure configuration containing a plurality of design variables;
calculating a plurality of sample point data, namely obtaining the macroscopic equivalent physical attributes of the lattice materials corresponding to different design variables;
establishing an interpolation model for mapping a mathematical relationship between design variables in the lattice material and physical attributes of the lattice material based on the plurality of sample point data;
setting design definition domains of multiple design variables for the dot matrix-solid material, including a dot matrix material definition domain and a solid material definition domain, and establishing a dot matrix-solid multi-material interpolation model according to an interpolation model for mapping mathematical relations between the design variables in the dot matrix material and physical attributes of the dot matrix material;
constructing a multivariate lattice-entity composite structure topological optimization mathematical model;
calculating a target function value corresponding to a current design variable and sensitivity information of the current design variable to a target function and a constraint function based on a dot matrix-entity multi-material interpolation model and a multivariable dot matrix-entity composite structure topological optimization mathematical model, and updating the design variable;
and judging the convergence of the optimization iteration according to the updated design variables, and finishing the optimization if the iteration is converged.
2. The method for optimizing the topology of the lattice-entity composite structure based on the multivariate design as recited in claim 1, wherein the convergence of the optimization iteration is judged according to the updated design variables, if the iteration converges, the optimization is completed, otherwise, the next step is performed;
judging whether the current design variable meets the updating condition of the material definition domain or not according to the updated design variable, if so, updating the material definition domain, otherwise, not updating; and the objective function value corresponding to the current design variable and the sensitivity information of the current design variable to the objective function and the constraint function are calculated again.
3. The method of claim 1, wherein the design variables are defined parameters of lattice material, and the defined parameters of lattice material include width parameter or length parameter or angle parameter or composition coefficient parameter of lattice material structure configuration;
in establishing the lattice material structure configuration, the variation range of the design variable is given.
4. The method of claim 1, wherein the macro-equivalent physical properties of the lattice material corresponding to different design variables comprise relative density of the lattice material and equivalent elastic matrix of the lattice material.
5. The method of claim 3, wherein the fitting relation of the interpolation model is determined according to the fitting constants calculated based on the sample points and the least square method and the defining parameters of the lattice material.
6. The method of claim 1, wherein the lattice material domain is a domain in which the design variables are between the minimum value of the parameters defining the predefined lattice material and the maximum value of the parameters defining the predefined lattice material.
7. The method as claimed in claim 1, wherein the solid material domain is defined by a plurality of design variables that are larger than the maximum value of the parameters defining the predefined lattice material, and the design variables are varied cooperatively within a range between the maximum value of the parameters defining the lattice material and the maximum value of the variation range of the design variables.
8. The method of claim 1, wherein the lattice-entity multi-material interpolation model is obtained based on a lattice material domain, an entity material domain, design variables, and an interpolation model mapping mathematical relationships between design variables and physical properties of lattice materials in lattice materials.
9. The method for lattice-entity composite topology optimization based on multivariate design as claimed in claim 1, wherein after the multivariate lattice-entity composite topology optimization mathematical model is constructed, the topology optimization structure parameters are defined and the optimization iteration initial design is set.
10. The method for optimizing the topology of the lattice-entity composite structure based on the multivariate design as recited in claim 1, wherein the method comprises the steps of calculating the material properties corresponding to the design variables at the time based on the lattice-entity multi-material interpolation model and the multivariate lattice-entity composite structure topology optimization mathematical model, performing finite element analysis, calculating the structural displacement, and calculating the objective function value corresponding to the current design variables;
and updating the design variables by using a moving progression method.
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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115935730A (en) * 2022-11-18 2023-04-07 华中科技大学 Multi-scale topological optimization method for five-mode metamaterial bone scaffold facing seepage performance
CN116415459A (en) * 2023-03-30 2023-07-11 之江实验室 Macro-micro cooperative topology design method of thin-wall structure and robot calf model

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115935730A (en) * 2022-11-18 2023-04-07 华中科技大学 Multi-scale topological optimization method for five-mode metamaterial bone scaffold facing seepage performance
CN116415459A (en) * 2023-03-30 2023-07-11 之江实验室 Macro-micro cooperative topology design method of thin-wall structure and robot calf model
CN116415459B (en) * 2023-03-30 2024-02-02 之江实验室 Macro-micro cooperative topology design method of thin-wall structure and robot calf model

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