CN111639460A - Design method of porous model adopting few structural units - Google Patents

Design method of porous model adopting few structural units Download PDF

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CN111639460A
CN111639460A CN202010413989.4A CN202010413989A CN111639460A CN 111639460 A CN111639460 A CN 111639460A CN 202010413989 A CN202010413989 A CN 202010413989A CN 111639460 A CN111639460 A CN 111639460A
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扈婧乔
李明
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Zhejiang University ZJU
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    • G06—COMPUTING; CALCULATING OR COUNTING
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    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/27—Design optimisation, verification or simulation using machine learning, e.g. artificial intelligence, neural networks, support vector machines [SVM] or training a model
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F18/00—Pattern recognition
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    • G06F18/23—Clustering techniques
    • G06F18/231—Hierarchical techniques, i.e. dividing or merging pattern sets so as to obtain a dendrogram
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
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    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
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    • G06F2113/00—Details relating to the application field
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Abstract

The invention discloses a design method of a porous model with a small number of structural units, which aims to design a porous structure with excellent performance, manufacturability and geometric effective communication under the constraint of a given design area, external load and material volume. Firstly, macroscopic free material distribution optimization is adopted, matrix eigenvalues of anisotropic elasticity tensor are considered, material properties are limited in a solvable and manufacturable range, and theoretical optimal solutions of all possible elastic continuous media are obtained; based on the method, the material space dimension is reduced by means of the proposed high-fidelity hierarchical clustering; embedding the microstructure into each macro unit through reverse homogenization, enabling the microstructure to meet the specified material property, simultaneously controlling the geometrical smoothness of adjacent microstructures, preventing the problem of force transmission damage caused by a fracture structure, and finally obtaining an ideal porous structure. The method greatly expands the optimization space of material properties and structural performance, and obtains a structure with more excellent design performance.

Description

Design method of porous model adopting few structural units
Technical Field
The invention relates to the technical field of structure optimization, in particular to a topological structure design method of a porous model adopting a small number of structural units.
Background
Porous models exist widely in nature, such as bones, plant rhizomes, and the like. These models, formed by natural selection for billions of years, generally have relatively low densities, not only save materials but also reduce overall weight; secondly, different porous structures can show excellent characteristics in the aspect of multifunctional application, have a plurality of structural functions of high specific stiffness and specific strength, and can prevent heat, insulate heat, absorb and reduce noise and the like; in the aspect of three-dimensional printing, the porous structure enables accurate preparation, and the progress of the incremental manufacturing technology also enhances the manufacturing capability of the complex porous structure on a micro scale, so that a basis is provided for deeper development of porous model research.
Due to limitations in designability, the design of conventional materials generally follows the principle from the material function to the actual requirements, namely: according to the possible coverage range of the material performance, the limited design parameters of the material are adjusted to meet the actual requirements of engineering. Therefore, in most cases, the design and selection of materials cannot be truly optimized. With the development of advanced composite materials with complex microstructures represented by porous materials, the designability of the materials is greatly improved, and various materials with excellent performance can be designed by directly utilizing optimization technology from the actual requirements of engineering.
Existing methods for designing porous structures either take into account explicit geometric description of the microstructure or optimize both macro-units and microstructure. These topological optimization methods contain only one design variable (usually young's modulus) per cell, which assumption simplifies the topological optimization calculations, but at the same time also greatly limits the optimization space for material properties and structural performance. In fact, for an isotropic material, the young modulus and the poisson ratio have two variables, and for a general anisotropic material, the two-dimensional and three-dimensional design variables have 6 and 21 variables respectively, and the variables are fully utilized to design a structure with more excellent performance. Therefore, the free material optimization takes all components of the elasticity tensor of the material as design variables, can obtain the theoretical optimal solution of all elastic continuous media, and can well solve the problem.
In addition, controlling the geometric connectivity of adjacent microstructures has been a key problem based on a homogenization topology optimization method. Each macro-unit is embedded in an independently optimized microstructure, and the structural fracture generated by adjacent microstructures can cause force transmission failure problems, thereby destroying the physical properties of the macro-structure. The existing connectivity control method is mostly based on pure geometric control, or fixed connection points, or geometric interpolation is adopted, and the physical performance of the structure is still limited.
Disclosure of Invention
In view of the problems of the porous structure design, the invention provides a porous model design method using a small number of structural units. The method comprises the steps of obtaining theoretical optimal solutions of all possible elastic continuous media by means of free material distribution, expanding optimization space of material properties and structural performance, further performing dimensionality reduction on the continuous solutions through hierarchical clustering, obtaining high-fidelity discrete solutions under any specified target material quantity, embedding microstructures into each macro unit through reverse homogenization, enabling the microstructures to meet specified material properties, and simultaneously performing geometric smoothness control on adjacent microstructures.
The technical scheme adopted by the invention is as follows:
a design method of a porous model with a small number of structural units comprises the following steps:
1) by utilizing free material optimization, the elasticity tensor D of the anisotropic material is taken as a design variable, and manufacturability constraint is applied to limit the material range at the same time, so that the theoretical optimal distribution of all possible elastic continuous media is obtained;
2) based on the theoretical optimal distribution, for any given target material type, high-fidelity hierarchical clustering is adopted to reduce the spatial dimension of the material;
3) and matching corresponding microstructures by means of inverse homogenization based on the obtained elasticity tensor of the target material, and simultaneously performing geometric smoothness control to obtain a geometrically effectively communicated porous structure.
In the above technical solution, the step 1) of optimizing the free material specifically includes:
finite element division is carried out on a design area by adopting a regular polyhedron shape or a regular polyhedron to obtain M finite elements, and a cone space formed by taking the elastic tensor D of the anisotropic material as a symmetrical semi-regular matrix
Figure BDA0002494269960000021
Inner design variables, which have 6 and 21 design variables in two-dimensional and three-dimensional cases, respectively; with structural compliance c (D, u) as the objective function, the following is defined:
c(D,u)＝FTu
where F is the nodal stress vector and u is the set of real numbers belonging to the d dimension
Figure BDA0002494269960000022
The global displacement vector of (2);
the free material optimization problem is represented as follows:
Figure BDA0002494269960000023
Figure BDA0002494269960000024
wherein DeIs the elastic tensor of finite element e, K (D) is the total stiffness matrix of the structure, Tr (D)e) Is a local unit trace of the elasticity tensor of the material, the upper and lower bounds of which are
Figure BDA0002494269960000031
Constraining the maximum sum of global traces to be T0(ii) a Furthermore, manufacturability constraints are introduced to limit the range of materials
De-I≥0，e＝1，...，M
Wherein I is an identity matrix and measures a constraint range, and the value is preferably 0.005 in the method. This constraint prevents the stiffness in either direction from going to 0 by limiting the minimum eigenvalue of the elasticity tensor to be greater than 0.
The step 2) of high-fidelity hierarchical clustering specifically comprises the following steps:
the method comprises the steps of giving a target material type number k, measuring the similarity of material tensors on different finite units by using Euclidean distance, using a classical machine learning method-a hierarchical clustering method, reducing material types and reducing material space dimension by continuously calculating class intervals-merging classes, and finally obtaining a discrete structure consisting of k materials; for newly generated class k materials, the material properties of the new class are averaged within the elastic material space classification.
The discrete method can ensure that the local trace and the global trace of the discrete structure are consistent with the continuous solution, ensure that all constraints of the original problem are met, and ensure that the structural rigidity (measured by the size of the structural flexibility c) cannot generate excessive loss due to the discrete, thereby obtaining the high-fidelity discrete solution.
The step 3) is to cooperate with an inverse homogenization matching microstructure, and specifically comprises the following steps:
the target elasticity tensor D of the current microstructure and the adjacent microstructures thereof can be obtained through the optimization of the steps0、Ddir 0Finite element division is carried out on the microstructure to obtain N microscopic finite elements, and the equivalent elasticity tensor D of the microstructure is minimizedH(ρ) and the target elasticity tensor D0The Frobenius norm between the two to inlay and match the microstructure; meanwhile, in order to control the geometric connectivity of adjacent microstructures, the equivalent elastic tensor D of the medium cells in four adjacent directionsdir H*(p) with a corresponding target Ddir 0*The frobenius norm of (a) is also optimized and defined as follows:
Ddir 0*＝(D0+Ddir 0)/2
the optimization problem is embodied in the form:
Figure BDA0002494269960000032
Figure BDA0002494269960000033
where ρ represents the microstructure material density, defined in a real number set in the N dimension
Figure BDA0002494269960000034
Internal, rhonFor the density of the nth unit, an objective function J optimizes physical performance and geometric connectivity at the same time, and w measures the weight of the connectivity in the objective function, wherein the value range is between [0, 1 ]]K (rho) is the microstructure total stiffness matrix uA(kl)，FA(kl)Is the local displacement vector and the equivalent stress vector, V (rho) is the volume of the material in the microstructure, V0The material volume fraction is constrained.
The invention has the beneficial effects that:
1) existing topological optimization methods involve only one design variable (usually young's modulus) per cell, greatly limiting the optimization space for material properties and structural performance. The method takes elasticity tensor as a design variable by means of free material distribution and takes a theoretical optimal solution as a basis to design a porous structure with more excellent performance.
2) The method combines the topological optimization problem with the classical machine learning method by constructing a new design space and introducing a hierarchical clustering method, thereby obtaining a high-fidelity discrete solution under any specified material variety quantity.
3) Controlling the geometric connectivity of adjacent microstructures has always been a key problem based on a homogenization topology optimization method. The existing connectivity control method is mostly based on pure geometric control, and the method innovatively starts from the physical performance of the structure, considers that the reverse homogenization of the microstructure is cooperatively carried out under the condition of ensuring the elasticity tensor of the matched target, and ensures the geometric smoothness between adjacent units.
Drawings
FIG. 1 is a schematic flow diagram of the process of the present invention; : a. designing an area; b. optimizing the free material to obtain optimal elasticity tensor distribution; c. obtaining a few optimized material distributions after hierarchical clustering; d. and (3) obtaining an integral structure through cooperative reverse homogenization.
Detailed Description
The invention is further described with reference to the following figures and specific embodiments.
The invention discloses a design method of a porous model with a small number of structural units, which aims to design a porous structure with excellent performance, manufacturability and geometric effective communication under the constraint of a given design area, external load and material volume. Firstly, macroscopic free material distribution optimization is adopted, matrix eigenvalues of anisotropic elasticity tensor are considered, material properties are limited in a solvable and manufacturable range, and theoretical optimal solutions of all possible elastic continuous media are obtained; based on the method, the material space dimension is reduced by means of the proposed high-fidelity hierarchical clustering; embedding the microstructure into each macro unit through reverse homogenization, enabling the microstructure to meet the specified material property, simultaneously controlling the geometrical smoothness of adjacent microstructures, preventing the problem of force transmission damage caused by a fracture structure, and finally obtaining an ideal porous structure. The method greatly expands the optimization space of material properties and structural performance, and obtains a structure with more excellent design performance. The method specifically comprises the following steps:
1) free material optimization
Finite element division is carried out on a design area by adopting a regular polyhedron shape or a regular polyhedron to obtain M finite elements, and a cone space formed by taking the fourth-order elasticity tensor D of the anisotropic material as a symmetrical semi-positive matrix
Figure BDA0002494269960000051
Inner design variables having 6 and 21 design variables in two-dimensional and three-dimensional cases, respectively, and a tensor D is expressed as D by Einstein notationijklI, j, k, l ═ 1, 2, and its matrix form is as follows:
Figure BDA0002494269960000052
the optimization problem is defined with structural compliance c (D, u) as the objective function as follows:
c(D，u)＝FTu
where F is the nodal stress vector and u is the set of real numbers belonging to the d dimension
Figure BDA0002494269960000053
The global displacement vector of (2).
The free material optimization problem can be expressed in the form:
Figure BDA0002494269960000054
Figure BDA0002494269960000055
wherein DeIs the elastic tensor of unit e, K (D) is the total stiffness matrix of the structure, Tr (D)e) Is a local unit trace of the elasticity tensor of the material, the upper and lower bounds of which are
Figure BDA0002494269960000056
Constraining the maximum sum of global traces to be T0。
The problem contains a nonlinear non-convex constraint condition, and the non-convex semi-definite Programming problem (SDP) is converted into a linear SDP problem by using the Schur's complementary theorem:
Figure BDA0002494269960000057
Figure BDA0002494269960000058
the equilibrium equation k (d) where the optimization variable γ is used to constrain forces, u ═ F.
The problem can be solved by adopting general numerical optimization software to obtain a globally optimal continuous solution.
In addition, manufacturability constraint is introduced to consider the matrix eigenvalue of the elasticity tensor, and the range of the material is effectively limited
De-I≥0，e＝1，...，M
Where I is the identity matrix, measures the range of constraints, and takes the value 0.005 in this method. This constraint prevents the stiffness in either direction from going to 0 by limiting the minimum eigenvalue of the elasticity tensor to be greater than 0.
2) Obtaining high-fidelity discrete solution by hierarchical clustering aiming at any given target material category quantity
The method comprises the steps of giving the number k of types of target materials, measuring the similarity of material tensors on different finite units by using Euclidean distance, using a classical machine learning method-a hierarchical clustering method, reducing the types of the materials and reducing the spatial dimension of the materials by continuously calculating the class spacing-merging classes, and finally obtaining a discrete structure consisting of the k types of materials. For newly generated class k materials, the material properties of the new class are averaged within the elastic material space classification.
The discrete method can ensure that the local trace and the global trace of the discrete structure are consistent with the continuous solution, ensure that all constraints of the original problem are met, and ensure that the structural rigidity (measured by the size of the structural flexibility c) cannot generate excessive loss due to the discrete, thereby obtaining the high-fidelity discrete solution.
3) Reverse homogenized matched microstructures with simultaneous geometric smoothness control
Finite element division is carried out on the microstructure to obtain N microscopic finite elements, and step 2) optimization is carried out to obtain a target elasticity tensor D of the current microstructure and adjacent microstructures of the current microstructure0、Ddir 0By minimizing the microstructure equivalent elastic tensor DH(ρ) and the target elasticity tensor D0And the Frobenius norm in between to inlay the matching microstructure. Meanwhile, in order to control the geometric connectivity of adjacent microstructures, the equivalent elastic tensor D of the media in four adjacent directionsdir H*(p) with a corresponding target Ddir 0*Optimization was also performed, defined as follows:
Ddir 0*＝(D0+Ddir 0)/2
the optimization problem is embodied in the form:
Figure BDA0002494269960000061
Figure BDA0002494269960000062
where ρ represents the microstructure material density, defined in a real number set in the N dimension
Figure BDA0002494269960000063
Internal, rhonFor the density of the nth unit, an objective function J optimizes physical performance and geometric connectivity at the same time, and w measures the weight of the connectivity in the objective function, wherein the value range is between [0, 1 ]]K (rho) is the microstructure total stiffness matrix uA(kl)，FA(kl)Is the local displacement vector and the equivalent stress vector, V (rho) is the volume of the material in the microstructure, V0The material volume fraction is constrained.
The following is a specific solution case performed by the method of the present invention:
fig. 1(a) is a two-dimensional cantilever beam case, the solution domain is discretized into a 10 × 30 square grid, the left end is fixed on the wall, and the lower endpoint of the right end is subjected to a vertically downward pulling force. The elasticity tensor distribution obtained by the optimization of the free material is shown as (b). Given the number of target material classes 7, the discrete solution distribution of hierarchical clustering is as (c). Based on this target elasticity tensor, the final result of inverse homogenizing the matched microstructure while ensuring geometric smoothness is shown in (d). The method can design a porous structure with more excellent performance while ensuring the geometric smoothness.

Claims (4)

1. A method for designing a porous model by using a small number of structural units is characterized by comprising the following steps:
1) by utilizing a free material optimization technology, the elasticity tensor D of the anisotropic material is taken as a design variable, and the manufacturability constraint is applied to limit the material range at the same time, so that the theoretical optimal distribution of all possible elastic continuous media is obtained;
2) based on the theoretical optimal distribution, for any given target material type, reducing the spatial dimension of the material by adopting a high-fidelity hierarchical clustering method, and forming a small number of target materials and the elasticity tensor thereof;
3) based on the obtained elasticity tensor of the target material, by means of a collaborative inverse homogenization method, collaborative inverse design of a plurality of microstructures is carried out simultaneously according to the position adjacent relation among the microstructures, physical properties are optimized, meanwhile, geometric smooth connection among adjacent units is guaranteed, and a porous structure with effective overall geometry and smooth structure is obtained.
2. The method for designing a porous model using a small number of structural units according to claim 1, wherein the step 1) is a free material optimization method, specifically as follows:
finite element division is carried out on a design area by adopting a regular polyhedron shape or a regular polyhedron to obtain M finite elements, and a cone space formed by taking the elastic tensor D of the anisotropic material as a symmetrical semi-regular matrix
Figure FDA0002494269950000011
Inner design variables, which have 6 and 21 design variables in two-dimensional and three-dimensional cases, respectively; with structural compliance c (D, u) as the objective function, the following is defined:
c(D，u)＝FTu
where F is the nodal stress vector and u is the set of real numbers belonging to the d dimension
Figure FDA0002494269950000012
The global displacement vector of (2);
the free material optimization problem is represented as follows:
Figure FDA0002494269950000013
Figure FDA0002494269950000014
wherein DeIs the elastic tensor of finite element e, K (D) is the total stiffness matrix of the structure, Tr (D)e) Is a local unit trace of the elasticity tensor of the material, the upper and lower bounds of which are
Figure FDA0002494269950000015
Constraining the maximum sum of global traces to be T0(ii) a Furthermore, manufacturability constraints are introduced to limit the range of materials
De-I≥0，e＝1，...，M
Where I is the identity matrix, measures a range of constraints that prevent the stiffness from going to 0 in either direction by limiting the minimum eigenvalue of the elasticity tensor to be greater than 0.
3. The method for designing a porous model with a small number of structural units according to claim 1, wherein the step 2) is high-fidelity hierarchical clustering, which comprises the following steps:
the method comprises the steps of giving a target material type number k, measuring the similarity of material tensors on different finite units by using Euclidean distance, using a classical machine learning method-a hierarchical clustering method, reducing material types and reducing material space dimension by continuously calculating class intervals-merging classes, and finally obtaining a discrete structure consisting of k materials; for newly generated class k materials, the material properties of the new class are averaged within the elastic material space classification.
4. The method for designing a porous model with a small number of structural units according to claim 1, wherein the step 3) is to inversely homogenize the matching microstructure as follows:
the target elasticity tensor D of the current microstructure and the adjacent microstructures thereof can be obtained through the optimization of the steps0、Ddir 0Finite element division is carried out on the microstructure to obtain N microscopic finite elements, and the equivalent elasticity tensor D of the microstructure is minimizedH(ρ) and the target elasticity tensor D0The Frobenius norm between the two to inlay and match the microstructure; while controlling the geometric communication between adjacent microstructuresEquivalent elastic tensor D of medium cell in four adjacent directionsdir H*(p) with a corresponding target Ddir 0*The frobenius norm of (a) is also optimized and defined as follows:
Ddir 0*＝(D0+Ddir 0)/2
the optimization problem is embodied in the form:
Figure FDA0002494269950000021
Figure FDA0002494269950000022
where ρ represents the microstructure material density, defined in a real number set in the N dimension
Figure FDA0002494269950000023
Internal, rhonFor the density of the nth unit, an objective function J optimizes physical performance and geometric connectivity at the same time, and w measures the weight of the connectivity in the objective function, wherein the value range is between [0, 1 ]]K (rho) is the microstructure total stiffness matrix uA(kl)，FA(kl)Is the local displacement vector and the equivalent stress vector, V (rho) is the volume of the material in the microstructure, V0The material volume fraction is constrained.
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Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20190091657A1 (en) * 2016-03-06 2019-03-28 Waters Technologies Corporation Hybrid material for chromatographic separations comprising a superficially porous core and a surrounding material
CN110197006A (en) * 2019-05-07 2019-09-03 浙江大学 Isotropic material Topology Optimization Method based on Second-order cone programming

Patent Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20190091657A1 (en) * 2016-03-06 2019-03-28 Waters Technologies Corporation Hybrid material for chromatographic separations comprising a superficially porous core and a surrounding material
CN110197006A (en) * 2019-05-07 2019-09-03 浙江大学 Isotropic material Topology Optimization Method based on Second-order cone programming

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
JINGQIAO HU 等: "Cellular structure design based on free material optimization under connectivity control", 《COMPUTER-AIDED DESIGN》 *

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