CN117473836A - Integrated design method for thin-wall-multi-class lattice filling structure - Google Patents
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Abstract
The invention provides an integrated design method of a thin-wall multi-class lattice filling structure, which is characterized in that a thin-wall layer and a filling layer are identified through three-step PDE filtering and Heaviside projection, continuous function mapping of unit cell attribute-mechanical property is established based on a multi-response Gaussian hidden variable model, quick prediction of the mechanical property of lattice unit cell is realized, a multi-scale material interpolation model of the thin-wall filling structure is constructed, and integrated design of the thin-wall multi-class lattice filling structure is realized through collaborative optimization of an integral structure and a filling lattice configuration. The invention realizes the collaborative optimization of the identification of the thin-wall layer in the structural design area and the configuration and the density of lattice unit cells with different configurations in the filling layer, expands the design freedom degree of the single-material filling thin-wall structure and improves the mechanical property of the filling thin-wall structure.
Description
Technical Field
The invention relates to the technical field of ultra-lightweight structure design methods, in particular to an integrated design method for a thin-wall multi-class lattice filling structure.
Background
The thin-wall multi-type lattice filling structure is a lightweight composite structure with excellent performance, and mainly comprises an external thin-wall layer and different lattice structures inside. Theory and research show that the thin-wall filling structure has the advantages of high specific stiffness, larger tensile strength, good buckling resistance, energy absorption and the like, and the mechanical property and the light weight level of the structure can be improved to the greatest extent through the collaborative design of the integral structure and the filling structure.
Topology optimization is used as a structural optimization method, the potential of the degree of freedom of design can be furthest mined, and a novel structure is often obtained in the conceptual design stage. At present, a thin-wall filling structure obtained through topological optimization is often solid filling or single filling configuration, cannot adapt to complex stress distribution, and has limited light weight potential. Through the cooperative optimization of the integral topology of the thin-wall structure and the multi-class filling configuration, the optimal matching of macro-micro mechanical properties can be realized, and the material-structure integration potential is fully excavated. The efficiency bottleneck caused by strong nonlinearity, high-efficiency calculation of the macroscopic mechanical properties of the multi-class microstructure, cooperative optimization of topological variables and configuration variables and the like due to thin-wall filling characteristic description is a core difficulty.
Disclosure of Invention
The invention constructs the integrated design method of the thin-wall multi-class lattice filling structure based on the multi-response Gaussian hidden variable lattice unit cell-mechanical property prediction model, the three-step filtering and the Heaviside projection method and the multi-scale material interpolation model of the thin-wall filling structure, and is expected to provide technical support for the high-performance and lightweight design of the carrier structure.
Specifically, the invention provides a thin-wall-multi-class lattice filling structure integrated design method, which comprises the following steps:
step 1: combining N to be adopted cls Lattice unit cell databases of more than or equal to 2 different configurations, and performing density sampling on lattice unit cells of all configurations by a test design method to obtain N data Calculating equivalent elastic matrix of sample lattice unit cell by adopting homogenization theory according to sample data, and constructing points based on lattice unit cell sample dataArray density field v and lattice configuration field z i (i=1,., N) and an equivalent elastic matrix, the response number is N resp ；
Step 2: establishing a finite element model according to design requirements, determining a non-design domain and a design domain, and selecting an overall topological field mu, a lattice density field v and a lattice configuration field z i (i=1,., n) is a design variable, a first stage optimization model is built that targets structural compliance, mass fraction is a constraint, initializing key parameters of the first-stage optimization model;
step 3: based on the lattice density field v and the lattice configuration field z i (i=1,., n) calculating a lattice unit cell equivalent elastic matrix D in the multi-response gaussian hidden variable prediction model lattice Wherein n represents the dimension of the multi-response gaussian hidden variable space;
step 4: based on the integral topological field mu, combining a three-step filtering method and a Heaviside projection to obtain a filling density fieldAnd thin-walled density field->
Step 5: according to lattice unit cell equivalent elastic matrix D lattice And filling density fieldCalculating a filling layer elastic matrix D infill Introducing a punishment function;
step 6: according to the filling density fieldThin-walled density field->And filling layer elastic matrix D infill Combining the thin-wall filling structure multi-scale material interpolation model to obtain an elastic matrix D and a mass density n of the units in the design domain:
step 7: carrying out finite element analysis by combining a macroscopic multi-scale material interpolation model to obtain the flexibility of the structure;
step 8: solving to obtain the sensitivity of the structure flexibility and the mass fraction with respect to the design variable;
step 9: according to the objective function and constraint conditions determined by the mathematical model of the optimization problem, submitting the sensitivity to a mobile asymptote algorithm solver to update the overall topological field mu, the lattice density field v and the lattice configuration field z i ；
Step 10: if the iteration number k is more than or equal to 200 or the maximum change of the design variable is less than 0.01 in the kth iteration step, the convergence condition is considered to be satisfied, and the iteration is stopped and the integral topology field mu and the lattice configuration field z are output i Otherwise, repeating the steps 3 to 9;
step 11: based on the existing integral topological field mu and lattice configuration field z i Establishing a second-stage optimization model by taking the lattice density field v as an optimization variable and the structural flexibility as an optimization target and the volume fraction as a constraint condition, and repeating the contents from the step 3 to the step 9;
step 12: if the maximum change of the design variable is smaller than 0.01 in the kth iteration step, the convergence condition is considered to be satisfied, the iteration is stopped and the result is output, otherwise, the step 11 is repeated.
Further, in step 1, the multi-response gaussian hidden variable prediction model is:
wherein h represents a compound represented by N resp A priori mean vector of 1, Y represents the constructed N resp ×N data A database matrix of the equivalent elastic matrix of the Viomonas;the representative regression coefficient matrix may be expressed as:
wherein H is represented by N data The a priori mean function h constitutes the transpose of the matrix:
r represents a co-correlation matrix of input parameters of the lattice unit cell database, and R * The co-correlation vectors representing the predicted points and the training set input parameters are respectively expressed as follows:
wherein, r (·, ·) represents a gaussian kernel function reflecting the degree of correlation of two vectors, expressed as follows:
r(s,s′)＝exp[-(v-v′) T Φ(v-v′)-(z(t)-z(t′)) T (z(t)-z(t′))]
where s denotes an input vector comprising a qualitative variable v and a quantitative variable t, s ' denotes an input vector comprising a qualitative variable v ' and a quantitative variable t ', a diagonal matrix Φ is set to avoid over-parameterization of the qualitative and quantitative variables, s and Φ are expressed as follows:
s＝[v,z(t) T ] T ＝[v,z 1 ,...,z i ] T
Φ＝diag(φ)，φ＝[φ 1 ,φ 2 ,…,φ p ] T
where v represents the quantitative variable of the input, i.e. the lattice unit cell density variable, then the diagonal matrix Φ dimension p=1; z (t) T The qualitative variables representing the inputs, i.e., lattice unit configuration variables, the lattice unit qualitative variables t map to a collection of i-dimensional hidden variable spaces.
Further, in step 2, the first-stage optimization model may be expressed as:
s.t.KU＝F
g mass ＝frac-frac * ≤0，
0≤μ≤1
0<v - ≤v≤v +
wherein s.t. represents a constraint condition, and T represents a transpose of the matrix; k represents the structural total stiffness matrix, and U and F represent global displacement and load vectors; u (u) e Representing cell node displacement vectors, k e Representing a cell stiffness matrix, e being a cell number; n represents the number of design domain units; mu (mu) (e) Cell density, v, representing the overall topological field (e) Representing the density of the cell lattice,cell variable values representing lattice configuration fields; i represents the hidden variable number, i=1, 2 in the two-dimensional hidden variable space; mu is mu (e) The vector of the composition, v is v (e) Component vector, z is->A vector of components; v - (v + ) Representing the lower and upper limit of the unit cell density, < >>The upper limit and the lower limit of the value of the ith hidden variable are represented; g mass Representing a quality score constraint; frac is the mass fraction, equal to the structural mass m (μ, ν) and the initialDesign domain total mass m 0 Ratio of; frac * Representing a preset quality score threshold.
Further, in step 4, the filtering expression is:
in the method, in the process of the invention,for Laplacian, τ is the density of the unit before filtering, and the filtered unit density ++can be obtained by applying filtering boundary conditions>R f Is the filter radius.
Further, in step 4, the e-th cell-filtered cell densityUnit density obtained after projection of Heaviside +.>The method comprises the following steps:
where beta is the sharpness of the projection, eta is the threshold of the projection,represents the unit density after the e-th unit filtering +.>The unit density obtained after projection;
the filling density field can be obtained through three filtering and projection operationsAnd thin-walled density field->
Further, in step 5, the filling layer elastic matrix D infill The expression form is as follows:
wherein D is 0 Representing a minimum elastic matrix that avoids singular global stiffness matrices; p is p 1 Representing a penalty factor.
Further, in step 5, the penalty function for the filling layer elastic matrix may be expressed as:
wherein λ=500; lattice configuration variable z= [ z ] 1 ,z 2 ] T ；z(t)＝[z 1 (t),z 2 (t)] T ,t＝1...N cls Representing N in two dimensions cls A unit cell-like configuration distribution position; gamma is the diagonal length of the smallest bounding rectangle of all cells.
Further, in step 6, the elastic matrix D and the mass density m of the cells in the design domain are:
wherein lambda is m And lambda is E Representing the mass ratio and the elastic modulus ratio of the filling material and the thin-wall layer material; ρ 0 Representing the density of the thin-wall layerThe method comprises the steps of carrying out a first treatment on the surface of the m represents the unit mass; p is p 2 Representing penalty factors; d (D) shell Representing a thin-wall layer elastic matrix; d (D) infill Representing the packed layer elastic matrix.
Further, in step 8, the flexibility c and the sensitivity of the structural mass m to the design parameters can be obtained from the following formula:
and->Expressed as:
wherein the method comprises the steps ofAnd->The method comprises the following steps of:
wherein A is a group of the total number of the,and->The method comprises the following steps of:
wherein e represents a unit number, i represents a hidden variable number; j represents the number of the training set sample, and the value range is interval [1, N data ]An integer thereon.
Further, in step 11, the second-stage optimization model may be expressed as:
s.t.KU＝F
g mass ＝frac-frac * ≤0，
0<v - ≤v≤v +
wherein v is - (c + ) Representing the lower and upper limits of lattice density; frac * Representing a predetermined quality score threshold; the values are the same as the first stage.
The beneficial effects achieved by the invention are as follows:
1. the invention realizes the collaborative optimization of the identification of the thin-wall layer in the structural design area and the configuration and the density of lattice unit cells with different configurations in the filling layer, expands the design freedom degree of the single-material filling thin-wall structure and improves the mechanical property of the filling thin-wall structure.
2. According to the invention, the equivalent elastic tensor of the multi-configuration lattice unit cell is predicted by adopting the multi-response Gaussian hidden variable model, so that the calculation time for evaluating the mechanical property of the lattice unit cell by using a traditional progressive homogenization method is remarkably reduced, and the calculation efficiency of the multi-scale filling thin-wall structure is improved.
3. The invention adopts the multi-response Gaussian hidden variable model to predict the equivalent elastic tensor of the multi-configuration lattice unit cell, and constructs the unified continuous function mapping of the qualitative variable, the quantitative variable and the mechanical property of the multi-configuration lattice unit cell, thereby ensuring the feasibility of the process of updating the design variable by the sensitivity analysis and solver.
4. The invention constructs a material interpolation model and a quality interpolation model of the multi-configuration lattice filling thin-wall structure facing the SIMP method, realizes the integrated design of at least six different configuration lattice filling thin-wall structures, and perfects the defect that the filling thin-wall structure is filled with only a single material in the prior art.
5. According to the invention, the thin-wall layer and the filling layer of the thin-wall filling structure are identified by performing three-step PDE filtering and Heaviside projection operation on the matrix density field, and the upper limit and the lower limit of the single cell density field are adjusted on the basis, so that the minimum rod diameter of lattice single cells in the multi-configuration lattice filling thin-wall structure can be further controlled, and the manufacturing requirement is met.
Drawings
FIG. 1 is a flow chart of an integrated design method of a multi-configuration lattice filling thin-wall structure constructed by the invention;
FIG. 2 is a schematic diagram of the configuration of 6 lattice unit cells with different configurations constructed according to the invention;
FIG. 3 is a schematic diagram of a two-dimensional simply supported beam structure design domain, load and boundary conditions provided by an embodiment of the present invention;
FIG. 4 is a schematic diagram depicting a thin-wall packing density field through a three-step PDE filtering and projection operation in accordance with the present invention;
FIG. 5 is a schematic diagram of the distribution of lattice unit cell configurations of different configurations after optimization in the design domain of the multi-configuration lattice filling thin-wall structure in the embodiment of the invention;
FIG. 6 is a schematic diagram of the distribution of equivalent density values of lattice unit cells after optimization in the design domain of a multi-configuration lattice filling thin-wall structure in an embodiment of the invention;
FIG. 7 is a schematic diagram of an optimized multi-configuration lattice filling thin-wall structure in an embodiment of the present invention.
Detailed Description
The technical scheme of the present invention will be described in more detail with reference to the accompanying drawings, and the present invention includes, but is not limited to, the following examples.
The embodiment provides an integrated design method for a multi-configuration lattice filling thin-wall structure, the flow of the design method is shown in figure 1, and the method specifically comprises the following steps:
step 1: combining N to be adopted cls Lattice unit cell databases of more than or equal to 2 different configurations, and performing density sampling on lattice unit cells of all configurations by a test design method to obtain N data Calculating an equivalent elastic matrix of a sample lattice unit cell by adopting a homogenization theory according to sample data, and constructing a lattice density field v and a lattice configuration field z based on the lattice unit cell sample data i (i=1,., N) and an equivalent elastic matrix, the response number is N resp 。
In the embodiment, the number of lattice unit cell configurations is 6, and lattice unit cell isotropy, namely equivalent elastic tensor response number N is selected resp =4, with gaussian hidden variable spatial dimension n=2.
The constructed multi-response Gaussian hidden variable prediction model of the equivalent elastic matrix can be expressed as follows:
wherein h represents a compound represented by N resp A priori mean vector of 1, Y represents the constructed N data ×N resp A database matrix of the equivalent elastic matrix of the Viomonas;the representative regression coefficient matrix may be expressed as:
wherein H is represented by N data The a priori mean vector h constitutes the transpose of the matrix:
and R represents a co-correlation matrix of input parameters of the lattice unit cell database, and R * Representing predicted points and training set input parametersThe co-correlation vectors of numbers are expressed in the following form:
wherein, r (·, ·) represents a gaussian kernel function reflecting the degree of correlation of two vectors, expressed as follows:
r(s,s′)＝exp[-(v-v′) T Φ(v-v′)-(z(t)-z(t′)) T (z(t)-z(t′))]
where s (s ') represents an input vector comprising a qualitative variable v (v') and a quantitative variable t (t '), z (·) represents a functional mapping of the projection of the qualitative variable t (t') into the hidden variable space, a diagonal matrix Φ is set to avoid over-parameterization of the qualitative and quantitative variables, s and Φ are expressed as follows:
s＝[v,z(t) T ] T ＝[v,z 1 ,...,z i ] T
Φ＝diag(φ)，φ＝[φ 1 ,φ 2 ,…,φ p ] T
where v represents the quantitative variable of the input, i.e. the lattice unit cell density variable, then the diagonal matrix Φ dimension p=1; z (t) T The qualitative variables representing the inputs, i.e., lattice unit configuration variables (lattice unit qualitative variables t map to a set of i-dimensional hidden variable spaces).
Step 2: establishing a finite element model according to design requirements, determining a non-design domain and a design domain, and selecting an overall topological field mu, a lattice density field v and a lattice configuration field z i (i=1,., n) is a design variable, a first stage optimization model is built that targets structural compliance, mass fraction is a constraint, initializing key parameters of the first-stage optimization model;
in this embodiment, the first-stage optimization model may be expressed as:
s.t.KU＝F
g mass ＝frac-frac * ≤0，
0≤μ≤1
0<v - ≤ν≤v +
wherein s.t. represents a constraint condition, and T represents a transpose of the matrix; k represents the structural total stiffness matrix, and U and F represent global displacement and load vectors; u (u) e Representing cell node displacement vectors, k e Representing a unit stiffness matrix, wherein e is the design domain unit number; n represents the number of design domain units; mu (mu) (e) Cell density, v, representing the overall topological field (e) Representing the density of the cell lattice,cell variable values representing lattice configuration fields; i represents the hidden variable number, i=1, 2 in the two-dimensional hidden variable space; mu is mu (e) The vector of the composition, v, represents the lattice density field, v is v (e) Component vector, z is->A vector of components; v - (v + ) Representing the lower and upper limit of the unit cell density, < >>The upper limit and the lower limit of the value of the ith hidden variable are represented; g mass Representing a quality score constraint; frac is the mass fraction, equal to the structural mass m (μ, v) and the total mass m of the initial design domain 0 Ratio of; frac * Representing a preset quality score threshold.
Step 3: based on the lattice density field v and the lattice configuration field z i (i=1,., n) calculate a lattice unit cell equivalent elastic matrix D lattice Wherein n represents the dimension of the multi-response gaussian hidden variable space;
step 4: based on the integral topological field mu, combining a three-step filtering method and a Heaviside projection to obtain a filling density fieldAnd thin-walled density field->
Firstly, filtering is adopted to avoid checkerboard phenomenon and grid dependence, and meanwhile, the acquisition of spatial gradient information is facilitated, and the solving format is as follows:
in the method, in the process of the invention,for Laplacian, τ is the density of the unit before filtering, and the filtered unit density ++can be obtained by applying filtering boundary conditions>R f Is the filter radius.
Based on filtering, combining with the Heaviside projection can obtain clear topological configuration of 0-1 distribution, and the unit density after the e unit filteringUnit density obtained after projection +.>The method comprises the following steps:
where beta is the sharpness of the projection, eta is the threshold of the projection,represents the unit density after the e-th unit filtering +.>The cell density obtained after projection.
The filling density field can be obtained through three filtering and projection operationsAnd thin-walled density field->Specifically, R is used for the design variable μ 1 Performing first filtering for radius to obtain unit density +.>Then pair->Projection is carried out to obtain density field->At R 1 For radius pair->Performing the second filtering to obtain the unit density->Then pair->Projection is carried out to obtain density field->By R 2 For radius pair->Filtering for the third time to obtain density field->Ask for->Is>(/>Gradient operator) and multiplied by a normalization factor +.>Obtaining normalized spatial gradient norm field +.>Then carrying out a third projection to obtain +.>
Step 5: according to the lattice unit cell equivalent elastic matrix D obtained in the step 3 lattice And the filling density field obtained in the step 4Calculating a filling layer elastic matrix D infill ：
Wherein D is 0 Representing a minimum elastic matrix that avoids singular global stiffness matrices; p is p 1 Representing a penalty factor.
Introducing a penalty function to guide the iterative direction of lattice configuration variables of the optimization process, wherein the penalty function can be expressed as:
wherein λ=500; lattice configuration variable z= [ z ] 1 ,z 2 ] T ；z(t)＝[z 1 (t),z 2 (t)] T ,t＝1...N cls Representing N in two dimensions cls A unit cell-like configuration distribution position; gamma is the diagonal length of the smallest bounding rectangle of all cells.
Step 6: filling density field obtained according to step 4Thin-walled density field->And the filling layer elastic matrix D obtained in the step 5 infill Combining the thin-wall filling structure multi-scale material interpolation model to obtain an elastic matrix D and a mass density m of units in a design domain:
wherein lambda is m And lambda is E Representing the mass ratio and the elastic modulus ratio of the filling material and the thin-wall layer material; ρ 0 Representing the density of the thin-wall layer; m represents the unit mass; p is p 2 Representing penalty factors; d (D) shell Representing a thin-wall layer elastic matrix; d (D) infill Representing the packed layer elastic matrix.
Step 7: and carrying out finite element analysis by combining a macroscopic multi-scale material interpolation model to obtain the flexibility of the structure.
Step 8: solving to obtain the sensitivity of the structure flexibility and the mass fraction with respect to the design variable;
from the derived sensitivity formula, the compliance c and the structural mass m are determined with respect to the global topological field μ (e) Lattice density field v (e) Lattice configuration field z i (e) The sensitivity of (2) can be obtained from the following formula:
in the formula, e represents a unit number, and i represents a hidden variable number.And->Expressed as:
wherein the method comprises the steps ofAnd->The method comprises the following steps of:
wherein A is a group of the total number of the,and->The method comprises the following steps of:
wherein j represents the sample number of the training set, and the value range is interval [1, N data ]An integer thereon.
In additionAnd->Can be developed by a chain rule respectively:
will beSubstituted into->Obtaining:
to this end, the structural flexibility and the structural mass are relative to the design variables μ, z i The sensitivity expression of v is derived.
Step 9: submitting the sensitivity to a moving asymptote algorithm (Method of Moving Asymptotes, MMA) solver to update the design variables μ, z according to the objective function and constraints determined by the mathematical model of the optimization problem i ,v；
Step 10: if the number of iterations k is greater than or equal to 200 or the maximum variation of the design variable is less than 0.01 at the kth iteration step, thenMeeting convergence condition, stopping iteration and outputting the integral topological field mu and lattice configuration field z i Otherwise, repeating the steps 3 to 9; the unit lattice configuration variable z to be output (e) Rounding to the t-th meeting the minimum two norms (e) The value z (t) of quasi-lattice configuration in hidden variable space (e) )：
Wherein t is (e) The class value of lattice unit distributed by the unit e is represented, and the range of the class value is interval [1, N ] cls ]An integer thereon.
Step 11: based on the existing integral topological field mu and lattice configuration field z i And (3) taking the lattice density field v as an optimization variable, establishing a mathematical model of the second-stage optimization problem by taking the structural flexibility as an optimization target and the volume fraction as a constraint condition, and repeating the contents of the steps 3 to 9.
The second-stage optimization model can be expressed as:
s.t.KU＝F
g mass ＝frac-frac * ≤0，
0<v - ≤v≤v +
wherein v is - (v + ) Representing the lower and upper limits of lattice density; frac * Representing a predetermined quality score threshold; the values are the same as the first stage.
Step 12: and if the maximum change of the design variable is smaller than 0.01 in the kth iteration step, considering that the convergence condition is met, stopping iteration and outputting a result.
One specific embodiment of the above scheme is as follows:
this embodiment is a general 2D simply supported beam, the constraint condition of which is shown in fig. 3, the gray part is a design domain, and the black area near the constraint point is a non-design domain. The filter boundary conditions are Dirichlet boundary conditions.
Example 2
Another embodiment of the present disclosure is as follows:
step 1: 6 isotropic lattice unit cells with different configurations are adopted, and the equivalent elastic matrix response number N resp =4; at [0,1]Uniformly sampling 20 lattice unit cells of each class in the density range, and totaling N data =120 samples, and construct a function map D of lattice unit cell configuration, density and its equivalent elastic tensor using a four-response gaussian two-dimensional hidden variable model lattice 。
Step 2: the design domain of the simply supported beams is determined and the finite element mesh is divided by 150×50. Elastic modulus E of the thin-walled body 0 =1, mass density ρ 0 =1, the ratio of the elastic modulus and the mass density of the filler to the material parameters of the thin-walled body is λ E =0.4 and λ m =0.7, poisson ratio v of thin-wall layer and filling is 0.3, penalty factor p 1 And p is as follows 2 Setting 1, setting the overall topological variable mu, the lattice density variable v and the lattice configuration variable z as design variables, and controlling the lattice unit cell density to [0.2,0.9 ]]Upper limit frac of structural mass fraction * At 0.4, a mathematical model of the optimization problem is built with compliance as the objective function (first stage).
Step 3: the lattice density field v and the lattice configuration field z i (i=1, 2) carrying in the constructed multi-response Gaussian hidden variable model, and calculating the lattice unit cell equivalent elastic matrix D lattice 。
Step 4: as shown in fig. 4, a packed density field is obtained by three steps of PDE filtering and Heaviside projectionAnd thin-walled density field->R is taken out respectively from the radius of twice filtering 1 ＝12、R 2 =5, the PDE filtering boundary condition is set as shown in fig. 3, whereby solving the PDE equation yields filtered +.> And->The initial value of the projection sharpness beta is 2, each iteration is multiplied by 2 in 30 steps, and the upper limit is 64; the projection threshold η remains at 0.5 at all times; in this embodiment, the parameters of the two projections are the same.
Step 5: penalty function is introduced and lattice unit cell equivalent elastic matrix D is combined lattice And filling density fieldCalculating a filling layer elastic matrix D infill ，；/>
Step 6: combining filled density fieldsThin-walled density field->And filling layer elastic matrix D infill And combining the thin-wall filling structure multi-scale material interpolation model to obtain an elastic matrix D and a mass density m of the units in the design domain. Wherein p is 1 、p 2 、λ E And lambda (lambda) m Has been initialized in step 2.
Step 7-8: finite element analysis is carried out by combining a macroscopic multi-scale material interpolation model, and the flexibility c and the structural mass m relative to the whole topological field mu are respectively calculated according to a deduced sensitivity formula (e) Lattice density field v (e) Lattice configuration field z i (e) Is subjected to sensitivity analysis.
Step 9-10: and submitting the sensitivity analysis result to an MMA solver for updating the design variables. If it isThe first stage iteration times are more than 200 or the maximum change of design variables is less than 0.01, the convergence condition is considered to be satisfied, and the iteration is stopped and the integral topological field mu and the lattice configuration field z are output i (i=1, 2); using a minimized two-norm-to-lattice configuration field z i (i=1, 2) rounding; otherwise, returning to the step 3 to repeat the process.
Step 11: based on rounded configuration field z i (i=1, 2) and the output overall topological field μ, creating a mathematical model that uses the lattice density field v as a design variable and the compliance as an objective function to create an optimization problem (second stage). The design variable range and the optimization parameters are the same as the first-stage optimization model.
Step 12: the first stage of interpolation model assembly, finite element analysis, sensitivity analysis, and optimization iteration process are repeated. If the maximum change of the design variable is smaller than 0.01, the convergence condition is considered to be satisfied, iteration is stopped, and an optimization result is output; otherwise, repeating the above process.
The present invention is not limited to the above embodiments, and those skilled in the art can implement the present invention in various other embodiments according to the examples and the disclosure of the drawings, so that the design of the present invention is simply changed or modified while adopting the design structure and concept of the present invention, and the present invention falls within the scope of protection.
Claims (10)
1. The integrated design method for the thin-wall multi-class lattice filling structure is characterized by comprising the following steps of:
step 1: combining N to be adopted cls Lattice unit cell databases of more than or equal to 2 different configurations, and performing density sampling on lattice unit cells of all configurations by a test design method to obtain N data Calculating an equivalent elastic matrix of a sample lattice unit cell by adopting a homogenization theory according to sample data, and constructing a lattice density field v and a lattice configuration field z based on the lattice unit cell sample data i (i=1,., N) and an equivalent elastic matrix, the response number is N resp ；
Step 2: root of Chinese characterEstablishing a finite element model according to design requirements, determining a non-design domain and a design domain, and selecting an overall topological field mu, a lattice density field v and a lattice configuration field z i (i=1,., n) is a design variable, a first stage optimization model is built that targets structural compliance, mass fraction is a constraint, initializing key parameters of the first-stage optimization model;
step 3: based on the lattice density field v and the lattice configuration field z i (i=1,., n) calculating a lattice unit cell equivalent elastic matrix D in the multi-response gaussian hidden variable prediction model lattice Wherein n represents the dimension of the multi-response gaussian hidden variable space;
step 4: based on the integral topological field mu, combining a three-step filtering method and a Heaviside projection to obtain a filling density fieldAnd thin-walled density field->
Step 5: according to lattice unit cell equivalent elastic matrix D lattice And filling density fieldCalculating a filling layer elastic matrix D infill Introducing a punishment function;
step 6: according to the filling density fieldThin-walled density field->And filling layer elastic matrix D infill Combining the thin-wall filling structure multi-scale material interpolation model to obtain an elastic matrix D and a mass density m of units in a design domain:
step 7: carrying out finite element analysis by combining a macroscopic multi-scale material interpolation model to obtain the flexibility of the structure;
step 8: solving to obtain the sensitivity of the structure flexibility and the mass fraction with respect to the design variable;
step 9: according to the objective function and constraint conditions determined by the mathematical model of the optimization problem, submitting the sensitivity to a mobile asymptote algorithm solver to update the overall topological field mu, the lattice density field v and the lattice configuration field z i ；
Step 10: if the iteration number k is more than or equal to 200 or the maximum change of the design variable is less than 0.01 in the kth iteration step, the convergence condition is considered to be satisfied, and the iteration is stopped and the integral topology field mu and the lattice configuration field z are output i Otherwise, repeating the steps 3 to 9;
step 11: based on the existing integral topological field mu and lattice configuration field z i Establishing a second-stage optimization model by taking the lattice density field v as an optimization variable and the structural flexibility as an optimization target and the volume fraction as a constraint condition, and repeating the contents from the step 3 to the step 9;
step 12: if the maximum change of the design variable is smaller than 0.01 in the kth iteration step, the convergence condition is considered to be satisfied, the iteration is stopped and the result is output, otherwise, the step 11 is repeated.
2. The integrated design method of thin-wall multi-class lattice filling structure according to claim 1, wherein in step 1, the multi-response gaussian hidden variable prediction model is:
wherein h represents a compound represented by N resp A priori mean vector of 1, Y represents the constructed N resp ×N data A database matrix of the equivalent elastic matrix of the Viomonas;the representative regression coefficient matrix is expressed as:
wherein H is represented by N data The a priori mean function h constitutes the transpose of the matrix:
r represents a co-correlation matrix of input parameters of the lattice unit cell database, and R * The co-correlation vectors representing the predicted points and the training set input parameters are respectively expressed as follows:
wherein, r (·, ·) represents a gaussian kernel function reflecting the degree of correlation of two vectors, expressed as follows:
r(s,s′)＝exp[-(v-v′) T Φ(v-v′)-(z(t)-z(t′)) T (z(t)-z(t′))]
where s denotes an input vector comprising a qualitative variable v and a quantitative variable t, s ' denotes an input vector comprising a qualitative variable v ' and a quantitative variable t ', a diagonal matrix Φ is set to avoid over-parameterization of the qualitative and quantitative variables, s and Φ are expressed as follows:
s＝[v,z(t) T ] T ＝[v,z 1 ,...,z i ] T
Φ＝diag(φ)，φ＝[φ 1 ,φ 2 ,…,φ p ] T
where v represents the quantitative variable of the input, i.e. the lattice unit cell density variable, then the diagonal matrix Φ dimension p=1; z (t) T The qualitative variables representing the inputs, i.e., lattice unit configuration variables, the lattice unit qualitative variables t map to a collection of i-dimensional hidden variable spaces.
3. The integrated thin-wall multi-class lattice filling structure design method according to claim 2, wherein in step 2, the first-stage optimization model is expressed as:
s.t.KU＝F
g mass ＝frac-frac * ≤0，
0≤μ≤1
0<v - ≤v≤v +
wherein s.t. represents a constraint condition, and T represents a transpose of the matrix; k represents the structural total stiffness matrix, and U and F represent global displacement and load vectors; u (u) e Representing cell node displacement vectors, k e Representing a cell stiffness matrix, w being a cell number; n represents the number of design domain units; mu (mu) (e) Cell density, v, representing the overall topological field (e) Representing the density of the cell lattice,cell variable values representing lattice configuration fields; i represents the hidden variable number, i=1, 2 in the two-dimensional hidden variable space; mu is mu (e) The vector of the components, v is v (e) Component vector, z is->A vector of components; v - (v + ) Representing the lower and upper limit of the unit cell density, < >>The upper limit and the lower limit of the value of the ith hidden variable are represented; g mass Representing a quality score constraint; frac is the mass fraction, equal to the structural mass m (μ, v) and the total mass m of the initial design domain 0 Ratio of; frac * Representing a preset quality score threshold.
4. The integrated design method for thin-wall multi-class lattice filling structure according to claim 3, wherein in step 4, the filtering expression form is:
in the method, in the process of the invention,for Laplacian, τ is the density of the unit before filtering, and the filtered unit density ++can be obtained by applying filtering boundary conditions>R f Is the filter radius.
5. The integrated thin-wall multi-class lattice filling structure design method according to claim 4, wherein in step 4, the e-th unit filtered unit densityUnit density obtained after projection of Heaviside +.>The method comprises the following steps:
where beta is the sharpness of the projection, eta is the threshold of the projection,represents the unit density after the e-th unit filtering +.>The unit density obtained after projection;
the filling density field can be obtained through three filtering and projection operationsAnd thin-walled density field->
6. The integrated design method of thin-wall multi-class lattice filling structure according to claim 5, wherein in step 5, the filling layer elastic matrix D infill The expression form is as follows:
wherein D is 0 Representing a minimum elastic matrix that avoids singular global stiffness matrices; p is p 1 Representing a penalty factor.
7. The integrated thin-wall multi-class lattice filling structure design method according to claim 6, wherein in step 5, the penalty function for the filling layer elastic matrix is expressed as:
wherein λ=500; lattice configuration variable z= [ z ] 1 ,z 2 ] T ；z(t)＝[z 1 (t),z 2 (t)] T ,t＝1...N cls Representing N in two dimensions cls A unit cell-like configuration distribution position; gamma is the diagonal length of the smallest bounding rectangle of all cells.
8. The integrated design method of thin-wall multi-class lattice filling structure according to claim 7, wherein in step 6, the elastic matrix D and mass density m of the cells in the design domain are:
wherein lambda is m And lambda is E Representing the mass ratio and the elastic modulus ratio of the filling material and the thin-wall layer material; ρ 0 Representing the density of the thin-wall layer; m represents the unit mass; p is p 2 Representing penalty factors; d (D) shell Representing a thin-wall layer elastic matrix; d (D) infill Representing the packed layer elastic matrix.
9. The integrated design method of thin-wall multi-class lattice filling structure according to claim 8, wherein in step 8, the flexibility c and the sensitivity of the structural mass m to the design parameters can be obtained by the following formula:
and->Expressed as:
wherein the method comprises the steps ofAnd->The method comprises the following steps of:
wherein A is a group of the total number of the,and->The method comprises the following steps of:
wherein e represents a unit number, i represents a hidden variable number; j represents the number of the training set sample, and the value range is interval [1, N data ]An integer thereon.
10. The integrated thin-wall multi-class lattice filling structure design method according to claim 9, wherein in step 11, the second-stage optimization model is expressed as:
s.t.KU＝F
g mass ＝frac-frac * ≤0，
0<v - ≤v≤v +
wherein v is - (v + ) Representing the lower and upper limits of lattice density; frac * Representing a predetermined quality score threshold; the values are the same as the first stage.
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