CN116756948A - Multi-material multi-scale optimization design method for multifunctional structure - Google Patents
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Abstract
The invention discloses a multi-material multi-scale optimization design method for a multifunctional structure, and relates to the field of engineering structure design. On a mesoscale, different relative density typical cells are optimally designed based on the multifunctional cells, macroscopic rigidity and density attributes of the cells are obtained through a homogenization method, an optimal design system is built on the macroscopic scale, macroscopic relative density distribution in a design domain is obtained through a SIMP method, relative density interval division is carried out, design domain multi-material filling is carried out through the different relative density typical cells and solid cells, a lattice-entity multi-material structure is obtained, a cell structure is replaced by a solid structure with the same attribute, the macroscopic optimized design domain is filled, structural member performance simulation verification and iterative optimization design are carried out, and finally the multi-scale optimized topological structure is obtained. The invention develops an optimized design flow with low calculation cost and high design efficiency based on the multifunctional cell structure, and can be directly applied to the multifunctional and lightweight design of the engineering structure.
Description
Technical Field
The invention relates to the technical field of engineering structural design, in particular to a multi-material multi-scale optimization design method of a multifunctional structure.
Background
With the continuous improvement of the performance of weaponry and the continuous extension of the operational radius, various devices of platforms such as aviation, aerospace and vehicle-mounted platforms have better impact resistance and vibration reduction performance to face more severe external environments while meeting the light design requirements, so that a large number of applications of multi-scale structures based on a lattice are promoted. The cell scale is called mesoscale, the structure scale is called macro scale, and the mesoscale and the structure scale are mutually coupled, namely, the material distribution in the mesoscopic cell can influence the performance of the macro structure, and the boundary condition change of the macro structure can change the stress level in the cell. Therefore, in order to realize reasonable distribution of mesoscale materials so as to achieve optimal macrostructure performance, a reasonable multiscale optimization design method is needed.
The conventional multi-scale optimization method can be classified into a random cell optimization method and a non-random cell optimization method according to the mesoscopic structure adopted. The non-random cell optimization method often realizes the relative density change of a cell structure by changing the internal characteristic parameters of a single cell structure, is based on a plurality of simple face-centered cubic or body-centered cubic cells, has limited vibration reduction and buffer performance improvement effects on a macroscopic structure, and often has the situation that the designed structure cannot be processed when the relative density is close to 0 and close to 1. In contrast, the random cell optimization method has the advantages of strong designability, good cell multifunctional expansibility and the like, but has the defects of unmatched connection at different cell interfaces, low design and calculation efficiency, incapability of realizing rapid engineering application and the like.
Disclosure of Invention
The invention aims at: aiming at the defects of the prior art, the invention provides the multi-material multi-scale optimization design method of the multifunctional structure, which overcomes the defect that the traditional optimization design method cannot be rapidly applied to severe platforms with environmental conditions such as aerospace and the like, develops an optimization design flow with low calculation cost and high design efficiency based on the multifunctional cell structure, and can be directly applied to the multifunctional and lightweight design of engineering structures.
The specific technical scheme is as follows:
the invention relates to a multi-material multi-scale optimization design method of a multifunctional structure, which comprises the following steps:
step 1, on a mesoscale, optimizing and designing typical cells with different relative densities and processability and connectivity based on a multifunctional cell structure with energy absorption and vibration reduction characteristics, and obtaining macroscopic rigidity attributes and density attributes of the typical cells by a homogenization method;
step 2, on a macro scale, determining an initial design domain to be optimized, taking the lowest frequency response amplitude as an optimization target, taking the volume fraction of materials in cells and the first-order modal frequency as constraint conditions, taking the distribution of the materials in the design domain as an optimization variable, constructing an optimization design system, and obtaining the macroscopic relative density distribution in the design domain by a solid isotropic material punishment method;
step 3, dividing reasonable relative density intervals according to relative density distribution in a design domain, and filling design domain multi-material by utilizing the typical cells and the solid cells with different relative densities designed in the step 1 to obtain a lattice-entity multi-material structural configuration;
and 4, filling the lattice-entity multi-material structure obtained in the step 3 into a macroscopically optimized design domain by using the macroscopic stiffness attribute and the density attribute of the cell obtained in the step 1 and replacing the cell structure with a solid structure with the same stiffness attribute and density attribute, and performing structural member performance simulation verification and iterative optimization design to finally obtain the multiscale optimized topological structure.
Furthermore, in the step 1, the multi-functional cell structure based on the energy absorption and vibration reduction characteristics comprehensively considers engineering constraints of reasonable distribution of materials in the cell and manufacturability of the cell structure, and realizes material distribution design by changing the size of the characteristic structure in the cell, so as to construct a typical cell structure with different relative densities.
Furthermore, in the step 1, a homogenization method is adopted to calculate macroscopic stiffness properties of the cell structure, and an equivalent stiffness matrix D of the cell structure is mainly solved e By strain of the cell structureFor the boundary conditions of the analysis, let the i-th element +.>Wherein D is not 0 and the values of the other elements are 0 e Is the ith column D of (2) e (i) Stress response of cell structure in current boundary state +.>The relation of->The ith column value of the equivalent stiffness matrix of the cell structure can therefore be solved by:
wherein V is the cell structure volume, N is the number of grid units of the cell structure, sigma n Is the average stress of the nth cell, deltaV n For the volume of the nth unit, sequentially solving for six times according to the mode to obtain the equivalent stiffness matrix D of the cell structure e 。
Further, in the step 2, the construction of the optimization design system is specifically:
taking material distribution in a target structural design domain as an optimization variable, taking the lowest frequency response amplitude as an optimization target, taking the volume consumption of the material and the first-order modal frequency as constraint conditions, and constructing an optimization design system, wherein the specific mathematical expression is as follows:
wherein:for the design variables, the relative density ρ of each cell is included i ；/>Is a frequency response; K. m, C are the total stiffness matrix, the total mass matrix and the resistance coefficient matrix of the structure respectively; u is structural displacement; v is the total volume of the material during the optimization; />Is the upper limit of the total volume of the structure; f (f) 1 The first-order modal frequency of the structure; />Is a preset lower modal frequency limit.
Further, in the step 2, macroscopic relative density distribution in the design domain is obtained by a punishment method of the solid isotropic material, which specifically comprises the following steps:
obtaining the relative density ρ of units by SIMP method i And cell stiffness matrix k i The mapping relation between the two is as follows:
in omega i Designing a domain for a unit; b is a strain matrix of the unit; d (D) A And D 0 Rigidity matrix of lattice cell and entity cell respectively, D A ·D 0 For both reaches Ha Maji dΩ represents the cell design field Ω i Is a differential unit of (a);
and assembling the unit stiffness matrix into a structure total stiffness matrix K, calculating to obtain a structure displacement U by a finite element method, solving sensitivity information of a topological optimization problem according to a chain rule, and updating design variables by using a moving asymptote method until convergence conditions are met, so as to obtain macroscopic relative density distribution in a design domain.
Further, the step 3 divides the macroscopically optimized design domain into a plurality of (ρ) cells based on the typical cells having different relative densities obtained in the step 1 m -a,ρ m +a]Relative density interval, where ρ m For the relative density of the m-th typical cell structure, a is the density interval coefficient, and the set of a plurality of sub-density intervals is ensured to be [0,1]。
Further, in the step 3, with respect to the design domain after the relative density interval is divided, the relative density obtained by the optimization design in the step 1 is ρ m Is filled into the corresponding (ρ) m -a,ρ m +a]In the density interval, in addition, for the area with the relative density approaching 0, in order to avoid the generation of interrupted material areas, a cell structure filling with smaller relative density is designed; and for the area with the relative density close to 1, solid cells are directly adopted for filling, so that a lattice-entity multi-material structure configuration is obtained, and the problem of weak connection caused by adopting different cell structures at any relative density position is effectively avoided.
The beneficial effects of the invention are as follows:
according to the multi-material multi-scale optimization design method for the multi-functional structure, provided by the invention, the multi-functional cell structure with energy absorption and vibration reduction characteristics is taken as a basic unit, the equivalent density change is realized by changing the internal structural parameters of the cells, the optimized macro structure is ensured to have excellent vibration reduction and buffering performance, and meanwhile, the problem of poor connectivity among cells caused by adopting a random cell structure is solved.
According to the multi-material multi-scale optimization design method for the multi-functional structure, disclosed by the invention, multi-material interpolation is carried out by combining cell-entity materials, so that the mechanical property of a macroscopic structure is effectively improved, and the problem that the cell processability of the relative density approaching 0 and 1 is poor in the traditional multi-scale optimization method is solved.
According to the multi-material multi-scale optimization design method for the multi-functional structure, provided by the invention, the optimized macro-scale material filling is carried out on a plurality of typical cell structures which are excellent in performance and convenient to process, the cell structures are replaced by solid structures with equivalent macro-stiffness properties by adopting a homogenization method, and performance verification and iterative optimization are carried out on the optimized target structure, so that the efficiency of optimization design and simulation calculation is improved.
According to the multi-material multi-scale optimization design method for the multifunctional structure, provided by the invention, factors such as optimization design efficiency, vibration reduction and buffering performance, manufacturability of the optimized structure and the like are comprehensively considered, and the requirements of light weight and mechanical environment adaptability design of various devices on platforms such as aviation, aerospace and vehicle are better met.
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For a clearer description of the technical solutions of embodiments of the present invention, the drawings that are needed in the embodiments will be briefly described below, it being understood that the following drawings only illustrate some embodiments of the present invention and should not be considered limiting in scope, and other related drawings can be obtained according to these drawings without inventive effort for a person skilled in the art, wherein:
FIG. 1 is a flow chart of a multi-material to scale optimization design method of the multi-functional structure of the present invention;
FIG. 2 is a schematic diagram of a cell model in an embodiment of the invention;
FIG. 3 is a schematic diagram of a lattice structure formed by the cell array shown in FIG. 2;
FIG. 4 is an optimized target structure in an embodiment of the invention;
FIG. 5 is a schematic diagram of an optimized multifunctional structure configuration in an embodiment of the present invention;
FIG. 6 is a graph comparing frequency response curves of target structures before and after multi-scale optimization in an embodiment of the invention.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the particular embodiments described herein are illustrative only and are not intended to limit the invention, i.e., the embodiments described are merely some, but not all, of the embodiments of the invention. The components of the embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations.
It should be noted that the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus.
Example 1
The invention relates to a multi-material multi-scale optimization design method of a multi-functional structure, as shown in figure 1, on the mesoscale, a typical cell structure with different relative densities is optimally designed based on the multi-functional cells taking into consideration the energy absorption and vibration reduction characteristics, and macroscopic density attributes and rigidity attributes of the typical cell structure are obtained through a homogenization method; on a macro scale, determining an initial design domain to be optimized, taking the lowest frequency response amplitude as an optimization target, taking the volume fraction of materials in a design domain and the first-order modal frequency as constraint conditions, taking the distribution of the materials in the design domain as an optimization variable, constructing an optimization design system, and obtaining the macro relative density distribution in the design domain by a solid isotropic material punishment (Solid isotropic material with penalization, SIMP) method; dividing reasonable relative density intervals according to relative density distribution in a design domain, and filling the design domain by utilizing typical cells and solid cells with different relative densities to obtain a lattice-entity multi-material structural configuration; and replacing a cell structure with a solid structure with the same density attribute and rigidity attribute, and performing structural member performance simulation verification and iterative optimization design to finally obtain the multi-scale optimized topological structure.
Referring to fig. 2 and 3, in this example, a cell structure with better connectivity and energy absorption and vibration reduction properties is adopted as a lattice unit. The cell structure has higher porosity and can reduce the weight by more than 55% at least compared with a solid structure; the vibration damping effect is good, a typical mass array-spring structure is formed by the central mass block and the bending piece, and a local resonance forbidden band effect can be generated in a vibration environment so as to damp vibration energy in a desired frequency band; the energy absorption effect is good, and the energy absorption device is a complex modified negative poisson ratio structure.
In consideration of cell structural strength and manufacturability, cells with relative densities of 0.1, 0.3, 0.5, 0.7 and solid cells are respectively designed to be used as filling materials in density intervals of 0-0.2, 0.2-0.4, 0.4-0.6, 0.6-0.8 and 0.8-1.0.
And calculating macroscopic rigidity attribute of the cell structure by adopting a homogenization method. Homogenizing the equivalent is mainly solving the equivalent stiffness matrix D of the cell structure e By strain of the cell structureFor the boundary conditions of the analysis, let the i-th element +.>The value of (D) is not 0, and the values of the other elements are 0 e Is the ith column D of (2) e (i) Stress response of cell structure in current boundary state +.>The relation of->The ith column value of the cell equivalent stiffness matrix can be solved by:
wherein V is the cell structure volume, N is the number of grid units of the cell structure, sigma n Is the average stress of the nth cell, deltaV n Is the firstThe volumes of n units are sequentially solved for six times according to the mode, and a cell structure equivalent stiffness matrix D can be obtained e 。
Referring to fig. 4, in this example, the macro material distribution optimization design is performed on the flat sample, which includes the following sub-steps:
1) Taking material distribution in a target structural design domain as an optimization variable, taking the lowest frequency response amplitude as an optimization target, taking the volume consumption of the material and the first-order modal frequency as constraint conditions, and constructing an optimization design system, wherein the specific mathematical expression is as follows:
wherein:for the design variables, the relative density ρ of each cell is included i ；/>Is a frequency response; K. m, C are the total stiffness matrix, the total mass matrix and the resistance coefficient matrix of the structure respectively; u is structural displacement; v is the total volume of the material during the optimization; />Is the upper limit of the total volume of the structure; f (f) 1 The first-order modal frequency of the structure; />Is a preset lower modal frequency limit.
This exampleIn the process, the liquid crystal display device comprises a liquid crystal display device,the value is 0.5%>And the value 900 is taken, and the optimization target is that the amplitude of the frequency response function of the frequency band of 15-2000 Hz is minimum.
2) Obtaining the relative density ρ of units by SIMP method i And cell stiffness matrix k i The mapping relation between the two is as follows:
in omega i Designing a domain for a unit; b is a strain matrix of the unit; d (D) A And D 0 Rigidity matrix of lattice cell and entity cell respectively, D A ·D 0 For both, dΩ represents the cell design field Ω (Hadamard product) of Ha Maji i Is a differential unit of (a);
3) And assembling the unit stiffness matrix into a structure total stiffness matrix K, and calculating by a finite element method to obtain the structure displacement U. According to the chained rule, solving the sensitivity information of the topological optimization problem, and updating the design variable by using a moving asymptote method (MMA) until the convergence condition is met, so as to obtain the macroscopic relative density distribution in the design domain.
Referring to fig. 5, the cells with relative densities of 0.1, 0.3, 0.5, 0.7 and solid (with relative density of 1) are embedded into the regions with relative densities of 0-0.2, 0.2-0.4, 0.4-0.6, 0.6-0.8 and 0.8-1.0 after macroscopic optimization design, so as to obtain the configuration of the multi-functional multi-material structure after multi-scale topological optimization.
For the optimized configuration, the solid structure with the same macroscopic rigidity attribute is used for replacing cell materials, and the simulation calculation of the structural mechanical property is performed, so that the calculation efficiency is improved. Referring to the comparison of the structure frequency response curves before and after the optimization shown in fig. 6, the amplitude of the optimized structure frequency response function is reduced by about 40%, the RMS value is reduced from 2.03×104 to 1.92×104, and the proposed multi-scale optimization design method is proved to have better effect.
The invention is not limited to the specific embodiments described above. The invention extends to any novel one, or any novel combination, of the features disclosed in this specification, as well as to any novel one, or any novel combination, of the steps of the method or process disclosed.
Claims (7)
1. The multi-material multi-scale optimization design method for the multifunctional structure is characterized by comprising the following steps of:
step 1, on a mesoscale, optimizing and designing typical cells with different relative densities and processability and connectivity based on a multifunctional cell structure with energy absorption and vibration reduction characteristics, and obtaining macroscopic rigidity attributes and density attributes of the typical cells by a homogenization method;
step 2, on a macro scale, determining an initial design domain to be optimized, taking the lowest frequency response amplitude as an optimization target, taking the volume fraction of materials in cells and the first-order modal frequency as constraint conditions, taking the distribution of the materials in the design domain as an optimization variable, constructing an optimization design system, and obtaining the macroscopic relative density distribution in the design domain by a solid isotropic material punishment method;
step 3, dividing reasonable relative density intervals according to relative density distribution in a design domain, and filling design domain multi-material by utilizing the typical cells and the solid cells with different relative densities designed in the step 1 to obtain a lattice-entity multi-material structural configuration;
and 4, filling the lattice-entity multi-material structure obtained in the step 3 into a macroscopically optimized design domain by using the macroscopic stiffness attribute and the density attribute of the cell obtained in the step 1 and replacing the cell structure with a solid structure with the same stiffness attribute and density attribute, and performing structural member performance simulation verification and iterative optimization design to finally obtain the multiscale optimized topological structure.
2. The multi-functional multi-material multi-scale optimization design method according to claim 1, wherein the multi-functional cell structure based on the energy absorption and vibration reduction characteristics in the step 1 comprehensively considers engineering constraints of reasonable distribution of materials in cells and manufacturability of the cell structure, realizes material distribution design by changing the size of the characteristic structure in the cells, and constructs typical cell structures with different relative densities.
3. The multi-functional multi-material multi-scale optimization design method according to claim 1, wherein in the step 1, a homogenization method is adopted to calculate macroscopic stiffness properties of the cell structure, and an equivalent stiffness matrix D of the cell structure is mainly solved e By strain of the cell structureFor the boundary conditions of the analysis, let the i-th element +.>Wherein D is not 0 and the values of the other elements are 0 e Is the ith column D of (2) e (i) Stress response to cell structure in current boundary stateThe relation of->The ith column value of the equivalent stiffness matrix of the cell structure can therefore be solved by:
wherein V is the cell structure volume, N is the number of grid units of the cell structure, sigma n Is the average stress of the nth cell, deltaV n For the volume of the nth unit, sequentially solving for six times according to the mode to obtain the equivalent stiffness matrix D of the cell structure e 。
4. The multi-material multi-scale optimization design method of the multi-functional structure according to claim 3, wherein in the step 2, an optimization design system is built specifically as follows:
taking material distribution in a target structural design domain as an optimization variable, taking the lowest frequency response amplitude as an optimization target, taking the volume consumption of the material and the first-order modal frequency as constraint conditions, and constructing an optimization design system, wherein the specific mathematical expression is as follows:
find
min
wherein:for the design variables, the relative density ρ of each cell is included i ；/>Is a frequency response; K. m, C are the total stiffness matrix, the total mass matrix and the resistance coefficient matrix of the structure respectively; u is structural displacement; v is the total volume of the material during the optimization; />Is the upper limit of the total volume of the structure; f (f) 1 The first-order modal frequency of the structure; />Is a preset lower modal frequency limit.
5. The multi-functional multi-material multi-scale optimization design method according to claim 4, wherein the step 2 is to obtain the macroscopic relative density distribution in the design domain by using a solid isotropic material punishment method, specifically:
obtaining the relative density ρ of units by SIMP method i And cell stiffness matrix k i The mapping relation between the two is as follows:
in omega i Designing a domain for a unit; b is a strain matrix of the unit; d (D) A And D 0 Rigidity matrix of lattice cell and entity cell respectively, D A ·D 0 For both reaches Ha Maji dΩ represents the cell design field Ω i Is a differential unit of (a);
and assembling the unit stiffness matrix into a structure total stiffness matrix K, calculating to obtain a structure displacement U by a finite element method, solving sensitivity information of a topological optimization problem according to a chain rule, and updating design variables by using a moving asymptote method until convergence conditions are met, so as to obtain macroscopic relative density distribution in a design domain.
6. The multi-functional multi-material multi-scale optimization design method according to claim 1, wherein the step 3 is based on the typical cells with different relative densities obtained in the step 1, and divides the macroscopically optimized design domain into a plurality of (ρ m -a,ρ m +a]Relative density interval, where ρ m For the relative density of the m-th typical cell structure, a is the density interval coefficient, and the set of a plurality of sub-density intervals is ensured to be [0,1]。
7. The multi-functional multi-material multi-scale optimization design method according to claim 6, wherein the step 3 is characterized in that the design domain after the relative density interval is divided is optimized in the step 1 to obtain the relative density ρ m Is filled into the corresponding (ρ) m -a,ρ m +a]Density regionIn addition, for the area with the relative density close to 0, in order to avoid the generation of interrupted material areas, a cell structure filling with smaller relative density is designed; and for the area with the relative density close to 1, solid cell filling is directly adopted to obtain the lattice-solid multi-material structure configuration.
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