CN110941924A - Multi-component system integration integrated multi-scale topology optimization design method - Google Patents
Multi-component system integration integrated multi-scale topology optimization design method Download PDFInfo
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Abstract
The invention belongs to the field of integrated design methods, and discloses a multi-scale topological optimization design method for integrated integration of a multi-component system. The method comprises the following steps: describing each component in the multiple components by adopting a function respectively, acquiring the density of each component according to a discrete material optimization method, mapping the density of each component to a grid obtained by dividing the macro structure design domain, interpolating the Young modulus of the support structure with the elastic modulus of each component in the grid mapped to the macro structure so as to obtain an interpolated equivalent elastic modulus, and homogenizing the macro units according to a stiffness matrix K aiming at the macro unitsHAnd macro-units in macro-scaleDisplacement field UiAnd constructing a multi-component system integration integrated multi-scale design optimization model, carrying out sensitivity analysis on design variables, and iteratively updating the design variables in a macro scale and a micro scale. The invention realizes the multi-scale optimization design of the integration of a multi-component system.
Description
Technical Field
The invention belongs to the field of integrated design methods, and particularly relates to a multi-scale topological optimization design method for integrated integration of a multi-component system.
Background
Conventional design methods select materials based on the structural design and then design the optimal structure for a given material. At present, more design freedom degrees are given to additive manufacturing, the structure is often designed into a porous structure or a lattice structure, and the periodic porous composite material can achieve the given strength and rigidity target while the weight and the efficiency are reduced, so that the weight reduction target is achieved on the premise of improving the performance of the whole structure. The multi-scale integrated design is that a macro structure is assumed to be composed of periodic micro-structure arrangement, the two-scale parallel design comprises a coupling process of two stages, and the external stage is that the optimal layout of the structure is obtained by the macro structure under boundary conditions, applied loads and constraint conditions; since each macro-cell is treated as an element with a pseudo-density, the cell can be subdivided into a representation of microstructures with an intermediate density, so the internal phase can be seen as a topological optimization of the microstructures to optimize the cell to obtain an effective elastic matrix. Multi-component systems integrate one or more components of a particular constant shape within a limited design area to meet desired functional requirements. The multi-component system integrates various performance indexes to re-optimize the layout of the embedded object, simplifies the manufacturing process and saves the cost. The multi-component system is embedded into a macro structure to find the optimal layout, the performance index of the whole structure is not only influenced by the macro factors, but also the micro material structure influences the optimal position of the embedded component and the optimal layout of the structure. Integrated applications of multi-component systems often require an overall lightweight design, but at the same time require higher performance goals.
Therefore, a multi-component system integration integrated model is needed to be established in the field to perform a multi-scale topology optimization design method, so that the optimal position and the optimal structure layout of an embedded component are obtained in the process of embedding a multi-component system into a macro structure, and the overall lightweight design and the higher performance goal are achieved.
Disclosure of Invention
Aiming at the above defects or improvement requirements of the prior art, the invention provides a multi-component system integration integrated multi-scale topological optimization design method, wherein the multi-component system integration integrated optimization design is combined, the multi-components are correspondingly mapped into a divided grid of a design domain through a function, meanwhile, a supporting material and the embedded multi-component elastic modulus are interpolated to obtain a supporting structure and an interpolated equivalent elastic modulus mapped to each component in the macro structure grid, then, a displacement field of a macro structure and a micro unit is obtained based on finite element analysis, a multi-component system integration integrated multi-scale design optimization model is constructed, then, sensitivity analysis is carried out on design variables in the macro scale and the micro scale, design variables in the macro scale and the micro scale are iteratively updated, and thus, the optimal positions of the multiple components and the optimal layout of the supporting structure are determined, the invention simultaneously considers a plurality of key factors such as material attributes and quantity of the components, mutual constraints among the components and between the components and a design domain, synthesis of different performance indexes, influence of macroscopic load and boundary conditions on the layout of a macroscopic structure and the like, and realizes that the optimal position and the optimal layout of the structure of the embedded component are obtained in the process of embedding the multi-component system into the macroscopic structure, thereby realizing the overall lightweight design and obtaining higher performance targets.
In order to achieve the aim, the invention provides a multi-component system integration integrated multi-scale topology optimization design method, which comprises the following steps:
s1, meshing a macro structure design domain of the design material structure to divide the macro structure design domain into a plurality of macro units, and respectively meshing the macro units to obtain a plurality of micro units of each macro unit;
s2, describing each component in the multiple components by adopting a function respectively, acquiring the density of each component according to a discrete material optimization method, and mapping the density of each component to a grid obtained by dividing the macro structure design domain to form an embedded component;
s3 interpolating the elastic modulus of each component in the grid mapped to the macrostructure with the support structure of the design material structure, thereby obtaining interpolated equivalent elastic moduli of each component in the grid mapped to the macrostructure and the support structure;
s4 determining a macro-cell homogenization stiffness matrix K for the latticed macro-structure and micro-cellsHAnd displacement field U of macro unit in macro scaleiAnd constructing an integrated multi-component system integration multi-scale design optimization model by combining the support structure and the interpolated equivalent elastic modulus mapped to each component in the macro-structure grid, then carrying out sensitivity analysis on design variables in the macro-scale and the micro-scale, and iteratively updating the design variables in the macro-scale and the micro-scale, thereby determining the optimal positions of the components and the optimal layout of the support structure.
Further preferably, in step S2, the density of each component is mapped into the grid by using a density point method in which a density variable and a grid centroid are combined; the discrete material optimization method obtains a calculation model of the density of each component as follows:
wherein: n is a radical ofcIs the number of elements, i, k is the center point of one of the cells and one of the elements, β is a rate parameter controlling the macro cell density towards 0 or 1, di，kIs the distance, r, between macro-unit i and module centroid kcIs the radius of the assembly.
More preferably, step S3 specifically includes: interpolating the elastic modulus of each component in the support structure of the design material structure and the grid mapped to the macrostructure by adopting a discrete material optimization method, thereby obtaining the interpolated equivalent elastic modulus of each component in the support structure and the grid mapped to the macrostructure, wherein the calculation model of the equivalent elastic modulus is as follows:
where ρ ismac，iIs the density of the element, p, corresponding to the support structure on a macroscopic scalec，i，kFor inserting the element density of the component of material k, EiIs the interpolated equivalent elastic modulus; e0Is the modulus of elasticity of the support structure; ec，kIs the modulus of elasticity of the component with insertion material k; n is a radical ofmIs the amount of material inserted into the assembly.
Preferably, in step S4, the multi-component system integration multi-scale design optimization model is:
Find：ρmac＝{ρmac，1，ρmac，2，…，ρmac，M}，ρmic＝{ρmic，1，ρmic，2，…，ρmic，Nc}
ρc＝{ρc，1，ρc，2，…，ρc，Nc}，Sk＝{Xk，1，Yk，1，Xk，2，Yk，2，…，Xk，Nc，Yk，Nc}
wherein C represents the compliance of the macrostructure, pmac，i(i ═ 1,2,3, …, M) denotes the relative density of the ith macro-unit on a macroscopic scale, ranging from ρmac，minTo 1; rhomic，e(e 1,2,3, …, N) represents the relative density of the e-th microscopic unit on the microscopic scale, ranging from ρmic，minTo 1; rhomax，miAnd ρmic，minThe relative density of the minimum element preset for avoiding singularity in the numerical calculation process; m is the number of macro units, and N is the number of micro units; rhoc，NcDenotes the NthcRelative density of individual components, NcIs a positive integer greater than 0, SkDesigning variables for the positions of the components, which are respectively: xk，Nc，Yk，Nc，θk，NcWherein X isk，NcExpressed as the abscissa, Y, of the center of mass of the assemblyk，NcIs the ordinate of the center of mass of the assembly; eiIs the interpolated equivalent elastic modulus of the support structure and each component mapped to the macrostructure grid, U is the displacement vector of the macrostructureiIs a displacement field of a macro unit, K is a rigidity matrix of a macro structure, KHA homogenizing stiffness matrix representing macro-units with an artificial density equal to 1, F being the external force to which the macro-structure is subjected, V representing the volume of the macro-structure, ViWhich represents the volume of the macro-unit,
denotes the volume of the microscopic unit, f1And is the volume fraction of the volume constraint of the macrostructure, f2Is the volume fraction of the microstructure volume constraint; g3Is a non-overlapping constraint, V, for preventing mutual interference between the embedded components and interference between the components and the supporting materialcIs the volume of the embedded component, LcIndicating the perimeter of all components.
As further preferred, LcThis can be obtained by measuring the intermediate density that occurs at the boundary of the embedded component, whose computational model is as follows:
Lc＝∫DDexp[-α(ρc，i，-0.5)m]dD
where ρ isc，iα and m are parameters that control density filtering for the density of the ith module.
More preferably, step S4 specifically includes the following steps:
s11 initializing and defining design parameters and optimization parameters;
s12 calculating the microscopic elastic matrix D of the microscopic unit by material interpolation method in microscopic scale, solving the rigidity matrix k of the microscopic unit according to the microscopic elastic matrix D, and then solving the microscopic unit by finite element analysis in microscopic scaleDisplacement field chi of observation uniti，
S13 displacement field χ according to microscopic unitiCalculating the equivalent elastic property D of the macro unit by using a homogenization methodH；
S14 equivalent elastic property D of microscopic element based on solutionHCalculating the actual elastic matrix D of the macro unit by adopting a material interpolation methodMAnd from the actual elastic matrix D of the macro-cellsMSolving a macro-unit homogenization rigidity matrix KHAnd a macro cell stiffness matrix KiThen, finite element analysis is carried out in a macroscopic scale, and the displacement field U of the macroscopic unit is calculatedi；
S15 displacement field U based on solved macro unitiCalculating an objective function C in the multi-component system integration integrated multi-scale design optimization model;
s16, carrying out sensitivity analysis on the target function C and each design variable in the constraint function in the multi-component system integrated multi-scale design optimization model according to chain calculation rules in macro and micro scales, and iteratively updating each design variable;
s17, according to the updated design variables, judging whether the objective function of the multi-component system integration multi-scale design optimization model is converged, if not, turning to the step S12, and if so, outputting the optimal positions of the components and the optimal layout of the supporting structure.
As a further preferred, the plurality of components are all cylindrical, and each of the plurality of components can be described by using an implicit function or an explicit function, respectively, where the model for describing one of the components by using the explicit function is:
wherein r iscIs the radius of the assembly; x is the number ofkThe abscissa is the center coordinate of the component; y iskIs the ordinate of the center coordinate of the assembly.
Generally, compared with the prior art, the above technical solution conceived by the present invention mainly has the following technical advantages:
1. the invention is based on the integrated optimization design of a multi-component system, correspondingly maps multiple components into a divided grid of a design domain through a function, interpolates a support material and the embedded elastic modulus of the multiple components simultaneously to obtain an interpolated equivalent elastic modulus of a support structure and each component mapped into a macro-structure grid, then obtains a displacement field of a macro structure and a micro unit based on finite element analysis, constructs an integrated multi-scale design optimization model of the multi-component system, then carries out sensitivity analysis on design variables in the macro scale and the micro scale, and iteratively updates the design variables in the macro scale and the micro scale to determine the optimal positions of the multiple components and the optimal layout of the support structure The integration of different performance indexes, the influence of macroscopic load and boundary conditions on the layout of a macroscopic structure and other key factors are realized, the optimal position and the optimal layout of the structure of an embedded component are obtained in the process of embedding the multi-component system into the macroscopic structure, and therefore the overall lightweight design is realized, and a higher performance target is obtained.
2. The invention adopts a density point method of combining density variable and grid centroid to map the density of each component into the grid, and simultaneously adopts a discrete material optimization method to interpolate the elastic modulus of each component in the grid of the designed material structure and the support structure mapped to the macrostructure, thereby better integrating each component and the support material to obtain the equivalent elastic modulus of the integrated material.
3. The integrated multi-scale design optimization model of the multi-component system simultaneously considers a plurality of key factors such as material attributes and quantity of components, mutual constraints among the components and between the components and the design domain, synthesis of different performance indexes, and influence of macroscopic load and boundary conditions on the macroscopic structure layout, and achieves the optimal position and structure layout of an embedded component in the process of embedding the multi-component system into the macroscopic structure, thereby achieving the overall lightweight design and obtaining higher performance targets.
Drawings
FIG. 1 is a flow chart of a multi-component system integration integrated multi-scale topology optimization design method according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of an initial design domain subject to external loading involved in an embodiment of the present invention;
FIG. 3 is a schematic diagram of an initial design domain embedding multiple circular components involved in an embodiment of the present invention;
figure 4 is a prior art optimized design for a single-scale cantilever beam having a C of 462.6082;
fig. 5 is an optimized design of the integrated cantilever beam of the multi-scale and multi-component system by using the method of the present invention, wherein the C of the single-scale cantilever beam is 218.82;
fig. 6 is a volume fraction iteration curve of a macro structure and a microstructure in the optimization design of the multi-scale and multi-component system integrated cantilever beam by using the method of the present invention, wherein (a) in fig. 6 is the volume fraction iteration curve of the microstructure, and (b) in fig. 6 is the volume fraction iteration curve of the macro structure;
FIG. 7 is a multi-scale topological optimization objective function iteration curve of the multi-component system in an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
As shown in fig. 1, the multi-component system integration-based multi-scale topology optimization design method specifically includes the following steps:
step one, carrying out meshing on a macro structure design domain of a design material structure to enable the macro structure design domain to be divided into a plurality of macro units, and respectively carrying out meshing on the plurality of macro units to obtain a plurality of micro units of each macro unit.
And step two, describing each component in the plurality of components by adopting a function respectively, acquiring the density of each component according to a discrete material optimization method, and mapping the density of each component to a grid obtained by dividing the macro structure design domain to form an embedded component.
And thirdly, interpolating the elastic modulus of the support structure of the design material structure and each component mapped to the grid of the macrostructure, thereby obtaining the interpolated equivalent elastic modulus of the support structure and each component mapped to the grid of the macrostructure.
Step four, determining a macro unit homogenization rigidity matrix K aiming at the macro structure and the micro unit after grid divisionHAnd displacement field U of macro unit in macro scaleiAnd constructing an integrated multi-component system integration multi-scale design optimization model by combining the support structure and the interpolated equivalent elastic modulus mapped to each component in the macro-structure grid, then carrying out sensitivity analysis on design variables in the macro-scale and the micro-scale, and iteratively updating the design variables in the macro-scale and the micro-scale, thereby determining the optimal positions of the components and the optimal layout of the support structure.
In the second step, the shape of the circular component can be expressed by using a display function, and the display function of the component can be expressed as:
wherein r iscIs the radius of the assembly circle; x is the number ofkThe abscissa is the coordinate of the center of the component; y iskIs the ordinate of the center coordinate of the component.
Mapping a plurality of component densities into a grid using a point density method in which density variables and grid centroids are co-located, for which the NthcThe relative density field of each component can be obtained by using a discrete material optimization method, wherein the component density is as follows:
where Nc is the number of components, i, k are the center points of the grid and components, respectively, β is a rate parameter that controls the macro cell density towards 0 or 1, di，kIs the distance between macro-cell i and component centroid k, and the formula is defined as:
wherein x isi、yiThe horizontal and vertical coordinates of the center of each Euler grid are taken; x is the number ofk，Nc，yk，NcAre respectively NthcThe abscissa and ordinate of the centroid of the individual embedded components.
The method of material interpolation is applied at the finite element level, here using the Discrete Material Optimization (DMO) method, specifically relating the modulus of elasticity of the support structure (support structure) to that of the embedded component, and modeling the equivalent modulus of elasticity as:
where ρ ismac，iIs the density of the element, p, corresponding to the support structure on a macroscopic scalec，i，kFor inserting the element density of the component of material k, EiIs the interpolated equivalent elastic modulus; e0Is the modulus of elasticity of the support structure; ec，kIs the modulus of elasticity of the component with insertion material k; n is a radical ofmIs the amount of material inserted into the assembly.
When the material of the plurality of inserted components is the same, i.e. NmWhen 1, the model of the equivalent elastic modulus can be simplified as the following equation:
where ρ ismac，iIs the density of elements corresponding to the support structure on a macroscopic scale; rhoc，iElement density corresponding to the insert assembly; e0Is the modulus of elasticity of the support structure; ecIs the modulus of elasticity of the insert assembly.
In the fourth step, the integrated multi-scale design optimization model of the multi-component system is as follows:
Find:ρmac＝{ρmac，1，ρmac，2，…，ρmac，M}，ρmic＝{ρmic，1，ρmic，2，…，ρmic，Nc}
ρc＝{ρc，1，ρc，2，…，ρc，Nc}，Sk＝{Xk，1，Yk，1，Xk，2，Yk，2，…，Xk，Nc，Yk，Nc}
wherein C represents the compliance of the macrostructure, pmac，i(i ═ 1,2,3, …, M) denotes the relative density of the ith macro-unit on a macroscopic scale, ranging from ρmac，minTo 1; rhomic，e(e 1,2,3, …, N) represents the relative density of the e-th microscopic unit on the microscopic scale, ranging from ρmic，minTo 1; rhomac，minAnd ρmic，minThe relative density of the minimum element preset for avoiding singularity in the numerical calculation process; m is the number of macro units, and N is the number of micro units; rhoc，NcDenotes the NthcRelative density of individual components, NcIs a positive integer greater than 0, SkDesigning variables for the positions of the components, which are respectively: xk，Nc，Yk，Nc，θk，NcWherein X isk，NcExpressed as the abscissa, Y, of the center of mass of the assemblyk，NcIs the ordinate of the center of mass of the assembly; eiIs the interpolated equivalent elastic modulus of the support structure and each component mapped to the macrostructure grid, U is the displacement vector of the macrostructureiIs a displacement field of a macro unit, K is a rigidity matrix of a macro structure, KHA homogenizing stiffness matrix representing macro-units with an artificial density equal to 1, F being the external force to which the macro-structure is subjected, V representing the volume of the macro-structure, ViWhich represents the volume of the macro-unit,
denotes the volume of the microscopic unit, g1And is the volume fraction of the macrostructure volume constraint, g2Is the volume fraction of the microstructure volume constraint; g3Is a non-overlapping constraint, V, for preventing mutual interference between the embedded components and interference between the components and the supporting materialcIs the volume of the embedded component, LcIndicating the perimeter of all components.
LcThis can be obtained by measuring the intermediate density that occurs at the boundary of the embedded component, whose computational model is as follows:
Lc＝∫Dexp[-α(ρc，i，-0.5)m]dD
where ρ isc，iα and m are parameters that control density filtering for the density of the ith module.
Preferably, in the fourth step, an integrated design model is established by establishing the integrated multi-component system in the multi-scale parallel design, and the fourth step specifically comprises the following steps:
(4.1) initially defining design parameters and optimization parameters, wherein the design parameters comprise: the length and width of the structural design domain, the number of units in the horizontal and vertical directions of the macro structure, the number of units in the horizontal and vertical directions of the micro structure, the material properties of the macro units and the micro units, and the like.
(4.2) calculating the microscopic elastic matrix D of the microscopic unit by adopting material interpolation in the microscopic scale, and solving the displacement field chi of the microscopic unit through microscopic structure finite element analysisiThe analysis was as follows:
D(ρmic，e)＝(ρmic，e)pD0
wherein D0Is the elastic modulus of the macrostructure. Finite element analysis of the microstructure is affected by periodic boundary conditions as follows:
kχi＝f
wherein:
wherein k represents the stiffness matrix of the microscopic elements χiRepresenting the displacement field of the microscopic element, b is the strain matrix.
(4.3) displacement field χ according to microscopic UnitiAnd calculating the equivalent elastic property D of the macro unit by using a homogenization methodHThe analysis process is as follows:
wherein, I is a 3 × 3 identity matrix in the two-dimensional plane, corresponding to the applied identity strain field, including horizontal identity strain, vertical identity strain and shear identity strain; chi shapeiA displacement field representing a microscopic element; b is a strain matrix.
(4.4) solving-based equivalent elastic property matrix D of microscopic elementsHCalculating the actual elastic matrix D of the macro unit by adopting a material interpolation methodMThe analysis process is as follows:
DM(ρmac，i，ρmic，e)＝(ρmac，i)pDH(ρmic，e)
(4.5) elastic matrix D based on Macro-unitsMCalculating a macro-unit homogenization rigidity matrix KHAnd a macro cell stiffness matrix KiThen, finite element analysis is carried out in a macroscopic scale, and the displacement field U of the macroscopic unit is calculatediThe analysis was as follows:
so KHCan be expressed as:
wherein B is a strain matrix and DHIs the equivalent elastic property matrix of the microscopic elements. So that the displacement field U of the macro-unitiThe calculation model of (a) is:
Ui＝Ki-1F
(4.6) displacement field U based on solved macro-unitsiAnd calculating an objective function C in the multi-component system integration multi-scale design optimization model:
wherein C is an objective function C in the multi-component system integration multi-scale design optimization model and represents the flexibility of a macro structure, U is a displacement vector of the macro structure, and U is a displacement vector of the macro structureiIs a displacement field of a macro unit, K is a rigidity matrix of a macro structure, KHA homogenizing stiffness matrix representing macro-units with an artificial density equal to 1, EiIs the interpolated equivalent modulus of elasticity for the support structure and each component mapped into the macrostructure grid, and M is the number of macro-units.
(4.7) carrying out sensitivity analysis on the objective function C and each design variable in the constraint function in the multi-component system integration integrated multi-scale design optimization model according to chain calculation rules in macro-scale and micro-scale, wherein the sensitivity analysis specifically comprises the following steps:
wherein the content of the first and second substances,
representing the objective function C on the macro scale for the ith designFirst order differential of the variables, C being the objective function;
representing the interpolated equivalent modulus of elasticity E of a support structure and of individual components mapped in said macrostructure meshiFirst order differentiation of the i design variables;
on a macroscopic scale, the density field of a discrete insert component represents the size, location of the component. The sensitivity of the objective function to the design variable of the embedded component is calculated by a chain rule:
wherein the content of the first and second substances,
one of the design variables representing the kth embedded component, i.e.
And
comprises the following steps:
based on the above steps, the method can be obtained
And
at the microscopic scaleIn degree, the objective function is relative to ρmic，eIs equal to the compliance pair rho of all macro-cellsmic，eThe sum of the sensitivities of (a), i.e.:
can be calculated based on the substep (4.3)
Thus, the objective function is for ρmic，eThe sensitivity of (a) is:
volume constraint g based on integrated model1And g2The sensitivity of the current macroscopic and microscopic volume constraints to design variables can be calculated as:
wherein, g1And is the volume fraction of the macrostructure volume constraint, g2Is the volume fraction of the microstructure volume constraint; viWhich represents the volume of the macro-unit,
representing the volume of the microscopic unit.
Non-interference constraint function g currently about multi-component systems3Design variables for components
The sensitivity of (A) is analyzed as follows, wherein
Wherein, g3Is a non-overlapping constraint for preventing interference between the embedded components and interference between the components and the support material.
(4.8) iteratively updating each design variable using a moving asymptote algorithm (MMA).
And (4.9) judging whether the objective function of the multi-component system integration integrated multi-scale design optimization model is converged or not according to the updated design variables, if not, turning to the step (4.2), and if so, outputting the optimal positions of the components and the optimal layout of the supporting structure.
Example 1
The topological optimization design method for multi-scale and multi-component system integration provided by the invention is described below by combining examples. In this example, the properties of the support structure define an elastic modulus E-10 and a poisson ratio μ -0.3; the properties of the embedded multicomponent material define the elastic modulus E4 and the poisson ratio μ 0.3. Initial design domain as shown in fig. 2, the macro structure size is defined as 120 × 40, and the finite cell is divided into 120 × 40; the structural size of the microstructure is 1, and the finite element grid dimension is defined as 80 multiplied by 80; the left boundary is fixedly constrained, and a concentrated load F is-5N at the midpoint of the right boundary; embedding circular components in the initial design domain as shown in fig. 3, the number of components is 4, the radius is 4, and the coordinates of the centers of mass of the components are (48,16), (48,24), (72,16), (72,24), respectively.
Single-scale macroscopic design optimization based on SIMP, as shown in fig. 4, with a volume fraction VOL of 0.5 and a total flexibility value C of 462.6082; for ease of comparison, all boundary conditions, design domains, material properties, finite element models, etc. are the same in multi-scale and multi-component systems. FIG. 5 shows the optimization result of the multi-component system integration design under multi-scale, the macroscopic volume fraction VOLmac0.5, volume fraction VOL of microstructure materialmac0.4, and the total flexibility value is 218.82. The invention provides a multi-scale and multi-componentCompared with the optimization result in fig. 4, the system integration design optimization method has lower structural flexibility, i.e. the overall structural rigidity is higher. The main reason is that the optimal macro structure topology, microstructure configuration and optimal layout of multiple components can be obtained by multi-scale parallel design, so that the multiple components can play the greatest bearing role in the whole structure, and the design space of topology optimization is greatly utilized. FIG. 6 is an iterative plot of macroscopic and microscopic volume fractions, from which it can be seen that the microscopic volume fraction has reached the constrained volume fraction at iteration 20 steps, and the macroscopic volume fraction also reaches the constrained volume fraction shortly after constraint 40 steps; fig. 7 shows an iteration curve of the integrated objective function, from which it can be seen that the objective function converges rapidly, and after 20 steps of iteration, the convergence rate gradually decreases, and the method can converge rapidly and reach a stable value.
The multi-scale and multi-component system integration integrated topological optimization design method provided by the invention is a design method of a system, a plurality of key factors such as material attributes and quantity of components, mutual constraints among the components and between the components and the design domain, synthesis of different performance indexes, influence of macroscopic load and boundary conditions on macroscopic structure layout and the like need to be considered at the same time, and the design method based on simulation, test and experience cannot be realized or has too high realization cost, and cannot find an optimal design scheme.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (7)
1. A multi-component system integration integrated multi-scale topology optimization design method is characterized by comprising the following steps:
s1, meshing a macro structure design domain of the design material structure to divide the macro structure design domain into a plurality of macro units, and respectively meshing the macro units to obtain a plurality of micro units of each macro unit;
s2, describing each component in the multiple components by adopting a function respectively, acquiring the density of each component according to a discrete material optimization method, and mapping the density of each component to a grid obtained by dividing the macro structure design domain to form an embedded component;
s3 interpolating the elastic modulus of each component in the grid mapped to the macrostructure with the support structure of the design material structure, thereby obtaining interpolated equivalent elastic moduli of each component in the grid mapped to the macrostructure and the support structure;
s4 determining a macro-cell homogenization stiffness matrix K for the latticed macro-structure and micro-cellsHAnd displacement field U of macro unit in macro scaleiAnd combining the support structure with the interpolated equivalent elastic modulus E mapped to each component in the macrostructure meshiThe method comprises the steps of constructing an integrated multi-scale design optimization model of the multi-component system, then carrying out sensitivity analysis on design variables in a macro scale and a micro scale, and iteratively updating the design variables in the macro scale and the micro scale, thereby determining the optimal positions of a plurality of components and the optimal layout of a supporting structure.
2. The method for multi-scale topological optimization design of integration and integration of multi-component system according to claim 1, wherein in step S2, the density of each component is mapped into the grid by using a density point method in which density variables and grid centroids are combined; the discrete material optimization method obtains a calculation model of the density of each component as follows:
where Nc is the number of components, i, k are the center points of one of the grids and one of the components, respectively, β is a rate parameter controlling the macro-cell density towards 0 or 1, di，kIs the distance, r, between macro-unit i and module centroid kcIs the radius of the assembly.
3. The multi-scale topological optimization design method for integration and integration of multi-component systems according to claim 1, wherein step S3 specifically comprises: interpolating the elastic modulus of each component in the support structure of the design material structure and the grid mapped to the macrostructure by adopting a discrete material optimization method, thereby obtaining the interpolated equivalent elastic modulus of each component in the support structure and the grid mapped to the macrostructure, wherein the calculation model of the equivalent elastic modulus is as follows:
where ρ ismac，iIs the density of the element, p, corresponding to the support structure on a macroscopic scalec，i，kFor inserting the element density of the component of material k, EiIs the interpolated equivalent elastic modulus; e0Is the modulus of elasticity of the support structure; ec，kIs the modulus of elasticity of the component with insertion material k; n is a radical ofmIs the amount of material inserted into the assembly.
4. The method for multi-scale topological optimization design of integration of multi-component system according to claim 1, wherein in step S4, the multi-scale design optimization model of integration of multi-component system is:
Find：ρmac＝{ρmac，1，ρmac，2，…，ρmac，M}，ρmic＝{ρmic，1，ρmic，2，…，ρmic，Nc}
ρc＝{ρc，1，ρc，2，…，ρc，Nc}，Sk＝{Xk，1，Yk，1，Xk，2，Yk，2，…，Xk，Nc，Yk，Nc}
Min：
Subject to：
wherein C represents the compliance of the macrostructure, pmac，i(i ═ 1,2, 3.., M) denotes the relative density of the ith macro-unit on a macro-scale, ranging from ρmac，minTo 1; rhomic，e(e 1,2, 3.., N) represents the relative density of the e-th microscopic unit on the microscopic scale, and the value ranges from ρmic，minTo 1; rhomac，minAnd ρmic，minThe relative density of the minimum element preset for avoiding singularity in the numerical calculation process; m is the number of macro units, and N is the number of micro units; rhoc，NcDenotes the NthcRelative density of individual components, NcIs a positive integer greater than 0, SkDesigning variables for the positions of the components, which are respectively: xk，Nc，Yk，Nc，θk，NcWherein X isk，NcExpressed as the abscissa, Y, of the center of mass of the assemblyk，NcIs the ordinate of the center of mass of the assembly; eiIs the interpolated equivalent elastic modulus of the support structure and each component mapped to the macrostructure grid, U is the displacement vector of the macrostructureiIs a displacement field of a macro unit, K is a rigidity matrix of a macro structure, KHA homogenizing stiffness matrix representing macro-units with an artificial density equal to 1, F being the external force to which the macro-structure is subjected, V representing the volume of the macro-structure, ViWhich represents the volume of the macro-unit,
denotes the volume of the microscopic unit, g1And is the volume fraction of the macrostructure volume constraint, g2Is the volume fraction of the microstructure volume constraint; g3Is a non-overlapping constraint, V, for preventing mutual interference between the embedded components and interference between the components and the supporting materialcIs the volume of the embedded component, LcIndicating the perimeter of all components.
5. The multi-scale topological optimization design method for integration and integration of multi-component system according to claim 4, wherein L iscThis can be obtained by measuring the intermediate density that occurs at the boundary of the embedded component, whose computational model is as follows:
Lc＝∫Dexp[-α(ρc，i，-0.5)m]dD
where ρ isc，iα and m are parameters that control density filtering for the density of the ith module.
6. The multi-scale topological optimization design method for integration and integration of multi-component systems according to claim 1, wherein step S4 specifically comprises the following steps:
s11 initializing and defining design parameters and optimization parameters;
s12 calculating the micro elastic matrix D of the micro unit by material interpolation method in micro scale, solving the rigidity matrix k of the micro unit according to the micro elastic matrix D, and then carrying out finite element analysis in micro scale to solve the displacement field χ of the micro uniti，
S13 displacement field χ according to microscopic unitiCalculating the equivalent elastic property D of the macro unit by using a homogenization methodH；
S14 equivalent elastic property matrix D based on solved microscopic unitsHCalculating the actual elastic matrix D of the macro unit by adopting a material interpolation methodMAnd from the actual elastic matrix D of the macro-cellsMSolving a macro-unit homogenization rigidity matrix KHAnd a macro cell stiffness matrix KiThen, finite element analysis is carried out in a macroscopic scale, and the displacement field U of the macroscopic unit is calculatedi；
S15 displacement field U based on solved macro unitiCalculating an objective function C in the multi-component system integration integrated multi-scale design optimization model;
s16, carrying out sensitivity analysis on the target function C and each design variable in the constraint function in the multi-component system integrated multi-scale design optimization model according to chain calculation rules in macro and micro scales, and iteratively updating each design variable;
s17, according to the updated design variables, judging whether the objective function of the multi-component system integration multi-scale design optimization model is converged, if not, turning to the step S12, and if so, outputting the optimal positions of the components and the optimal layout of the supporting structure.
7. The multi-scale topological optimization design method for integration and integration of multi-component system according to claim 1,
the plurality of components are all circular, and can be described by adopting implicit functions or explicit functions respectively, wherein the model for describing one of the components by adopting the explicit functions is as follows:
wherein r iscIs the radius of the assembly; x is the number ofkThe abscissa is the center coordinate of the component; y iskIs the ordinate of the center coordinate of the assembly.
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