CN111062166B - Three-period minimum curved surface porous structure topology optimization method based on variable density method - Google Patents
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Abstract
The invention discloses a three-cycle minimum curved surface porous structure topology optimization method based on a variable density method, which comprises the following steps: step 1: inputting a three-period minimum curved function expression, a design domain range and an optimized target density; step 2: generating density distribution and generating a density mapping grid by using a variable density topological optimization method according to the design domain range and the target density; and step 3: generating a double-equivalent-parameter three-period minimum curved surface according to the density implicit grid; and 4, step 4: extracting the intersecting outlines of six covers of the double-equivalent-parameter three-cycle minimal curved surface to generate a cover outline triangular mesh; and 5: tracking and judging the triangular mesh of the cover profile, deleting the illegal mesh, and generating the legal triangular mesh of the six covers; step 6: and combining the cover triangular mesh with the double-equivalent-parameter three-cycle minimum curved surface, and outputting an STL file with an optimized three-cycle minimum curved surface porous structure. The optimal design of the three-cycle minimum curved surface porous structure of the Sheet configuration is realized.
Description
Technical Field
The invention relates to the technical field of Computer Aided Design (CAD) and Computer Aided Engineering (CAE), in particular to a topological optimization method of a three-cycle minimum curved surface porous structure based on a variable density method.
Background
The porous structure is a complex topological structure widely existing in nature, has the excellent characteristics of high porosity and large specific surface area, and has wide application in the industrial field. Researchers have designed and developed modeling methods for various solid structures for a long time, which belong to basic core technologies in the aviation industry and the automobile industry, but the design and research on porous structures are still in the primary stage. For the design of the porous structure, there are several technical schemes such as lattice structure, honeycomb structure, foam structure, etc. The design methods of the porous structures have various defects generally, the lattice structure is formed by connecting rod pieces, parametric design can be conveniently carried out, and the connecting position of the rod pieces is a potential position where stress concentration occurs; the honeycomb structure belongs to a two-dimensional porous structure and is sensitive to the stress direction; foam structure is similar to natural structure, but precise control of the pores remains one of the design challenges.
In recent years, three-cycle extremely small curved surfaces have attracted increasing attention from researchers due to their excellent curved surface characteristics. The three-period extremely-small curved surface is a smooth curved surface with zero surface average curvature, researchers in some leaves and animal wings find similar structures, and holes which are smooth inside and are communicated provide an ideal solution for application in the industry. And the three-cycle extremely-small curved surface has a definite mathematical expression, the size and the distribution of the holes can be accurately controlled, the smoothness and the continuity of the curved surface can be still ensured while the non-uniform holes are generated, and the occurrence of stress concentration is reduced as much as possible. The three-period extremely-small curved surface porous structure mainly has two configurations, namely a Skelton configuration and a Sheet configuration. The three-cycle minimum curved surface divides the space into two parts, and the two parts are respectively extracted to generate a Skelton configuration structure; and if the structure between f (x, y, z) ═ c and f (x, y, z) ═ c is extracted, the Sheet configuration structure is generated. Due to the relative complexity of Sheet structure generation, most of the current research is directed to a three-cycle, extremely-small curved surface porous structure in a Skelton configuration.
The performance optimization of the porous structure is the key point of the current academic research, and in fact, most of the pores of the porous structure widely existing in nature are distributed in a non-uniform form, and how to reasonably design the distribution of the pores to play the maximum role is the research key point in the current CAE field. Topology optimization is an optimization technology which is rapidly developed in recent years and is widely applied to the fields of mechanics and heat. The variable density method is a classical optimization method of topological optimization, the method assumes that the density of a material can be changed linearly, meanwhile, the attribute of the material is directly related to the density, and in a given design domain, the optimal material density distribution is found by continuously calculating an objective function and updating the material distribution in the design domain. The optimization method has natural similarity with the porous structure, the key for determining the mechanical property of the porous structure is the relative density of the porous structure, and the aim of optimizing the mechanical property can be fulfilled by optimizing the density distribution of the porous structure.
At present, researchers have few optimized researches on the mechanical properties of three-cycle extremely-small curved surfaces, and most researches stay in the aspect of experimental tests on the porous properties of the three-cycle extremely-small curved surfaces. Feng et al proposed a method for optimizing a three-cycle minimum curved porous structure unit (see Feng J, Fu J, Shang C, et al. Sandwich panel Design and performance optimization based on three-dimensional temporal minimum surfaces [ J ]. Computer-air designed Design,2019,115:307-322.), but the method only aims at parameter optimization of the unit structure and does not relate to distribution optimization in a region range. Li et Al tried a Topology optimization method for three-cycle very small surfaces (see Li D, Liao W, Dai N, et Al. optimal Design and modeling of magnetic-based functional scaled cellular structures for Additive Manufacturing [ J ]. Computer-aid Design, 2018,104:87-99.), but this method was for three-cycle very small surface porous structures in Skelton configuration, which has been demonstrated to be inferior in Skelton configuration to three-cycle very small surface porous structures in Sheet configuration (see Al-Ketan O, Rowston R, Al-rule. polar-mechanical property correlation of 3D scaled structure, skin Manual, sampled and magnetic fibrous porous structures [ 183, 167J. 19: 8.). Although the three-cycle extremely-small curved surface of the Sheet configuration has excellent performance, the design process is complex and is difficult to combine with the topological optimization technology, and the research progress aiming at the three-cycle extremely-small curved surface porous structure optimization of the Sheet configuration is not seen at present.
According to literature analysis, most of the current three-cycle minimum curved surface porous structure optimization researches belong to experimental analysis, time consumption is serious when finite element analysis optimization is directly carried out on a porous structure, most of the few methods for carrying out three-cycle minimum curved surface porous structure optimization by using a topological optimization technology are directed at another Skelton configuration, the performance is worse than that of a Sheet configuration, and the performance advantage of the three-cycle minimum curved surface is not completely utilized. In addition, no literature on topological optimization of three-cycle extremely-small curved surface porous structures is found.
Disclosure of Invention
In order to solve the defects that the performance of the existing three-cycle minimum curved surface porous structure is simple to optimize, and the results of part of topology optimization methods are not perfect, the invention provides a method which is stable and reliable, and realizes the optimization design of the three-cycle minimum curved surface porous structure of the Sheet configuration by using a topology optimization theory and a graphical geometry tracking algorithm.
The technical scheme of the invention is as follows:
a three-cycle extremely-small curved surface porous structure topological optimization method based on a variable density method comprises the following steps:
step 1: inputting three-period minimum surface function expression f (x, y, z) as c, c as equivalent parameter, and designing domain range x as [ x ∈ [ x [ ]min,xmax]，y∈[ymin,ymax]，z∈[zmin,zmax]Optimizing a target density rho;
step 2: generating density distribution in a design domain and generating a density mapping grid by using a variable density topological optimization method according to the design domain range and the target density;
and step 3: generating a double-equivalent-parameter three-period minimum curved surface according to the density implicit grid;
and 4, step 4: extracting the intersecting outlines of six covers of the double-equivalent-parameter three-cycle minimal curved surface to generate a cover outline triangular mesh;
and 5: tracking and judging the triangular mesh of the cover profile, deleting the illegal mesh, and generating the legal triangular mesh of the six covers;
step 6: and combining the cover triangular mesh with the double-equivalent-parameter three-cycle minimum curved surface, and outputting an STL file with an optimized three-cycle minimum curved surface porous structure.
The invention has the beneficial effects that at least:
the density distribution in the design domain is generated by using a topological optimization method based on a variable density method, the tight combination of the porous structure in the topological optimization technology is realized directly according to the corresponding relation between the density and the parameters of the three-period minimum curved surface porous structure, and the time-consuming finite element analysis stage of the porous structure is avoided. In addition, by utilizing a geometric tracking processing algorithm, the generation of a three-cycle minimum curved surface of the Sheet configuration is realized, the output model has reliable closure, and the grid defects such as gaps and the like can not occur. The method is stable and reliable, and can effectively generate the topological optimization structure of the three-period minimum curved surface porous structure.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a flow chart of a three-cycle minimal curved surface porous structure topology optimization method based on a variable density method;
FIG. 2 is a schematic diagram of a calculation result of topology optimization by a variable density method;
FIG. 3 is a schematic diagram of a density mapping grid generation principle;
FIG. 4 is a diagram illustrating the generation result of a dual-equivalent-parameter three-period minimum curved surface;
FIG. 5 is a schematic diagram of the generation of intersecting contours on six covers;
FIG. 6 is a schematic diagram of a contour triangulated mesh generation on a cover;
FIG. 7 is a schematic view of a cover contour triangle mesh tracking judgment;
FIG. 8 is a diagram of a cover triangular mesh with illegal triangle processing completed;
FIG. 9 is the data of the relationship between the density and the parameters of the curved surface in example G;
FIG. 10 shows the porous structure generated by the topology optimization of the curved surface according to example G.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be further described in detail with reference to the accompanying drawings and examples. It should be understood that the detailed description and specific examples, while indicating the scope of the invention, are intended for purposes of illustration only and are not intended to limit the scope of the invention.
The invention provides a three-cycle minimal curved surface porous structure topology optimization method based on a variable density method, which is shown in a flow chart shown in figure 1 and comprises the following specific implementation steps:
step 101: inputting three-period minimum surface function expression f (x, y, z) as c, c as equivalent parameter, and designing domain range x as [ x ∈ [ x [ ]min,xmax]，y∈[ymin,ymax]，z∈[zmin,zmax]And optimizing the target density rho.
Step 102: and according to the design domain range and the target density, generating density distribution in the design domain by using a topological optimization method based on a variable density method and generating a density mapping grid.
The variable density method assumes that the density and the physical property of the material unit have certain correlation and can be expressed by a functional formula, in the topological optimization iterative computation, the density of each material unit is used as a design variable, the density of each unit is continuously updated in an iterative manner, the correlation performance is computed, and finally the optimal material density distribution condition is obtained.
In the invention, the specific method for generating the density mapping grid is as follows: calculating to obtain the density value of each discrete point in the design domain range by using a topology optimization method based on a variable density method, as shown in fig. 2, sequentially connecting the discrete points to generate a quadrilateral mesh 201 as shown in fig. 3, subdividing the quadrilateral mesh twice, subdividing each quadrilateral into four uniform quadrilaterals 202 by each subdivision calculation, calculating the density value of each vertex by using a linear interpolation method, and finally generating to obtain a density mapping mesh.
Step 103: and generating the double-equivalent-parameter three-cycle minimum curved surface according to the density mapping grid.
The specific steps for generating the double-equivalent-parameter three-period extremely-small curved surface comprise:
step 103-1: calculating to obtain density and c according to experimental fitting2Calculating the density value rho of each vertex of the density mapping gridiCorresponding to ci 2And record ci 2Maximum value of cimax 2Minimum value cimin 2；
Step 103-2: according to the expression f (x, y, z) ═ ciGenerating triangular mesh curved surface S by linear interpolation of Marching Cube algorithm 1301, as shown in fig. 4;
the method utilizes a large number of Cube units to judge the position relation with an implicit surface, the judgment can be carried out by substituting the vertex coordinates of a Cube into an implicit function, the intersection part of the surface and the Cube is represented by a triangular mesh, and finally the original implicit surface is approximated by a triangular mesh surface calculated by a large number of interpolation.
Step 103-3: according to the expression f (x, y, z) ═ ciGenerating triangular mesh curved surface S by Marchang Cube algorithm linear interpolation 2302, as shown in fig. 4.
Step 104: and extracting the intersecting outlines of six covers of the three-period minimum curved surface with double equivalent parameters to generate a cover outline triangular mesh.
The specific steps of generating the cover contour triangular mesh comprise:
step 104-1: dividing six cover function expressions x into xmin、x＝xmax、y＝ymin、 y＝ymax、z＝zmin、z＝zmaxDistribution brought triangular mesh curved surface S1The expression f (x, y, z) ═ ciTriangular mesh curved surface S2The expression f (x, y, z) ═ ciAnd calculating to obtain a triangular mesh curved surface S1Triangular mesh curved surface S2Intersecting contours on the six covers, as shown in FIG. 5;
step 104-2: the limitation of the minimum angle and the maximum edge in the delaunay triangulation algorithm is cancelled, and delaunay triangles in the surface area are generated on the six covers by taking the intersecting contour as a constraint to be used as a cover contour triangular mesh, as shown in fig. 6.
The delaunay triangulation algorithm is a common triangulation method for a plane area point set, under the condition that singularity does not occur, the sum of the minimum angles of triangulation is larger than the sum of the minimum angles of triangles formed by any non-delaunay triangulation, and in addition, no other node is included in the circumscribed circle of any triangle in delaunay triangulation. The delaunay triangularization algorithm is integrated into various algorithm libraries as a mature graphic algorithm, and the delaunay triangularization is realized by adopting an open source CGAL library.
Step 105: and tracking and judging the triangular mesh of the cover profile, deleting the illegal mesh, and generating the legal triangular mesh of the six covers.
The specific steps of generating six cover legal triangular meshes comprise:
step 105-1: establishing a triangular bad edge array, traversing each triangle of the cover contour triangular mesh, substituting the geometric coordinate of the center point of the triangle into a functional formula f (x, y, z), and if the calculation result is less than cimax 2And is greater than cimin 2Skipping the triangle, if the calculated result is larger than cimax 2Or less than cimin 2As shown in fig. 7, the triangle 501 is an illegal triangle, the triangle is deleted from the cover contour triangular mesh, and the edge 502 of the triangle that is not recorded in the bad edge array is recorded in the bad edge array;
step 105-2: traversing the bad edge array, searching a triangle which shares the same edge with the bad edge in the cover contour triangular mesh, deleting the triangle from the cover contour triangular mesh, and recording the edge of the triangle which is not recorded in the bad edge array;
step 105-3: step 105-2 is iteratively performed until the number of edges in the bad edge array is 0.
Step 106: as shown in fig. 8, the cover triangular mesh is combined with the dual-equivalence-parameter three-cycle minimal surface, and the optimized three-cycle minimal surface porous structure STL file is output.
Typical embodiments of the present invention are as follows:
the embodiment adopts a three-period minimum-curvature surface G surface, the functional expression of which is f (x, y, z) ═ sin (0.3 pi x) cos (0.3 pi y) + sin (0.3 pi z) cos (0.3 pi x) + sin (0.3 pi y) cos (0.3 pi z) ═ 0, and the design domain range x epsilon [0,60 pi z ∈ 0]，y∈[0,30]，z∈[0,10]The optimization target density ρ is 0.4. The relationship between the G surface parameters and the density is fitted by an experimental method as shown in FIG. 9P is shownG＝0.66×(c2)0.52According to the density distribution calculated by the topological optimization method based on the variable density method, the density distribution can be substituted into the fitting formula rhoGAnd calculating to obtain corresponding structural parameters. When a three-cycle extremely-small curved surface porous structure with a Sheet configuration is generated, in order to keep the generated structure consistent with the shape of a design domain, a part with the density of a G curved surface of 0 is replaced by the minimum wall thickness of a G curved surface porous structure, and the final generated result is shown in FIG. 10. The method realizes the generation of the three-cycle minimal curved surface porous structure with the Sheet configuration according to the topology optimization structure.
The above-mentioned embodiments are intended to illustrate the technical solutions and advantages of the present invention, and it should be understood that the above-mentioned embodiments are only the most preferred embodiments of the present invention, and are not intended to limit the present invention, and any modifications, additions, equivalents, etc. made within the scope of the principles of the present invention should be included in the scope of the present invention.
Claims (7)
1. A three-cycle extremely-small curved surface porous structure topological optimization method based on a variable density method is characterized by comprising the following steps:
step 1: inputting three-period minimum surface function expression f (x, y, z) as c, c as equivalent parameter, and designing domain range x as [ x ∈ [ x [ ]min,xmax]，y∈[ymin,ymax]，z∈[zmin,zmax]Optimizing a target density rho;
step 2: according to the design domain range and the target density, generating density distribution in the design domain by using a topological optimization method based on a variable density method and generating a density mapping grid;
and step 3: generating a double-equivalent-parameter three-period extremely-small curved surface according to the density mapping grid, comprising the following steps of:
step 3.1: density and c calculated from experimental fitting2Calculating the density value rho of each vertex of the density mapping gridiCorresponding to ci 2And record ci 2Maximum value of cimax 2Minimum value cimin 2；
Step 3.2: according to the expression f (x, y, z) ═ ciGenerating triangular mesh curved surface S by Marchang Cube algorithm linear interpolation1；
Step 3.3: according to the expression f (x, y, z) ═ ciGenerating triangular mesh curved surface S by Marchang Cube algorithm linear interpolation2；
And 4, step 4: extracting the intersecting outlines of six covers of the double-equivalent-parameter three-cycle minimal curved surface to generate a cover outline triangular mesh;
and 5: tracking and judging the triangular mesh of the cover profile, deleting the illegal mesh, and generating the legal triangular mesh of the six covers;
step 6: and combining the legal triangular mesh of the cover with the double-equivalent-parameter three-cycle minimum curved surface, and outputting an STL file with an optimized three-cycle minimum curved surface porous structure.
2. The topological optimization method of the three-cycle minimal curved surface porous structure based on the variable density method as claimed in claim 1, wherein in the step 2, the specific method for generating the density mapping grid is as follows: calculating by using a topology optimization method based on a variable density method to obtain the density value of each discrete point in a design domain, sequentially connecting the discrete points to generate a quadrilateral mesh, subdividing the quadrilateral mesh twice, subdividing each quadrilateral into four uniform quadrilaterals by each subdivision calculation, calculating the density value of each vertex by using a linear interpolation method, and finally generating to obtain a density mapping mesh.
3. The topological optimization method for the porous structure of the three-cycle extremely-small curved surface based on the variable density method as claimed in claim 1, wherein the step 4 of generating the cover contour triangular mesh comprises the following specific steps:
step 4.1: dividing six cover function expressions x into xmin、x＝xmax、y＝ymin、y＝ymax、z＝zmin、z＝zmaxDistribution brought into curved surface S1The expression f (x, y, z) ═ ciCurved surface S2The expression f (x, y, z) ═ ciIs calculated to obtainTriangular mesh surface S1Triangular mesh curved surface S2Intersecting contours on the six covers;
step 4.2: and (4) eliminating the limitation of minimum angles and maximum edges in the Delaunay triangulation algorithm, and generating Delaunay triangles in the surface area as a cover contour triangular mesh on six covers by taking the intersecting contour as a constraint.
4. The method for topological optimization of a porous structure with a three-cycle minimal curved surface based on a variable density method as claimed in claim 1, wherein the specific steps of generating six cover legal triangular meshes in step 5 comprise:
step 5.1: establishing a triangular bad edge array, traversing each triangle of the cover contour triangular mesh, substituting the geometric coordinate of the center point of the triangle into a functional formula f (x, y, z), and if the calculation result is less than cimax 2And is greater than cimin 2Skipping the triangle, if the calculated result is larger than cimax 2Or less than cimin 2If the triangle is an illegal triangle, deleting the triangle from the cover contour triangular mesh, and recording the edge of the triangle which is not recorded in the bad edge array;
step 5.2: traversing the bad edge array, searching a triangle which shares the same edge with the bad edge in the cover contour triangular mesh, deleting the triangle from the cover contour triangular mesh, and recording the edge of the triangle which is not recorded in the bad edge array;
step 5.3: and (5.2) iterating until the number of the edges in the bad edge array is 0.
5. The variable density method-based three-cycle infinitesimal surface porous structure topology optimization method of claim 1, wherein when the three-cycle infinitesimal surface G surface is used, the function expression is f (x, y, z) ═ sin (0.3 π x) cos (0.3 π y) + sin (0.3 π z) cos (0.3 π x) + sin (0.3 π y) cos (0.3 π z) ═ 0;
and designing a domain range x belonging to [0,60 ], y belonging to [0,30], z belonging to [0,10], and optimizing the target density rho to be 0.4.
6. The variable density method-based three-cycle extremely-small curved surface porous structure topological optimization method of claim 5, wherein the relationship between G curved surface parameters and density is fitted by an experimental method as follows:
ρG＝0.66×(c2)0.52。
7. the method for optimizing the topology of the three-cycle minimal curved surface porous structure based on the variable density method as claimed in claim 5, wherein when the three-cycle minimal curved surface porous structure with the Sheet configuration is generated, in order to keep the generated structure consistent with the shape of the design domain, the part with the G curved surface density of 0 is replaced by the G curved surface porous minimum wall thickness.
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