Multi-scale lofting curved surface structure design method and application thereof

Abstract

The invention discloses a multi-scale lofting curved surface structure design method which comprises the following steps: defining at least two single connected domains on a lofting curved surface structure, obtaining a corresponding mapping parameter domain and calculating affine stretching transformation; combining conformal mapping and affine stretching transformation to obtain Taihmiller space mapping, and calculating a stretching quotient; calculating an interpolation shape between the two single connected domains based on a scalability quotient, establishing a rapid prediction model of the mechanical properties of the gradient micro-structure through a generalized homogenization method to obtain volume fraction distribution of a sample micro-structure unit in a lofting curved surface body configuration, and calculating a corresponding global level set function to obtain the mechanical properties of the gradient micro-structure. And inversely mapping the global level set function back to the lofting curved surface body configuration to realize the multi-scale lofting curved surface structure design of gradient filling. The problem that an ideal lofting curved surface body configuration is difficult to obtain due to the fact that an existing interactive design multi-scale lofting curved surface needs tedious iteration and interpolation method embedding and cannot be controlled can be solved.

Description

Technical Field

[0001] The present application relates to the technical field of structural optimization design, and in particular to a multi-scale lofted surface structure design method, a multi-scale lofted surface structure design device, an electronic device, and a computer-readable storage medium. Background Art

[0002] Lofting is the process of using multiple two-dimensional objects as the cross-section of the structure and sweeping them along a certain direction to form a complex three-dimensional object. It is often used in the generation of complex curved surfaces such as blades, and is a commonly used surface and curved body generation operation in mechanical structure design. Traditional lofting operations are mainly oriented towards continuous parametric curves, such as NURBS curves, and curve masking calculations are achieved through order increase and node refinement. Although the calculation method for continuous parametric lofting surface generation is simple in principle, it is not easy to achieve a robust and convenient lofting function; in addition, interactive design of lofting surfaces is a tedious iterative process that depends on the number, shape and position of the lofted surfaces. Since the interpolation method is embedded in the computer and cannot be controlled by the designer, it will take a lot of time to obtain the ideal lofted surface configuration.

[0003] In addition, the introduction of multi-scale design has greatly improved the function and performance of the structure, but the combination with the complex surface structure generated by lofting is limited, and the relationship between the geometric characteristics and physical properties of the microstructure filled in the special-shaped hexahedral unit has not yet been established. Therefore, how to achieve fast and stable multi-scale lofted surface structure design is an urgent problem to be solved. Summary of the invention

[0004] In order to overcome the defects of the above-mentioned prior art, an embodiment of the present invention provides a multi-scale lofted surface structure design method and application, which can solve the problem that the existing interactive design of multi-scale lofted surfaces requires cumbersome iterations, and the interpolation method is embedded and cannot be controlled, resulting in difficulty in obtaining an ideal lofted surface body configuration.

[0005] On the one hand, an embodiment of the present invention proposes a multi-scale lofting surface structure design method, including: defining at least two simply connected domains on the surface structure to be lofted, obtaining a mapping parameter domain corresponding to each of the simply connected domains, and calculating the affine stretching transformation of the mapping parameter domains corresponding to any two of the simply connected domains; compounding the conformal mappings corresponding to the two simply connected domains and the affine stretching transformation between the two simply connected domains to obtain a Teichmuller space mapping between the two simply connected domains, and calculating a stretch quotient between the two simply connected domains based on the Teichmuller space mapping; calculating the Beltrami coefficient between the two simply connected domains based on the stretch quotient, and obtaining a to an interpolation shape between two of the simply connected domains; according to the interpolation shape, the surface structure to be lofted is mapped to a unit cube structure, and the lofted surface body configuration is calculated according to the unit cube structure; the equivalent mechanical properties of a finite number of sample microstructure units are calculated by a generalized homogenization method, and a rapid prediction model of the mechanical properties of a gradient microstructure with a variable volume fraction ratio is established by fitting sample data with a continuous function; according to the rapid prediction model, the target volume fraction distribution of the finite number of sample microstructure units in the lofted surface body configuration is obtained, and the corresponding global level set function is calculated, and the global level set function is inversely mapped back to the lofted surface body configuration, so as to realize the multi-scale lofted surface structure design with gradient filling.

[0006] In one embodiment of the present invention, the Beltrami coefficient between the two simply connected domains is calculated based on the stretch quotient, including: making the stretch quotient change monotonically and linearly to determine the introduced variable between the two simply connected domains, setting an interpolation function to establish a relationship in which the stretch quotient changes with the change of the introduced variable, and obtaining the Beltrami coefficient corresponding to the stretch quotient according to the interpolation function.

[0007] In one embodiment of the present invention, the simply connected domain includes multiple triangular facets in the form of triangulation, and the interpolation shape between two simply connected domains is obtained based on the Beltrami coefficients, including: obtaining the surface shape of each triangular facet after mapping according to the Beltrami coefficients, and assembling and stitching all the mapped surface shapes to obtain the interpolation shape between the two simply connected domains.

[0008] In one embodiment of the present invention, the method of obtaining the surface shape of each triangular face after mapping according to the Beltrami coefficient includes: obtaining the vertex set and connection relationship of the triangular face to determine the original triangulation mesh and the target triangulation mesh composed of the triangular facets on the two simply connected domains; using a piecewise linear function to approximate the mapping of the original triangulation mesh to the target triangulation mesh to obtain the linear change of each triangular facet in the original triangulation mesh mapped to the target triangulation mesh; and calculating the area of ​​each triangular facet and the Beltrami coefficient according to the linear change to construct the surface shape of each triangular facet after mapping.

[0009] In one embodiment of the present invention, the mapping of the surface structure to be lofted to a unit cube structure according to the interpolation shape, and calculating the lofted surface body configuration according to the unit cube structure, includes: arranging the mapping parameter domain corresponding to the simply connected domain according to the surface structure to be lofted, and normalizing it in the direction of the distribution arrangement; performing local spline interpolation on the normalized mapping parameter domain according to the interpolation shape to form a parameter curved edge hexahedron region; normalizing the parameter curved edge hexahedron region in the direction of the local spline interpolation to obtain the unit cube structure corresponding to the surface structure to be lofted, and calculating the lofted surface body configuration according to the unit cube structure.

[0010] In one embodiment of the present invention, the normalized mapping parameter domain is subjected to local spline interpolation to form a parametric curved hexahedron region, including: setting interpolation points to perform local spline interpolation on the mapping parameter domain to obtain a spline curve, and then calculating the expansion quotient of the mapping parameter domain mapped to the parametric curved hexahedron region; interpolating the amplitude angle of the mapping parameter domain mapped to the parametric curved hexahedron region according to the expansion quotient to obtain the Beltrami coefficient of the mapping parameter domain, and calculating each intermediate cross-sectional shape of the mapping parameter domain mapped to the parametric curved hexahedron region according to the Beltrami coefficient; and calculating the modeling equation of the parametric curved hexahedron region according to the intermediate cross-sectional shapes.

[0011] In one embodiment of the present invention, the target volume fraction distribution of the finite number of sample microstructure units in the lofted surface body configuration is obtained according to the rapid prediction model, and the corresponding global level set function is calculated, including: establishing a structural optimization formula with minimum flexibility as the goal under the global volume fraction constraint of the lofted surface body configuration according to the rapid prediction model, and calculating local level set functions according to the target volume fraction distribution after optimization by the structural optimization formula, and assembling the local level set functions into a global level set function.

[0012] On the other hand, an embodiment of the present invention further proposes a multi-scale lofting surface structure design device, comprising: a simply connected domain mapping module, used to define at least two simply connected domains on the surface structure to be lofted, obtain a mapping parameter domain corresponding to each of the simply connected domains, and calculate the affine stretching transformation of the mapping parameter domain corresponding to any two of the simply connected domains; a stretch quotient calculation module, used to compound the conformal mappings corresponding to the two simply connected domains and the affine stretching transformation between the two simply connected domains, to obtain a Teichmuller space mapping between the two simply connected domains, and calculate the stretch quotient between the two simply connected domains based on the Teichmuller space mapping; an interpolation shape calculation module, used to calculate the Beltrami coefficient between the two simply connected domains based on the stretch quotient, and obtain the shape according to the Beltrami coefficient. The interpolation shape between the two simply connected domains; a lofted surface body configuration calculation module, used to map the surface structure to be lofted to a unit cube structure according to the interpolation shape, and calculate the lofted surface body configuration according to the unit cube structure; a performance prediction module, used to calculate the equivalent mechanical properties of a finite number of sample microstructure units by a generalized homogenization method, and use a continuous function to fit the sample data to establish a rapid prediction model for the mechanical properties of a gradient microstructure with a variable volume fraction ratio; a lofted surface structure design module, used to obtain the target volume fraction distribution of the finite number of sample microstructure units in the lofted surface body configuration according to the rapid prediction model, and calculate the corresponding global level set function, and inversely map the global level set function back to the lofted surface body configuration to achieve gradient-filled multi-scale lofted surface structure design.

[0013] On the other hand, an embodiment of the present invention further proposes an electronic device, comprising: a memory and one or more processors connected to the memory, the memory storing a computer program, and the processor being used to execute the computer program to implement the multi-scale lofting surface structure design method as described in any one of the above embodiments.

[0014] On the other hand, an embodiment of the present invention further proposes a computer-readable storage medium, which stores computer-executable instructions, and the computer-executable instructions are used to execute the multi-scale lofting surface structure design method as described in any one of the above embodiments.

[0015] It can be seen from the above that, compared with the prior art, the above embodiments of the present invention can have at least one or more of the following beneficial effects:

[0016] The simply connected domain on the lofted surface structure is conformally mapped to the complex plane and the affine stretch transformation of the mapping parameter domain is calculated. The affine stretch transformation is combined with the conformal mapping to obtain the Teichmuller space mapping and the stretch quotient is calculated. The corresponding Beltrami coefficient is calculated based on the stretch quotient to obtain the interpolation shape between the simply connected domains, and the continuous and uniform transition between the two plane curved edge regions is achieved. The mechanical properties of the microstructure filled in the hexahedral unit are calculated by the generalized homogenization method, and an optimization model is established and solved to obtain the volume fraction distribution (pseudo-density field) of the microstructure in the parameter domain. According to the volume fraction of each unit, the local level set function of each unit is obtained in combination with the optimization model, which is further assembled into a global level set function. The global level set function is inversely mapped back to the lofted surface body configuration to realize the design of multi-scale lofted surface structure with gradient filling. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0018] Figure 1 A flowchart of a multi-scale lofted surface structure design method provided by an embodiment of the present invention;

[0019] Figure 2 A schematic diagram of the mapping principle between two adjacent lofted surfaces provided by an embodiment of the present invention;

[0020] Figure 3 A schematic diagram of a process for constructing a curved surface shape of a triangular patch mapping according to Beltrami coefficients provided in an embodiment of the present invention;

[0021] Figure 4 A schematic diagram of the principle of conformal parameterization of a lofted surface provided by an embodiment of the present invention;

[0022] Figure 5 A schematic diagram of the principles of lofted surface modeling and parameterization provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0023] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. The present invention will be described below with reference to the accompanying drawings and in combination with the embodiments.

[0024] In order to enable ordinary technicians in the field to better understand the technical solution of the present invention, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments, and all should belong to the protection scope of the present invention.

[0025] It should be noted that the terms "first", "second", etc. in the specification and claims of the present invention and the above-mentioned drawings are applicable to distinguishing similar objects, and are not necessarily used to describe a specific order or sequence. It should be understood that the terms used in this way can be interchangeable where appropriate, so that the embodiments of the present invention described herein can be implemented in an order other than those illustrated or described herein. In addition. The terms "including" and "having" and any variations thereof are intended to cover non-exclusive inclusions, for example, a process, method, system, product or device that includes a series of steps or units is not necessarily limited to those steps or units that are clearly listed, but may include other steps or units that are not clearly listed or are proprietary to these processes, methods, products or devices.

[0026] It should also be noted that the division of multiple embodiments in the present invention is only for the convenience of description and should not constitute a special limitation. The features in various embodiments can be combined and referenced to each other without contradiction.

[0027] like Figure 1 As shown, the first embodiment of the present invention proposes a multi-scale lofting surface structure design method, for example, including: step S1, defining at least two simply connected domains on the surface structure to be lofted, obtaining a mapping parameter domain corresponding to each of the simply connected domains, and calculating the affine stretching transformation of the mapping parameter domains corresponding to any two of the simply connected domains; step S2, compounding the conformal mappings corresponding to the two simply connected domains and the affine stretching transformation between the two simply connected domains to obtain a Teichmuller space mapping between the two simply connected domains, and calculating the stretch quotient between the two simply connected domains based on the Teichmuller space mapping; step S3, calculating the Beltrami coefficient between the two simply connected domains based on the stretch quotient, and calculating the Beltrami coefficient according to the Beltrami coefficient The interpolation shape between the two simply connected domains is obtained; step S4, mapping the surface structure to be lofted to the unit cube structure according to the interpolation shape, and calculating the lofted surface body configuration according to the unit cube structure; step S5, calculating the equivalent mechanical properties of a finite number of sample microstructure units by a generalized homogenization method, and using a continuous function to fit the sample data to establish a rapid prediction model for the mechanical properties of a gradient microstructure with a variable volume fraction ratio; step S6, obtaining the target volume fraction distribution of the finite number of sample microstructure units in the lofted surface body configuration according to the rapid prediction model, and calculating the corresponding global level set function, inversely mapping the global level set function back to the lofted surface body configuration, and realizing the multi-scale lofted surface structure design with gradient filling.

[0028] Specific, combined Figure 2 As shown, for example, define two plane curved edges Jordan region S on the surface structure to be lofted mand S n , S m and S n By selecting corner points to transform into topological quadrilaterals, two topological quadrilaterals are conformally mapped to the rectangular domain R m and R n Inside, the ratio of the vertical and horizontal coordinates of the rectangle is a m and a n That is S m and S n Further use of a m and a n Calculate from S m and S n The stretch quotient of the two rectangular domains is obtained by [m，n] , establish a linear function of the stretch quotient with respect to parameter t, and calculate the Beltrami coefficient of each stretch quotient value corresponding to parameter t. Then, the interpolation shape is obtained based on the Beltrami coefficient corresponding to t. When parameter t is within the definition domain, the interpolation shape changes from S m and S n Continuous transition.

[0029] Furthermore, let the mapping corresponding to the parameter scalar t be The interpolation area corresponding to t is triangulated as The interpolation parameter t ranges from t∈[0,1]. m and S n Smooth shape interpolation between the two must meet the following conditions:

[0030] (1) When t∈[0,1], f t＝0 =Id and f t＝1 =f. For the interpolation of the mapping, it is necessary to satisfy the mapping to S when t = 0. m Mapped to itself, i.e. f t＝0 :S m →S m , when t = 1, the mapping is S m To S n The Teichmüller map, that is, f t＝1 =f:S m →S n .

[0031] (2) Point-wise bounded non-conformal distortion. Let point p be S m A point in the S m, K(p) is the non-conformal distortion of point p, that is, the stretch quotient. And when the interpolation is obtained, the mapping at time t is When is the expansion quotient at this time, then Special note, if in f [m，n] :S m →S n In the special case that K(f)≡1 exists for any point, i.e., the mapping f is a conformal mapping, then for have Right now This also requires conformal mapping.

[0032] (3) Symmetry. Let f -1 :S m →S n is the inverse process of the mapping f, and It is from S n To S m is the interpolation result at time t calculated by the same method, then In other words, whether from S m To S n Or from S n To S m , the interpolation results for t are also one-to-one corresponding, and there is

[0033] (4) Smoothness and harmony. [0，1]×z It is C ∞ - continuous, for all z∈Ω, the derivative exists and is bounded. And for t∈[0,1], is a harmonic function, that is

[0034] Therefore, in step S1, for example, in the two Jordan regions S m and S n Find the corresponding counterclockwise corner points on and Constructed with S m and S n The topological quadrilateral (S m ,g,{p m，0 , p m，1 , p m，2 , p m，3}) and (S n ,g,{p n，0 , p n，1 , p n，2 , p n，3}). The two topological quadrilaterals are conformally mapped to R m and R n In two plane rectangles, we get The two rectangles R on m =g m (Sm ) and R n =g n (S n ):

[0035] g m :S m (p m，0 , p m，1 , p m，2 , p m，3 ) → R m (0, a m , a m +i,i)

[0036] g n :S n (p n，0 , p n，1 , p n，2 , p n，3 ) → R n (0, a n , a n +i,i)

[0037] In order to obtain continuous shape interpolation, we need to find a harmonic mapping that satisfies:

[0038]

[0039] in, and is the boundary of two topological quadrilaterals, Δ is the Laplacian operator defined on the complex plane. [m，n] In the above example, the ordinate of each point in the rectangle does not change, but the abscissa is transformed proportionally, that is:

[0040]

[0041] Therefore, if Figure 2 As shown, for the composite mapping Due to g m and g n -1 is a conformal mapping, and h [m，n] is harmonic, so f [m，n] is also harmonic. In addition, due to g m and g n -1 is a conformal mapping, then the mapping f [m，n] The conformal distortion and h [m，n] The conformal distortion of is the same. At the same time, it can be calculated The Beltrami coefficient is:

[0042]

[0043] This means:

[0044]

[0045] Similarly, we can get:

[0046]

[0047] Because h [m，n] : R m →R n is a stretching affine transformation, whose stretch quotient is K z (f) = κ mn , that is, a n / a m (a n >a m ) or a m / a n (a n ≤a m ), so f [m，n] The Beltrami coefficient can be calculated as:

[0048]

[0049] At this point, the Teichmüller distance of the mapping, that is, the maximum quasi-conformal expansion between two rectangular domains (further developed to the extremal mapping in two plane curve domains), is uniquely determined. m and S n The Teichmüller distance d between them is:

[0050]

[0051] In step S2, for two conformal mappings, a stretched affine transformation (Teichmüller mapping) g m , g n -1 and h [m，n] The compound obtained from S m and S n Teichmüller Mapping:

[0052]

[0053] The mapping f can be calculated [m，n] After obtaining the expansion quotient K(f), we introduce the control variable t∈[0,1] into the two Jordan regional interpolation problems and select a suitable interpolation function to establish the relationship between the expansion quotient and the variable t. t(f), calculate the corresponding Beltrami coefficient and the relationship between the variable t μ t (f).

[0054] For two Jordan regions S on the complex plane m and S m ,according to Figure 2 and the mapping relationship f given in step S1 [m，n] ,have

[0055] g m :S m (p m，0 , p m，1 , p m，2 , p m，3 ) → R m (0, a m , a m +i,i)

[0056] g n :S n (p n，0 , p n，1 , p n，2 , p n，3 ) → R n (0, a n , a n +i,i)

[0057] Computable mapping f [m，n] The expansion quotient is

[0058]

[0059] In step S3, when t∈[0,1], if the expansion quotient changes monotonically with the change of t, we can first assume that the change is linear, that is, K t (f) = (K(f)-1)·t+1, and according to K t (f) The Beltrami coefficient of the mapping at this time can be obtained as:

[0060]

[0061] Define a normalization coefficient To satisfy:

[0062]

[0063] And order Then the Beltrami coefficient at time t can be defined as:

[0064]

[0065] The Beltrami coefficient corresponding to any moment can be calculated based on the mapping and the t value at any moment.

[0066] Furthermore, for example, two Jordan regions S are input in triangulated form m and S n , S m To S n The mapping relationship For the convenience of calculation, for S m and S n The triangulation representation of S needs to unify the index order in its connection relationship, that is, for S m ={X m , F m} and S n ={X n , F n}, there is F m =F n , so S m and S n The connection relationship of the midpoints is uniformly represented by F. If the triangulation of the input surface does not correspond, the mapping relationship between the two surfaces can be used to represent the connection relationship between the midpoints. [m，n] S m The coordinate points on S are mapped to n On, and continue to use S m The triangulation index F on m .

[0067] The above steps provide a method to calculate the expansion quotient and Beltrami coefficient corresponding to any intermediate time t through the mapping relationship between two topological quadrilaterals. Theoretically, given the Beltrami coefficient μ t (f), there must be a smooth mapping f t Corresponding to it. However, in the actual operation process, it is impossible to obtain the expected piecewise linear mapping by directly solving the Beltrami equation. At the same time, since in the actual calculation of piecewise mapping, the Beltrami coefficient depends on the triangle face (i.e., the connection relationship F) rather than the coordinate point, the obtained equation cannot be solved for the discrete triangulation grid whose number of points is often only half of the number of faces. Therefore, for example, the solution of the mapping is divided into two steps: first, the shape of each triangle face after mapping is calculated under the specified stretch quotient and Beltrami coefficient, and then all the triangle faces are assembled and stitched together to obtain the target interpolation result.

[0068] Furthermore, using S m = {V m , F} to represent the region S m The triangulation grid on which X mis a set of triangle vertices, and F represents the connection relationship of the triangles. f=u+iv is the original mesh S m = {V m , F} to the target grid S n = {V n , F}, where the grid S m and S n The number of vertices is the same and they have the same connectivity F. Use piecewise linear functions to approximate the mapping f. Specifically, f is in each triangle F = [v i , v j , v k ] is a linear function:

[0069] f| F (x, y) = (a F x+b F y+p F )+i(c F x+d F y+q F )

[0070] where v i , v j and v k are the vertices of triangle F in counter-clockwise order.

[0071] like Figure 3 As shown, for each triangle, its Beltrami coefficient can be assumed to be μ t (F) = ρ t (F)+iτ t (F). If the point before mapping is v = g + ih, and the point after mapping is w = s + it, that is:

[0072]

[0073] Then for each triangle, according to the linear change, we can get:

[0074]

[0075] Among them, there are:

[0076]

[0077] Among them, A F is the area of ​​the triangle. At the same time, the Beltrami coefficient can be calculated on this triangle:

[0078]

[0079] Assume e kis the longest side of this triangle. For the corresponding local triangle, when assuming F′=[v i ′ v j ′v k ′], and v i ′=(0,0),v j '=(1,0), then we can establish the equation:

[0080]

[0081] To solve for v′ kx and v′ kx .

[0082] Furthermore, after obtaining the shape of each triangle, all triangles need to be stitched together to form a new interpolated shape. In order to ensure that the shape of the triangle does not change, for example, each triangle is rotated and scaled proportionally so that all triangles can be seamlessly stitched together to form a new triangulated mesh.

[0083] Specifically, on the two-dimensional complex plane, the rotation and scaling matrices of a vector can be written as follows:

[0084]

[0085] Among them, c F′ and F′ At least one entry is non-zero. Using the mapping The expression of the local transformation on the triangle F′ can be obtained as:

[0086]

[0087] Where E represents the triangle boundary relationship, and E = {{i, j}, {j, k}, {k, i}}. Substituting the matrix form in the above expression into the expression, the optimized column form of each item in the coordinate transformation matrix can be obtained as follows:

[0088]

[0089] in,

[0090]

[0091] The optimization formula given by the above formula is a least squares problem, and its solution can be written as:

[0092]

[0093] remember are the vertices of the triangulated mesh after stitching, then It can be obtained by solving the following optimization problem:

[0094]

[0095] The column form is about the variables The quadratic positive definite optimization problem can be written in classic matrix form, namely

[0096]

[0097] in, is the assembled coefficient matrix, each row of which corresponds to a summed term in the column form; the variable is a 2N×1 vector, where N is the number of coordinate points in the triangulation:

[0098]

[0099] b is a column vector of (M+2C)×1, where M is the number of terms to be summed in the first half of the formula, and C is the number of fixed points. From the matrix form, it can be seen that this optimization problem can be solved by the conjugate gradient method after some vertices are given. So far, the Teichmüller mapping between the Jordan regions on the two complex planes and the corresponding expansion quotient of each triangular facet are obtained, and the expansion quotient is interpolated and the corresponding Beltrami coefficient is calculated; further, according to the Beltrami coefficient interpolated from the intermediate state, the shape of each triangular facet is solved, and then all triangles are ARAP stitched, and the shape of the intermediate transition surface and the corresponding triangulation are obtained by solving a quadratic positive definite optimization problem.

[0100] Furthermore, for S 0 To S n The expansion quotient of each of these n+1 Jordan regions to the Teichmüller mapping of the previous region is:

[0101]

[0102] At the same time, it can be obtained that each lofting area S i For the first region S 0 The expansion quotient is:

[0103]

[0104] To facilitate calculation and representation, this section defines the mapping relative scaling quotient That is, we do not discuss the relationship between the side lengths of the mapped rectangles, but only focus on the relative proportions. That is:

[0105]

[0106] like Figure 4As shown, if only lofting is performed between two plane curve domains, according to the interpolation method given in the above steps, let the expansion quotient K t (f) It is feasible to solve the intermediate state shape as a linear function of t. However, for multiple planar domains S k , k∈{0, 2, ..., n}, if the linear interpolation method is still used, the surface at the interpolation endpoint will be discontinuous. However, if the traditional continuous interpolation method is used for the stretch quotient, including polynomial interpolation, spline interpolation, etc., the following situations will easily occur: (1) If the stretch quotients mapped between multiple regions are the same, in order to ensure that the interpolation function passes through the specified data points, jitter will occur; (2) The global interpolation method is prone to more exceedance points, that is, the interpolation function value in some intervals will be higher or lower than the maximum or minimum interpolation point, resulting in inaccurate shape interpolation.

[0107] Therefore, for the lofted structure, in the process of continuous shape interpolation calculation, the fewer the above two situations occur, the better. Based on this, for example, the local interpolation method of the spline curve is introduced into the interpolation of the expansion quotient K(f) and the deflection angle (arg(μ)) / 2.

[0108] In step S4, the lofted surface body modeling and parameterization method is as follows: Figure 5 As shown in the figure, the lofted surface is conformally mapped to the rectangular parameter domain, and all the lofted surfaces are arranged according to the normalized T number distribution; secondly, the U-direction interpolation is completed by the local spline interpolation method to form a parametric curved edge hexahedron region; then, the continuous parameters in the U direction are normalized to obtain the mapping of the lofted surface body to the unit cube; finally, the continuous and smooth lofted surface body configuration is calculated based on this mapping. In order to facilitate parameterized calculations, the height area is normalized, so the value range of t is t∈[0,1], and t=0 corresponds to S 0 , t = 1 corresponds to s n .by Perform local spline interpolation on the interpolation points to obtain the curve Then the expansion quotient corresponding to each isoparametric surface is calculated as It can be concluded that:

[0109]

[0110] In order to obtain the specific Beltrami coefficient of each triangle, it is also necessary to interpolate the argument of the mapping of each lofted surface, that is:

[0111]

[0112] Therefore, we have:

[0113]

[0114] At this point, the Beltrami coefficient corresponding to the single parameter t is obtained. The shape of each intermediate cross section can be calculated by the shape interpolation method given in step S4.

[0115] In step S5, based on the principle of the 2D generalized periodic lattice structure equivalent performance homogenization mapping method, a 3D generalized periodic lattice equivalent performance calculation method and its corresponding matrix operation form are given to overcome the problem of inaccurate performance characterization of microstructures in special-shaped units in three-dimensional multi-scale design problems. Assume that the mapping function expression of the periodic structure in the parameter domain is:

[0116] X=F(u)

[0117] Right now:

[0118]

[0119] In the above formula, X=[x 1 , x 2 , x 3 ] T Cartesian coordinates in the mapped physical space, u = (u, υ, ω) T is the parameter coordinate in the parameter domain, and the Jacobi matrix corresponding to the mapping is:

[0120]

[0121] Under this condition, the weak form of the microscopic equilibrium equation is:

[0122]

[0123] in, is the inverse of the Jacobian matrix, which represents the mapping from the parameter coordinate system u to the Cartesian coordinate system X. represents the corresponding tensor component, represents the generalized displacement in the microscopic parameter coordinate system, which is periodic in this coordinate system. ijkl is the flexibility tensor.

[0124] The solution is that the generalized homogenization equivalent performance of any geometric periodic microstructure is:

[0125]

[0126] in, can be written as:

[0127]

[0128] In the above formula, δ mjis the Kronecker symbol, so the weak form of the microscopic equilibrium equation and the generalized homogenization equivalent performance can be written as:

[0129]

[0130] Will Written as a three-dimensional second-order tensor coordinate transformation formula, that is:

[0131]

[0132] Where M jm and All refer to the components of the tensor and satisfy Define the displacement vector as Then the above formula can be written as:

[0133]

[0134] Writing the tensor form as a matrix form, we have:

[0135]

[0136] Where R is the strain transformation matrix, and the displacement vector It can be obtained through v, is the strain operator defined in microscopic parameter coordinates, R, Sum Operator They are:

[0137]

[0138]

[0139] The weak form of the microscopic equilibrium equation and the generalized homogenization equivalent performance are written in matrix form, respectively, and we have:

[0140] C ijkl ε iij ε kl →ε T Cε

[0141] The generalized displacement is χ is Therefore, the matrix form of the equilibrium equation and the equivalent elastic tensor is:

[0142]

[0143] In matrix form, CH = (DH) -1So far, the equivalent physical properties of the microstructure filled in the irregular unit have been obtained. By matching the coefficients in the elastic matrix of the microstructure with the volume fraction of the microstructure, and interpolating the coefficients in the elastic matrix using a cubic polynomial, we can obtain:

[0144]

[0145] Among them, D solid is the elastic matrix of the solid material, that is, D solid =D pqrs , assume that the elastic matrix of each microstructure satisfies:

[0146]

[0147] Here, “⊙” represents the Hadamard product. is a matrix with 36 coefficients, when the microstructure filling domain is determined, can be uniquely calculated by the above formula. And when the deflection angle is fixed, It will only be affected by the volume fraction, so each item in the matrix can be constructed with respect to the volume fraction function and reassemble it into a coefficient matrix, namely:

[0148]

[0149] If we assume is a The polynomial function, the highest degree of the polynomial is m, the above relationship can be expressed as:

[0150]

[0151] In step S6, the lofted surface structure is filled with gradients of the same family of microstructures with different volume fractions, which is completed on the basis of analyzing the microstructure and fitting and predicting its performance using a polynomial function. i The optimization process is driven by the results of finite element analysis and sensitivity analysis. The distribution of the microstructure volume fraction is changed in the parameter domain until the set iteration stop condition is met. The volume fraction value of each unit at the end of the iteration is output and the corresponding height parameter h is calculated. i The optimized column formula formed by the above content is shown below.

[0152]

[0153] In the formula, represents the pseudo-density field formed by the volume fraction of the microstructure in the parameter domain, C represents the flexibility value of the overall structure, and are the upper and lower bounds of the microstructure volume fraction, respectively. is the overall stiffness matrix of the structure assembled according to the equivalent elastic matrix of the gradient microstructure, and U and F are the displacement vector and external load vector of the node, respectively. i The volume of the design domain for each microstructure, is the volume of the overall macroscopic design domain, and v i and satisfy V is the volume fraction constraint of the optimization design, and is the optimized volume fraction control function, That is, the volume fraction of the overall structure is ensured to be no greater than the specified overall volume fraction V. Since the volume occupied by each unit in the lofted surface structure obtained by mapping is different, each microstructure must be accumulated when calculating the volume fraction of the overall structure. Represents the volume fraction within each microstructure unit, that is, the unit pseudo-density. In order to obtain the coefficient matrix of the microstructure elastic tensor, since the parameterized model and mesh of the swept surface structure have been calculated before the optimization design, the geometric shape of each microstructure unit is fixed, so the elastic tensor matrix of the microstructure is composed of volume fraction Unique control, which can be further interpolated by a polynomial function.

[0154] In summary, the multi-scale lofted surface structure design method proposed in the first embodiment of the present invention is to conformally map the simply connected domain on the lofted surface structure to the complex plane and calculate the affine stretch transformation of the mapping parameter domain, composite the affine stretch transformation with the conformal mapping to obtain the Teichmuller space mapping and calculate the stretch quotient, calculate the corresponding Beltrami coefficient based on the stretch quotient, thereby obtaining the interpolation shape between the simply connected domains, and realize the continuous and uniform transition of the two plane curved edge regions, and calculate the mechanical properties of the microstructure filled in the hexahedral unit by the generalized homogenization method, establish an optimization model and complete the solution, and obtain the volume fraction distribution (pseudo-density field) of the microstructure in the parameter domain, and obtain the local level set function of each unit according to the volume fraction of each unit combined with the optimization model, which is further assembled into a global level set function, and the global level set function is inversely mapped back to the lofted surface body configuration to realize the multi-scale lofted surface structure design with gradient filling.

[0155] In addition, the second embodiment of the present invention also proposes a multi-scale lofted surface structure design device, including: a simply connected domain mapping module, a stretch quotient calculation module, an interpolation shape calculation module, a lofted surface body configuration calculation module, a performance prediction module and a lofted surface structure design module.

[0156] Among them, the simply connected domain mapping module is used to define at least two simply connected domains on the surface structure to be lofted, obtain the mapping parameter domain corresponding to each of the simply connected domains, and calculate the affine stretch transformation of the mapping parameter domain corresponding to any two of the simply connected domains. The stretch quotient calculation module is used to combine the conformal mappings corresponding to the two simply connected domains and the affine stretch transformation between the two simply connected domains to obtain the Teichmuller space mapping between the two simply connected domains, and calculate the stretch quotient between the two simply connected domains based on the Teichmuller space mapping. The interpolation shape calculation module is used to calculate the Beltrami coefficient between the two simply connected domains based on the stretch quotient, and obtain the interpolation shape between the two simply connected domains according to the Beltrami coefficient. The lofted surface body configuration calculation module is used to map the surface structure to be lofted to the unit cube structure according to the interpolation shape, and calculate the lofted surface body configuration according to the unit cube structure. The performance prediction module is used to calculate the equivalent mechanical properties of a finite number of sample microstructure units by generalized homogenization method, and to establish a rapid prediction model of mechanical properties of gradient microstructures with variable volume fraction ratios by fitting sample data with continuous functions. The lofted surface structure design module is used to obtain the target volume fraction distribution of the finite number of sample microstructure units in the lofted surface body configuration according to the rapid prediction model, and to calculate the corresponding global level set function, and to inversely map the global level set function back to the lofted surface body configuration, so as to realize the multi-scale lofted surface structure design with gradient filling.

[0157] The multi-scale lofting surface structure design method implemented by the multi-scale lofting surface structure design device disclosed in the second embodiment of the present invention is as described in the first embodiment above, so it will not be described in detail here. Optionally, the various modules in the second embodiment and the above-mentioned other operations or functions are respectively for implementing the method described in the first embodiment, and the beneficial effects of the multi-scale lofting surface structure design device provided by this embodiment are the same as the beneficial effects of the multi-scale lofting surface structure design method provided by the first embodiment above, and for the sake of brevity, they are not repeated here.

[0158] The third embodiment of the present invention further proposes an electronic device, for example, comprising: at least one processing unit, and at least one storage unit, wherein the storage unit stores a computer program, and when the computer program is executed by the processing unit, the processing unit executes the method described in the first embodiment, and the beneficial effects of the electronic device provided by this embodiment are the same as the beneficial effects of the multi-scale lofting surface structure design method provided by the first embodiment.

[0159] The fourth embodiment of the present invention also provides a computer-readable storage medium, on which a computer program is stored. When the program is executed by a processor, the steps of the above method are implemented, and the beneficial effects of the computer-readable storage medium provided by this embodiment are the same as the beneficial effects of the multi-scale lofting surface structure design method provided by the first embodiment.

[0160] Among them, computer-readable storage media may include, but are not limited to, any type of disk, including floppy disks, optical disks, DVDs, CD-ROMs, microdrives and magneto-optical disks, ROMs, RAMs, EPROMs, EEPROMs, DRAMs, VRAMs, flash memory devices, magnetic or optical cards, nanosystems (including molecular memory ICs), or any type of medium or device suitable for storing instructions and / or data.

[0161] It should be noted that, for the aforementioned method embodiments, for the sake of simplicity, they are all expressed as a series of action combinations, but those skilled in the art should be aware that the present application is not limited by the described order of actions, because according to the present application, certain steps can be performed in other orders or simultaneously. Secondly, those skilled in the art should also be aware that the embodiments described in the specification are all preferred embodiments, and the actions and modules involved are not necessarily required by the present application.

[0162] In the above embodiments, the description of each embodiment has its own emphasis. For parts that are not described in detail in a certain embodiment, reference can be made to the relevant descriptions of other embodiments.

[0163] In the several embodiments provided in the present application, it should be understood that the disclosed devices can be implemented in other ways. For example, the device embodiments described above are only schematic, such as the division of the units, which is only a logical function division. There may be other division methods in actual implementation, such as multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the mutual coupling or direct coupling or communication connection shown or discussed can be through some service interfaces, and the indirect coupling or communication connection of devices or units can be electrical or other forms.

[0164] The units described as separate components may or may not be physically separated, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed on multiple network units. Some or all of the units may be selected according to actual needs to achieve the purpose of the solution of this embodiment.

[0165] In addition, each functional unit in each embodiment of the present application may be integrated into one processing unit, or each unit may exist physically separately, or two or more units may be integrated into one unit. The above-mentioned integrated unit may be implemented in the form of hardware or in the form of software functional units.

[0166] If the integrated unit is implemented in the form of a software functional unit and sold or used as an independent product, it can be stored in a computer-readable memory. Based on this understanding, the technical solution of the present application is essentially or the part that contributes to the prior art or all or part of the technical solution can be embodied in the form of a software product, and the computer software product is stored in a memory, including a number of instructions to enable a computer device (which can be a personal computer, server or network device, etc.) to perform all or part of the steps of the method described in each embodiment of the present application. The aforementioned memory includes: U disk, read-only memory (ROM), random access memory (RAM), mobile hard disk, disk or optical disk and other media that can store program code.

[0167] A person skilled in the art may understand that all or part of the steps in the various methods of the above embodiments may be completed by entering a program to instruct the relevant hardware. The program may be stored in a computer-readable memory, and the memory may include: a flash drive, a read-only memory (ROM), a random access memory (RAM), a magnetic disk or an optical disk, etc.

[0168] The above is only an exemplary embodiment of the present disclosure, and the scope of the present disclosure cannot be limited thereto. That is, any equivalent changes and modifications made according to the teachings of the present disclosure are still within the scope of the present disclosure. After considering the specification and practicing the disclosure here, those skilled in the art will easily think of the implementation scheme of the present disclosure. This application is intended to cover any modification, use or adaptation of the present disclosure, which follows the general principles of the present disclosure and includes common knowledge or customary technical means in the technical field not recorded in the present disclosure. The description and examples are regarded as exemplary only, and the scope and spirit of the present disclosure are defined by the claims.

[0169] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0170] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A multi-scale lofted surface structure design method, characterized in that: include: Define at least two simply connected domains on the surface structure to be lofted, obtain a mapping parameter domain corresponding to each of the simply connected domains, and calculate an affine stretching transformation of the mapping parameter domains corresponding to any two of the simply connected domains; Combining the conformal mappings corresponding to the two simply connected domains and the affine stretching transformation between the two simply connected domains to obtain a Teichmuller space mapping between the two simply connected domains, and calculating the stretch quotient between the two simply connected domains based on the Teichmuller space mapping; Calculating a Beltrami coefficient between the two simply connected domains based on the stretch quotient, and obtaining an interpolation shape between the two simply connected domains according to the Beltrami coefficient; Mapping the surface structure to be lofted to a unit cube structure according to the interpolation shape, and calculating the configuration of the lofted surface body according to the unit cube structure; The equivalent mechanical properties of a finite number of sample microstructure units are calculated by the generalized homogenization method, and a rapid prediction model for the mechanical properties of gradient microstructures with variable volume ratios is established by fitting the sample data with a continuous function. The target volume fraction distribution of the finite number of sample microstructure units in the lofted surface volume configuration is obtained according to the rapid prediction model, and the corresponding global level set function is calculated. The global level set function is inversely mapped back to the lofted surface volume configuration to realize the multi-scale lofted surface structure design with gradient filling.

2. The multi-scale lofted surface structure design method according to claim 1, characterized in that: The Beltrami coefficient between the two simply connected domains is calculated based on the stretch quotient, including: The stretch quotient is made to change monotonically and linearly to determine the introduced variable between the two simply connected domains, an interpolation function is set to establish a relationship in which the stretch quotient changes with the change of the introduced variable, and the Beltrami coefficient corresponding to the stretch quotient is obtained according to the interpolation function.

3. The multi-scale lofted surface structure design method according to claim 1, characterized in that: The simply connected domain includes a plurality of triangular facets in a triangulated form, and the interpolation shape between two simply connected domains is obtained based on the Beltrami coefficients, including: The curved surface shape after mapping of each triangular facet is obtained according to the Beltrami coefficients, and all the mapped curved surface shapes are assembled and stitched to obtain the interpolation shape between the two simply connected domains.

4. The multi-scale lofted surface structure design method according to claim 3, characterized in that: The step of obtaining the surface shape of each triangular face after mapping according to the Beltrami coefficients includes: Acquire the vertex set and connection relationship of the triangular facets to determine an original triangulation mesh and a target triangulation mesh composed of the triangular facets on the two simply connected domains; Using a piecewise linear function to approximate the mapping of the original triangulated mesh to the target triangulated mesh, and obtaining a linear change of each triangular patch in the original triangulated mesh mapped to the target triangulated mesh; The area of ​​each triangular facet and the Beltrami coefficient are calculated according to the linear change to construct the surface shape after each triangular facet is mapped.

5. The multi-scale lofted surface structure design method according to claim 1, characterized in that: The step of mapping the surface structure to be lofted to a unit cube structure according to the interpolation shape, and calculating the lofted surface body configuration according to the unit cube structure, comprises: Arranging the mapping parameter domain corresponding to the simply connected domain according to the surface structure to be lofted, and normalizing it according to the direction of the distribution arrangement; Performing local spline interpolation on the normalized mapping parameter domain according to the interpolation shape to form a parameter curved-edge hexahedron region; The parameter curved-edge hexahedron region is normalized according to the direction of local spline interpolation to obtain the unit cube structure corresponding to the surface structure to be lofted, and the lofted surface body configuration is calculated based on the unit cube structure.

6. The multi-scale lofted surface structure design method according to claim 5, characterized in that: The method of performing local spline interpolation on the normalized mapping parameter domain to form a parameter curved-edge hexahedron region includes: Setting interpolation points to perform local spline interpolation on the mapping parameter domain to obtain a spline curve, and then calculating the expansion and contraction quotient of the mapping parameter domain to the parameter curved hexahedron region; Interpolating the argument of the mapping parameter domain to the parameter curved hexahedron region according to the expansion and contraction quotient to obtain the Beltrami coefficient of the mapping parameter domain, and calculating each intermediate cross-sectional shape of the mapping parameter domain to the parameter curved hexahedron region according to the Beltrami coefficient; The modeling equation of the parametric curved hexahedron region is calculated based on the intermediate cross-sectional shape.

7. The multi-scale lofted surface structure design method according to claim 1, characterized in that: The method of obtaining the target volume fraction distribution of the finite number of sample microstructure units in the lofted curved surface configuration according to the fast prediction model and calculating the corresponding global level set function includes: According to the rapid prediction model, a structural optimization formula with minimum flexibility as the goal under the global volume fraction constraint of the lofted surface body configuration is established, and the local level set function is calculated according to the target volume fraction distribution after optimization by the structural optimization formula, and the local level set function is assembled into a global level set function.

8. A multi-scale lofting surface structure design device, characterized in that: include: A simply connected domain mapping module, used to define at least two simply connected domains on the surface structure to be lofted, obtain a mapping parameter domain corresponding to each of the simply connected domains, and calculate an affine stretching transformation of the mapping parameter domains corresponding to any two of the simply connected domains; a stretch quotient calculation module, used for compounding the conformal mappings respectively corresponding to the two simply connected domains and the affine stretch transformation between the two simply connected domains to obtain a Teichmuller space mapping between the two simply connected domains, and calculating the stretch quotient between the two simply connected domains based on the Teichmuller space mapping; An interpolation shape calculation module, used for calculating the Beltrami coefficient between the two simply connected domains based on the expansion quotient, and obtaining the interpolation shape between the two simply connected domains according to the Beltrami coefficient; A lofted surface body configuration calculation module, used for mapping the surface structure to be lofted to a unit cube structure according to the interpolation shape, and calculating the lofted surface body configuration according to the unit cube structure; The performance prediction module is used to calculate the equivalent mechanical properties of a finite number of sample microstructure units by generalized homogenization method, and to establish a rapid prediction model for the mechanical properties of gradient microstructures with variable volume ratios by fitting sample data with continuous functions; The lofted surface structure design module is used to obtain the target volume fraction distribution of the finite number of sample microstructure units in the lofted surface body configuration according to the fast prediction model, and calculate the corresponding global level set function, and inversely map the global level set function back to the lofted surface body configuration to realize the multi-scale lofted surface structure design with gradient filling.

9. An electronic device, characterized in that: include: A memory and one or more processors connected to the memory, the memory storing a computer program, and the processor being used to execute the computer program to implement the multi-scale lofting surface structure design method as described in any one of claims 1 to 7.

10. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores computer-executable commands, and the computer-executable commands are used to execute the multi-scale lofting surface structure design method as described in any one of claims 1-7.

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