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

Abstract

The application discloses a kind of multi-scale lofting curved surface structure design methods, comprising: defining at least two single-connected domains on lofting curved surface structure, obtain corresponding mapping parameter domain and calculate affine stretch transformation;According to the composition of conformal mapping and affine stretch transformation obtains Teiximieller space mapping and calculates scale quotient;Based on scale quotient, interpolation shape between two single-connected domains is calculated, and the fast prediction model of the mechanical property of gradient microstructure is established by generalized homogenization method, to obtain the volume fraction distribution of sample microstructure unit in lofting curved surface body configuration, and the corresponding global level set function is calculated, the global level set function is inversely mapped back to lofting curved surface body configuration, and the multi-scale lofting curved surface structure design of gradient filling is realized.It can solve the problem that the ideal lofting curved surface body configuration is difficult to obtain due to the tedious iteration required by the existing interactive design multi-scale lofting curved surface, and the interpolation method is embedded and cannot be controlled.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of structural optimization design, and particularly relates to a multi-scale lofting curved surface structure design method, a multi-scale lofting curved surface structure design device, an electronic device and a computer readable storage medium. BACKGROUND

[0002] Lofting is a process of forming a complex three-dimensional object by sweeping a plurality of two-dimensional object as a cross section of a structure along a certain direction, which is commonly used in the generation of complex curved surfaces such as blades, and is a commonly used curved surface and curved surface body generation operation in mechanical structure design. The traditional lofting operation is mainly aimed at continuous parameterized curves such as NURBS curves, and the curve masking calculation is realized by order raising and node refinement. Although the continuous parameterized lofting curved surface generation calculation method is simple in principle, it is not easy to realize a robust and convenient lofting function. In addition, the interactive design lofting curved surface is a tedious iterative process, which depends on the number, shape and position of the lofting surface. Since the interpolation method is embedded in the computer, the designer cannot control it, so a large amount of time will be consumed to obtain an ideal lofting curved surface body configuration.

[0003] Moreover, the introduction of multi-scale design has greatly improved the function and performance of the structure, but the combination with the complex curved surface structure generated by lofting is limited, and the correlation between the geometric characteristics of the microstructure filled in the special hexahedral element and the physical performance has not been established. Therefore, how to realize the fast and stable multi-scale lofting curved surface body structure design is a problem to be solved at present. SUMMARY

[0004] In order to overcome the defects of the prior art, the embodiments of the present application provide a multi-scale lofting curved surface structure design method and application, which can solve the problem that the existing interactive design multi-scale lofting curved surface needs a tedious iteration, the interpolation method is embedded and cannot be controlled, and it is difficult to obtain an ideal lofting curved surface body configuration.

[0005] In one aspect, the embodiment of the present application provides a multi-scale lofted curved surface structure design method, comprising: defining at least two simply connected domains on a to-be-lofted curved surface structure, obtaining a mapping parameter domain corresponding to each of the simply connected domains, and calculating an affine stretching transformation of the mapping parameter domain corresponding to any two of the simply connected domains; composing the conformal mapping 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 scaling 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 scaling quotient, and obtaining an interpolation shape between the two simply connected domains according to the Beltrami coefficient; mapping the to-be-lofted curved surface structure to a unit cube structure according to the interpolation shape, and calculating a lofted curved surface body configuration according to the unit cube structure; calculating the equivalent mechanical properties of a limited number of sample microstructure units by a generalized homogenization method, and establishing a fast prediction model of the mechanical properties of the gradient microstructure with variable volume fraction ratio by fitting the sample data with a continuous function; obtaining a target volume fraction distribution of the limited number of sample microstructure units in the lofted curved surface body configuration according to the fast prediction model, and calculating a corresponding global level set function, mapping the global level set function back to the lofted curved surface body configuration to realize multi-scale lofted curved surface structure design with gradient filling.

[0006] In one embodiment of the present application, the calculation of the Beltrami coefficient between the two simply connected domains based on the scaling quotient comprises: linearly changing the scaling quotient monotonically to determine an introduced variable between the two simply connected domains, setting an interpolation function to establish a relationship between the scaling quotient and the introduced variable, and obtaining the Beltrami coefficient corresponding to the scaling quotient according to the interpolation function.

[0007] In one embodiment of the present application, the simply connected domain comprises a plurality of triangular facets in the form of triangular subdivision, and the calculation of the interpolation shape between the two simply connected domains based on the Beltrami coefficient comprises: obtaining a mapped curved surface shape of each of the triangular facets according to the Beltrami coefficient, assembling and stitching all the mapped curved surface shapes to obtain the interpolation shape between the two simply connected domains.

[0008] In one embodiment of the present application, the method for obtaining the curved surface shape of each triangular facet after mapping according to the Bézier coefficients comprises: obtaining a vertex set and a connection relationship of the triangular facet to determine an original triangular subdivision mesh and a target triangular subdivision mesh composed of the triangular facet on two single-connected domains; approximating the mapping of the original triangular subdivision mesh to the target triangular subdivision mesh by using a piecewise linear function to obtain a linear variation of each triangular facet in the original triangular subdivision mesh to the target triangular subdivision mesh; and calculating the area of each triangular facet and the Bézier coefficients according to the linear variation to construct the curved surface shape of each triangular facet after mapping.

[0009] In one embodiment of the present application, the method for mapping the surface structure to be lofted to a unit cube structure according to the interpolation shape and calculating a lofted surface body configuration according to the unit cube structure comprises: arranging the mapping parameter domain corresponding to the single-connected domain according to the surface structure to be lofted and performing normalization processing according to the direction of the arrangement; completing local spline interpolation on the normalized mapping parameter domain according to the interpolation shape to form a parameter curved hexahedron region; performing normalization processing on the parameter curved hexahedron region according to 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 application, the method for completing local spline interpolation on the normalized mapping parameter domain to form a parameter curved hexahedron region comprises: setting an interpolation point to perform local spline interpolation on the mapping parameter domain to obtain a spline curve, and further calculating a scaling quotient of the mapping parameter domain to the parameter curved hexahedron region; interpolating an amplitude angle of the mapping parameter domain to the parameter curved hexahedron region according to the scaling quotient to obtain a Bézier coefficient of the mapping parameter domain, and calculating each intermediate state cross-sectional shape of the mapping parameter domain to the parameter curved hexahedron region according to the Bézier coefficient; and calculating a modeling equation of the parameter curved hexahedron region according to the intermediate state cross-sectional shape.

[0011] In one embodiment of the present application, the method for obtaining a target volume fraction distribution of the limited number of sample microstructure units in the lofted surface body configuration according to the fast prediction model and calculating a corresponding global level set function comprises: establishing a structure optimization formula with a minimum flexibility as a target under a global volume fraction constraint of the lofted surface body configuration according to the fast prediction model, and calculating a local level set function according to the target volume fraction distribution after optimization of the structure optimization formula, and assembling the local level set function into a global level set function.

[0012] In another aspect, the embodiments of the present application also provide a multi-scale lofted curved surface structure design device, comprising: a simply connected domain mapping module, configured to define at least two simply connected domains on a curved 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 domain corresponding to any two of the simply connected domains; a stretch ratio calculation module, configured to compose the conformal mapping 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 a stretch ratio between the two simply connected domains based on the Teichmuller space mapping; an interpolation shape calculation module, configured to calculate a Beltrami coefficient between the two simply connected domains based on the stretch ratio, and obtain an interpolation shape between the two simply connected domains according to the Beltrami coefficient; a lofted curved surface body configuration calculation module, configured to map the curved surface structure to be lofted to a unit cube structure according to the interpolation shape, and calculate a lofted curved surface body configuration according to the unit cube structure; a performance prediction module, configured to calculate equivalent mechanical properties of a limited number of sample microstructure units by a generalized homogenization method, and establish a fast prediction model of mechanical properties of gradient microstructures with variable volume fraction ratio by fitting sample data with a continuous function; and a lofted curved surface structure design module, configured to obtain a target volume fraction distribution of the limited number of sample microstructure units in the lofted curved surface body configuration according to the fast prediction model, and calculate a corresponding global level set function, map the global level set function back to the lofted curved surface body configuration, and realize multi-scale lofted curved surface structure design with gradient filling.

[0013] In another aspect, the embodiments of the present application also provide 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 configured to execute the computer program to implement the multi-scale lofted curved surface structure design method according to any one of the above embodiments.

[0014] In another aspect, the embodiments of the present application also provide a computer readable storage medium, the computer readable storage medium storing computer executable instructions, and the computer executable instructions being configured to execute the multi-scale lofted curved surface structure design method according to any one of the above embodiments.

[0015] From the above, the above embodiments of the present application can have at least one or more of the following beneficial effects compared with the prior art:

[0016] The interpolation shape between single-connected domains is obtained by conformal mapping single-connected domains on lofted curved surface structure to complex plane, calculating affine stretch transformation of mapping parameter domain, combining affine stretch transformation with conformal mapping to obtain Teichmuller space mapping and calculating stretch quotient, calculating corresponding Beltrami coefficient based on the stretch quotient, realizing continuous and uniform transition of two planar curved edge regions, calculating mechanical properties of microstructure filled in hexahedral element by generalized homogenization method, establishing optimization model and completing solving, obtaining volume fraction distribution (pseudo density field) of microstructure in parameter domain, obtaining local level set function of each element according to the volume fraction of each element, combining the optimization model to further assemble into global level set function, inversely mapping the global level set function back to the lofted curved surface body configuration, and realizing gradient filling of multi-scale lofted curved surface structure design. BRIEF DESCRIPTION OF DRAWINGS

[0017] The accompanying drawings, which are included to provide a further understanding of the application and are incorporated in and constitute a part of this application, illustrate embodiments of the application and together with the description serve to explain the application. In the drawings:

[0018] Figure 1 A flow chart of the multi-scale lofted curved surface structure design method provided by the embodiment of the application is shown in the figure.

[0019] Figure 2 A mapping principle diagram between two adjacent lofted curved surfaces provided by the embodiment of the application is shown in the figure.

[0020] Figure 3 A process diagram of constructing triangular facet mapping of curved surface shape according to the Beltrami coefficient provided by the embodiment of the application is shown in the figure.

[0021] Figure 4 A principle diagram of lofted surface conformal parameterization provided by the embodiment of the application is shown in the figure.

[0022] Figure 5 A principle diagram of lofted curved surface body modeling and parameterization provided by the embodiment of the application is shown in the figure. DETAILED DESCRIPTION

[0023] It should be noted that the embodiments and features in the embodiments of the application can be combined with each other without conflict. The embodiments of the application will be described below with reference to the accompanying drawings and in combination with the embodiments.

[0024] In order for those skilled in the art to better understand the technical solutions of the present application, the technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, not all the embodiments of the present application, and all should belong to the protection scope of the present application.

[0025] It should be noted that the terms "first", "second", etc. in the specification and claims of the present application and in the above drawings are used to distinguish similar objects, and do not necessarily have to be used to describe a specific order or sequence. It should be understood that the terms thus used can be interchanged under appropriate circumstances, so that the embodiments of the application described herein can be implemented in an order other than that illustrated or described herein. In addition, the terms "include" and "have" and any variations thereof are intended to cover non-exclusive inclusion, for example, a process, method, system, product or device including a series of steps or units does not have to be limited to only those steps or units clearly listed, but can include other steps or units not clearly listed or inherent to these processes, methods, products or devices.

[0026] It should also be noted that the division of the various embodiments in the present application is only for the convenience of description, and should not constitute a special limitation. The features in the various embodiments can be combined with each other under the condition of no contradiction, and can be mutually referenced.

[0027] As shown in Figure 1 The first embodiment of the present application proposes a multi-scale lofting curved surface structure design method, for example, including: step S1, defining at least two simply connected domains on the lofting curved surface structure, obtaining the mapping parameter domain corresponding to each of the simply connected domains, and calculating the affine stretching transformation of the mapping parameter domain corresponding to any two of the simply connected domains; step S2, composing the conformal mapping corresponding to two of the simply connected domains and the affine stretching transformation between the two simply connected domains to obtain the Teichmuller space mapping between the two simply connected domains, and calculating the stretching 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 stretching quotient, and obtaining the interpolation shape between the two simply connected domains according to the Beltrami coefficient; step S4, mapping the lofting curved surface structure corresponding to the unit cube structure according to the interpolation shape, and calculating the lofting curved 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 establishing a fast prediction model of the mechanical properties of the gradient microstructure with variable volume fraction ratio by fitting the sample data with a continuous function; step S6, obtaining the target volume fraction distribution of the finite number of sample microstructure units in the lofting curved surface body configuration according to the fast prediction model, and calculating the corresponding global level set function, 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 with gradient filling.

[0028] Specifically, as shown in Figure 2 For example, two planar curved Jordan regions S mand S n , will S m and S n By selecting corner points to transform them into topological quadrilaterals, the two topological quadrilaterals are conformally mapped onto the rectangular domain R. m and R n Inside, the ratio of the rectangle's x and y coordinates, a m and a n That is, S m and S n The conformal mode. Further utilizing a m and a n Calculate from S m and S n The scaling quotient is obtained, and the stretching affine transformation h between the two rectangular domains is obtained. [m，n] A linear function of the scaling quotient with respect to parameter t is established, and the Beltrami coefficient is calculated using the scaling quotient value corresponding to each parameter t. Then, the interpolation shape is determined based on the Beltrami coefficient corresponding to t. When parameter t is within its 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 region corresponding to t is represented by triangulation as follows: The interpolation parameter t takes values ​​in the range [0, 1]. For two Jordan regions S m and S n Smooth shape interpolation between these elements must satisfy the following conditions:

[0030] (1) When t∈[0,1], we have f t＝0 =Id and f t＝1 =f. For the interpolation of the mapping, it must satisfy that when t=0, the mapping is to transform S. 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 mapping, i.e., f t＝1 =f:S m →S n .

[0031] (2) Pointwise bounded nonconformal distortion. Let point p be S. m One point in the middle, i.e., p∈ S m, K(p) is the non-conformal distortion at point p, i.e., the scaling quotient. And when the mapping at time t is interpolated, it is... When, the corresponding ones are If it is the expansion quotient at this time, then there is It should be noted that if in f [m，n] :S m →S n There is a special case where K(f) ≡ 1 for any point, meaning that the mapping f is a conformal mapping. Therefore, for... have Right now At this point, it is also necessary to maintain a conformal mapping.

[0032] (3) Symmetry. Let f -1 :S m →S n This is the inverse process of mapping f, and From S n To S m If the interpolation result at time t is obtained using the same method, then we have: In other words, regardless of whether it is from S m To S n Or from S n To S m The interpolation results obtained for t are also one-to-one correspondences, and have

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

[0034] Therefore, in step S1, for example, in the two Jordan regions S respectively m and S n Find the corresponding counterclockwise distributed corner points on the top, that is... and Construct S m and S n Based on 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}). Map the two topological quadrilaterals to R using conformal mapping. m and R n Within two planar rectangles, we obtain the result in the complex plane. 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] To obtain continuous shape interpolation, a harmonic mapping must be found that satisfies:

[0038]

[0039] in, and Let Δ be the boundary between two topological quadrilaterals, and let Δ be the Laplacian operator defined on the complex plane. The stretching affine transformation h... [m，n] In the rectangle, the ordinate of each point remains unchanged, while the abscissa changes proportionally, that is:

[0040]

[0041] Therefore, as Figure 2 As shown, for composite mapping Due to g m and g n -1 It is a conformal mapping, and h [m，n] It is harmonized, therefore f [m，n] It is also harmonized. Furthermore, due to g m and g n -1 If it is a conformal mapping, then the mapping f [m，n] Conformal distortion and h [m，n] The conformal distortion is the same. At the same time, it can be calculated that... The Beltrami coefficient is:

[0042]

[0043] This means:

[0044]

[0045] Similarly, we have:

[0046]

[0047] Since h [m，n] : R m → R n is a stretching affine transformation with stretch quotient K z (f) = K mn , i.e. a n / a m (a n > a m ) or a m / a n (a n ≤ a m ), the Beltrami coefficient of f [m，n] can be calculated as:

[0048]

[0049] At this time, the Teichmüller distance between the two rectangular domains, i.e. the maximum quasi-conformal dilation between two rectangular domains (further developed to the extremal mapping between two planar curve domains), is uniquely determined. S m The Teichmüller distance d between S n and S m is:

[0050]

[0051] In step S2, for two conformal mappings, a stretching affine transformation (Teichmüller mapping) g n , g -1 [m，n] and h m are composed to obtain the Teichmüller mapping from S n and S [m，n] :

[0052]

[0053] After calculating the stretch quotient K(f) of the mapping f t(f) After that, the Beltrami coefficient corresponding to t is calculated t (f).

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

[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] The scaling quotient of the mapping f [m，n] can be calculated as

[0058]

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

[0060]

[0061] Define a normalization coefficient such that:

[0062]

[0063] Let then the Beltrami coefficient at time t can be defined as:

[0064]

[0065] ​According to the mapping and the value of t at any time, the Beltrami coefficient corresponding to the time can be calculated.

[0066] Further, for example, input two Jordan regions S m and S n in the form of triangular subdivision m to S n mapping relationship For the convenience of calculation, the index order in the connection relationship of the triangular subdivision representation of S m and S n needs to be unified, that is, for S m ={X m , F m} and S n ={X n , F n}, F m =F n , so the connection relationship of the nodes in S m and S n will be unified as F in the following. If the triangular subdivision of the input surface does not correspond, the coordinate points on S m can be mapped to S n according to the mapping relationship f [m，n] between the two surfaces, and the triangular subdivision index F m on S m is used.

[0067] The above steps give a method for calculating the scaling quotient and Beltrami coefficient corresponding to any time t through the mapping relationship of two topological quadrilaterals. In theory, after the Beltrami coefficient μ t (f) is given, there must be a smooth mapping f t corresponding to it. However, in the actual operation process, the expected piecewise linear mapping cannot be obtained by directly solving the Beltrami equation. At the same time, since the Beltrami coefficient depends on the triangular patch (i.e. the connection relationship F) rather than the coordinate points in the actual calculation of piecewise mapping, for a discrete triangular subdivision grid with the number of points being only half of the number of surfaces, the obtained equation cannot be solved. Therefore, for example, the solution of the mapping is divided into two steps: first, the shape of each triangular patch after mapping under the specified scaling quotient and Beltrami coefficient is solved, and then all triangular patches are assembled and stitched to obtain the target interpolation result.

[0068] Further, S m ={V m , F} is used to represent the triangular subdivision grid on the region S m , where X mIt is the set of vertices of triangles, and F represents the connection relationship between the triangles. f = u + iv is from the original mesh S. m ={V m ,F}to the target meshS n ={V n The mapping of F}, where the grid S m and S n The number of vertices is the same, and they have the same connection relationship F. A piecewise linear function is used to approximate the mapping f, specifically, f 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 It is the counterclockwise sorted vertex of triangle F.

[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 the mapping is v = g + ih, and the point after the mapping is w = s + it, then:

[0072]

[0073] Then, for each triangle, we can obtain the following based on linear transformation:

[0074]

[0075] Among them are:

[0076]

[0077] Among them, A F This is the area of ​​the triangular facet. Simultaneously, the Beltrami coefficient can be calculated on this triangle as:

[0078]

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

[0080]

[0081] Solve for v′ kx and v′ kx .

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

[0083] Specifically, in the two-dimensional complex plane, the rotation and scaling matrices of a vector can be written in the following form:

[0084]

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

[0086]

[0087] Where E represents the boundary relationship of the triangle, E = {{i, j}, {j, k}, {k, i}}. Substituting the matrix form from the above into the expression, we can obtain the optimized formulation of each term in the coordinate transformation matrix as follows:

[0088]

[0089] in,

[0090]

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

[0092]

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

[0094]

[0095] The formula is about variables The quadratic positive definite optimization problem can be written in classic matrix form, i.e.

[0096]

[0097] in, The coefficient matrix is ​​assembled, with each row corresponding to a summation term in the column; variables Let N be a 2N×1 vector, where N is the number of coordinate points in the triangulation:

[0098]

[0099] b is a (M+2C)×1 column vector, where M is the number of terms summed in the first half of the equation, and C is the number of fixed points. From the matrix form, it can be seen that this optimization problem can be solved using the conjugate gradient method given some vertices. Thus, we have obtained the Teichmüller mapping between the Jordan regions on the two complex planes and the scaling quotient of each corresponding triangular facet, and calculated the corresponding Beltrami coefficients by interpolating the scaling quotients. Further, based on the Beltrami coefficients interpolated from the intermediate states, we solve for the shape of each triangular facet, and then perform ARAP stitching on all triangles. By solving a quadratic positive definite optimization problem, we obtain the shape of the intermediate transition surface and the corresponding triangulation.

[0100] Furthermore, regarding S 0 To S n For these n+1 Jordan regions, the scaling factor of each region's Teichmüller mapping to the previous region is:

[0101]

[0102] At the same time, we can obtain the S of each staking area. i For the first region S 0 The scaling factor is:

[0103]

[0104] For ease of calculation and representation, this section defines the mapping relative scaling quotient. This means we don't discuss the side length relationships of the mapped rectangles, but only focus on their relative proportions. In other words:

[0105]

[0106] like Figure 4As shown, if lofting is only performed between two planar curve domains, following the interpolation method given in the above steps, let the scaling factor K... t (f) It is feasible to solve for the shape of the intermediate state 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 endpoints will be discontinuous. However, if the traditional continuous interpolation method, including polynomial interpolation, spline interpolation, etc., is used for the scaling factor, the following situations are likely to occur: (1) If the scaling factor is the same between multiple regions, 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 overshoot 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 lofted structures, the fewer of the above two types of situations occur during continuous shape interpolation calculations, the better. Based on this, for example, the local interpolation method of spline curves can be introduced into the interpolation of the scaling quotient K(f) and the deflection angle (arg(μ)) / 2.

[0108] In step S4, the lofted surface modeling and parameterization method is as follows: Figure 5 As shown, the lofted surfaces are conformally mapped onto the rectangular parameter domain, and all lofted surfaces are arranged according to the distribution of the normalized T-number. Next, interpolation in the U-direction is performed using local spline interpolation to form a parametric curved hexahedral region. Then, the continuous parameters in the U-direction are normalized to obtain the mapping from the lofted surface to the unit cube. Finally, the configuration of the continuous and smooth lofted surface is calculated based on this mapping. To facilitate parametric calculation, the height region is normalized; therefore, the range of t is t∈[0,1], and t=0 corresponds to S. 0 t=1 corresponds to s n .by Curves are obtained by performing local spline interpolation at the interpolation points. Then, the scaling quotient corresponding to each isoparametric surface is calculated as follows: Therefore, we can conclude that:

[0109]

[0110] To obtain the specific Beltrami coefficients for each triangle, it is also necessary to interpolate the argument of the mapping for each lofted surface, i.e.:

[0111]

[0112] Therefore, we have:

[0113]

[0114] Thus, the Beltrami coefficients corresponding to the single parameter t are obtained, and the shape of each intermediate state section can be calculated by the shape interpolation method given in step S4

[0115] In step S5, according to 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 representation of microstructure in a three-dimensional heterogeneous unit in a multi-scale design problem. Assuming that the mapping function expression of the periodic structure in the parameter domain is:

[0116] X = F(u)

[0117] That is:

[0118]

[0119] In the above formula, X = [x1, x2, x3] T The Cartesian coordinates in the physical space after mapping, u = (u, v, w) T The parameter coordinates in the parameter domain, and the corresponding Jacobian matrix of the mapping is:

[0120]

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

[0122]

[0123] Where, is the inverse of the Jacobian matrix, indicating the mapping of the parameter coordinate system u to the Cartesian coordinate system X, Indicates the corresponding tensor component, Indicates the generalized displacement in the microscopic parameter coordinate system, which is periodic in this coordinate system, C ijkl The compliance tensor.

[0124] Solving can obtain the generalized homogenization equivalent performance of any geometric periodic microstructure:

[0125]

[0126] Where, Can be written as:

[0127]

[0128] In the above formula, δ mj is 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 This can be written as a three-dimensional second-order tensor coordinate transformation formula, namely:

[0131]

[0132] In the formula, M jm and Both refer to the components of a tensor, and satisfy the following conditions: Define the displacement vector as The above formula can then be written as:

[0133]

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

[0135]

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

[0137]

[0138]

[0139] If we express the weak form of the microequilibrium equation and the equivalent performance of the generalized homogenization in matrix form, we have:

[0140] C ijkl ε iij ε kl →ε T Cε

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

[0142]

[0143] In matrix form, we have CH = (DH) -1 Thus, the equivalent physical properties of the microstructure filling within the irregular unit were obtained. By mapping the coefficients of the elasticity matrix of the microstructure to the volume fraction of the microstructure, and interpolating the coefficients of the elasticity matrix using a cubic polynomial, the following can be obtained:

[0144]

[0145] where D solid is the elasticity matrix of the solid material, i.e. D solid = D pqrs , and let the elasticity matrix of each microstructure satisfy:

[0146]

[0147] where "⊙" denotes the Hadamard product. is a 36-coefficient matrix, which can be uniquely calculated from the above equation when the microstructure filling domain is determined, and when the deflection angle is fixed, will only be affected by the volume fraction, so each item in the matrix can be constructed as a function of the volume fraction and reassembled into a coefficient matrix, i.e.

[0148]

[0149] If it is assumed that is a polynomial function of , the highest order of the polynomial is m, and the above relationship can be expressed as:

[0150]

[0151] In step S6, the lofted surface structure is filled with a gradient of the same family of microstructures with different volume fractions, and the performance of the microstructure is fitted and predicted based on the analysis of the microstructure and the polynomial function. Since the height parameter h i of the local level set function is one-to-one corresponding to the volume fraction of the microstructure, the optimization process is driven by the results of the finite element analysis and the sensitivity analysis, and the volume fraction value of each element at the time of iteration completion is output by changing the distribution of the volume fraction of the microstructure in the parameter domain until the set iteration stopping condition is met, and the corresponding height parameter h i is calculated. The optimization equation formed by the above content is shown in the following equation.

[0152]

[0153] In the equation, denotes the pseudo-density field formed by the volume fraction of the microstructure in the parameter domain, C denotes the flexibility value of the overall structure, and are the upper and lower bounds of the volume fraction of the microstructure, respectively. is the overall stiffness matrix of the structure assembled according to the equivalent elasticity matrix of the gradient microstructure, and U and F are the displacement vector and the external load vector of the node, respectively.v i is the volume of each microstructure design domain, is the volume of the whole macroscopic design domain, and v i and satisfies V is the volume fraction constraint of the optimization design, and is the volume fraction control function of the optimization, that is, it ensures that the volume fraction of the whole structure is not greater than the specified whole volume fraction V, and since the volume occupied by each unit in the lofted surface structure obtained by mapping is different, the volume fraction of the whole structure must be calculated by accumulating each microstructure. represents the volume fraction in each microstructure unit, that is, the pseudo density of the unit. is the coefficient matrix of the calculated microstructure elastic tensor, and since the parametric model of the swept surface structure and its grid have been calculated before the optimization design, the geometry of each microstructure unit is fixed, so the elastic tensor matrix of the microstructure is calculated by the volume fraction which is uniquely controlled and can be further interpolated by a polynomial function.

[0154] In summary, the multi-scale lofted surface structure design method of the first embodiment of the present application maps the simply connected domain on the lofted surface structure to the complex plane and calculates the affine stretching transformation of the mapping parameter domain, composes the affine stretching transformation and the conformal mapping to obtain the Teichmuller space mapping and calculates the stretching quotient, calculates the corresponding Beltrami coefficient based on the stretching quotient, thereby obtaining the interpolation shape between the simply connected domains, realizes the continuous and uniform transition of the two planar curved edge regions, and calculates the mechanical properties of the hexahedral unit filled with the microstructure by the generalized homogenization method, establishes an optimization model and completes the solution, obtains the volume fraction distribution (pseudo density field) of the microstructure in the parameter domain, obtains the local level set function of each unit according to the volume fraction of each unit, and further assembles it into a global level set function, maps the global level set function back to the lofted surface body configuration, and realizes the gradient filling of the multi-scale lofted surface structure design.

[0155] In addition, the second embodiment of the present application also provides a multi-scale lofted surface structure design device, which comprises a simply connected domain mapping module, a stretching 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] The system includes several modules: a simple connected domain mapping module, a lofted surface structure, and a lofted surface volume configuration module. The simple connected domain mapping module defines at least two simple connected domains on the surface structure to be lofted, obtains the mapping parameter domain corresponding to each simple connected domain, and calculates the affine stretching transformation between any two simple connected domains. A scaling quotient calculation module combines the conformal mappings corresponding to the two simple connected domains with the affine stretching transformation between the two simple connected domains to obtain the Teichmüller space mapping between the two simple connected domains, and calculates the scaling quotient between the two simple connected domains based on the Teichmüller space mapping. An interpolation shape calculation module calculates the Beltrami coefficients between the two simple connected domains based on the scaling quotients, and obtains the interpolation shape between the two simple connected domains based on the Beltrami coefficients. A lofted surface volume configuration calculation module maps the surface structure to be lofted to a unit cube structure based on the interpolation shape, and calculates the lofted surface volume configuration based on the unit cube structure. The performance prediction module calculates the equivalent mechanical properties of a finite number of sample microstructural units using a generalized homogenization method, and establishes a rapid prediction model for the mechanical properties of gradient microstructures with variable volume fractions by fitting the sample data with a continuous function. The lofted surface structure design module obtains the target volume fraction distribution of the finite number of sample microstructural units in the lofted surface configuration based on the rapid prediction model, calculates the corresponding global level set function, and inversely maps the global level set function back to the lofted surface configuration to achieve gradient-filled multi-scale lofted surface structure design.

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

[0158] The third embodiment of the present invention also proposes an electronic device, for example including: 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 performs the method described in the first embodiment, and the beneficial effects of the electronic device provided in this embodiment are the same as the beneficial effects of the multi-scale lofted surface structure design method provided in the first embodiment.

[0159] The fourth embodiment of the present application also provides a computer readable storage medium, which stores a computer program, and the computer program is executed by a processor to implement the steps of the method, and the computer readable storage medium has the same beneficial effects as the multi-scale lofting curved surface structure design method of the first embodiment.

[0160] The computer readable storage medium can include, but is not limited to, any type of disk, including floppy disks, optical disks, DVDs, CD-ROMs, micro-drives, 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 foregoing method embodiments, in order to simply describe, they are all described as a series of action combinations, but those skilled in the art should know that the present application is not limited to the order of the actions described, because according to the present application, certain steps can be performed in other order or at the same time. Secondly, those skilled in the art should know that the embodiments described in the specification all belong to preferred embodiments, and the actions and modules involved are not necessarily necessary for the present application.

[0162] In the above embodiments, the description of each embodiment has its own focus, and the parts not described in detail in a certain embodiment can be referred to the related description of other embodiments.

[0163] In the several embodiments provided by the present application, it should be understood that the disclosed device can be implemented in other ways. For example, the device embodiments described above are only schematic. The division of the units is only a logical function division. There can be another division manner for actual implementation, for example, a plurality of units or components can be combined or integrated into another system, or some features can be ignored or not executed. In addition, the coupling or direct coupling or communication connection between the units shown or discussed can be indirect coupling or communication connection through some other services, devices or units, and can be electrical or other forms.

[0164] The units described as separate components can or can not be physically separate, and the components shown as units can or can not be physical units, that is, they can be located in one place, or can be distributed on a plurality of network units. According to actual needs, some or all of the units can be selected to achieve the purpose of the embodiment.

[0165] In addition, each of the function units in the embodiments of the present application can be integrated in one processing unit, or each unit can be physically present separately, or two or more units can be integrated in one unit. The integrated unit can be realized in the form of hardware or in the form of a software function unit.

[0166] When the integrated unit is realized in the form of a software function unit and sold or used as an independent product, it can be stored in a computer readable memory. Based on such understanding, the technical solutions of the present application essentially or the part that contributes to the prior art or the whole or part of the technical solutions can be embodied in the form of a software product. The computer software product is stored in a memory and includes several instructions for causing a computer device (which can be a personal computer, a server or a network device, etc.) to execute all or part of the steps of the methods described in the embodiments of the present application. The aforementioned memory includes: a U disk, a read-only memory (Read-Only Memory, ROM), a random access memory (Random Access Memory, RAM), a mobile hard disk, a magnetic disk or an optical disk, and various media that can store program codes.

[0167] A person of ordinary skill in the art can understand that all or part of the steps in the various methods of the above embodiments can be instructed by a program to be completed by relevant hardware, and the program can be stored in a computer readable memory, which can include a flash disk, a read-only memory (Read-Only Memory, ROM), a random access memory (Random Access Memory, RAM), a magnetic disk or an optical disk, etc.

[0168] The above is only exemplary embodiments of the present disclosure, which cannot limit the scope of the present disclosure. 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. Those skilled in the art will easily think of embodiments of the present disclosure after considering the specification and practicing the disclosure herein. The present application is intended to cover any variations, uses or adaptive changes of the present disclosure, which follow the general principles of the present disclosure and include common knowledge or conventional technical means in the technical field not recorded in the present disclosure. The specification and examples are only considered as exemplary, and the scope and spirit of the present disclosure are defined by the claims.

[0169] Each of the technical features of the above embodiments can be combined arbitrarily. In order to make the description simple, not all possible combinations of the technical features in the above embodiments are described, however, as long as the combinations of the technical features do not exist contradictory, they should be considered as the scope recorded in the present disclosure.

[0170] Those skilled in the art can easily understand that the above description is only the preferred embodiment of the present application, and is not intended to limit the present application. Any modification, equivalent replacement and improvement made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A method for designing multi-scale lofted surface structures, characterized in that, include: Define at least two simply connected domains on the surface structure to be lofted, obtain the mapping parameter domain corresponding to each simply connected domain, and calculate the affine stretching transformation of the mapping parameter domain between any two simply connected domains. The conformal mappings corresponding to the two simply connected domains and the affine stretching transformation between the two simply connected domains are combined to obtain the Teschmüller space mapping between the two simply connected domains, and the stretching quotient between the two simply connected domains is calculated based on the Teschmüller space mapping. Based on the scaling quotient, the Beltrami coefficient between the two simply connected domains is calculated, and the interpolation shape between the two simply connected domains is obtained according to the Beltrami coefficient. The lofted surface structure is mapped to a unit cube structure according to the interpolation shape, and the lofted surface volume configuration is calculated according to the unit cube structure. The equivalent mechanical properties of a finite number of microstructural units are calculated using the generalized homogenization method, and a rapid prediction model of the mechanical properties of gradient microstructures with variable volume fraction 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 configuration is obtained based on the fast prediction model, and the corresponding global level set function is calculated. The global level set function is then mapped back to the lofted surface configuration to realize the gradient-filled multi-scale lofted surface structure design.

2. The multi-scale lofting surface structure design method according to claim 1, characterized in that, The calculation of the Beltrami coefficient between the two simply connected components based on the scaling quotient includes: The scaling quotient is monotonically linearly varied to determine the introduced variable between the two simply connected domains. An interpolation function is set to establish the relationship between the scaling quotient and the introduced variable. The Beltrami coefficient corresponding to the scaling quotient is obtained according to the interpolation function.

3. The multi-scale lofting surface structure design method according to claim 1, characterized in that, The simply connected domain comprises multiple triangular facets in the form of triangulation, and obtaining the interpolation shape between two simply connected domains based on the Beltrami coefficient includes: The surface shape after mapping each of the triangular facets is obtained based on the Beltrami coefficients. All the mapped surface shapes are assembled and stitched together to obtain the interpolated shape between the two simply connected domains.

4. The multi-scale lofting surface structure design method according to claim 3, characterized in that, The step of obtaining the surface shape after mapping each triangular facet based on the Beltrami coefficient includes: Obtain the vertex set and connection relationship of the triangular facets to determine the original triangulation mesh and the target triangulation mesh composed of the triangular facets on the two simply connected domains; The mapping from the original triangulation mesh to the target triangulation mesh is approximated by a piecewise linear function, resulting in a linear transformation of each triangular facet in the original triangulation mesh onto the target triangulation mesh. The area of ​​each triangular facet and the Beltrami coefficient are calculated based on the linear transformation to construct the surface shape mapped from each triangular facet.

5. The multi-scale lofting surface structure design method according to claim 1, characterized in that, The step of mapping the lofted surface structure to a unit cube structure according to the interpolation shape, and calculating the lofted surface volume configuration according to the unit cube structure, includes: The mapping parameter domains corresponding to the simply connected domains are arranged according to the distribution of the surface structure to be lofted, and then normalized according to the direction of the distribution arrangement. Local spline interpolation is performed on the normalized mapping parameter domain according to the interpolation shape to form a parametric curved hexahedral region; The parametric curved hexahedral region is normalized according to the direction of local spline interpolation to obtain the unit cube structure corresponding to the lofted surface structure, and the lofted surface configuration is calculated based on the unit cube structure.

6. The multi-scale lofting surface structure design method according to claim 5, characterized in that, The normalized mapping parameter domain undergoes local spline interpolation to form a parametric curved hexahedral region, including: Interpolation points are set to perform local spline interpolation on the mapping parameter domain to obtain spline curves, and then the scaling quotient of the mapping parameter domain to the parametric curvilinear hexahedral region is calculated. The Beltrami coefficient of the mapping parameter domain is obtained by interpolating the angle of the mapping parameter domain to the parametric curvilinear hexahedral region based on the scaling factor, and the intermediate cross-sectional shape of each state of the mapping parameter domain to the parametric curvilinear hexahedral region is calculated based on the Beltrami coefficient. The modeling equation for the parametric curved hexahedral region is calculated based on the intermediate cross-sectional shape.

7. The multi-scale lofting surface structure design method according to claim 1, characterized in that, The step of obtaining the target volume fraction distribution of the finite number of sample microstructural units in the lofted surface configuration based on the fast prediction model, and calculating the corresponding global level set function, includes: Based on the fast prediction model, a structural optimization formula is established for the lofted surface configuration under the global volume fraction constraint with minimum compliance as the objective. The local level set function is calculated based on 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: The single-connected domain mapping module is used to define at least two single-connected domains on the surface structure to be lofted, obtain the mapping parameter domain corresponding to each single-connected domain, and calculate the affine stretching transformation of the mapping parameter domain between any two single-connected domains. The scaling quotient calculation module is used to combine the conformal mappings corresponding to the two simply connected domains and the affine stretching transformation between the two simply connected domains to obtain the Teichmüller space mapping between the two simply connected domains, and to calculate the scaling quotient between the two simply connected domains based on the Teichmüller space mapping. An interpolation shape calculation module is used to calculate the Beltrami coefficient between two simply connected domains based on the scaling quotient, and to obtain the interpolation shape between the two simply connected domains based on the Beltrami coefficient. The lofted surface configuration calculation module is used to map the lofted surface structure to a unit cube structure according to the interpolation shape, and to calculate the lofted surface 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 using the generalized homogenization method, and to establish a fast prediction model of the mechanical properties of gradient microstructures with variable volume fraction by fitting the sample data with a continuous function. 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 configuration according to the fast prediction model, calculate the corresponding global level set function, and inversely map the global level set function back to the lofted surface configuration to realize the gradient-filled multi-scale lofted surface structure design.

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, the processors executing the computer program to implement the multi-scale lofted surface structure design method as described in any one of claims 1-7.

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

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