Three-dimensional printing flexible surface dynamic deformation device and method based on fractal structure

Abstract

The invention provides a three-dimensional printing flexible surface dynamic deformation device and method based on a fractal structure, and belongs to the technical field of intelligent materials and flexible structures, and the device comprises a flexible surface unit created based on the fractal structure, a steering engine and a unit body connecting structure; the flexible surface unit is an N-level unit constructed based on a fractal structure; the method comprises the following steps: carrying out equilateral triangle basic gridding on a flattened target curved surface by adopting a first-level unit, and carrying out triangular hierarchical layout on a basic grid according to a Gaussian curvature and a conformal scaling factor of the target curved surface to obtain an initial gridding layout; and optimizing the rationalized gridding layout by adopting a multi-energy item optimization target to obtain an optimized gridding layout. The unit gaps are adjusted through layered splicing of the units, dynamic area changes are achieved, curvature differences are formed, and the problem that a traditional homogeneous structure is low in deformation freedom degree is solved. Through the re-layout of the units after fractal classification, the fitting precision of the complex curved surface is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of intelligent materials and flexible structures, and particularly relates to a three-dimensional printing flexible surface dynamic deformation device and method based on a fractal structure. Background Art

[0002] With the continuous progress of material science and digital manufacturing technology, the rapid development of artificial intelligence technology has made the human demand for natural interaction more and more urgent. Flexible media with real-time deformation ability can provide a more flexible and dynamic interaction interface for intelligent devices, meeting the requirements of future intelligent life for the flexibility and adaptability of devices. The flexible media with real-time deformation can quickly change its own shape through external stimuli, thus realizing revolutionary applications in fields such as engineering, architecture, aerospace, etc. For example, in the engineering field, it can be used as the skin of an intelligent robot to imitate the active deformation ability of biological muscles and enhance the interaction ability between the robot and the environment; in the consumer electronics field, devices that can dynamically adjust the surface shape can improve the user experience; in the architecture field, stylized building skins that can automatically adjust their shapes according to environmental changes (such as light and temperature) can be designed; in the aerospace field, the aerodynamic performance can be optimized by programming to control the deformation of the wing surface, improving fuel efficiency and flight stability. The flexible media with real-time deformation ability provides new ideas for solving key problems such as energy efficiency, environmental adaptability, and human-computer interaction, and is the core foundation for the development of future intelligent and flexible systems.

[0003] In the prior art, the research on programmable deformation media with real-time deformation ability mainly focuses on three aspects: material progress, driving innovation, and manufacturing technology. For example, the introduction of intelligent materials (such as shape memory alloys (SMA), electroactive polymers (EAP), and liquid crystal elastomers (LCE), etc.) has been able to achieve controllable deformation; the innovation of deformation driving methods includes mechanical structure driving, electromagnetic driving, pneumatic driving, etc. However, these methods rely on new materials and new technologies, resulting in high R & D and manufacturing costs. Although the 3D printing flexible surface has the ability to deform, the uniformly distributed units cannot achieve complex deformation, and the 4D printing technology cannot solve the problem of multi-degree-of-freedom deformation. Therefore, at the present stage, as a deformation medium, how to achieve high-degree-of-freedom deformation feedback without complex driving is the core problem to be solved. Summary of the Invention

[0004] In view of the deficiencies of the prior art, a three-dimensional printing flexible surface dynamic deformation device and method based on a fractal structure are proposed. According to the fractal principle of the Sierpinski triangle, multi-level self-similar unit bodies (such as first-level, second-level, and third-level) are constructed. The dynamic area change is realized by adjusting the unit spacing, and the curvature difference of each part of the flexible surface is changed to form a curved surface. At the same time, the servo drive and the connection structure are combined to control the unit spacing, realizing programmable real-time deformation. Through the optimization of the fractal unit layout, the curved surface can be accurately fitted, and the deformation ability is enhanced. The multi-level modular design enables damaged units to be replaced individually, effectively reducing the maintenance cost. Through the deep integration of fractal geometry and dynamic deformation control, the present invention solves the core problems of low degrees of freedom, poor accuracy, and high cost of traditional flexible surfaces, providing a highly adaptable and low-cost dynamic deformation solution for fields such as robots and intelligent buildings, and having both theoretical innovation and engineering practical value.

[0005] In a first aspect, the present application proposes a three-dimensional printing flexible surface dynamic deformation device based on a fractal structure, including:

[0006] A flexible surface unit created based on a fractal structure, a servo motor, and a unit body connection structure;

[0007] The flexible surface unit is an N-level unit constructed based on a fractal structure, where N is greater than or equal to 2. Multiple first-level units are spliced into second-level units, multiple second-level units are spliced into third-level units, and multiple N-1-level units are spliced into N-level units;

[0008] The first-level unit includes: a main unit body and a driven unit body, and the main unit is connected to the driven unit body through a unit body connection structure;

[0009] The servo motor is installed in the main unit body and is used to drive the driven unit body to generate changes in rotation and displacement.

[0010] The main unit body is a unidirectionally closed regular triangular prism shell. Three servo limit grooves are evenly distributed at three corner points in the regular triangular prism shell. The servo limit grooves are used to install servo motors. Each of the three side surfaces of the regular triangular prism shell has a first U-shaped groove, and the first U-shaped groove is used to install the unit body connection structure.

[0011] The driven unit body is a unidirectionally closed regular triangular prism shell of the same size as the main unit body. Each of the three side surfaces of the regular triangular prism shell has a second U-shaped groove, and the second U-shaped groove is an integrally formed limiting structure.

[0012] The unit connection structure is a crankshaft connecting rod structure. The crankshaft connecting rod structure is a truncated body defined by a filleted rectangle with equal cross-sections. Both ends of the crankshaft connecting rod structure are semi-open circular buckles. The semi-open circular buckle at one end of the crankshaft connecting rod structure is connected to the active unit body, and the semi-open circular buckle at the other end of the crankshaft connecting rod structure is connected to the driven unit body.

[0013] In a second aspect, the present application proposes a three-dimensional printing flexible surface dynamic deformation method based on a fractal structure, which is implemented by using the three-dimensional printing flexible surface dynamic deformation device described in the first aspect, and includes:

[0014] Flatten the target surface by using a boundary-first flattening algorithm, and display the Gaussian curvature and conformal scaling factor of the target surface on the flattened target surface;

[0015] Use first-level elements to perform equilateral triangle-based meshing on the flattened target surface, and perform triangular hierarchical layout on the basic grid according to the Gaussian curvature and conformal scaling factor of the target surface to obtain an initial meshed layout;

[0016] Rationalize the initial meshed layout to minimize unnecessary deformation and distortion;

[0017] Use an optimization objective with multiple energy terms to optimize the rationalized meshed layout to obtain an optimized meshed layout, and print the target surface using the optimized meshed layout. The optimization objective with multiple energy terms includes: weighted summation of geometric approximation energy, physical feasibility energy, and fractal error compensation energy.

[0018] The step of using first-level elements to perform equilateral triangle-based meshing on the flattened target surface, and performing triangular hierarchical layout on the basic grid according to the Gaussian curvature and conformal scaling factor of the target surface to obtain an initial meshed layout includes:

[0019] On the flattened target surface, take the corresponding point of the maximum conformal scaling factor as the center point, and use the center point as the starting point for equilateral triangle meshing in the first first-level element, and perform equilateral triangle meshing of equal size outward to obtain a basic grid;

[0020] Perform feature weighted fusion on the grayscale map of Gaussian curvature and the grayscale map of conformal scaling factor to obtain a feature value grayscale map with Gaussian curvature and conformal scaling factor and the corresponding gray value of the feature value grayscale map;

[0021] Perform spectral clustering processing on the grayscale map with Gaussian curvature and conformal scaling factor features, and divide it into N different regions according to the gray value corresponding to the feature value grayscale map. In different regions, perform meshing with equilateral triangles of the same size as the first-level elements or N-level elements to obtain an initial meshed layout.

[0022] The target surface is meshed with equilateral triangles of the same size as the first-level units or N-level units in different regions to obtain an initial meshed layout, including:

[0023] If the gray values of all pixels in the Nth type of region on the target surface are within the maximum value range, the flattened target surface is meshed with N-level units into equilateral triangles;

[0024] If the gray values of all pixels in the first type of region on the target surface are within the minimum value range, the flattened target surface is meshed with first-level units into equilateral triangles;

[0025] If the gray values of all pixels in the second to N-1th types of regions on the target surface are between the minimum value range and the maximum value range, the flattened target surface is meshed with the corresponding units from the second-level units to the N-1th level units into equilateral triangles according to the magnitude order of the gray values corresponding to the eigenvalue gray map.

[0026] The initial meshed layout is rationalized to minimize unnecessary deformation and distortion, including:

[0027] Calculate the contact force of each first-level unit in the initial meshed layout;

[0028] Optimize the contact force of each first-level unit by using the least squares method to obtain a rationalized meshed layout.

[0029] The geometric approximation energy is calculated as follows:

[0030]

[0031] where E geo is the geometric approximation energy, x i is the coordinate of the ith vertex on the target surface, c i is the coordinate of the projection point of the vertex x i on the target surface, and M is the number of vertices.

[0032] The physical feasibility energy is calculated as follows:

[0033]

[0034] where E physics is the physical feasibility energy, θ ij is the relative rotation angle of the ith and jth adjacent units around their common edge, is the target relative rotation angle of the ith and jth adjacent units around their common edge, θ max is the maximum allowable rotation angle of the connection structure, and γ is the penalty weight for excessive rotation angle.

[0035] The fractal error compensation energy is calculated as follows:

[0036]

[0037] Where E fractal is the fractal error compensation energy, r k is the side length scaling ratio from fractal level L k to L k+1 , ΔX k is the measured or simulated layer manufacturing error, η is the regularization weight in fractal energy optimization, and k is the level of the fractal.

[0038] Beneficial effects:

[0039] The present application proposes a three-dimensional printing flexible surface dynamic deformation device and method based on a fractal structure, and the beneficial effects include:

[0040] 1. Fractal structure design: The Sierpinski triangle fractal principle is adopted to construct multi-level self-similar units (such as first-level, second-level, and third-level). By adjusting the unit gaps through hierarchical splicing of the units, dynamic area changes are achieved to form curvature differences, breaking through the problem of low degrees of freedom of deformation in traditional homogeneous structures. The nested characteristics of the fractal structure enable the design of the flexible surface to have high scalability, and the first-level, second-level, third-level, and even more levels of fractal units can be dynamically switched, supporting multi-scale deformation coordination from micro to macro.

[0041] 2. Driving and connection mechanism: A modular fractal driving network is proposed. Multiple low-level units (such as first-level units) are spliced to form high-level units (such as second-level and third-level units), forming a hierarchical network of fractal driving to achieve coordinated deformation between units; a mechanical linkage design uses a crank and connecting rod to transmit power, greatly reducing the number of required servos and lowering the hardware complexity and cost. Based on the fractal hierarchical network, the servos are programmed and controlled to support the instant switching of surface deformation, realizing a "programmable flexible surface". Combining natural forces (gravity, wind force) and artificial forces (pneumatic push rods, thimble thrust) as external tensile forces, vertically acting on the flexible surface expansion unit gaps, reduces the active driving energy consumption.

[0042] 3. Curved surface construction algorithm: The conformal mapping and fractal reconstruction algorithms are integrated to realize an automated design process (flattening → meshing → layout → optimization) from the target surface to the flexible surface. According to the distribution of the scaling factors, the fractal unit levels are planned on the flattened surface. Through the re-layout of the units after fractal grading, the local deformation requirements exceeding the conformal scaling factor are compensated, improving the fitting accuracy of complex surfaces, establishing a quantitative relationship between the unit spacing and the conformal scaling factor, and realizing multi-degree-of-freedom deformation from a plane to a curved surface by driving the spacing change. Description of the drawings

[0043] Figure 1 The Sierpinski triangle fractal principle of the prior art;

[0044] Figure 2 Schematic diagram of the three - level unit structure in the embodiment of the present application; among them, (a) is the schematic diagram of the active unit, (b) is the schematic diagram of the unit connection structure, and (c) is the schematic diagram of the driven unit;

[0045] Figure 3 Schematic diagram of the crank - connecting rod structure at the active structure in the embodiment of the present application; among them, (a) is the top view of the crank - connecting rod structure, and 3(b) is the three - dimensional structure diagram of the crank - connecting rod structure;

[0046] Figure 4 Schematic diagram of the connection between units in the embodiment of the present application;

[0047] Figure 5 Schematic diagram of the two - level unit structure in the embodiment of the present application;

[0048] Figure 6 Schematic diagram of the three - level unit structure in the embodiment of the present application;

[0049] Figure 7 Schematic diagram of the unit structure in the embodiment of the present application;

[0050] Figure 8 Flowchart of a three - dimensional printing flexible surface method based on a fractal structure in the embodiment of the present application;

[0051] Figure 9 Schematic diagram of the Gaussian curvature of the target surface in the embodiment of the present application;

[0052] Figure 10 Schematic diagram of the conformal scaling factor of the surface in the embodiment of the present application;

[0053] Figure 11 Schematic diagram of the weighted fusion of Gaussian curvature and conformal scaling factor and the sub - region clustering diagram in the embodiment of the present application, where (a) is the schematic diagram of the weighted fusion of Gaussian curvature and conformal scaling factor, and (b) is the spectral clustering sub - region diagram of Gaussian curvature and conformal scaling factor;

[0054] Figure 12 Basic layout diagram of the unit in the embodiment of the present application;

[0055] Figure 13 Initial layout diagram in the embodiment of the present application;

[0056] Figure 14 Schematic diagram of the rotation optimization in the embodiment of the present application; among them, (a) is the schematic diagram before the least - squares optimization, and (b) is the schematic diagram after the least - squares optimization;

[0057] Figure 15Schematic diagram of the optimization of the embodiments of the present application;

[0058] Among them, 1 - active unit body, 2 - unit body connection structure, 3 - driven unit body, 1-1 - first U-shaped groove, 1-2 - servo, 3-1 - second U-shaped groove. Specific implementation manners

[0059] The following further describes in detail the specific implementation manners of the present application in conjunction with the accompanying drawings and embodiments.

[0060] Embodiment 1:

[0061] This embodiment proposes a three-dimensional printing flexible surface dynamic deformation device based on a fractal structure, which is applicable to fields such as robots, consumer electronics, building skins, and aerospace, and realizes high-degree-of-freedom dynamic deformation control, including:

[0062] Flexible surface units created based on a fractal structure, servo 1-2, and unit body connection structure 2;

[0063] The flexible surface unit is an N-level unit constructed based on a fractal structure, where N is greater than or equal to 2. Multiple first-level units are spliced into second-level units, multiple second-level units are spliced into third-level units, and multiple N-1-level units are spliced into N-level units;

[0064] The first-level unit includes: an active unit body 1 and a driven unit body 3, and the active unit is connected to the driven unit body 3 through the unit body connection structure 2;

[0065] The servo 1-2 is installed in the active unit body 1 and is used to drive the driven unit body 3 to generate rotational and displacement changes.

[0066] The active unit body 1 is a one-way closed regular triangular prism shell. Three servo limit grooves are evenly distributed at three corner points in the regular triangular prism shell. The servo limit grooves are used to install the servo 1-2. Each of the three side surfaces of the regular triangular prism shell has a first U-shaped groove 1-1, and the first U-shaped groove 1-1 is used to install the unit body connection structure 2.

[0067] The driven unit body 3 is a one-way closed regular triangular prism shell of the same size as the active unit body 1. Each of the three side surfaces of the regular triangular prism shell has a second U-shaped groove 3-1, and the second U-shaped groove 3-1 is an integrally formed limit structure.

[0068] The unit body connection structure 2 is a crankshaft connecting rod structure. The crankshaft connecting rod structure is a truncated body defined by a filleted rectangle with equal cross-sections. Both ends of the crankshaft connecting rod structure are semi-open circular buckles. One end of the semi-open circular buckle of the crankshaft connecting rod structure is connected to the active unit body 1, and the other end of the semi-open circular buckle of the crankshaft connecting rod structure is connected to the driven unit body 3.

[0069] In this embodiment, the fractal theory is a mathematical theory that presents geometric properties with a fractional dimension. Its core is fractal geometry. The geometry after fractal has self-similarity and infinite divisibility, making it easier to mesh the surface in the form of a numerical algorithm and thus being widely used in the construction of flexible surfaces. Based on the fractal concept, the Sierpinski triangle fractal structure further proposes a hierarchical structure. Starting from an equilateral triangle, by dividing it into four congruent equilateral triangles and continuously iterating to generate smaller equilateral triangles, a self-similar fractal structure is formed, as shown in Figure 1 . Based on the infinite cutting fractal principle of the Sierpinski triangle, the appropriate division level is selected according to the size of the flexible surface and the accuracy requirement for retaining surface details of the target surface. In this way, the size of the unit body at each level of the flexible surface can be accurately determined. In this embodiment, the unit bodies of the flexible surface are designed from small to large as first-level unit, second-level unit, and third-level unit triangular structures. In fact, more levels of units can be designed according to specific needs, and the division standard is determined by the accuracy of retaining the changes in surface morphology details. This embodiment only describes the case of three-level units. In this embodiment, four first-level units are spliced into a second-level unit, and four second-level units are spliced into a third-level unit. Generally, the first-level unit, second-level unit, and third-level unit are set from small to large for the flexible surface unit, and the three-level triangles can more accurately construct the surface.

[0070] The triangle of the first-level unit, as the smallest basic unit, mainly has two structures, the active unit body 1 structure and the driven unit body 3 structure. The active unit body 1 structure is a unidirectionally closed regular triangular prism shell. Inside the shell, 3 servo limit slots are evenly distributed at three corner points for placing small servo motors 1-2. Each of the three side surfaces of the regular triangular prism shell has a U-shaped groove for installing a crankshaft connecting rod structure as shown in Figure 3 shown. Figure 3 (a) is the top view of the crankshaft connecting rod structure. Figure 3 (b) is the three-dimensional structure diagram of the crankshaft connecting rod structure. One end of this structure is connected to the servo motor 1-2, as shown in Figure 2 (a). The driven unit body 3 structure is a unidirectionally closed regular triangular prism shell of the same size as the active unit body 1 structure. Similarly, each of the three side surfaces of the shell has a U-shaped groove, but different from the active unit body 1 structure, the groove is an integrally formed limit structure, as shown in Figure 2 (c). The connection structure between the active unit body 1 and the driven unit body 3 is a frustum defined by a rounded rectangle with equal cross-sections. The two ends are semi-open circular buckles, which are respectively connected to the active unit body 1 and the driven unit body 3, as shown in Figure 2 (b).

[0071] When the servo 1-2 of the active unit body 1 operates, the crankshaft connecting rod structure connected to it realizes rotation and displacement changes, thereby driving the connecting structure to drive the driven unit to move. By controlling the angle of the servo 1-2, the gap change between the active unit body 1 and the driven unit body 3 can be controlled. The depth of the groove and the rotation angle of the connecting structure determine the maximum rotation angle and the maximum gap between the two unit bodies, as Figure 4 shown. According to the Sierpinski triangle fractal theory, by driving the servo 1-2 of the active unit body 1 to have a zero gap with the adjacent driven unit body 3, a secondary unit body structure is formed, and at the same time, a spacing can be achieved with the adjacent secondary units, as Figure 5 . The thickness of the triangular unit body main structure of the secondary unit remains unchanged, but the side length of the regular triangular prism becomes twice the original. The connecting structure remains the same size, but the number of connecting structures on each side of the regular triangular prism becomes twice the original. Similarly, for the triangular unit body structure of the tertiary unit, by driving the servo 1-2 of the active unit body 1 to have a gap of 0 with the surrounding driven unit bodies 3, a primary unit body structure composed of 16 primary unit bodies is realized, as Figure 6 shown.

[0072] In the overall structure, the thickness H body of the unit body structure is defined, and the thickness of the connecting structure is H pin , as Figure 7 shown. In order to further accurately calculate and control each unit body, the regular triangle in the structure is parameterized. Therefore, for each unit body, there are a total of six vertices as six parameters, which are expressed as:

[0073]

[0074] Among them, V represents the unit body, s ∈ {f, b}, f represents the front of the unit body, b represents the back of the unit body, i ∈ [1, M] represents a certain unit body among unit body 1 - unit body M, and t ∈ {1, 2, 3} represents the side of the regular triangular face in the unit body. Through these parameters, the geometric structure of each unit body can be accurately controlled. Triangles of different levels can form different surface curvatures in space through their gaps and rotation angles with each other. Based on this, a differentiated surface form is established. By controlling the servo 1-2 of the active unit body 1, it is possible to switch between the triangles of the primary unit, secondary unit, and tertiary unit at any time, thereby forming an instant dynamic surface change.

[0075] Example 2:

[0076] This example proposes a three-dimensional printing flexible surface dynamic deformation method based on a fractal structure, which is realized by using a three-dimensional printing flexible surface dynamic deformation device described in Example 1, as Figure 8 shown, including:

[0077] Step S1: Flatten the target surface using the boundary - first flattening algorithm, and display the Gaussian curvature and conformal scaling factor of the target surface on the flattened target surface;

[0078] Step S2: Use first - order elements to perform equilateral - triangle - based meshing on the flattened target surface, and perform triangular hierarchical layout on the basic mesh according to the Gaussian curvature and conformal scaling factor of the target surface to obtain an initial meshing layout;

[0079] Step S3: Rationalize the initial meshing layout to minimize unnecessary deformation and distortion;

[0080] Step S4: Use an optimization objective with multiple energy terms to optimize the rationalized meshing layout to obtain an optimized meshing layout, and print the target surface using the optimized meshing layout. The optimization objective with multiple energy terms includes: the weighted sum of geometric approximation energy, physical feasibility energy, and fractal error compensation energy.

[0081] In this embodiment, in step S1, the boundary - first flattening algorithm is used to flatten the target surface, and the Gaussian curvature and conformal scaling factor of the target surface are displayed on the flattened target surface, including:

[0082] Step S1.1: Calculate the Gaussian curvature of the target surface and visualize it on the flattened plane;

[0083] Step S1.2: Calculate the conformal scaling factor of the surface and visualize it on the flattened plane;

[0084] Step S1.3: Use the boundary - first flattening algorithm to flatten the target surface, and display the Gaussian curvature and conformal scaling factor information on the flattened target surface;

[0085] In this embodiment, the flexible - surface construction idea is to use conformal mapping and boundary - first flattening algorithm in conformal geometry to visualize the curvature information and conformal scaling factor of the target surface on the surface, and then flatten the surface into a plane.

[0086] The construction of the flexible surface starts from the target surface, and the target surface can be described by Gaussian curvature. The Gaussian curvature of the surface can be represented by the conformal scaling factor to reflect the change relationship before and after surface mapping. Based on these two pieces of information of the surface, a flexible surface can be constructed.

[0087] First, input the target surface Through software calculation, the Gaussian curvature K can be obtained and visualized on the surface. The visualization of the Gaussian curvature can well describe the geometric properties of a given surface. Through conformal mapping, the information of the Gaussian curvature K is represented by the conformal scaling factor λ. If we set Let \(\alpha\) be an arbitrary region in the complex plane. There exists a mapping, which can be of any complex geometric shape, such that \(f:\alpha\rightarrow\mathbb{R}\). 3 Let \(df\) denote the Jacobian matrix or differential of \(f\). Then it can represent how the vectors in \(\mathbb{R}\) are transformed by \(f\) to \(\mathbb{R}\). 2 Let \(df\) denote the Jacobian matrix or differential of \(f\). Then it can represent how the vectors in \(\mathbb{R}\) are transformed by \(f\) to \(\mathbb{R}\). 3 If at every point, there is only a positive conformal scaling factor \(\lambda\) between the two inner products \(X\cdot Y\) and \(df(X)\cdot df(Y)\), that is, there is a proportional relationship between the inner products \(X\cdot Y\) and \(df(X)\cdot df(Y)\), then \(f\) is conformal. Analyzed from the perspective of geometric vectors, a conformal mapping must preserve angles because angles can be represented by inner products. The fact that \(\lambda\) is positive ensures that it is never zero, that is, angles can always be clearly represented and exist. Regarding \(\lambda\) as the conformal scaling factor of the target surface, and \(\lambda:\alpha\rightarrow\mathbb{R}\). + It can be expressed as \(\lambda = e^{\varphi}\), where \(\varphi\) can be any function \(\varphi:\lambda\rightarrow\mathbb{R}\) (not just positive). \(\varphi\) is called the logarithmic scaling factor because \(\varphi=\log(\lambda)\). The Gaussian curvature of the target surface can be expressed as φ It can be expressed as \(\lambda = e^{\varphi}\), where \(\varphi\) can be any function \(\varphi:\lambda\rightarrow\mathbb{R}\) (not just positive). \(\varphi\) is called the logarithmic scaling factor because \(\varphi=\log(\lambda)\). The Gaussian curvature of the target surface can be expressed as

[0088]

[0089] where \(\Delta\) represents the Laplace operator on the plane, and \(\Delta\) f represents the Laplace - Beltrami operator on the new surface. It should be noted that when the mapping degenerates to a plane automorphism (\(K = 0\)), this equation simplifies to the harmonic equation \(\Delta\) φ \(\varphi=0\). At this time, \(\varphi\) degenerates to a classical harmonic function.

[0090] By calculating the conformal scaling factor of the surface as above and visualizing it, the scale change of the target surface during the conformal mapping process is intuitively described. Using the Boundary First Flattening (BFF) algorithm, first process the boundary part of the surface, flatten it to a two - dimensional plane, and then optimize the internal points based on this fixed boundary, so that the entire surface is smoothly flattened, and the curvature and conformal scaling factor information are retained on the plane, providing a basis for the layout of different - level unit cells of the flexible surface in the next step, such as Figure 9 is the flattened graph of the Gaussian curvature, Figure 10 is the flattened graph of the conformal scaling factor.

[0091] In this embodiment, step S2: Perform primary equilateral - triangle - based meshing on the flattened target surface, and perform triangular hierarchical layout on the basic mesh according to the Gaussian curvature and conformal scaling factor of the target surface to obtain an initial meshed layout, including:

[0092] On the flattened target surface, taking the corresponding point of the maximum value of the conformal scaling factor as the center point, using the center point as the starting point for meshing the first-level equilateral triangle, and performing equilateral triangle meshing of the same size outward to obtain the basic grid layout. Feature weighted fusion is performed on the surface Gaussian curvature grayscale map and the conformal scaling factor grayscale map, followed by normalization processing to obtain the eigenvalue grayscale map with Gaussian curvature and conformal scaling factor, as well as the grayscale values corresponding to the eigenvalue grayscale map, as shown in Figure 11 (a). Spectral clustering is performed on this feature grayscale map and divided into N regions according to the grayscale values corresponding to the eigenvalue grayscale map, as shown in Figure 11 (b). The grayscale eigenvalue within the first-level equilateral triangle grid is identified, and outward meshing is performed with equilateral triangles of the same size as the first-level unit or the Nth level to obtain the initial meshing layout.

[0093] Performing meshing with equilateral triangles of the same size as the first-level unit or the Nth level in different regions to obtain the initial meshing layout includes:

[0094] If the grayscale values of all pixels within the Nth region on the target surface are within the maximum value range, then the Nth-level unit is used to perform equilateral triangle meshing on the flattened target surface;

[0095] If the grayscale values of all pixels within the first region on the target surface are within the minimum value range, then the first-level unit is used to perform equilateral triangle meshing on the flattened target surface;

[0096] If the grayscale values of all pixels within the second to N-1th regions on the target surface are between the minimum value range and the maximum value range, then according to the size order of the grayscale values corresponding to the eigenvalue grayscale map, any level of unit from the second-level unit to the N-1th level unit is used to perform equilateral triangle meshing on the flattened target surface.

[0097] In this embodiment, equilateral triangle meshing is performed on the planar topological structure of the flattened target surface. Eigenvalue clustering analysis is performed based on the Gaussian curvature and conformal scaling factor information presented by the flattened equilateral triangle topological structure. By reconstructing the flexible surface with different levels of unit structures, the target surface is accurately fitted.

[0098] First, as shown in Figure 12 , equilateral triangle meshing is performed on the flattened topological structure. Taking the corresponding point of the maximum value of the conformal scaling factor as the center point, using the center point as the starting point for meshing the first-level equilateral triangle, and performing equilateral triangle meshing of the same size outward. And the meshed equilateral triangle grid is used as the layout basis for the flexible surface unit body.

[0099] In the research of surface geometry analysis and mesh generation, Gaussian curvature and conformal scaling factor are two key characteristic parameters. Gaussian curvature describes the local curvature of the surface and is a global invariant, which plays an important role in fields such as physical bionics and structural optimization. The conformal scaling factor, on the other hand, is used to measure the scale change of the surface in conformal mapping and can characterize the uniformity of local deformation. In practical applications, in order to more effectively combine these two characteristics, it is necessary to perform appropriate normalization on them so that they are within the same numerical range for unified analysis. Therefore, first, the grayscale images of Gaussian curvature and conformal scaling factor need to be normalized, and their grayscale values are mapped to the interval [0,1]. This can eliminate the numerical scale differences of different characteristics, make the subsequent weighted processing more reasonable, and ensure the balanced contribution of different characteristics to the final result.

[0100] After the normalization process, spectral clustering can be used to perform weighted fusion on Gaussian curvature and conformal scaling factor. Spectral clustering is a graph theory-based method that obtains clustering results by calculating the similarity matrix and Laplacian matrix and performing eigenvalue decomposition on them. In this study, we can construct a feature graph containing Gaussian curvature and conformal scaling factor and perform weighted processing on it to enhance the influence of a certain feature on clustering. The relative contributions of the two are adjusted by a weight factor α, and the calculation method of the composite feature graph is as follows:

[0101] compositeFeatureMap = α·gaussianCurvature+(1-α)·conformalFactor

[0102] where compositeFeatureMap is the grayscale image of the eigenvalue with Gaussian curvature and conformal scaling factor, gaussianCurvature is the grayscale image of Gaussian curvature, and conformalFactor is the grayscale image of the conformal scaling factor. When α = 0.5, the Gaussian curvature and conformal scaling factor have equal weights on the final feature, and adjusting the value of α can bias towards a certain feature to meet different research needs. Through the analysis of eigenvalues and eigenvectors, spectral clustering can automatically identify regions with similar geometric properties and perform reasonable partitioning, better analyzing non-linear structures and complex surface morphologies.

[0103] After spectral clustering is completed, we can use its results to divide the surface into regions, that is, divide the surface into N sub-regions according to the characteristic similarity of Gaussian curvature and conformal scaling factor. The number of divided region categories is consistent with the mesh grading. The geometric properties within these regions tend to be consistent, enabling the meshing scheme to more accurately adapt to the surface morphology within each region. For meshing, equilateral triangle meshes are a better choice because they can ensure the regularity of mesh division and reduce unnecessary deformation errors. Therefore, within each clustering region, we can use equilateral triangle meshes for subdivision so that the meshes can be reasonably distributed over the entire surface.

[0104] Specifically, first determine the initial equilateral triangle elements in the first-level mesh and identify the change trend of the gray feature values therein. Then, based on the feature distribution of the first-level elements, expand outward level by level to form a multi-level mesh division. During the expansion process, each new layer of mesh elements should maintain geometric similarity with the previous layer and ensure scale consistency as much as possible. This means that the new meshes should be equal in shape and size to the first-level elements or the Nth-level elements, thereby constructing a structurally stable meshing layout. Finally, through the above method, we can generate a set of meshing layouts with Gaussian curvature and conformal scaling factor characteristics on the surface. Not all surfaces can be approximated. When there are conical singularity surfaces on the target surface, the Boundary First Flattening algorithm can be combined to divide the surface at the conical singularity and then analyze it. As Figure 13 。

[0105] In this embodiment, step S3: Rationalize the initial meshing layout to minimize unnecessary deformation and distortion, including:

[0106] Step S3.1: Calculate the contact force of each first-level element in the initial meshing layout;

[0107] Step S3.2: Optimize the contact force of each first-level element using the least squares method to obtain a rationalized meshing layout.

[0108] In this embodiment, to complete the hierarchical layout of the flexible structure plane, it is necessary to construct the target surface according to the 3D printing structure. In this process, if the contact forces between the unit cells are unbalanced, deformation and distortion are likely to occur, affecting the physical stability of the structure and leading to the failure of the design or the degradation of the structure's function. Therefore, it is necessary to minimize unnecessary deformation and distortion through the balance calculation of contact forces and externally applied forces to ensure the correct interaction between the unit cells. In addition, using the least squares method to optimize the contact force calculation helps to achieve dynamic balance and ensure the stability of the overall structure. Finally, through the static equilibrium conditions and moment calculation, the perfect matching between the unit cells is ensured, improving the performance and reliability of the flexible surface, as Figure 14 shown, where Figure 14 (a) is a schematic diagram before least squares optimization, Figure 14 (b) is a schematic diagram after least squares optimization.

[0109] Denote the flexible surface composed of the unit cell structure as In the process of converting between the plane and the curved surface through a simple driving force, some unit cells will be stacked together by extrusion, and some unit cells will be stretched between adjacent unit cells through the connection structure. Eventually, the unit cells will come into contact with each other directly or indirectly, generating contact forces that balance the elastic forces. Using this equilibrium condition, the contact force is calculated as a least squares problem. Denote N as the number of vertices in j , and its coordinates are x

[0110] minimize∑∥f c ∥ 2

[0111] subject to static equilibrium

[0112] where f c is the total contact force of any unit.

[0113] The static equilibrium is expressed as:

[0114]

[0115] For each i-th unit cell, and represent the total contact force and the externally applied force respectively, and represent the total moments generated by the contact force and the externally applied force with respect to an arbitrarily given point respectively. Since the point selected is arbitrary, in fact, the moments are calculated with respect to the preselected front vertex of each unit. Given a point r, the moment calculation formula is: and In specific implementation, in addition to the unit automatically switching the unit level under the drive of the servo, an external force needs to be applied, that is, the external tensile force acts vertically on the gap between the flexible surface deployment units. The setting of the tensile force will vary according to different application scenarios. For example, under the boundary fixed condition, natural forces such as gravity, wind force, and air flow are used to achieve the tensile force perpendicular to the flexible surface; artificial forces such as the thrust of the thimble-type push rod structure and the expansion force of pneumatic are used to generate the tensile force perpendicular to the flexible surface.

[0116] In this embodiment, step S4: An optimization target with multiple energy terms is adopted to optimize the rationalized grid layout to obtain an optimized grid layout, and the optimized grid layout is used to print the target surface. The optimization target with multiple energy terms includes: the weighted sum of geometric approximation energy, physical feasibility energy, and fractal error compensation energy, as Figure 15 shown.

[0117] During the construction of the 3D printed flexible surface, due to the influence of gravity on the unit body, the actually manufactured unit bodies will have differences in terms of shape accuracy, vertex proximity, and smoothness.

[0118] To optimize these differences, in response to the above problems, based on the Shape-up algorithm, a multi-energy-term optimization framework is proposed. By introducing the optimization target with multiple energy terms, a systematic optimization framework covering multiple aspects such as geometric approximation energy, physical feasibility energy, and fractal error compensation energy is constructed. Through weight setting, the priority of each energy term can be flexibly adjusted according to specific requirements, so as to achieve better balance and control during the optimization process. It can quantify the influence of different design goals, thereby ensuring that the finally generated flexible surface not only meets the design requirements but also has good physical properties.

[0119] Conformal mapping provides the three-dimensional position on the design surface for the vertices of each unit body. However, due to the changes of some unit bodies, some results need to be further optimized to meet the following requirements: 1) All triangular unit bodies are rigid, that is, they have a specified side length; 2) There is no excessive torsion between the triangles; 3) The vertices remain close to the input design surface. To meet these requirements, we optimize the vertex positions by minimizing the objective function. This objective function measures the distance of each vertex from the corresponding vertex of the basic target shape to the 3D model. This optimization problem is a constrained non-linear optimization problem, where the soft constraints represent the desired properties of the final assembly, and the hard constraints are used to capture manufacturing limitations and driving factors:

[0120]

[0121] where E geo is the geometric approximation energy term, which measures the deviation degree of the vertex from the design surface; Ephysics is the physical coordination energy term that controls the deformation coordination and material safety between units; E fractal is the fractal error energy term that suppresses the inter-layer propagation of manufacturing errors. The weights w1, w2, and w3 control the trade-off between these objectives. x i is a vertex; c i is the projection point coordinate of vertex x i on the target surface; L ij is the designed side length to ensure the structural integrity of the printing unit; θ ij is the relative rotation angle of adjacent units around the common edge; θ max is the maximum allowable rotation angle to prevent plastic deformation of the material; x m is the coordinate of the optimized m-th vertex in three-dimensional space; c m is the coordinate of the nearest projection point of the optimized m-th vertex on the target design surface; δ print is the maximum vertex offset threshold allowed by the 3D printing process.

[0122] Geometric approximation energy. It is mainly used to measure the deviation between the actual coordinates and the expected coordinates of each vertex in the current geometric data. By minimizing E geo , the vertices in the geometric data can be made as close as possible to the predetermined target positions, thus achieving the optimization and adjustment of the geometric shape. This energy function helps to guide the vertices to move towards the target positions, ensuring that the final geometric form meets the expected requirements and effectively controlling the geometric constraints during the deformation process.

[0123]

[0124] where x i represents the coordinate of the i-th vertex on the surface, c i represents the projection point coordinate of vertex x i on the target surface. The target surface is a triangular mesh, and the point-face nearest point algorithm is used: traverse all triangular patches, calculate the nearest distance from x i to each face, and take the point corresponding to the minimum value. M is the number of vertices.

[0125] Physical feasibility energy. The first part is used to measure the deviation between the angle between connection points and the target angle. Measuring the deviation between the actual angle and the target angle, by minimizing this part, the angles of the system can be made as close as possible to the target. The second part penalizes the deviation exceeding the maximum allowable angle to ensure that the system angles do not exceed the preset maximum value. Through these two parts, the physical energy realizes angle optimization and ensures that the system follows physical constraints to avoid unreasonable deformations:

[0126]

[0127] where, θ ij is the relative rotation angle of the i-th and j-th adjacent units around their common edge; θ max is the maximum allowable rotation angle of the connection structure, determined by the mechanical properties of the material; γ is the penalty weight for excessive rotation angle, used to strictly limit θ ij > θ max , empirically set: usually take γ = 10 3 ~10 5 , ensuring that the objective function increases sharply when exceeding the limit.

[0128]

[0129] where, the vertex coordinates of the adjacent unit bodies are {x i , x j , x k} and {x j , x k , x l} sharing the common edge L jk ; is the target relative rotation angle of the i-th and j-th adjacent units around their common edge, determined by the geometric curvature of the target surface;

[0130]

[0131] κ g is the geodesic curvature which is the degree of bending of a curve on a surface and can be calculated through the parameterization of the target surface.

[0132]

[0133] L ij is the length of the common edge, φ i is the interior angle of the neighborhood of the common edge.

[0134] Fractal error compensation energy. The first part measures the deviation between each scale factor and the target scale. By minimizing this energy, it can be ensured that all scale factors are close to 0.5, thus achieving the required scale control, avoiding too large or too small scales, and ensuring the uniformity and rationality of the shape or structure. The second part measures the magnitude of geometric deformation and controls the deformation of the structure. By minimizing these two parts, the fractal energy optimizes the scale distribution and maintains the structural stability.

[0135]

[0136] r k is the fractal level from L k to L k+1The side length scaling ratio is defined according to the Sierpinski fractal theory. The theoretical value is 0.5, and the actual value is measured or adjusted through optimization. k is the level of the fractal, identifying different levels in the fractal structure. The manufacturing error ΔX of the measured or simulated level k . The attenuation coefficient ∈ = f(material stiffness, layer spacing) (indicating a 50% attenuation during error transfer). η is the regularization weight in the fractal energy optimization.

[0137] While maintaining the fractal scaling ratio, suppress the cross - level transfer of manufacturing errors.

[0138] Due to the manufacturing error ΔX k cannot be accurately measured in advance.

[0139] The printed layer L k → Measurement error ΔX k → Correction of the sub - layer L k+1 Design

[0140]

[0141] σ is the forgetting factor (usually taking values from 0.95 to 0.99), used to balance the weights of new and old data.

[0142] In this embodiment, in the final configuration, the unit cells are in close contact through stretching, bending extrusion, etc. The close - contact energy supports the magnitude of the minimum normal contact force. Assuming infinite friction between the unit cell structures, this structural compactness is defined as the part where the sum of the magnitudes of the normal contact forces of each pair of contacting unit cell structures in the flexible surface is less than f min .

[0143] Each embodiment in this application is described in a progressive manner. For the same or similar parts between embodiments, reference can be made to each other. The key point of each embodiment is to illustrate the differences from other embodiments.

[0144] The protection scope of this application is not limited to the above - mentioned embodiments. Obviously, those skilled in the art can make various changes and deformations to this disclosure without departing from the scope and spirit of this disclosure. If these changes and deformations fall within the scope of the claims of this disclosure and their equivalent technologies, the intention of this disclosure also includes these changes and deformations.

Claims

1. A three-dimensional printing flexible surface dynamic deformation device based on a fractal structure, characterized in that Comprising: Flexible surface units, servo motors, and unit connection structures created based on a fractal structure; The flexible surface units are N-level units constructed based on a fractal structure, where N is greater than or equal to 2. Multiple first-level units are spliced into second-level units, multiple second-level units are spliced into third-level units, and multiple (N - 1)-level units are spliced into N-level units; The first-level unit includes: an active unit body and a driven unit body. The active unit is connected to the driven unit body through a unit connection structure; The servo motor is installed in the active unit body and is used to drive the driven unit body to produce changes in rotation and displacement.

2. The three-dimensional printing flexible surface dynamic deformation device based on a fractal structure according to claim 1, characterized in that The active unit body is a unidirectionally closed regular triangular prism shell. Three servo limit grooves are evenly distributed at three corner points in the regular triangular prism shell. The servo limit grooves are used to install servo motors. Each of the three side surfaces of the regular triangular prism shell has a first U-shaped groove, and the first U-shaped groove is used to install the unit connection structure; The driven unit body is a unidirectionally closed regular triangular prism shell of the same size as the active unit body. Each of the three side surfaces of the regular triangular prism shell has a second U-shaped groove, and the second U-shaped groove is an integrally formed limiting structure.

3. The three-dimensional printing flexible surface dynamic deformation device based on the fractal structure according to claim 1, characterized in that The unit connection structure is a crankshaft connecting rod structure. The crankshaft connecting rod structure is a truncated body defined by a filleted rectangle with equal cross-sections. Both ends of the crankshaft connecting rod structure are semi-open circular buckles. The semi-open circular buckle at one end of the crankshaft connecting rod structure is connected to the active unit body, and the semi-open circular buckle at the other end of the crankshaft connecting rod structure is connected to the driven unit body.

4. The three-dimensional printing flexible surface dynamic deformation method based on a fractal structure is implemented by using the three-dimensional printing flexible surface dynamic deformation device based on a fractal structure described in any one of claims 1 to 3, and is characterized in that, Comprising: Flatten the target surface using the boundary-first flattening algorithm, and display the Gaussian curvature and conformal scaling factor of the target surface on the flattened target surface; Use first-level units to perform equilateral triangle-based meshing on the flattened target surface, and perform triangular hierarchical layout on the basic grid according to the Gaussian curvature and conformal scaling factor of the target surface to obtain an initial meshed layout; Rationalize the initial meshed layout to minimize unnecessary deformation and distortion; Use an optimization objective with multiple energy terms to optimize the rationalized meshed layout to obtain an optimized meshed layout, and print the target surface using the optimized meshed layout. The optimization objective with multiple energy terms includes: weighted summation of geometric approximation energy, physical feasibility energy, and fractal error compensation energy.

5. The method for dynamically deforming a flexible surface of three-dimensional printing based on a fractal structure according to claim 4, wherein The step of using first-level units to perform equilateral triangle-based meshing on the flattened target surface, and performing triangular hierarchical layout on the basic grid according to the Gaussian curvature and conformal scaling factor of the target surface to obtain an initial meshed layout includes: On the flattened target surface, taking the corresponding point of the maximum value of the conformal scaling factor as the center point, and using the center point as the starting point for equilateral triangle meshing in the first-level unit, and performing equilateral triangle meshing of the same size outward to obtain a basic grid; Perform feature weighted fusion on the grayscale image of the Gaussian curvature and the grayscale image of the conformal scaling factor to obtain a feature value grayscale image with Gaussian curvature and conformal scaling factor and the grayscale value corresponding to the feature value grayscale image; Perform spectral clustering on the grayscale image with Gaussian curvature and conformal scaling factor features, and divide it into N different regions according to the grayscale values corresponding to the eigenvalue grayscale image. In different regions, perform meshing with equilateral triangles of the same size as the first-level unit or the N-level unit to obtain the initial meshing layout.

6. The method for dynamically deforming a three-dimensional printed flexible surface based on a fractal structure according to claim 5, characterized in that, The meshing with equilateral triangles of the same size as the first-level unit or the N-level unit in different regions to obtain the initial meshing layout includes: If the grayscale values of all pixels in the Nth region on the target surface are within the maximum value range, use the N-level unit to perform equilateral triangle meshing on the flattened target surface; If the grayscale values of all pixels in the first region on the target surface are within the minimum value range, use the first-level unit to perform equilateral triangle meshing on the flattened target surface; If the grayscale values of all pixels in the second to N-1th regions on the target surface are between the minimum value range and the maximum value range, according to the size order of the grayscale values corresponding to the eigenvalue grayscale image, use the corresponding units from the second-level unit to the N-1th level unit to perform equilateral triangle meshing on the flattened target surface.

7. The three-dimensional printing flexible surface dynamic deformation method based on the fractal structure according to claim 4, characterized in that Rationalize the initial meshing layout to minimize unnecessary deformation and distortion, including: Calculate the contact force of each first-level unit in the initial meshing layout; Optimize the contact force of each first-level unit using the least squares method to obtain the rationalized meshing layout.

8. The method for three-dimensional printing flexible surface dynamic deformation based on a fractal structure according to claim 4, characterized in that The geometric approximation energy is calculated as follows: Among them, E geo is the geometric approximation energy, x i is the coordinate of the i-th vertex on the target surface, c i is the coordinate of the projection point of vertex x i on the target surface, and M is the number of vertices.

9. The method for dynamically deforming a flexible surface of 3D printing based on a fractal structure according to claim 4, wherein The physically feasible performance energy is calculated as follows: Among them, E physics is the physically feasible energy, θ ij is the relative rotation angle of the i-th and j-th adjacent units around their common edge, is the target relative rotation angle of the i-th and j-th adjacent units around their common edge, θ max is the maximum allowable rotation angle of the connection structure, and γ is the penalty weight for excessive rotation angle.

10. The method for three-dimensional printing flexible surface dynamic deformation based on a fractal structure according to claim 4, wherein The fractal error compensation energy is calculated as follows: Among them, E fractal is the fractal error compensation energy, r k is the fractal level L k to L k+1 of the side length scaling ratio, ΔX k is the measured or simulated level manufacturing error, η is the regularization weight in the fractal energy optimization, and k is the level of the fractal.

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