A method, system, device and medium for constructing and optimizing a discrete developable surface
By constructing parallelogram elements within a quadrilateral mesh and applying constraints of isogonality, shape similarity, and equidistance, the stability and consistency issues in the construction and optimization of developable surfaces are resolved, enabling an efficient and low-cost manufacturing process.
Patent Information
- Application Number
- CN202610333634.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-03-18
- Publication Date
- 2026-06-23
AI Technical Summary
Existing methods for constructing and optimizing developable surfaces suffer from insufficient stability during numerical solutions, slow iterative convergence, and inconsistent shapes and sizes of discrete surface elements, leading to local stress concentration, increased manufacturing costs, a lack of a unified modeling and solution framework, and complex operation.
A checkerboard pattern of parallelogram units is constructed within a quadrilateral grid, and isogonality, shape similarity, and equidistant constraints are applied. By introducing interpolation points and scaling parameters, higher-order geometric relationships are transformed into secondary constraints, which are uniformly encoded into a unified planar unit. A half-side data structure is used to manage the grid relationships, and the solution is obtained by combining the Gauss-Newton iterative algorithm.
It achieves stable construction and optimization of discrete developable surfaces, reduces the number of molds and manufacturing complexity, improves manufacturing accuracy and efficiency, and reduces costs. It is applicable to fields such as architecture and curtain wall design and manufacturing.
Smart Images
Figure CN122263226A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the fields of discrete differential geometry, computational geometry and computer-aided design, and specifically to a method, system, device and medium for constructing and optimizing discrete developable surfaces. Background Technology
[0002] Developable surfaces are a class of surfaces that can be unfolded onto a plane at equal intervals without stretching or compression. Typical examples include cylindrical surfaces, conical surfaces, and composite surfaces formed by combining them. In computational geometry, the essential geometric characteristic of these surfaces is that their Gaussian curvature is zero everywhere.
[0003] With the rapid development of computer-aided design and digital manufacturing technologies, developable surfaces have shown significant advantages in effectively avoiding in-plane stretching and compression during material processing, and have important application value in engineering fields such as sheet metal processing, bending forming, composite material layup, and building curtain wall design. Against this backdrop, geometric modeling methods based on discrete representations have gradually become the mainstream technical approach in related research and applications because they are easier to combine with numerical computation and engineering implementation.
[0004] Existing research typically employs polygonal meshes, especially triangular or quadrilateral meshes, to discretize continuous surfaces and approximates the theoretical characteristics of developable surfaces by imposing constraints on the mesh's geometric properties. For example, common methods include constraining the Gaussian curvature at mesh vertices to zero in a discrete sense, or equivalently, restricting the sum of the dihedral angles of adjacent faces at interior points to be equal to... This is to characterize the developability conditions of the surface. In addition, to meet practical needs such as engineering design and shape control, existing technologies usually require numerical optimization solutions for the developability of discrete surfaces based on given target shapes or boundary conditions.
[0005] Defects and shortcomings of existing technology: First, existing methods for constructing and optimizing developable surfaces generally suffer from insufficient stability during numerical solutions. Their optimization models typically contain highly nonlinear and non-convex constraints, which can lead to slow convergence of the iterative process, or even solution degeneration or numerical instability.
[0006] Secondly, the discrete surface elements (such as triangles or quadrilaterals) constructed by existing methods lack uniform constraints in shape and size, and there are often significant differences between different elements, which can easily lead to local stress concentrations during surface development or shaping. This phenomenon not only affects the smoothness of structural development, but also increases manufacturing costs in actual engineering processing, thereby weakening the core value of developable surfaces in the manufacturing field.
[0007] Finally, at the methodological level, existing technologies for discretizing modeling, constructing, and optimizing developable surfaces typically employ separate technical solutions, lacking a unified modeling and solution framework with a clear structure and logical coherence. This fragmentation in construction logic significantly increases the operational complexity and implementation cost in engineering applications, hindering the promotion and large-scale application of the method. Summary of the Invention
[0008] To overcome the problems of high implementation cost and low manufacturing accuracy in the actual engineering manufacturing process of the existing technology, the purpose of this invention is to provide a method, system, device and medium for constructing and optimizing discrete developable surfaces.
[0009] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A method for constructing and optimizing discrete developable surfaces includes the following steps: Obtain the quadrilateral mesh; design parameters for the quadrilateral mesh, including scale and angle parameters; Based on the quadrilateral mesh, parallelogram units distributed in a checkerboard pattern are constructed inside the quadrilateral mesh to establish a developable surface geometric model; wherein, the constraints of the developable surface geometric model include: checkerboard pattern constraint with embedded parallelograms, isogonality constraint, shape similarity constraint and isometry constraint. Solve the geometric model of the developable surface to obtain a discrete developable surface with congruent parallelograms; encode the geometric features of the discrete developable surface into a unified planar unit so that each surface can be processed and shaped using the same processing parameters or the same set of molds.
[0010] Furthermore, the constraints of the checkerboard pattern with embedded parallelograms are determined through the following process: Take any quadrilateral unit as the control quadrilateral, and arrange the vertices of the control quadrilateral in sequence as the first vertex. Second vertex The third vertex With the fourth vertex ; Controlling each side of the quadrilateral superior, Take values in sequence , Take values in sequence By introducing a proportional parameter Construct interpolation points on the edge The constraints are: (1) The four interpolation points obtained by connecting them constitute - Parallelogram; scale parameter Replace with ,get: (2) The interpolation points obtained by connecting them constitute - Parallelogram; in, (3) As the first auxiliary variable, It is the second auxiliary variable.
[0011] Furthermore, the isoangular constraint is: (4) in, The included angle parameter, As the first vertex, The second vertex, The third vertex, It is the fourth vertex.
[0012] Furthermore, shape similarity constraints are determined through the following process: Based on the interpolation points, a correspondence is established between the ratio of adjacent side lengths and the ratio of diagonal lengths of a parallelogram, resulting in the following similarity expression: (5) in, Let represent the ratio of adjacent sides and the ratio of diagonals of a parallelogram, respectively. The ratio of the diagonals of a parallelogram From the included angle parameter The ratio of adjacent sides of a parallelogram Determine the ratio of the diagonals of the parallelogram. The following relationship must be satisfied: (6) The shape similarity constraint is: (7) in, The length of the first diagonal. The length of the second diagonal. The length of the first diagonal The corresponding first unit direction vector, The length of the second diagonal The corresponding second unit direction vector, .
[0013] Furthermore, the isometry constraint is: (9) in, The length of the first diagonal of the quadrilateral. The length of the second diagonal of the quadrilateral. It is the inner product of the two diagonals of the quadrilateral.
[0014] Furthermore, the two diagonals of a quadrilateral satisfy the following relationship: (8) in, , The lengths of the two diagonals of the quadrilateral, specifically... The length of the first diagonal of the quadrilateral. The length of the second diagonal of the quadrilateral. It is the dot product of the two diagonals of the quadrilateral. For noodles The first vertex in the middle, For noodles The second vertex in the middle, For noodles The third vertex in the middle, For noodles The fourth vertex in the middle.
[0015] Furthermore, the objective energy function of the developable surface geometric model is: , in, As the first weight, As the second weight, As the third weight, As the fourth weight, As the fifth weight, The energy term for the interpolation points on the edge. For proportional parameters The energy term within the range of values, The energy term is constrained by isoangularity. The energy term is constrained by shape similarity. The energy term is constrained by equality. For smooth energy functions, To fit the energy function.
[0016] A system for constructing and optimizing discrete developable surfaces, comprising: The quadrilateral mesh acquisition module is used to acquire quadrilateral meshes; the design parameters of the quadrilateral meshes include scale parameters and included angle parameters; The developable surface geometry model creation module is used to create a developable surface geometry model based on a quadrilateral mesh. The constraints of the developable surface geometry model include: checkerboard pattern constraints with embedded parallelograms, isogonality constraints, shape similarity constraints, and isometric constraints. The solver module is used to solve the geometric model of the developable surface to obtain a discrete developable surface with congruent parallelograms; the geometric features of the discrete developable surface are encoded into a unified planar element, so that each surface can be processed and shaped using the same processing parameters or the same set of molds.
[0017] An electronic device includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor, when executing the computer program, implements the method for constructing and optimizing the discrete developable surface.
[0018] A computer-readable storage medium storing a computer program, characterized in that the computer program, when executed by a processor, implements the method for constructing and optimizing the discrete developable surface.
[0019] Compared with the prior art, the present invention has the following beneficial effects: This invention constructs parallelogram-shaped units arranged in a checkerboard pattern within a quadrilateral mesh. Through constraints based on isogonality, shape similarity, and equidism, all parallelogram-shaped units within the checkerboard pattern are geometrically congruent. Thus, the discrete developable surface is composed of a set of planar units with consistent geometric shapes. The geometric features of the surface are stably encoded into these unified planar units, allowing each surface to be processed and shaped using the same processing parameters or a single set of molds. Compared to traditional freeform surface structures composed of numerous non-isomorphic surface pieces, this invention significantly reduces the number of molds and manufacturing complexity, lowers processing and construction costs, shortens the manufacturing cycle, improves the feasibility of engineering implementation, and achieves high accuracy. This invention integrates checkerboard pattern construction, isogonal and equidistant geometric constraints, and the engineering solution process into a unified technical system, realizing a complete engineering process from input quadrilateral mesh to output discrete developable surface. This invention is applicable to fields such as the design of curved building structures and curtain wall units, modeling of expandable shells and foldable structures, design of sheet metal bending and sheet metal forming, design of engineering shells and cladding structures, and manufacturing of prefabricated structures based on planar sheets.
[0020] Furthermore, based on the proposed discretization model of developable surfaces—discrete equiangular Chebyshev nets—this invention transforms the original high-order geometric relations into a constraint expression of no higher than quadratic by introducing interpolation points, scaling parameters, and auxiliary variables on the edges of the quadrilateral mesh. These constraints are then uniformly incorporated into the objective function as energy terms for solution, effectively reducing the nonlinearity in the engineering modeling and solution process. Furthermore, a guided projection solution method based on Gauss-Newton iteration is employed, ensuring good stability and computational efficiency in the solution process during the construction of developable surfaces. Attached Figure Description
[0021] Figure 1 This is a flowchart of the method for constructing and optimizing discrete developable surfaces according to the present invention; Figure 2 A schematic diagram illustrating the overall process of geometric modeling and solution methods for developable surfaces; Figure 3 for In the same quadrilateral - Parallelogram and - Parallelogram; where (a) is - Parallelogram, (b) is - Parallelogram; Figure 4 It features a checkerboard pattern with embedded parallelograms; Figure 5 This is a schematic diagram of the process for constructing developable surfaces based on three shape constraints. Figure 6 This is a schematic diagram illustrating the process of constructing a developable surface based on a reference surface. Figure 7 This is a schematic diagram of the system for constructing and optimizing discrete developable surfaces according to the present invention. Detailed Implementation
[0022] To facilitate understanding of the present invention, a more complete description will be given below with reference to the accompanying drawings. Preferred embodiments of the invention are shown in the drawings. However, the invention can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a thorough and complete understanding of the disclosure of the invention.
[0023] See Figure 1This invention proposes a method for constructing and optimizing discrete developable surfaces, comprising: obtaining a quadrilateral mesh; the design parameters of the quadrilateral mesh include scale parameters and included angle parameters; based on the quadrilateral mesh, constructing parallelogram elements distributed in a checkerboard pattern within the quadrilateral mesh to establish a geometric model of the developable surface; wherein the constraints of the geometric model of the developable surface include: checkerboard pattern constraints with embedded parallelograms, isoangular constraints, shape similarity constraints, and equidistant constraints; solving the geometric model of the developable surface to obtain a discrete developable surface with congruent parallelograms; encoding the geometric features of the discrete developable surface into a unified planar element, enabling each surface to be processed and shaped using the same processing parameters or the same set of molds. This method constructs the discrete developable surface by introducing a discrete isoangular Chebyshev mesh, and performs geometric constraint modeling and numerical optimization on the input quadrilateral mesh. This invention integrates the construction method of discrete developable surfaces, the constraint modeling process, and the optimization solution process into a unified technical framework, forming a complete, logically clear, and easily implemented overall technical solution. (See also...) Figure 2 Specifically, it includes the following steps: S1: Establish the geometric model of the developable surface; In the smoothness theory of differential geometry, developable surfaces are usually characterized by their parameterization. Any smooth surface can be described by two families of parametric curves. When these two families of parametric curves maintain a fixed angle at every point on the surface and the curve spacing along each parameter direction remains unchanged, the corresponding surface has zero Gaussian curvature, thus forming a class of smooth developable surfaces. In theoretical analysis, it has a clear mathematical definition, but it depends on continuous parameters and smooth structure, making it difficult to directly use for the design and manufacture of engineering surfaces spliced from a finite number of planar components.
[0024] In practical engineering applications, curved surfaces typically need to be represented in the form of discrete meshes and composed of several machinable planar elements. Traditional continuous description methods for smooth developable surfaces are difficult to directly guide the geometric construction and assembly constraints of discrete elements. Based on the study of the isoangular and iso-spacing geometric characteristics of smooth developable surfaces, this invention proposes a developable geometric modeling method for discrete quadrilateral meshes.
[0025] Specifically, this invention introduces an embedded parallelogram structure into a quadrilateral mesh of a smooth developable surface, and applies isoangular and iso-spacing constraints to the quadrilateral mesh, thereby geometrically forming a discrete isoangular Chebyshev mesh, thus achieving modeling of the geometry of the developable surface at the discrete level. The specific steps include: S2: Obtain the quadrilateral mesh (i.e., the initial mesh, either designed or existing). Design parameters for quadrilateral grids, including scale parameters With angle parameter .
[0026] Determining the constraints of the developable surface geometry model includes steps S2-S5.
[0027] S3: Determine the constraints of the checkerboard pattern with embedded parallelograms; When constructing a checkerboard pattern with embedded parallelograms, the quadrilateral grid is first used as the control grid, and the construction of edge interpolation points is performed for each quadrilateral cell in the grid.
[0028] Specifically, any quadrilateral unit is used as the control quadrilateral, and its vertices are arranged in sequence as the first vertex. Second vertex The third vertex With the fourth vertex .
[0029] Furthermore, such as Figure 3 As shown in (a), on each side of the control quadrilateral superior( Take values in sequence , Take values in sequence By introducing a proportional parameter Construct interpolation points on the edge The constraints are: (1) Connecting the four interpolation points in the above order forms a parallelogram. The parallelogram constructed in this way is defined as follows: - Parallelogram.
[0030] Similarly, such as Figure 3 As shown in (b), when the scaling parameter Replace with ,Right now: (2) The interpolation points obtained by connecting them in the above order also form a parallelogram. The parallelogram constructed in this way is defined as follows: - Parallelogram.
[0031] Furthermore, to ensure the stability of the numerical solution process, the scaling parameter is limited. The range of values for which a first auxiliary variable is introduced. With the second auxiliary variable And constrained by the following mathematical relation: (3) This ensures the proportional parameters Always in The effective range.
[0032] like Figure 4 As shown, when performing edge interpolation construction on the entire quadrilateral mesh, construction is performed alternately in adjacent quadrilateral cells. - Parallelogram and - Parallelograms are used to create parallelogram cells that form a regular checkerboard pattern within the overall mesh. This construction method embeds a set of proportional parameters within the quadrilateral mesh without altering the original topology and overall geometry of the control quadrilateral mesh. A uniformly controlled parallelogram checkerboard pattern.
[0033] Formulas (1) to (3) above constitute a checkerboard pattern constraint with embedded parallelograms.
[0034] To uniformly constrain the geometry of parallelogram units in the checkerboard pattern, the present invention sets triple shape control constraints in subsequent steps S4 to S6.
[0035] S4: Determine the isogonal constraint; To ensure that the discrete mesh meets the equiangular conditions required for equiangular Chebyshev meshes, this invention further applies equiangular constraints to the quadrilateral mesh based on the checkerboard pattern construction.
[0036] Specifically, for any quadrilateral unit in the quadrilateral grid of the checkerboard pattern, its two diagonals are the first diagonal. With the second diagonal By applying a uniform included angle constraint to the two diagonals of all controlling quadrilateral elements, the following isoangular constraint is established: (4) in, The included angle parameter is used to ensure that the included angle of the diagonals of each control quadrilateral unit in the entire control quadrilateral grid remains consistent. Furthermore, as can be seen from the interpolation points on the sides defined by formulas (1) and (2), the interior angles of the parallelogram units in the checkerboard pattern also remain consistent.
[0037] Thus, the two diagonals of all control quadrilateral elements in the control quadrilateral mesh form two families of directions, and the included angle between them remains constant throughout the discrete mesh, thereby corresponding to an isoangular parametric curve structure in a discrete sense.
[0038] S5: Determine shape similarity constraints; To further constrain the geometry of parallelogram units in a checkerboard pattern, ensuring they possess consistent shape characteristics while maintaining isoangularity, this invention introduces a similarity constraint in addition to the isoangularity constraint. This similarity constraint is achieved by controlling the proportional relationship between adjacent sides and diagonals of the parallelogram.
[0039] Specifically, based on the definition of interpolation points on the sides in formulas (1) and (2), a correspondence between the ratio of adjacent side lengths and the ratio of diagonal lengths of a parallelogram can be established, thus obtaining the following similarity expression: (5) in, To represent the ratio of adjacent sides and the ratio of diagonals of a parallelogram, the adjacent side ratios of parallelograms are uniformly set. and diagonal ratio This ensures that all parallelograms in the checkerboard pattern maintain similar shapes. Combining this with the aforementioned isogonal constraint, i.e., formula (5), the ratio of the diagonals of the parallelograms... It can be determined by the included angle parameter The ratio of adjacent sides of a parallelogram It is determined that the ratio of the diagonals of the parallelogram satisfies the following relationship: (6) Therefore, the similarity condition of all parallelogram units in the checkerboard pattern can be equivalently represented by formulas (4) and (6). These constraints jointly guarantee the geometric similarity of each parallelogram unit in the checkerboard pattern, and further, by adjusting the ratio of adjacent sides of the parallelograms... and diagonal ratio The values of can be used to construct parallelogram units with different geometric shapes.
[0040] To facilitate subsequent numerical optimization and constraint linearization, this invention further introduces a method to control the lengths of the two diagonals of the quadrilateral (i.e., the length of the first diagonal). The length of the second diagonal and the length of the first diagonal The corresponding first unit direction vector The length of the second diagonal The corresponding second unit direction vector Specifically, the shape similarity constraint is defined as follows: (7) in, .
[0041] S6: Determine the equidistant constraints; To further address the issue of inconsistent geometric shapes among elements in discrete surfaces, this invention introduces equidistant constraints on the basis of the aforementioned checkerboard construction and isogonal constraints, ensuring that the parallelogram elements generated in the checkerboard pattern remain geometrically identical.
[0042] As can be seen from the construction of interpolation points on the edges defined by formulas (1) and (2), when all parallelogram units in the checkerboard pattern are congruent, the two diagonals of the corresponding control quadrilateral should satisfy the following relationship: (8) in, , The lengths of the two diagonals of the quadrilateral, specifically... The length of the first diagonal of the quadrilateral. The length of the second diagonal of the quadrilateral. It is the inner product of the two diagonals of the quadrilateral. For noodles The first vertex in the middle, For noodles The second vertex in the middle, For noodles The third vertex in the middle, For noodles The fourth vertex in the middle, combined with formula (4), yields the following expression: (9) As can be seen from the above relationship, under the premise of satisfying the isoangularity constraint, by applying the constraint condition shown in formula (9), the diagonal directions of the two families in the quadrilateral mesh can be kept equidistantly distributed throughout the entire discrete mesh. This property corresponds to the equidistant condition required by the isoangular Chebyshev mesh in the discrete sense, and therefore formula (9) is defined as the equidistant constraint.
[0043] Under the combined effect of isoangular and isometric constraints, the quadrilateral mesh forms a discrete isoangular Chebyshev mesh in a discrete sense, thereby making the constructed discrete surface conform to the geometric characteristics of a developable surface, while maintaining the congruence of all parallelogram units in the checkerboard pattern.
[0044] The developable surface geometric model is then optimized and solved through steps S7 to S10.
[0045] S7: Manage mesh geometry using a half-edge data structure; This invention achieves efficient management of mesh geometric relationships through a half-side data structure. Its core objective is to reduce variable redundancy and improve the efficiency of constraint calculation and optimization iteration. The specific design is as follows: 1) Half-edge definition and interpolation point association: For any directed half-edge in a quadrilateral mesh The corresponding interpolation point along the edge is defined as Its reverse half The interpolation points are automatically set to Therefore, the same point is only visited once, and no variable is visited repeatedly.
[0046] 2) When extracting the embedded parallelogram checkerboard pattern from the quadrilateral mesh, if the edge interpolation points are constructed for each quadrilateral face in the manner described in formulas (1) and (2), the same mesh edge will be repeatedly processed in adjacent quadrilaterals, resulting in the same interpolation point being stored multiple times. This causes the number of interpolation point variables introduced to reach four times the number of quadrilateral mesh faces, significantly increasing the computational scale. To reduce the number of variables, edge interpolation points are stored only on the directed half-edges originating from a set of non-adjacent mesh points selected from the quadrilateral mesh. and its corresponding proportional parameters By using the above method, the total number of interpolation point variables is reduced to the number of sides of a quadrilateral grid, while the scaling parameter exists only as a globally independent variable throughout the entire grid. This effectively reduces the scale of optimization variables and improves the stability and computational efficiency of the numerical optimization process.
[0047] 3) Furthermore, all unknowns involved in the constraints of steps S3-S6 can be unified and integrated into a single optimization variable. ,in, The node variables representing all vertices in the quadrilateral mesh. This represents the interpolation point variables located on the non-repeating edges of the quadrilateral grid, stored using a half-edge data structure in design 2). As input design parameters used to control the chessboard grid structure and parallelogram shape, As an auxiliary variable introduced in the constraint modeling process.
[0048] S8: Establish the target energy function; To facilitate the unified solution of the aforementioned discrete geometric constraints (i.e., checkerboard pattern constraints with embedded parallelograms, isogonal constraints, shape similarity constraints, and equidistant constraints) using numerical optimization methods, this invention transforms the constraints in steps S2-S5 into an optimizable objective energy function form.
[0049] Energy terms of edge interpolation points improved by step S7 for: , in, This represents the index of a group of non-adjacent vertices selected from the half-side data structure within the half-side data structure. The index of a family of non-adjacent vertices selected in the half-data structure. The index of adjacent points in the half-side data structure. A family of non-adjacent grid points in a quadrilateral grid. A quadrilateral grid is a set of non-adjacent grid points. Adjacent grid points.
[0050] The proportional parameter in formula (3) The energy term within the range of values for: .
[0051] Equation (4) Energy term of moderate angular constraint for: , For the first The first, second, third, and fourth vertices of the quadrilateral are arranged in sequence.
[0052] The energy term of the shape similarity constraint in formula (8) for:
[0053] , Let be the lengths of the two diagonals of the f-th quadrilateral, i.e. Let f be the length of the first diagonal of the f-th quadrilateral. Let f be the length of the second diagonal of the f-th quadrilateral. for The auxiliary variable is introduced to reduce the algebraic degree of the relevant energy terms.
[0054] The energy term of the equality constraint in formula (9) for:
[0055] To ensure the mesh is as smooth as possible during optimization, this invention applies smoothness constraints to the control mesh and the polylines containing the interpolation points. By constraining the discrete second-order difference of any three consecutive points, the mesh maintains good visual continuity and engineering usability while preserving geometric constraints. Smooth energy function. for: , in, These can be represented as any three consecutive points in the grid. As the midpoint, and This represents two edge points.
[0056] In addition, such as Figure 6 As shown, the surface is optimized by reading in a reference surface with the same topology, ensuring that the optimized mesh closely fits the reference surface and the energy function when sliding along the surface. for: , in, To optimize the grid points of the quadrilateral grid, For the reference surface's grid points, Grid points of the reference surface The normal vector of the tangent plane of the reference surface at that location.
[0057] By combining the above energy terms, a unified target energy function is constructed. : , in, As the first weight, As the second weight, As the third weight, As the fourth weight, As the fifth weight, A value of 0 or 1 is used to control whether the corresponding constraint is enabled. , The value is between 0.0005 and 0.005, serving as a soft constraint.
[0058] S9: Guided Projection The algorithm solves the problem iteratively. Through steps S2 to S7, the present invention transforms all kinds of geometric constraints involved in the construction of developable surfaces into constraint equations of degree no higher than the second, thereby significantly reducing the nonlinearity of the overall optimization problem and providing a foundation for stable and efficient numerical solutions.
[0059] Based on this, the present invention employs a guided projection method based on a Gauss-Newton iterative framework. Guided Projection The algorithm iteratively minimizes the constructed overall objective energy function and jointly updates the grid vertex variables and edge interpolation point variables.
[0060] S10: Output a discrete developable surface with congruent parallelograms; Example 1 See Figure 5 For the initial mesh, a checkerboard pattern constraint with embedded parallelograms is applied to obtain the scaling parameters. =0.3 pairs of parallelogram checkerboard pattern, and proportional parameters =0.5 pairs of parallelogram checkerboard patterns, the obtained proportional parameters =0.3 pairs of parallelogram checkerboard pattern, and proportional parameters =0.5 pairs of parallelogram checkerboard patterns with set included angle parameters Below, based on the overall target energy function (energy terms of interpolation points on the edges) Energy terms with isoangular constraints By applying angular constraints, a surface with an isoangular parallelogram is obtained. This is based on the global objective energy function (the energy term of the interpolation points on the edges). Energy terms constrained by shape similarity By applying shape similarity constraints, surfaces with similar parallelograms are obtained. This is based on the overall objective energy function (the energy term of the interpolation points on the edges). Energy terms with equality constraints By applying isometry constraints, Gaussian curvature grayscale images are obtained respectively: a 60° congruent parallelogram developable surface at t=0.3, a 60° congruent parallelogram developable surface at t=0.5, and a 90° congruent parallelogram developable surface at t=0.3. The darker the color in the Gaussian curvature grayscale image, the greater the curvature.
[0061] Example 2 See Figure 6 Based on the surface to be optimized and the reference surface, constraints such as checkerboard pattern with embedded parallelograms, isogonality constraints, shape similarity constraints, and isometry constraints are applied, and the overall objective function is solved. This yields developable surfaces with congruent parallelograms at t=0.3, developable surfaces with congruent parallelograms at t=0.5, and Gaussian curvature grayscale images. In the Gaussian curvature grayscale image, the darker the color, the greater the curvature.
[0062] like Figure 5 and Figure 6 As shown, this invention applies different degrees of influence to different geometric constraints by adjusting the weight parameters corresponding to each energy function, thereby gradually obtaining an optimized structure that meets the expected requirements. During the optimization process, nonlinear geometric constraints are linearized in each iteration, and the constraint relationships between grid vertex variables and edge interpolation point variables are uniformly organized and managed based on a half-edge data structure. The positions of each grid vertex and interpolation point are iteratively updated through a numerical optimization algorithm, causing the discrete grid to gradually converge to a geometric shape that satisfies the developability constraints, ultimately outputting an optimized discrete developable surface.
[0063] Example 3 See Figure 7In one embodiment of the present invention, a system for constructing and optimizing discrete developable surfaces is also provided, comprising: The quadrilateral mesh acquisition module is used to acquire quadrilateral meshes; the design parameters of the quadrilateral meshes include scale parameters and included angle parameters; The developable surface geometry model creation module is used to create a developable surface geometry model based on a quadrilateral mesh. The constraints of the developable surface geometry model include: checkerboard pattern constraints with embedded parallelograms, isogonality constraints, shape similarity constraints, and isometric constraints. The solver module is used to solve the geometric model of the developable surface to obtain a discrete developable surface with congruent parallelograms.
[0064] Example 4 In one embodiment of the present invention, an electronic device is also provided, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the method for constructing and optimizing the discrete developable surface.
[0065] Example 5 In one embodiment of the present invention, a computer-readable storage medium is also provided, the computer-readable storage medium storing a computer program, which, when executed by a processor, implements the method for constructing and optimizing discrete developable surfaces.
[0066] This invention integrates checkerboard pattern construction, isoangular and equidistant geometric constraints, and the engineering solution process into a unified technical system. It also utilizes a half-side data structure to manage mesh topology, geometric variables, and constraints, achieving a complete engineering workflow from inputting a quadrilateral mesh to outputting a discrete developable surface. This integrated technical solution has a clear structure and coherent logic, making it suitable for applications in areas such as architectural curved structure and curtain wall unit design, developable shell and foldable structure modeling, sheet metal bending and forming design, engineering shell and cladding structure design, and prefabricated structure manufacturing based on planar sheet metal.
[0067] The optimized discrete developable surface obtained by this invention possesses excellent geometric controllability and clear engineering manufacturability, and can be widely applied in related technical fields of generating complex curved surfaces from planar materials. For example, in the design and construction of building curtain wall systems, freeform roofs, and spatial shell structures, the discrete developable surface obtained by this invention is composed of a set of parallelogram units with completely identical geometric shapes, which can serve as a geometric reference model for building skins or structural components, demonstrating significant technical advantages in the engineering manufacturing and assembly stages. Since the entire surface is composed of repeated congruent parallelogram units, all facets can be mass-produced using uniform processing parameters or the same set of molds, eliminating the need for customizing surface components for different locations, thereby effectively reducing the number of molds, lowering processing and construction costs, and shortening the manufacturing cycle. Compared to traditional freeform surface structures composed of a large number of non-isomorphic facets, this invention has significant advantages in manufacturing efficiency, cost control, and engineering feasibility.
Claims
1. A method for constructing and optimizing a discrete developable surface, characterized in that, Includes the following steps: Obtain a quadrilateral mesh; the design parameters for the quadrilateral mesh include scale parameters and included angle parameters; Based on the quadrilateral mesh, parallelogram units distributed in a checkerboard pattern are constructed inside the quadrilateral mesh to establish a developable surface geometric model; wherein, the constraints of the developable surface geometric model include: checkerboard pattern constraint with embedded parallelograms, isogonality constraint, shape similarity constraint and isometry constraint. Solve the geometric model of the developable surface to obtain a discrete developable surface with congruent parallelograms; encode the geometric features of the discrete developable surface into a unified planar unit so that each surface can be processed and shaped using the same processing parameters or the same set of molds.
2. The method of constructing and optimizing a discrete developable surface of claim 1, wherein, The constraints of the checkerboard pattern with embedded parallelograms are determined through the following process: Take any quadrilateral unit as the control quadrilateral, and arrange the vertices of the control quadrilateral in sequence as the first vertex. Second vertex The third vertex With the fourth vertex ; Controlling each side of the quadrilateral superior, Take values in sequence , Take values in sequence By introducing a proportional parameter Construct interpolation points on the edge The constraints are: ，（1） The four interpolation points obtained by connecting them constitute - Parallelogram; scale parameter Replace with ,get: （2） The interpolation points obtained by connecting them constitute - Parallelogram; in, （3） As the first auxiliary variable, It is the second auxiliary variable.
3. The method for constructing and optimizing discrete developable surfaces according to claim 1, characterized in that, The isoangular constraint is: （4） in, The included angle parameter, As the first vertex, The second vertex, The third vertex, It is the fourth vertex.
4. The method for constructing and optimizing discrete developable surfaces according to claim 1, characterized in that, Shape similarity constraints are determined through the following process: Based on the interpolation points, a correspondence is established between the ratio of adjacent side lengths and the ratio of diagonal lengths of a parallelogram, resulting in the following similarity expression: （5） in, Let represent the ratio of adjacent sides and the ratio of diagonals of a parallelogram, respectively. The ratio of the diagonals of a parallelogram From the included angle parameter The ratio of adjacent sides of a parallelogram Determine the ratio of the diagonals of the parallelogram. The following relationship must be satisfied: （6） The shape similarity constraint is: （7） in, The length of the first diagonal. The length of the second diagonal. The length of the first diagonal The corresponding first unit direction vector, The length of the second diagonal The corresponding second unit direction vector, .
5. The method for constructing and optimizing discrete developable surfaces according to claim 4, characterized in that, The isometry constraint is: （9） in, The length of the first diagonal of the quadrilateral. The length of the second diagonal of the quadrilateral. It is the inner product of the two diagonals of the quadrilateral.
6. The method for constructing and optimizing discrete developable surfaces according to claim 5, characterized in that, The two diagonals of a quadrilateral satisfy the following relationship: （8） in, , The lengths of the two diagonals of the quadrilateral, specifically... The length of the first diagonal of the quadrilateral. The length of the second diagonal of the quadrilateral. It is the dot product of the two diagonals of the quadrilateral. For noodles The first vertex in the middle, For noodles The second vertex in the middle, For noodles The third vertex in the middle, For noodles The fourth vertex in the middle.
7. The method for constructing and optimizing discrete developable surfaces according to claim 1, characterized in that, The objective energy function of the developable surface geometric model is: ， in, As the first weight, As the second weight, As the third weight, As the fourth weight, As the fifth weight, The energy term for the interpolation points on the edge. For proportional parameters The energy term within the range of values, The energy term is constrained by isoangularity. The energy term is constrained by shape similarity. The energy term is constrained by equality. For smooth energy functions, To fit the energy function.
8. A system for constructing and optimizing discrete developable surfaces, characterized in that, include: The quadrilateral mesh acquisition module is used to acquire quadrilateral meshes. The developable surface geometry model creation module is used to create a developable surface geometry model based on a quadrilateral mesh. The design parameters of the quadrilateral mesh include scale parameters and included angle parameters. The constraints of the developable surface geometry model include: checkerboard pattern constraints with embedded parallelograms, isogonality constraints, shape similarity constraints, and isodistance constraints. The solver module is used to solve the geometric model of the developable surface to obtain a discrete developable surface with congruent parallelograms; the geometric features of the discrete developable surface are encoded into a unified planar element, so that each surface can be processed and shaped using the same processing parameters or the same set of molds.
9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the method for constructing and optimizing discrete developable surfaces as described in any one of claims 1 to 7.
10. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by a processor, it implements the method for constructing and optimizing discrete developable surfaces as described in any one of claims 1 to 7.