A method and apparatus for topology optimization of three-dimensional curved composite material thin shell structures
By using conformal geometric mapping and shell element theory, the topology optimization of three-dimensional curved surfaces is reduced to a two-dimensional plane, which solves the problem of low computational efficiency in existing methods. It generates an optimal stiffening layout that conforms to the stress characteristics of thin shells, and realizes efficient and lightweight design of complex curved surface structures.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- HUAZHONG UNIV OF SCI & TECH
- Filing Date
- 2026-02-14
- Publication Date
- 2026-06-05
AI Technical Summary
Existing topology optimization methods for three-dimensional curved composite thin-shell structures are computationally inefficient, making them difficult to apply in engineering and unable to accurately describe the stress characteristics of thin-shell structures.
The dimensionality of the three-dimensional surface optimization problem is reduced to a two-dimensional plane by conformal geometric mapping. The optimization model is constructed using shell element theory, the stiffness matrix is constructed using Kirchhoff-Love shell theory, and the optimal stiffened topology distribution is generated by iterative solution using the moving asymptote method.
It significantly improves optimization efficiency, and the generated topology conforms to the actual stress state of thin shells, has engineering rationality and reliability, and is suitable for lightweight design of complex curved surface structures.
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Figure CN122154299A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the technical field of engineering structure optimization design, and more specifically, relates to a topology optimization method and device for a three-dimensional curved surface composite material thin shell structure. Background Technology
[0002] With the increasing urgency of lightweight structures in aerospace, new energy vehicles, and other fields, three-dimensional curved thin-shell structures made of composite materials are highly favored due to their excellent specific strength and specific stiffness. In the design of such structures, how to efficiently improve their mechanical properties by arranging internal stiffeners or reinforcing ribs is a core engineering problem. Topology optimization technology provides a powerful theoretical tool for achieving the optimal arrangement of stiffeners.
[0003] However, existing topology optimization methods for three-dimensional curved thin shells are typically performed directly on a full three-dimensional solid element model. This leads to high computational costs and unnecessary modeling complexity, and fails to fully utilize the geometric features of the thin shell structure. Specifically, on the one hand, performing topology optimization directly on complex three-dimensional curved surfaces results in a large number of design variables and low solution efficiency; on the other hand, standard solid element topology optimization methods struggle to naturally and accurately describe and optimize a stiffening layout that conforms to the stress characteristics of the thin shell and has clear engineering significance. Therefore, there is an urgent need for an optimization method that can balance computational efficiency and engineering practicality. The key lies in reasonably reducing the topology optimization problem of three-dimensional curved thin shells to its two-dimensional mid-surface at the mechanical level, and constructing a dedicated element model that can accurately reflect the mechanical behavior of the shell structure (such as in-plane and bending coupling stiffness), thereby directly and efficiently solving for the optimal stiffening topology distribution scheme on the three-dimensional curved thin shell. Summary of the Invention
[0004] To address the aforementioned deficiencies or improvement needs of existing technologies, this invention provides a topology optimization method and apparatus for three-dimensional curved surface composite material thin shell structures, aiming to solve the problem of low computational efficiency when directly performing three-dimensional solid topology optimization on three-dimensional curved surface composite material thin shells.
[0005] To achieve the above objectives, according to one aspect of the present invention, a topology optimization method for a three-dimensional curved surface composite material thin shell structure is provided, comprising the following steps: S1. The geometric model of the three-dimensional curved surface composite thin shell structure to be optimized is geometrically discretized and the corresponding STL format file is exported in the form of triangular patches. Based on the obtained STL format file, the conformal geometry mapping algorithm is used to map the three-dimensional curved surface mesh to the parameterized coordinates of the two-dimensional plane parameter domain, thereby obtaining the mapping relationship of the three-dimensional curved surface composite thin shell structure from the three-dimensional curved surface to the two-dimensional plane. S2, based on the shell element theory, the two-dimensional planar parameter domain is divided into finite element meshes to obtain finite element meshes, and the shell element stiffness matrix of the finite element mesh is further constructed. S3, based on the shell element stiffness matrix, in the two-dimensional plane parameter domain, with the relative density and orientation of the shell elements as design variables, minimizing the overall structural flexibility as the optimization objective, and material usage as the constraint, a topology optimization model based on conformal mapping finite element mesh is established. S4. The design variables are updated based on the moving asymptote method and the topology optimization model. S5, based on the updated design variables and mapping relationships, is inverted back to the original three-dimensional surface, thereby obtaining the three-dimensional configuration of the optimal material distribution of the three-dimensional surface composite thin shell structure.
[0006] Furthermore, the STL format file describes the surface geometry in the form of triangular facet discretization. Based on the triangular facets, a conformal geometry mapping algorithm is used to conformally map the three-dimensional surface mesh to the parameterized coordinates of the two-dimensional plane parameter domain.
[0007] Furthermore, establish each 3D mesh vertex ( x i , y i , z i The coordinates of the nodes of the corresponding orthogonal grid in the two-dimensional parametric plane. u i , v i The one-to-one correspondence between the three-dimensional curved surface and the two-dimensional plane is obtained to obtain the mapping relationship of the three-dimensional curved surface composite material thin shell structure from the three-dimensional curved surface to the two-dimensional plane.
[0008] Furthermore, the obtained two-dimensional planar parametric domain is meshed using finite element methods to regenerate a regular quadrilateral finite element mesh. For each mesh element in the quadrilateral finite element mesh, based on the Kirchhoff-Love shell theory, the shell element stiffness matrix of the mesh element in the two-dimensional parametric coordinate system is derived. The stiffness matrix of the shell element From the membrane element stiffness matrix With bending stiffness matrix Composed of components, these structures are used to reflect the in-plane and out-of-plane mechanical behavior of the original three-dimensional curved composite material thin shells.
[0009] Furthermore, the mathematical expression of the topology optimization model is:
[0010] In the formula, c For structural flexibility, Fand U These are the equivalent load vector and displacement vector mapped to the two-dimensional plane parametric domain, respectively. K For those that depend on design variables r and i The overall stiffness matrix, and V 0 represents the optimized material volume and the initial volume of the design domain, respectively. V f For the specified volume fraction, r min For minimum relative density, i min and i max This is the constraint range for the material orientation angle; Let be the relative density of the e-th unit; The main direction of the element material in the e-th element is denoted as .
[0011] Furthermore, based on the conformal mapping relationship established in step S1, the coordinates of each two-dimensional finite element element and its corresponding design variable values are mapped. r e , i e The corresponding position and region on the original 3D surface mesh are associated back through inverse mapping.
[0012] Furthermore, on the three-dimensional curved surface, the relative density of the elements is filtered according to the set density threshold to generate the material distribution topology configuration, and combined with the mapped orientation angle field, the three-dimensional configuration of the optimal material distribution is output.
[0013] Furthermore, the moving asymptote method is used to iteratively solve the topology optimization model to obtain the relative density of the shell elements and the orientation of the elements. The sensitivity of the objective function to the design variables is then calculated, and the design variables are updated based on the obtained sensitivity results.
[0014] The present invention also provides a topology optimization system for a three-dimensional curved surface composite thin shell structure. The system includes a memory and a processor. The memory stores a computer program, and the processor executes the computer program to perform the topology optimization method for the three-dimensional curved surface composite thin shell structure as described above.
[0015] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the topology optimization method for the three-dimensional curved surface composite thin shell structure as described above.
[0016] In summary, compared with the prior art, the topology optimization method and equipment for three-dimensional curved surface composite thin shell structures provided by this invention have the following advantages: 1. This invention transforms the complex three-dimensional surface optimization problem into an efficient computation within a two-dimensional planar parameter domain through conformal geometric mapping. This dimensionality reduction fundamentally solves the inherent problems of traditional three-dimensional solid topology optimization methods, such as the enormous scale of design variables, high computational costs, and difficulty in applying them to practical engineering designs. This dedicated framework significantly improves optimization efficiency, enabling rapid and lightweight design of complex geometric surfaces and providing a simple and feasible new approach for engineering applications.
[0017] 2. The conformal mapping method employed in this invention preserves local angles and shape characteristics to the maximum extent during the mapping of a three-dimensional surface to a two-dimensional parametric plane. This geometric characteristic is crucial for the optimization of composite material structures. Fiber-reinforced composite materials exhibit strong directionality, often several times greater along the fiber direction than in the direction perpendicular to the fiber. Therefore, conformal mapping ensures that the force transmission paths obtained in the two-dimensional domain, closely related to the material's principal directions, are accurately maintained in terms of angular relationships and continuity when inverted and mapped back to the original three-dimensional surface. Consequently, the resulting three-dimensional topological configuration is not only mathematically optimal but also conforms to physical realities in its force flow transmission, significantly improving the engineering rationality and structural performance of the optimization results.
[0018] 3. This invention deeply embeds Kirchhoff-Love shell theory into the topology optimization model, and the constructed element stiffness matrix accurately accounts for both the in-plane (membrane) stiffness and the out-of-plane (bending) stiffness of the thin-shell structure. This is fundamentally different from the simplified "plane stress element" model commonly used in traditional methods, which only considers in-plane load-bearing capacity. Plane stress elements cannot reflect the bending stiffness and buckling stability of thin shells, leading to distorted mechanical models in the optimization of curved structures. The resulting topological configurations often fail or perform poorly under bending loads. In contrast, the optimization model based on shell elements in this method fully conforms to the mechanical nature of thin shells, ensuring that the optimization process is carried out on a scientifically sound mechanical basis. This directly generates a material stiffening layout that highly matches the actual stress state of the thin shell, making the results both theoretically rigorous and engineering reliable. Attached Figure Description
[0019] Figure 1 This is a flowchart of a topology optimization method for a three-dimensional curved surface composite thin shell structure provided in an embodiment of the present invention; Figure 2 The conformal mapping technique mentioned in the embodiments of this invention maps the STL file of a three-dimensional surface to a design domain composed of rectangular units in a two-dimensional parametric plane; Figure 3It is the result of topology optimization of rectangular shell elements within a two-dimensional planar design domain; Figure 4 This is a schematic diagram showing the back-mapping of optimization results in the two-dimensional planar parameter domain back to the original three-dimensional surface; Figure 5 These are other case studies demonstrating the calculations performed using this method; Figure 6 (a), (b), (c) and (d) in the figure are the optimization results of the automobile engine hood provided in the embodiment of the present invention; Figure 7 (a), (b), (c) and (d) are schematic diagrams of the hull bottom optimization below the waterline; Figure 8 (a), (b), (c) and (d) in the diagram are schematic diagrams of aircraft wing skin optimization. Detailed Implementation
[0020] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.
[0021] This invention provides a topology optimization method for three-dimensional curved surface composite thin shell structures. The topology optimization method reduces the three-dimensional curved surface to a two-dimensional plane through conformal geometry mapping and constructs an optimization model based on shell element theory, thereby efficiently and directly solving for the optimal stiffening or material distribution topology.
[0022] The topology optimization method mainly includes the following steps: S1. The geometric model of the three-dimensional curved surface composite thin shell structure to be optimized is geometrically discretized and the corresponding STL format file is exported in the form of triangular patches. Based on the obtained STL format file, the conformal geometric mapping algorithm is used to map the three-dimensional curved surface mesh to the parameterized coordinates of the two-dimensional plane parameter domain, thereby obtaining the mapping relationship of the three-dimensional curved surface composite thin shell structure from the three-dimensional curved surface to the two-dimensional plane.
[0023] Step S1 includes the following sub-steps: S1.1 Obtain the geometric model of the three-dimensional curved surface composite thin shell structure to be optimized and perform geometric discretization. Export it as a standard, finely meshed STL format file. The STL format file describes the surface geometry in the form of triangular facet discretization.
[0024] S1.2. Based on the triangular facets, a conformal geometry mapping algorithm is used to conformally map the 3D curved surface mesh to the parameterized coordinates of the 2D planar domain. This process establishes the coordinates of each 3D mesh vertex ( x i , y i , z i The coordinates of the nodes of the corresponding orthogonal grid in the two-dimensional parametric plane. u i , v i The one-to-one correspondence between the three-dimensional curved surface and the two-dimensional plane is obtained to obtain the mapping relationship of the three-dimensional curved surface composite material thin shell structure from the three-dimensional curved surface to the two-dimensional plane.
[0025] S2, based on the shell element theory, performs finite element meshing on the two-dimensional planar parameter domain to obtain the finite element mesh, and further constructs the shell element stiffness matrix of the finite element mesh.
[0026] Step S2 includes the following steps: S2.1. Perform finite element mesh generation on the obtained two-dimensional planar parameter domain and regenerate a regular quadrilateral finite element mesh.
[0027] S2.2 For each mesh element in the quadrilateral finite element mesh, based on the Kirchhoff-Love shell theory, derive the shell element stiffness matrix of the mesh element in the two-dimensional parametric coordinate system. The stiffness matrix of this shell element. From the membrane element stiffness matrix With bending stiffness matrix Constructed in a composite form, these elements are used to reflect the in-plane and out-of-plane mechanical behavior of the original three-dimensional curved composite material thin shell. For composite materials, the stiffness matrix of this shell element is... The principal direction of the element material in the e-th element. i e The function.
[0028] S3, based on the shell element stiffness matrix, establishes a topology optimization model based on conformal mapping finite element mesh in the two-dimensional plane parameter domain, with the relative density and orientation of the shell elements as design variables, minimizing the overall structural flexibility as the optimization objective, and material usage as the constraint.
[0029] The mathematical expression for the topology optimization model is:
[0030] In the formula, c For structural flexibility, F and UThese are the equivalent load vector and displacement vector mapped to the two-dimensional plane parametric domain, respectively. K For those that depend on design variables r (Unit relative density) and i The overall stiffness matrix (in the direction of the element material). and V 0 represents the optimized material volume and the initial volume of the design domain, respectively. For the specified volume fraction, For minimum relative density, i min and i max This defines the constraint range for the material orientation angle. The subscript 'e' is the element index, indicating the e-th element. Let be the relative density of the e-th unit; The main direction of the element material in the e-th element is denoted as .
[0031] S4. The moving asymptote method is used to iteratively solve the topology optimization model to obtain the relative density of the shell elements and the orientation of the elements. The sensitivity of the objective function to the design variables is further calculated, and the design variables are updated based on the obtained sensitivity results.
[0032] Step S4 includes the following sub-steps: S4.1 Constructing the Lagrangian function with equilibrium equation constraints Based on the stationarity condition of the Lagrange function, the structural compliance objective function is analytically derived. c relative density r e and the main direction of unit material i e Sensitivity.
[0033] S4.2 In each optimization iteration step, first solve the equilibrium equation. Obtain the displacement field U Then, using the analytical expression derived in step S4.1, the sensitivity of all elements to density is calculated in parallel. c / r e With respect to fiber angle c / i e .
[0034] S4.3 Input the objective function value, constraint function value and their corresponding sensitivity information into the MMA solver. The MMA solver then constructs and solves a series of convex approximation subproblems, thereby synchronously updating the density variables and material orientation angle variables of all elements.
[0035] S4.4 Repeat the above analysis and update process until the preset convergence criterion is met, and output the converged design variable field.
[0036] S5, based on the updated design variables and mapping relationships, is inverted back to the original three-dimensional surface, thereby obtaining the three-dimensional configuration of the optimal material distribution of the three-dimensional surface composite thin shell structure.
[0037] Step S5 includes the following sub-steps: S5.1 After optimization convergence, extract the optimal relative density value of each finite element in the two-dimensional planar parameter domain. r e With material orientation angle i e ; S5.2. Based on the conformal mapping relationship established in step S1, assign the coordinates of each two-dimensional finite element element and its corresponding design variable values ( r e , i e The corresponding positions and regions on the original 3D surface mesh are associated back through inverse mapping. S5.3 On a three-dimensional curved surface, according to a set density threshold (e.g.) r (Fibers <0.3 are not displayed) The relative density of the units is filtered to generate a clear material distribution (solid and porous regions) topology. Combined with the mapped orientation angle field, the output is a three-dimensional curved thin shell optimization result that can be directly used to guide subsequent layup design.
[0038] The present invention will be further described in detail below with reference to specific embodiments.
[0039] Please see Figure 1 This invention provides a topology optimization method for a three-dimensional curved surface composite material thin shell structure, the specific implementation steps of which are as follows: Step 1: Surface Discretization and Parametric Mapping (1) First, in computer-aided design software, create or import a model of the three-dimensional curved composite thin-shell structure to be optimized (in this example, a curved cantilever beam surface). Then, export the geometric data of the model as a high-precision STL file in the form of a fine mesh. The STL file discretizes the original three-dimensional surface in the form of a set of triangular facets.
[0040] (2) Based on the obtained STL file, the triangular facets are obtained by using a conformal mapping algorithm to obtain the parametric coordinates (u, v) of each 3D vertex on the 2D parametric plane. Assume... A free surface defined in Euclidean space, u ,v Let represent an orthogonal coordinate system in two-dimensional parameter space. For any freeform surface belonging to a shell, this mapping relationship can be expressed as:
[0041] In the formula This represents the inverse mapping operator.
[0042] Therefore, the orthogonal curvilinear coordinate system of the three-dimensional curved shell can be easily obtained through conformal parameterization:
[0043] This conformal mapping method allows for the convenient generation of uniform meshes in a two-dimensional parameter space, enabling subsequent topology optimization analysis to be performed directly on the uniform meshes in the two-dimensional parameter space.
[0044] Step 2: Construction of Two-Dimensional Planar Design Domain Shell Element Model (3) Within the two-dimensional planar rectangular parameter domain obtained in step one, perform independent finite element mesh generation. Since the parameter domain is a regular planar region, a structured quadrilateral mesh can be directly generated. Each two-dimensional quadrilateral finite element element will serve as the basic design unit for subsequent topology optimization.
[0045] (4) For each quadrilateral element in the two-dimensional plane parameter domain, construct its element stiffness matrix based on the Kirchhoff-Love thin shell theory.
[0046] The stiffness characteristics of an element are determined by the element stiffness matrix. Characterization, the stiffness matrix of this element This forms the basis for sensitivity analysis and iterative calculations in subsequent topology optimization. For composite materials, It includes the principal orientation angle of the unit material. i e The function of element stiffness matrix. It is composed of independent in-plane stiffness components and bending stiffness components, that is:
[0047] in, The in-plane stiffness matrix reflects the element's ability to resist in-plane tensile and shear deformation, and its calculation formula is as follows:
[0048] In the formula, This is the in-plane strain-displacement relationship matrix. Let be the in-plane elastic matrix of the composite shell element, and be the element angle. i e The function; The unit area is denoted as .
[0049] The bending stiffness matrix reflects the element's ability to resist bending deformation, and its calculation formula is as follows:
[0050] In the formula, This is the bending strain-displacement relationship matrix. The bending elasticity matrix of the composite shell element is also the principal direction of the element material. i e The function.
[0051] matrix and Determined by the properties of the composite material and the element orientation. For an initially given material elastic matrix... D 0, through coordinate transformation matrix R ( i The initial material elasticity matrix is obtained. D 0 can be represented as:
[0052] parameter D xx These represent the elastic coefficients that constitute the stiffness matrix of orthotropic composite materials. The in-plane elastic matrix of a composite shell element. It can be represented as:
[0053] coordinate transformation matrix R ( i This can be represented as:
[0054] Bending elasticity matrix of composite shell unit With in-plane elastic matrix D m The relationship is: , t The thickness of the thin-shell structure is assumed to be a dimensionless unit thickness of 1 in this embodiment.
[0055] The element stiffness matrix of the shell element constructed accordingly It accurately describes the stiffness behavior of composite thin shells under in-plane and out-of-plane loads, laying a correct mechanical model foundation for scientific and efficient topology optimization in the two-dimensional plane parameter domain.
[0056] Step 3: Establishment of a two-dimensional topology optimization model Based on the aforementioned two-dimensional planar parameter domain and shell element model, a topology optimization model for composite material curved surface thin shells is established. This embodiment aims to minimize structural flexibility (i.e., maximize overall stiffness) and optimizes material distribution and fiber orientation under material usage constraints.
[0057] (5) Each shell element in the two-dimensional design domain is associated with two independent design variables. The density variable is one of these variables. r e Characterizes the retention state of material within a unit cell, with values continuously varying between 0 (representing pores) and 1 (representing solid material). The principal direction of the unit cell material. i e The orientation of the main fiber in the composite material region represented by the unit is characterized, and its value varies continuously within a given angular range.
[0058] (6) Based on the SIMP (Solid Isotropic Material with Penalization) method and the material orientation interpolation model, the following topology optimization model is established:
[0059] In the formula, c For structural flexibility; F and U These are the equivalent load vector and displacement vector mapped to the two-dimensional planar domain, respectively. K For those that depend on design variables r (Unit relative density) and i The global stiffness matrix (in the direction of the element material) is derived from the element stiffness matrix. The assembly is complete. Based on the model from step two, Interpolation and penalty depend on design variables: Where P is the penalty factor, defined as 3, used to force the intermediate density values to converge towards either 0 or 1. r min The minimum relative density is defined as 0.001 to avoid singularities in the stiffness matrix; and V 0 represents the optimized material volume and the initial volume of the design domain, respectively; The specified volume fraction; i min and i max This is the constraint range for the material orientation angle; N is the total number of elements in the design domain; Subscript eFor cell index, indicating the first cell. e Units.
[0060] This topology optimization model describes a continuum topology optimization problem in a two-dimensional planar parameter domain, with element density and orientation angle as design variables, minimum structural flexibility as the objective, and equilibrium equations and material volume as constraints.
[0061] Step 4: Optimization Solution and Sensitivity Analysis This step iteratively solves the aforementioned topology optimization model using the moving asymptote method. Its core lies in efficiently calculating the sensitivity of the objective function and constraint function to the design variables.
[0062] (7) Sensitivity analysis was performed using the adjoint variable method. First, the Lagrangian function for the equilibrium equation constraints was constructed:
[0063] in, l It is a Lagrange multiplier.
[0064] The objective function c applies to any design variable ( r e , i e The sensitivity of the design variable can be obtained by taking the partial derivative of the Lagrange function with respect to that variable. Specifically, the sensitivity of the design variable... r and i Differentiate:
[0065]
[0066] For the e-th unit, = V e Structural flexibility c For the density of the e-th unit r e The sensitivity can be written as:
[0067] Among them, the element stiffness matrix The membrane stiffness matrix has been explicitly defined in the above derivation. and bending stiffness matrix The combination of elements. Therefore, the element stiffness matrix. The derivative with respect to density can be expressed as the membrane stiffness matrix. and bending stiffness matrix derivative with respect to density and . can be written as:
[0068]
[0069] In the formula and These are the strain-displacement matrices for the membrane stiffness matrix and the bending stiffness matrix, respectively.
[0070] Similarly, structural flexibility c For the principal material orientation angle of the e-th element i e The sensitivity can be written as:
[0071] Therefore, the element stiffness matrix The derivative with respect to the element angle can be expressed as the membrane stiffness matrix. and bending stiffness matrix The derivative with respect to the unit angle, and , can be written as:
[0072]
[0073] for In more detail, it can be broken down as follows:
[0074] (8) The moving asymptote method is used as the optimization solver. In each iteration step: a Solve the equilibrium equations. Obtain the displacement field U ; b Based on the above analytical formula, the objective function sensitivity of all units is calculated in parallel. and and volume constraint sensitivity .
[0075] c Set the objective function value of the current iteration. c constraint functions All of its sensitivity information is input into the MMA solver.
[0076] d Based on this information, the MMA solver constructs an independent convex approximation subproblem for each design variable and solves the subproblem, thereby synchronously updating the density variables of all elements. r e and direction angle variables i e .
[0077] e Repeat the steps. a to d The process continues until the convergence criterion is met. The convergence criterion is set as follows: the relative change in the objective function value between two adjacent iterations is less than a given tolerance of 0.001, and the volume constraint is satisfied.
[0078] At this point, the optimization iteration is complete, and the converged design variable field is output. r e , i e}
[0079] Step 5: Result Mapping and 3D Configuration Reconstruction After the optimization iteration converges, the design variable field is obtained in the two-dimensional planar parameter domain. This step aims to accurately map these results back to the original three-dimensional physical space and reconstruct a three-dimensional topological configuration that can be used in engineering applications.
[0080] (9) Extract the optimal design variable values and the optimal relative density values of each element in the two-dimensional plane parameter domain from the converged optimization model. and the optimal value of the orientation angle of the element material These data constitute two scalar fields defined on a two-dimensional parametric grid.
[0081] (10) Perform inverse mapping using the conformal mapping relationship established in step one. For each finite element in the two-dimensional planar parameter domain, based on the parameter coordinates of the finite element ( u i , v i Find its unique corresponding position on the original 3D surface mesh. x i , y i , z i ).
[0082] This process essentially involves "pasting back" the material distribution and fiber orientation information, defined on the "flattened" two-dimensional plane, onto the three-dimensional surface based on the original geometric correspondence. Due to the conformal mapping property, the fiber orientation angles represented on the two-dimensional plane... After being mapped back to a 3D surface, its directional relationship with respect to the local surface is maintained, ensuring continuity in a mechanical sense.
[0083] Thus, the complete design process was completed, starting from the three-dimensional curved surface, undergoing efficient optimization in the two-dimensional domain, and then returning to the three-dimensional curved surface. This resulted in an optimized design of a three-dimensional composite material curved thin shell that meets both lightweight and stiffness requirements and has a clear reinforcement layout for engineering manufacturing.
[0084] Figure 2 to Figure 4 The system presents case studies of topology optimization using this method on three curved cantilever beams with different geometric specifications. The optimization results show that for each curved surface configuration, this method can generate a material distribution topology with clear engineering semantics. The optimized density field clearly reveals the optimal material aggregation paths, which naturally form a stiffening layout highly adapted to the surface geometry and load boundary conditions. Crucially, the element material orientation angle field, simultaneously optimized with the density field, exhibits a highly uniform and convergent distribution. This smoothness and consistency of the orientation field provides intuitive and precise guidance for the fiber placement of composite materials in actual manufacturing, ensuring a seamless transformation from theoretical optimization results to engineering manufacturing. The comparison of these three cases fully demonstrates that this method does not generate a uniform topology, but rather intelligently captures and responds to the geometric features of the original surface (such as curvature distribution and aspect ratio), thereby "tailor-making" unique and mechanically sound stiffening configurations for different surfaces, showcasing its sensitivity to surface geometric changes and design adaptability.
[0085] Furthermore, Figure 5 This study demonstrates the successful application of our method to more complex boundary conditions (such as multi-point constraints) and more irregular, continuous surface configurations. The optimization results in these cases not only exhibit clear rib topology and convergent direction fields, but more importantly, they verify the strong scalability and robustness of our method in handling diverse engineering boundary conditions and complex continuous surfaces. Whether it's a simple cantilever beam or a skinned structure with complex geometry, the "parametric mapping-shell element optimization" framework constructed by our method remains stable and efficient, generating designs with both high structural performance and high engineering interpretability. This signifies that our method has transcended mere academic algorithms, becoming a core tool capable of addressing the diverse design needs in practical engineering and providing an integrated solution from conceptual design to manufacturing guidance for three-dimensional curved composite thin-shell structures.
[0086] Finally, the surface topology optimization method proposed in this invention can be extended to engineering applications. Figure 6 to Figure 8The method has been demonstrated in applications in automobiles, ships, and aircraft. Specifically, in the design of lightweight automotive components, it achieves efficient material distribution and clear load path representation; in the optimization of curved hull skin structures, it adapts to complex fluid loads and spatial constraints, significantly improving structural stiffness and stability; and in the curved surface configurations of aircraft wings and fuselages, it successfully coordinates aerodynamic shape and internal stiffener topology, achieving significant weight reduction while meeting stringent aerodynamic performance requirements. These successful cross-disciplinary applications further demonstrate that this method possesses broad engineering adaptability and powerful cross-scale optimization capabilities, providing complete technical support from concept generation to detailed design for various curved composite structures, and has significant engineering promotion value and industrialization prospects.
[0087] The present invention also provides a topology optimization system for a three-dimensional curved surface composite thin shell structure. The system includes a memory and a processor. The memory stores a computer program, and the processor executes the computer program to perform the topology optimization method for the three-dimensional curved surface composite thin shell structure as described above.
[0088] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the topology optimization method for the three-dimensional curved surface composite thin shell structure as described above.
[0089] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A topology optimization method for a three-dimensional curved surface composite material thin shell structure, characterized in that, The steps are as follows: S1. The geometric model of the three-dimensional curved surface composite thin shell structure to be optimized is geometrically discretized and the corresponding STL format file is exported in the form of triangular patches. Based on the obtained STL format file, the conformal geometry mapping algorithm is used to map the three-dimensional curved surface mesh to the parameterized coordinates of the two-dimensional plane parameter domain, thereby obtaining the mapping relationship of the three-dimensional curved surface composite thin shell structure from the three-dimensional curved surface to the two-dimensional plane. S2, based on the shell element theory, the two-dimensional planar parameter domain is divided into finite element meshes to obtain finite element meshes, and the shell element stiffness matrix of the finite element mesh is further constructed. S3, based on the shell element stiffness matrix, in the two-dimensional plane parameter domain, with the relative density and orientation of the shell elements as design variables, minimizing the overall structural flexibility as the optimization objective, and material usage as the constraint, a topology optimization model based on conformal mapping finite element mesh is established. S4. The design variables are updated based on the moving asymptote method and the topology optimization model. S5, based on the updated design variables and mapping relationships, is inverted back to the original three-dimensional surface, thereby obtaining the three-dimensional configuration of the optimal material distribution of the three-dimensional surface composite thin shell structure.
2. The topology optimization method for three-dimensional curved surface composite thin-shell structures as described in claim 1, characterized in that: The STL format file describes the surface geometry in the form of triangular facet discretization. Based on the triangular facets, a conformal geometry mapping algorithm is used to map the three-dimensional surface mesh to the parameterized coordinates of the two-dimensional plane parameter domain in a conformal manner.
3. The topology optimization method for a three-dimensional curved surface composite thin-shell structure as described in claim 2, characterized in that: Establish each 3D mesh vertex ( x i , y i , z i The coordinates of the nodes of the corresponding orthogonal grid in the two-dimensional parametric plane. u i , v i The one-to-one correspondence between the three-dimensional curved surface and the two-dimensional plane is obtained to obtain the mapping relationship of the three-dimensional curved surface composite material thin shell structure from the three-dimensional curved surface to the two-dimensional plane.
4. The topology optimization method for three-dimensional curved surface composite thin-shell structures as described in claim 1, characterized in that: The obtained two-dimensional planar parametric domain is meshed using finite element methods to regenerate a regular quadrilateral finite element mesh. For each mesh element in the quadrilateral finite element mesh, the shell element stiffness matrix in the two-dimensional parametric coordinate system is derived based on the Kirchhoff-Love shell theory. The stiffness matrix of the shell element From the membrane element stiffness matrix With bending stiffness matrix Composed of components, these structures are used to reflect the in-plane and out-of-plane mechanical behavior of the original three-dimensional curved composite material thin shells.
5. The topology optimization method for three-dimensional curved surface composite thin-shell structures as described in claim 1, characterized in that: The mathematical expression for the topology optimization model is: In the formula, c For structural flexibility, F and U These are the equivalent load vector and displacement vector mapped to the two-dimensional plane parametric domain, respectively. K For those that depend on design variables ρ and θ The overall stiffness matrix, and V 0 represents the optimized material volume and the initial volume of the design domain, respectively. V f For the specified volume fraction, ρ min For minimum relative density, θ min and θ max This is the constraint range for the material orientation angle; Let be the relative density of the e-th unit; The main direction of the element material in the e-th element is denoted as .
6. The topology optimization method for a three-dimensional curved surface composite thin-shell structure as described in claim 5, characterized in that: Based on the conformal mapping relationship established in step S1, the coordinates of each two-dimensional finite element element and its corresponding design variable values are mapped. ρ e , θ e The corresponding position and region on the original 3D surface mesh are associated back through inverse mapping.
7. The topology optimization method for three-dimensional curved surface composite thin-shell structures as described in claim 6, characterized in that: On a three-dimensional curved surface, the relative density of the elements is filtered according to a set density threshold to generate a material distribution topology. Combined with the mapped orientation angle field, the three-dimensional configuration of the optimal material distribution is output.
8. The topology optimization method for a three-dimensional curved surface composite thin-shell structure as described in any one of claims 1-7, characterized in that: The moving asymptote method is used to iteratively solve the topology optimization model to obtain the relative density and orientation of the shell elements. The sensitivity of the objective function to the design variables is then calculated, and the design variables are updated based on the obtained sensitivity results.
9. A topology optimization system for a three-dimensional curved surface composite material thin shell structure, characterized in that: The system includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it performs the topology optimization method for the three-dimensional curved surface composite thin shell structure according to any one of claims 1-8.
10. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the topology optimization method for a three-dimensional curved surface composite thin-shell structure as described in any one of claims 1-8.