A flexible mechanism topology optimization method and device based on unstructured grid geometry explicit projection

Abstract

The present application belongs to the technical field of flexible structure topology optimization, and discloses a flexible mechanism topology optimization method and equipment based on unstructured grid geometry explicit projection, which comprises the following steps: (1) defining the control point coordinates and width of a Bezier curve as design variables, and constructing an explicit mapping relationship function; (2) mapping the explicit mapping relationship function into a discrete level set matrix, extracting a two-dimensional body grid based on the discrete level set matrix, and constructing a three-dimensional prism grid; (3) based on the three-dimensional prism grid, taking the displacement maximization of a predetermined output point of a flexible mechanism to be optimized as an objective function, constructing a topology optimization model comprising a global material volume constraint, a minimum cavity size constraint, a stress constraint and a minimum solid size constraint controlled by a lower limit of the width variable, and iteratively solving the topology optimization model to obtain the optimal topology configuration of the flexible mechanism to be optimized. The present application eliminates the boundary stress calculation distortion caused by the traditional fixed grid.

Description

Technical Field

[0001] This invention belongs to the technical field of flexible structure topology optimization, and more specifically, relates to a method and device for topology optimization of flexible mechanisms based on explicit projection of unstructured mesh geometry. Background Technology

[0002] Flexible mechanisms are a class of integral mechanisms that utilize the elastic deformation of flexible components to transmit motion, force, or energy. Compared to traditional rigid linkage mechanisms, they possess inherent advantages such as frictionless, wear-free, backlash-free, and lubrication-free operation, and are currently widely used in precision instruments, aerospace, microelectromechanical systems (MEMS), and soft robotics industries. In particular, planar spring structures, due to their ability to achieve specific stiffness characteristics and guiding functions through a compact planar topology, occupy a central position in precision vibration isolation, flexible support, and high-precision positioning platforms. In recent years, the development of processing technologies such as precision wire cutting, laser cutting, and additive manufacturing has solved the manufacturing challenges of planar springs with complex curved boundaries and fine topological features, thus bringing considerable design freedom to novel high-performance flexible structures.

[0003] Planar spring structures typically require slender beam systems with high aspect ratios or complex meandering features to meet specific flexibility requirements. For such structures to be effective, the design must simultaneously optimize topological connections, skeleton geometry, and cross-sectional dimension distribution. However, traditional topology optimization methods based on discrete element density present a typical high-dimensional non-convex problem when dealing with this type of issue. Due to the lack of explicit geometric feature descriptions, traditional methods are prone to getting trapped in local optima when searching for slender, clear spring paths from chaotic gray-scale elements, often generating topologically simple, fragmented, or island-like structures that are unable to evolve into high-performance complex spring configurations. Furthermore, the jagged boundaries caused by fixed meshes severely distort the accuracy of calculations for local properties such as stress, and make it difficult to precisely control minimum solid size and void size, making it difficult to directly use the optimization results for subsequent manufacturing. Summary of the Invention

[0004] In view of the above-mentioned defects or improvement needs of the existing technology, the present invention provides a topology optimization method and device for flexible mechanisms based on explicit projection of unstructured mesh geometry. It aims to solve the problems that existing topology optimization methods for flexible mechanisms are prone to getting trapped in local optima and that the serrated boundaries caused by fixed meshes result in serious distortion of the calculation accuracy of local properties such as stress, and it is difficult to achieve precise control of minimum solid size and minimum void size.

[0005] To achieve the above objectives, according to one aspect of the present invention, a topology optimization method for flexible mechanisms based on explicit projection of unstructured mesh geometry is provided, comprising the following steps: (1) Using the variable-width Bézier curve as the topological configuration skeleton of the flexible mechanism to be optimized, the coordinates of the control points of the Bézier curve and the width of the control points are defined as design variables, and an explicit mapping relationship function from the geometric parameter space to the physical density field is constructed. (2) The explicit mapping relation function is mapped to a discrete level set matrix using a structured background mesh. A two-dimensional body mesh is extracted based on the discrete level set matrix, and the two-dimensional body mesh is stretched in the normal direction to construct a three-dimensional prism mesh. The node mechanical boundary conditions of the three-dimensional prism mesh are automatically reconstructed based on the geometric position tolerance. (3) Based on the three-dimensional prism mesh, with the displacement maximization of the predetermined output point of the flexible mechanism to be optimized as the objective function, a topology optimization model is constructed that includes global material volume constraints, minimum void size constraints, stress constraints, and minimum solid size constraints controlled by the lower limit of the width variable. The topology optimization model is then iteratively solved to obtain the optimal topology configuration of the flexible mechanism to be optimized.

[0006] Furthermore, the central skeleton curve of the flexible mechanism to be optimized is described by a Bézier curve, and the corresponding parametric equation is defined as a linear combination of the Bernstein basis function and the coordinates of the control points.

[0007] Furthermore, the shortest distance between the background grid nodes and the Bézier curve is calculated based on the design variables, and a discretized level set matrix is ​​constructed by combining the width of the control points.

[0008] Furthermore, a geometric position tolerance is set, the boundary nodes of the body-fitted mesh are traversed, and the node set that satisfies the fixed boundary, the setting area of ​​external load, and the setting area of ​​virtual load is selected. Constraints are applied to the nodes in the node set in the stiffness matrix of the three-dimensional prism mesh.

[0009] Furthermore, the mathematical expression for the topology optimization model is:

[0010] In the formula, The virtual unit force vector at the predetermined output point. For displacement vectors, For volume constraint functions, The minimum void size constraint function, Let be the stress constraint function. and The preset minimum and maximum control point width thresholds, and These are the upper and lower limits of the control point coordinate range.

[0011] Furthermore, volume constraints are implemented by calculating the ratio of the total volume of the solid region to the volume of the initial design domain:

[0012] In the formula, For the initial design domain volume, This represents the upper limit of the material's volume fraction.

[0013] Furthermore, P-norm mapping and power function stress relaxation are introduced to aggregate spatially distributed local stresses into a single global scalar index, thereby achieving effective control over the global stress level.

[0014] Furthermore, based on the theory of continuum shape sensitivity analysis and the chain rule, analytical sensitivity formulas for the objective function and each constraint function with respect to geometric design variables are derived.

[0015] The present invention also provides a topology optimization system for flexible mechanisms based on explicit projection of unstructured mesh geometry. 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 flexible mechanisms based on explicit projection of unstructured mesh geometry as described above.

[0016] 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 flexible mechanism topology optimization method based on explicit projection of unstructured mesh geometry as described above.

[0017] In summary, compared with the prior art, the topology optimization method and device for flexible mechanisms based on explicit projection of unstructured mesh geometry provided by this invention have the following advantages: 1. Compared to traditional density-based topology optimization methods, which are prone to structural discrepancies, configuration breaks, or the creation of isolated regions, this invention utilizes the explicit geometric continuity of Bézier curves to fundamentally guarantee the intrinsic connectivity of flexible mechanisms. This method can automatically evolve high-performance, long-path meandering configurations based on sensitivity derivation, effectively improving the energy storage and transfer efficiency of flexible mechanisms. Simultaneously, by leveraging the precise description of geometric boundaries using unstructured body-fitted meshes, this invention eliminates the distortion in boundary stress calculations caused by traditional fixed meshes.

[0018] 2. Multidimensional Manufacturing Constraints Ensure Machinability: The minimum solid size and cavity size proposed in this invention achieve robust dual constraints on both the minimum solid size and the minimum cavity size. This mechanism ensures that the generated reed has sufficient structural stability while eliminating unmachinable micro-holes, allowing the optimization results to be directly applied to precision machining processes and possessing high engineering practical value.

[0019] 3. Since node indices are rearranged during the iteration process of unstructured body-fitted meshes, a search strategy based on geometric topological invariance is adopted. A geometric position tolerance is set, the boundary nodes of the body-fitted mesh are traversed, and the node sets that satisfy the requirements of fixed boundaries, external load settings, and virtual load settings are selected. Constraints are then applied to the nodes in the node sets within the stiffness matrix of the 3D prism mesh, achieving automated conversion of the mechanical analysis model.

[0020] 4. To control the lightweighting of the structure and meet the preset material usage requirements, this invention implements volume constraints by calculating the ratio of the total volume of the solid region to the volume of the initial design domain. This volume constraint ensures that the material consumption of the final evolved flexible mechanism configuration does not exceed a set proportion while meeting performance requirements. Attached Figure Description

[0021] Figure 1 This is a flowchart of a topology optimization method for flexible mechanisms based on explicit projection of unstructured mesh geometry, provided by an embodiment of the present invention. Figure 2 This is a schematic diagram of the design space and working conditions provided in the embodiments of the present invention; Figure 3 This is a schematic diagram of the initial Bézier curve configuration; Figure 4 This is a schematic diagram of the initial structure mesh generation result; Figure 5 This is a schematic diagram of the optimized Bézier curve configuration; Figure 6 This is a schematic diagram of the optimized structural mesh generation results. Detailed Implementation

[0022] 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.

[0023] This invention discloses a topology optimization method for flexible mechanisms based on explicit projection of unstructured mesh geometry. This method is a system design approach that can simultaneously consider the parallel optimization of configuration topology, geometric dimensions, and physical performance. Within a full-scale framework, this method utilizes Bézier curves to parametrically represent the structural skeleton, significantly reducing the dimensionality of design variables. This effectively helps the optimizer escape local optima traps. Combined with adaptive body-fitting unstructured mesh technology, it eliminates the influence of jagged boundaries on stress analysis and achieves control over minimum solid size and minimum void size. Ultimately, it generates innovative planar spring configurations with smooth boundaries, superior performance, and direct manufacturability.

[0024] Please see Figure 1 The optimization method mainly includes the following steps: (1) Using the variable-width Bézier curve as the topological skeleton of the flexible mechanism to be optimized, the coordinates of the control points of the Bézier curve and the width of the control points are defined as design variables, and an explicit mapping function from the geometric parameter space to the physical density field is constructed.

[0025] A variable-width Bézier curve is used as the basic geometric skeleton to describe the topological configuration of the flexible mechanism to be optimized. The coordinates and width of the control points on the Bézier curve are defined as design variables, constructing an explicit mapping relationship from the geometric parameter space to the physical density field. This method describes the central skeleton curve of the flexible mechanism to be optimized through an order Bézier curve, whose parametric equation is defined as a linear combination of Bernstein basis functions and control point coordinates.

[0026]

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[0028]

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[0031]

[0032] In the formula, The coordinates of control points other than the start and end points of the curve. To control the width of the point, For the Bessel interpolation polynomial, The curve parameters range from 0 to 1. The final set of curve-related design variables is as follows: .

[0033] (2) The explicit mapping relation function is mapped to a discrete level set matrix using a structured background mesh. A two-dimensional body mesh is extracted based on the discrete level set matrix, and the two-dimensional body mesh is stretched in the normal direction to construct a three-dimensional prism mesh. The node mechanical boundary conditions of the three-dimensional prism mesh are automatically reconstructed based on the geometric position tolerance.

[0034] The Bézier curve described in (1) is mapped to a discrete level set matrix using a structured background mesh. A two-dimensional unstructured body-fitted mesh is generated using a body-fitting mesh algorithm and stretched into a three-dimensional prismatic mesh along the thickness direction of the flexible mechanism. In each generation of calculation, the shortest distance between the background mesh nodes and the Bézier curve is calculated based on the design variables, and a discrete level set matrix is ​​constructed by combining the width of the control points.

[0035]

[0036]

[0037]

[0038]

[0039] In the formula, Represents any node in the background grid. The level set function value at that point, These are points sampled at equal intervals on the Bézier curve. This is the interpolation width at the current nearest sampling point. It is achieved through the smoothing Heaviside function. Mapping the horizontal field to a physical density field Control the size of the smoothing interval.

[0040] Since node indices are rearranged during iteration in unstructured body-fitted meshes, a search strategy based on geometric topological invariance is adopted. A geometric position tolerance is set, and the boundary nodes of the body-fitted mesh are traversed. Node sets that satisfy the requirements of fixed boundaries, external load settings, and virtual load settings are selected. Constraints are then applied to the nodes in the node sets within the stiffness matrix of the 3D prism mesh, achieving automated conversion of the mechanical analysis model.

[0041] (3) Based on the three-dimensional prism mesh, with the displacement maximization of the predetermined output point of the flexible mechanism to be optimized as the objective function, a topology optimization model is constructed that includes global material volume constraints, minimum void size constraints, stress constraints, and minimum solid size constraints controlled by the lower limit of the width variable. The topology optimization model is then iteratively solved to obtain the optimal topology configuration of the flexible mechanism to be optimized.

[0042] The topology optimization model takes maximizing the displacement of the predetermined output point of the flexible mechanism to be optimized as the objective function, and includes global material volume constraints, minimum void size constraints, stress constraints, and minimum solid size constraints.

[0043] The mathematical expression for the topology optimization model is:

[0044] In the formula, The virtual unit force vector at the predetermined output point. For displacement vectors, For volume constraint functions, The minimum void size constraint function, Let be the stress constraint function. and The preset minimum and maximum control point width thresholds, and These are the upper and lower limits of the control point coordinate range.

[0045] To control the lightweighting of the structure and meet the preset material usage requirements, this invention implements volume constraints by calculating the ratio of the total volume of the solid region to the volume of the initial design domain:

[0046] In the formula, For the initial design domain volume, This is the upper limit for the material volume fraction. This volume constraint ensures that the material consumption of the final evolved flexible mechanism configuration does not exceed the set proportion while meeting performance requirements.

[0047] To address the challenge of fabricating tiny holes in flexible mechanisms, this invention introduces a hole size constraint mechanism. A minimum desired hole size lower limit is set. Select the threshold that satisfies the topological interval. The nearest extreme point on the center line of the Bézier curve and Let the extreme points be... and The spatial distance is The actual geometric gap is Construct a one-sided quadratic penalty function to penalize holes that do not satisfy size constraints:

[0048] To prevent yielding failure of flexible mechanisms during long-stroke motion, this invention introduces P-norm mapping and power function stress relaxation to aggregate spatially distributed local stresses into a single global scalar index, thereby achieving effective control over the global stress level.

[0049]

[0050]

[0051] In the formula, For local Von Mises stress, For global stress index, For norm penalty parameters, This represents the allowable stress value for the material. This minimum void size constraint ensures that the optimized mechanism maximizes output displacement while keeping the stress in critical areas within a safe range.

[0052] Based on the theory of continuum shape sensitivity analysis and the chain rule, analytical sensitivity formulas for the objective function and each constraint function with respect to geometric design variables are derived, achieving accurate mapping of mechanical gradient information from the unstructured finite element space to the Bessel geometric parameter space. To efficiently handle large-scale mechanical equilibrium constraints, this invention employs the adjoint variable method to construct the adjoint equation, solves the adjoint equation to obtain the adjoint variables, and substitutes them into the solution to obtain the sensitivity formula.

[0053] in, Let be the accompanying variable. The sensitivity solutions for the remaining constraints are as follows:

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[0057] The solution to the topology optimization model is performed according to the following procedure: (1) Initialize the coordinates of the control points of the Bézier curve, the width variable of the control points, the material properties and the background mesh parameters.

[0058] (2) Construct a discrete horizontal field based on the geometric positional relationship between the background grid nodes and the Bézier curve, and obtain the physical density field based on the discrete horizontal field.

[0059] (3) Generate a two-dimensional body mesh based on the physical density field, and stretch the two-dimensional body mesh in the normal direction to construct a three-dimensional prism mesh; reconstruct the mechanical boundary conditions at the nodes of the three-dimensional prism mesh according to the geometric topological invariance criterion.

[0060] (4) Based on the body mesh and the reconstructed mechanical boundary, assemble the overall stiffness matrix and solve the finite element equations constructed based on the assembled overall stiffness matrix to obtain the displacement field, and then calculate the objective function value and the values ​​of various constraint functions.

[0061] (5) Based on the adjoint variable method and the chain rule, conduct analytical sensitivity analysis, calculate the sensitivity of the objective function and the constraint function, and use the MMA method to iteratively update the design variables.

[0062] (6) Determine whether the change in the current configuration meets the set convergence criteria. If it does, output the topology of the flexible mechanism with the best performance and the manufacturing size constraint. Otherwise, return to step (2) to continue optimization.

[0063] The present invention also provides a topology optimization system for flexible mechanisms based on explicit projection of unstructured mesh geometry. 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 flexible mechanisms based on explicit projection of unstructured mesh geometry as described above.

[0064] 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 flexible mechanism topology optimization method based on explicit projection of unstructured mesh geometry as described above.

[0065] The present invention will be further described in detail below with reference to specific embodiments.

[0066] Please see Figure 1 This invention provides a topology optimization method for flexible mechanisms based on explicit projection of unstructured mesh geometry, which includes the following steps: Step 1: Geometric explicit parametric modeling based on variable-width Bézier curves First, a variable-width Bézier curve is used as the basic geometric framework to describe the topological configuration of the planar reed structure. The coordinates and width of the control points of the Bézier curve are defined as design variables, thus constructing an explicit mapping relationship from the geometric parameter space to the physical density field. The Bézier curve describes the central skeleton curve of a planar reed structure, and its parametric equation is defined as a linear combination of the Bernstein basis functions and the coordinates of the control points:

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[0069] in, Here are the parameters of the Bézier curve, and their range is [value range missing]. , For the first Coordinates of control points ( ), where the curve starts and the end point Keep it fixed to limit the boundary position of the reed. For the first Width of each control point. Summarize the design variable matrix. :

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[0072] Step 2: Discrete level set construction and body-fitted unstructured mesh generation A fixed structured background mesh is laid out within the design domain to assist in the discretization of the geometric configuration. In each optimization iteration, the geometric relationship between the background mesh nodes and the Bézier curves defined in step one is calculated based on the current design variables, and a discretized level set matrix is ​​constructed.

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[0075] in, The interpolation width at the nearest sampling point. Represents the coordinates of any node in the background grid. The coordinates of points equidistantly sampled on the Bézier curve are given, and the number of samples is set appropriately according to the design domain size to ensure the accuracy of curve fitting.

[0076] Further employing a smoothing Heaviside function Mapping the level set matrix to the density matrix:

[0077] Subsequently, a two-dimensional body-fitted mesh is adaptively generated based on the zero isosurface of the level set matrix, and the two-dimensional body-fitted mesh is normal-stretched to construct a three-dimensional prismatic mesh. Since the number, position, and index of nodes in the unstructured mesh change during the iteration process, a search strategy based on geometric topological invariance is required. By setting geometric position tolerances, the boundary nodes are traversed and the set of nodes that satisfy the definitions of fixed constraints, external loads, and virtual loads is selected. Constraints are then applied to the nodes in the set of nodes in the stiffness matrix of the three-dimensional prismatic mesh to achieve automated reconstruction of the mechanical analysis model.

[0078] Step 3: Assemble the overall stiffness matrix and solve the finite element equations. After determining the optimization model and the design variables for the current iteration step, the global stiffness matrix is ​​assembled based on the unstructured triangular prism mesh generated in step two and the material property markers provided by the density matrix obtained in step two. Combined with the reconstructed mechanical boundary conditions from step two, a system of linear equations (i.e., finite element equations) is established:

[0079] in, The global stiffness matrix is ​​assembled based on design variables and body-fitted mesh. Let be the nodal displacement vector to be solved, containing displacement information for all degrees of freedom of the structure under load. This is the equivalent nodal force vector automatically applied to the corresponding boundary nodes based on the external load definition. The displacement field of the current configuration is obtained by solving this finite element equation. This allows us to calculate the specific value of the objective function and provide basic data for subsequent stress calculations and sensitivity analysis.

[0080] Step 4: Construct a multi-constraint topology optimization model (1) Set optimization goals The optimization objective is set as maximizing the output displacement of the flexible mechanism under a specific working load to improve the flexibility of the reed structure. The objective function value is calculated based on the displacement field obtained in step three.

[0081] To design a variable summary vector, This is the virtual unit force vector for the output position.

[0082] (2) Set global volume constraints and minimum solid size constraints To achieve lightweight structure and ensure minimal solid features during fabrication, volume proportion constraints and direct dimensional control based on width variables are applied simultaneously. The volume constraint function is:

[0083] in, For the initial design domain volume, This is the upper limit of the material volume fraction. It is the physical density function obtained by mapping the horizontal ensemble field through a smooth Heaviside function. The minimum solid size constraint limits the width variable of the control points of the Bézier curve. accomplish:

[0084] in, and The method uses preset minimum and maximum control point width thresholds to avoid complex geometric feature recognition and directly ensures the manufacturability of the reed from the parameter definition level.

[0085] (3) Set minimum void size constraint Based on the element density matrix obtained in step two and the displacement field obtained in step three, calculate the void size constraint function value:

[0086] in This is the lower limit of the desired minimum cavity size radius. For a true geometric gap, write:

[0087] in, and It is in accordance with the topological interval threshold Under the premise that the point is closest to the extreme value on the center line, The geometric distance between the two extreme points is denoted as .

[0088] (4) Set stress constraints The VonMises stress of the displacement field unit obtained in step three is calculated, and then aggregated into a global scalar index using the P-norm norm. Finally, the stress constraint function is calculated. : (39)

[0089] Step 5: Derivation of analytical sensitivity based on the adjoint variable method The sensitivity of the objective function to geometric parameters is derived using the adjoint variable method and the chain rule. Substituting the displacement field of the current iteration step calculated in step three, the solution can be obtained:

[0090]

[0091] in, The sensitivity of the stiffness matrix of the body-fitted mesh to the nodal coordinates of the unstructured mesh is given. The partial derivatives of the background horizontal field with respect to the coordinates and width of the control points of the Bézier curve. The specific calculations are as follows:

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[0093] The sensitivity of the volume constraint function, the minimum void size constraint function, and the stress constraint function can be written as follows:

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[0098] Step 6: Variable update and convergence determination based on MMA algorithm Based on the objective and constraint functions calculated in step four, and the sensitivity information of the objective and constraint functions obtained in step six, the MMA method is used to iteratively update the two sets of design variables: the coordinates and width of the control points of the Bézier curve. This update process fully utilizes the sensitivity information, enabling the geometric skeleton to evolve towards maximizing displacement while satisfying various physical and manufacturing constraints. After each update, it automatically determines whether the current configuration meets the set convergence criteria, typically based on the change in the objective function or the degree of satisfaction of the constraint functions. If converged, the optimal, smooth-boundary, and manufacturing-size-constrained complex planar reed topology is directly output, including its final geometric control parameters and the corresponding body-fitted unstructured mesh layout. If not converged, the updated design variables are substituted into the next round of calculation, returning to step two to reconstruct the level set matrix and generate a new body-fitted mesh, until the current configuration satisfies the convergence criteria.

[0099] Numerical Examples like Figures 2 to 6 As shown, the method proposed in this invention is explained in detail through a displacement maximization example of a planar spring structure. To simplify calculations, all numerical values ​​used in the example are dimensionless. The design space and its operating conditions are as follows: Figure 2 As shown, its geometric dimensions are length ,Width ,thickness The left side is completely fixed, and the sum of the z-axis values ​​applied to all points along the z-axis in the middle of the right side is... The force. The design domain is divided into Background mesh, material Young's modulus Poisson's ratio The optimization objective is to satisfy the upper limit of material volume fraction. Under the constraints of stress constraints and minimum void size, achieve the predetermined output point displacement. Maximize.

[0100] Regarding parameter optimization, the width of the control points for the Bézier curve... Restricted to Within this range, the minimum solid size of the reed is directly controlled. Stress constraints employ the P-norm aggregation operator, with its penalty parameter... The initial allowable stress threshold is set to 50 and dynamically adjusted during iterations to ensure the structural strength reliability under large deformations. The influence radius of the void size constraint is set to 2 to eliminate small and unmanufacturable void features generated during configuration evolution.

[0101] During optimization, design variables include the coordinates of the free control points of the Bézier curve and the width of the control points, which are updated iteratively using the MMA optimizer. For example... Figure 3 As shown, the initial configuration is a simple straight line or preset curved skeleton, and the body-fitting mesh generation result is as follows. Figure 4 As shown, the structure evolves automatically based on the sensitivity gradient as iterations proceed. Figure 5 , Figure 6 The final optimized complex planar reed configuration is shown. The generated reed exhibits a smooth, meandering feature, and its boundaries precisely fit the unstructured mesh, effectively avoiding the jagged boundaries and configuration breakage problems commonly found in traditional density methods.

[0102] The results show that the optimized reed meets the requirements. While maintaining volume constraints, the maximum stress was 2173.5676, and the p-norm, after aggregation, was 24.8553, which is below the given threshold of 25, achieving relatively accurate stress control. The displacement of the output point increased from the initial design of 26.9434 to the final 59.2801, nearly doubling. Due to the introduction of the control point width as a design variable, the thickness of the reed exhibited a non-uniform distribution, especially with the automatic increase in width of the stress concentration area at the corners. This reflects the adaptive adjustment made by the structure to balance flexibility requirements with local stress concentration. Furthermore, the minimum void size constraint ensured that there were no processing difficulties in the structure, allowing the results of this example to be directly used for precision wire cutting without secondary manual correction.

[0103] This embodiment demonstrates that the topology optimization method based on explicit projection of unstructured mesh geometry disclosed in this invention can effectively solve the problems of difficult configuration evolution, distorted stress calculation, and difficulty in integrating manufacturing constraints in the design of flexible mechanisms. By combining Bézier curve parameterization with body-fitted mesh analysis, not only is the calculation accuracy improved, but the design freedom of complex planar reed structures is also greatly expanded.

[0104] 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 flexible mechanisms based on explicit projection of unstructured mesh geometry, characterized in that, The steps are as follows: (1) Using the variable-width Bézier curve as the topological configuration skeleton of the flexible mechanism to be optimized, the coordinates of the control points of the Bézier curve and the width of the control points are defined as design variables, and an explicit mapping relationship function from the geometric parameter space to the physical density field is constructed. (2) The explicit mapping relation function is mapped to a discrete level set matrix using a structured background mesh. A two-dimensional body mesh is extracted based on the discrete level set matrix, and the two-dimensional body mesh is stretched in the normal direction to construct a three-dimensional prism mesh. The node mechanical boundary conditions of the three-dimensional prism mesh are automatically reconstructed based on the geometric position tolerance. (3) Based on the three-dimensional prism mesh, with the displacement maximization of the predetermined output point of the flexible mechanism to be optimized as the objective function, a topology optimization model is constructed that includes global material volume constraints, minimum void size constraints, stress constraints, and minimum solid size constraints controlled by the lower limit of the width variable. The topology optimization model is then iteratively solved to obtain the optimal topology configuration of the flexible mechanism to be optimized.

2. The topology optimization method for flexible mechanisms based on explicit projection of unstructured mesh geometry as described in claim 1, characterized in that: The central skeleton curve of the flexible mechanism to be optimized is described by a Bézier curve, and the corresponding parametric equation is defined as a linear combination of the Bernstein basis function and the coordinates of the control points.

3. The topology optimization method for flexible mechanisms based on explicit projection of unstructured mesh geometry as described in claim 1, characterized in that: The shortest distance between the background grid nodes and the Bézier curve is calculated based on the design variables, and a discretized level set matrix is ​​constructed by combining the width of the control points.

4. The topology optimization method for flexible mechanisms based on explicit projection of unstructured mesh geometry as described in claim 1, characterized in that: Set geometric position tolerances, traverse the boundary nodes of the body-fitted mesh and filter out the node sets that satisfy the setting regions of fixed boundaries, external loads and virtual loads, and apply constraints to the nodes in the node sets in the stiffness matrix of the three-dimensional prism mesh.

5. The topology optimization method for flexible mechanisms based on explicit projection of unstructured mesh geometry as described in any one of claims 1-4, characterized in that: The mathematical expression for the topology optimization model is: In the formula, The virtual unit force vector at the predetermined output point. For displacement vectors, For volume constraint functions, The minimum void size constraint function, Let be the stress constraint function. and The preset minimum and maximum control point width thresholds, and These are the upper and lower limits of the control point coordinate range.

6. The topology optimization method for flexible mechanisms based on explicit projection of unstructured mesh geometry as described in claim 5, characterized in that: Volume constraints are implemented by calculating the ratio of the total volume of the solid region to the volume of the initial design domain. In the formula, For the initial design domain volume, This represents the upper limit of the material's volume fraction.

7. The topology optimization method for flexible mechanisms based on explicit projection of unstructured mesh geometry as described in claim 5, characterized in that: By introducing P-norm mapping and power function stress relaxation, spatially distributed local stresses are aggregated into a single global scalar index, thereby achieving effective control over the global stress level.

8. The topology optimization method for flexible mechanisms based on explicit projection of unstructured mesh geometry as described in any one of claims 1-4, characterized in that: Based on the theory of continuum shape sensitivity analysis and the chain rule, we derive the analytical sensitivity formulas of the objective function and each constraint function with respect to geometric design variables.

9. A topology optimization system for flexible mechanisms based on explicit projection of unstructured mesh geometry, 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 flexible mechanisms based on explicit projection of unstructured mesh geometry as described in 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 flexible mechanism topology optimization method based on explicit projection of unstructured mesh geometry as described in any one of claims 1-8.

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