Design optimization method for free curve characteristics

Abstract

The invention discloses a free curve feature design optimization method, which comprises the following steps of: constructing a free curve feature region containing a central bus, and setting the direction and the position of the central bus to obtain a free curve feature region; a two-dimensional coordinate transformation formula is adopted to transform the free curve characteristic area from a global coordinate system to a local coordinate system with a central bus as an X axis, and bending control of the structure in any direction is achieved; a height function is utilized to describe changes of a longitudinal boundary, an implicit modeling mode is adopted to construct a region boundary with a longitudinal coordinate smaller than a height function value, and adjustability of the longitudinal form of a feature region is achieved by controlling parameters of the height function. The method has the advantages that through free curve feature modeling, flexible expression of a complex structure form and precise boundary control are achieved; meanwhile, by taking control parameters as design variables and combining an iterative optimization algorithm, the structural layout is optimized, the performance is improved, the boundary is ensured to be clear, and dual optimization of the structural performance and the design freedom degree is achieved.

Description

Technical Field

[0001] This invention relates to the field of computer-aided design and structural optimization technology, and in particular to a design optimization method for free curve features. Background Technology

[0002] The design optimization of free-form curve features refers to introducing a novel feature unit with high geometric deformation capability into structural optimization design. This feature is expressed as a two-dimensional structure with a controllable shape through mathematical models, implicit modeling, and Boolean operations. Geometric parameters are used as design variables to participate in the topology optimization process. This method can precisely control structural morphological changes, resulting in clear structural boundaries, strong manufacturability, and support for automatic splitting and merging of the structure during optimization, thereby improving the flexibility and performance of the optimization results.

[0003] Existing feature-driven optimization methods mainly employ fixed morphological features, such as circles, ellipses, or their variants. These shapes lack sufficient geometric flexibility and cannot effectively represent complex structural morphologies, especially when representing curved elongated structures (such as beams and flow channels) or micro-channels (such as capillaries). Furthermore, traditional methods struggle to achieve spontaneous splitting and merging of structural regions, limiting the diversity of structural topologies. Optimization results often contain redundant or inefficient regions, making it difficult to achieve optimal structural performance and design freedom. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a design optimization method for free curve features.

[0005] To solve the above-mentioned technical problems, the technical solution of the present invention is: a design optimization method for free curve features, comprising the following steps: S1. Construct a free curve feature region containing the central generatrix. By setting the direction and position of the central generatrix, use a two-dimensional coordinate transformation formula to transform the free curve feature region from the global coordinate system to a local coordinate system with the central generatrix as the X-axis, thereby realizing the bending control of the structure in any direction. S2 uses a height function to describe the change of the vertical boundary, uses an implicit modeling method to construct the boundary of the region with a ordinate smaller than the height function value, and achieves the adjustability of the vertical shape of the feature region by controlling the height function parameter; S3, construct a width function to describe the lateral boundary, and perform a Boolean intersection operation between the width function and the height function through a continuously differentiable KS function to generate a two-dimensional free curve feature profile with a closed boundary; S4 expresses the height and width functions using B-spline interpolation, and constructs a continuous smooth curve through a set of adjustable control points, thereby achieving flexible and variable modeling of the free curve feature contour boundary. S5, set the control parameters of each function in the free curve feature as structural optimization design variables, establish a feature-driven topology optimization model based on the structural performance objective function and constraints, and use the design variables as optimization independent variables; S6 employs an iterative optimization algorithm to update design variables. By gradually adjusting the geometric control parameters of the free curve features, it achieves the splitting, combination, and movement of feature regions during the structural optimization process, resulting in a structural layout with clear boundaries and optimized performance.

[0006] Preferably, step S1 includes: Determine the center generatrix parameters of the free curve feature region. The generatrix is ​​a straight line passing through the center point of the region, and set the direction angle of the generatrix. Establish a local coordinate system with the generatrix as the X-axis, and use a two-dimensional coordinate transformation formula to convert points in the global coordinate system into points in the local coordinate system; Construct the initial shape region of the free curve feature in the local coordinate system, and define the longitudinal and lateral boundaries of the region through the height function and the width function; Based on the local coordinate system and the definition of the central generatrix, the feature region is registered in the structural modeling system, and the generatrix orientation angle, center position coordinates, and boundary control parameters are set as optimization variables.

[0007] Preferably, step S2 includes: Construct a height function in a local coordinate system with the central generatrix as the X-axis to define the longitudinal boundary; An implicit modeling approach is used to define regions with ordinates less than the height function value as the internal regions of the feature. The height function is expressed by interpolation using B-spline functions, and a height curve with continuous derivatives is formed using multiple control points; The ordinates of the control points are incorporated into the structural optimization model as design variables, and the dynamic changes of the longitudinal boundaries of the feature region are achieved through iterative updates using optimization algorithms.

[0008] Preferably, step S3 includes: A width function to describe the lateral boundary is constructed in the local coordinate system, and the range of boundary variation in the lateral direction is defined by implicit modeling. The width function and the height function are intersected by a Boolean operation using a continuously differentiable KS function to generate a closed two-dimensional free curve feature profile. The two-dimensional free curve feature profile is used as a unit component in the structural optimization model, and the synchronous change of the profile boundary in the horizontal and vertical directions is controlled by iterative updating of design variables.

[0009] Preferably, step S4 includes: B-spline interpolation is used to model the height function and the width function respectively, and a continuous smooth boundary representation composed of control points, node vectors and B-spline basis functions is constructed. The position of each control point is set as an optimization design variable, and the control point values ​​are updated during the iteration process through an optimization algorithm, so as to achieve adjustable modeling and morphological evolution control of the free curve feature contour boundary.

[0010] Preferably, step S5 includes: Extract the control point parameters of the height and width functions from the features of the free curve, and use the position of each control point as a structural optimization design variable; A topology optimization model is constructed with the objective function of minimizing structural compliance, and constraints such as upper limit of volume, minimum spacing, and upper and lower limits of control points are set. By inputting control points as optimization variables into the optimization algorithm, the optimization evolution of the structural topology is achieved through sensitivity calculation and iterative updates.

[0011] Preferably, all design variables of the free curve feature are initialized. Each free curve feature is defined by multiple control points. Before optimization, the position of each control point is initially set, and an objective function is established. Compliance is represented as the relationship between displacement and force, and the calculation expression is as follows: In the formula, It refers to structural compliance, i.e., the objective function. It is a displacement vector. It is the stiffness matrix. It is the transpose of the displacement vector. It is an optimized region. It is a volume element; In each iteration of optimization, the design variables are updated using an optimization algorithm. For each free curve feature, the coordinates of its control points are adjusted through the optimization algorithm. The updated control points will drive changes in the structural morphology and gradually optimize the objective function. The update formula for each design variable is as follows: In the formula, and These are all updated control point coordinates. and These are the coordinates of the control points for the current free curve feature, within the height function. Indicates the first The ordinates of the control points, and in the width function... Indicates the first The x-coordinates of the control points It's the learning rate. and It is the objective function For design variables and The partial derivatives; After each optimization iteration, the boundary of the free curve feature of the structure will change accordingly based on the updated design variables; The updated control points will affect the boundary morphology of the feature region. Based on the updated free curve features, the new structural compliance will be calculated. It then compares the result with that of the previous iteration.

[0012] The beneficial effects of this invention are: This invention effectively solves the problem that fixed-form features cannot express complex structural morphologies in traditional optimization methods by introducing a design optimization method based on free-form curve features. By constructing a free-form curve feature region containing a central generatrix and employing two-dimensional coordinate transformation, the free-form curve feature is transferred from the global coordinate system to a local coordinate system with the generatrix as the X-axis, enabling flexible control of the structure's curvature in any direction. Combining implicit modeling of height and width functions with B-spline interpolation, this invention achieves precise control of the structural boundary, clearly expressing complex shapes such as curved elongated structures and micro-channels. Furthermore, by adjusting the parameters of the control points, flexible boundary variations are achieved, enhancing the geometric freedom and flexibility of the design.

[0013] This invention also establishes a feature-driven topology optimization model based on structural performance as the objective function and constraints by setting the control parameters in the free curve features as structural optimization design variables. During the optimization process, an iterative optimization algorithm is used to gradually adjust the control parameters, enabling the structure to spontaneously split, merge, and move during topology optimization, thereby obtaining a structural layout with optimized performance and clear boundaries. This method avoids the occurrence of redundant regions and inefficient structures, effectively improves the design freedom and performance optimization level of the structure, and ensures that the optimization results achieve the best balance between performance and manufacturing feasibility. Attached Figure Description

[0014] Figure 1 This is a flowchart of a design optimization method for free curve features according to the present invention; Figure 2 This is a schematic diagram of a free curve feature according to the present invention; Detailed Implementation

[0015] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings. It should be noted that these descriptions are for the purpose of aiding understanding the present invention, but do not constitute a limitation thereof. Furthermore, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0016] The working principle of this invention: A design optimization method for free curve features, comprising the following steps: S1. Construct a free curve feature region containing the central generatrix. By setting the direction and position of the central generatrix, use a two-dimensional coordinate transformation formula to transform the free curve feature region from the global coordinate system to a local coordinate system with the central generatrix as the X-axis, thereby realizing the bending control of the structure in any direction. This step aims to achieve precise control and expression of structural morphology in any direction, overcoming the shortcomings of traditional feature shapes in expressing bending. This method can be used in feature-driven structural optimization design processes, providing controllable, deformable, and manufacturable initial structural units for subsequent optimization algorithms.

[0017] Determine the center generatrix parameters of the free curve feature region. This center generatrix is ​​the geometrical axis of symmetry of the feature region, defined as a straight line passing through the center point of the region, with a unique direction and length. In practical modeling, this can be achieved by setting the coordinates of the center point. and busbar direction angle Determine the absolute position and orientation of the generatrix in the global coordinate system. The orientation angle is... For the bus and the global The angle between axes. This generatrix is ​​used to express the main extension direction of the structural form. For example, it can be used to simulate common linear structural units such as bridges, flow channels, and truss components. The setting of its direction directly affects the stress distribution path and functional achievement of the structure. Therefore, this generatrix is ​​not only an auxiliary line for the geometric configuration, but also the dominant path for the structural function.

[0018] Establish a local coordinate system based on the central generatrix. This local coordinate system is centered at the center point of the free curve characteristic region. With the origin as the reference point and the direction of the busbar as the reference point... An orthogonal two-dimensional coordinate system is established using axes to ensure that the feature region has a clear reference orientation in its own defined coordinate space. The specific transformation process is as follows: Let any point in the current global coordinate system... Transform it to a point in the local coordinate system. The two-dimensional rigid coordinate transformation formula used is: This transformation matrix is ​​an orthogonal rotation matrix that preserves the scale and shape of the region, only aligning the region with respect to its center point. Through this transformation process, the feature region of any free curve is precisely located along the generatrix direction and obtained a normalized expression in the local coordinate system, greatly facilitating the definition of subsequent parametric functions (such as height and width functions) in the local coordinate system.

[0019] In the local coordinate system, an initial shape region for the free curve feature is constructed. At this point, the feature region is confined within a rectangular area defined by a given height and width function, with the local X-axis as the generatrix. By default, this region can be approximated as a two-dimensional band symmetrically distributed around the generatrix, with its length aligned with the generatrix and its width perpendicular to it. This band is not a rigid boundary but rather a boundary framework that can be deformed using functions. Within this region, the range of variation for the longitudinal and transverse boundaries will be defined using height and width functions, respectively, to further shape the region's form.

[0020] Based on the established local coordinate system and central generatrix definition, the feature region is registered and its functional dependencies are bound in the structural modeling system. The system adds this feature region as an independent control unit to the design variable set and sets its generatrix orientation angle. Center position coordinates The boundary control parameters are set as optimization variables in sync, forming the basic characteristic units in the structural topology evolution process. Especially during topology optimization iterations, when the structure needs local turning or extension, only the direction parameters of the generatrix need to be adjusted to achieve automatic rotation and repositioning of the characteristic region, maintaining the continuity of the structural topology and functional consistency. This approach breaks the limitation of "rigid units cannot turn" in traditional optimization methods, allowing the structure to freely evolve its bending path according to stress or functional requirements, achieving high-performance, manufacturable structural design results.

[0021] In summary, this step, by introducing a geometric transformation strategy based on the busbar, establishes a consistent geometric expression model of "structure-function" in the early stage of structural optimization. This not only enables flexible modeling of free curve features in any direction, but also provides a unified and computable geometric framework for subsequent parameter control, boundary interpolation, and optimization iteration, significantly improving the accuracy and efficiency of structural design.

[0022] S2 uses a height function to describe the change of the vertical boundary, uses an implicit modeling method to construct the boundary of the region with a ordinate smaller than the height function value, and achieves the adjustability of the vertical shape of the feature region by controlling the height function parameter; This step aims to achieve high-degree-of-freedom deformation control of features in local structural optimization by mathematically expressing and dynamically adjusting the longitudinal boundary morphology of the free-form curve feature region. This method employs implicit modeling, coupled with an adjustable function control strategy, providing fundamental support for boundary evolution, feature fusion, and fine-grained structural control in the subsequent topology evolution process.

[0023] A modeling framework for the longitudinal boundary of the free curve feature region is established in a local coordinate system with the central generatrix as the X-axis. This local coordinate system is obtained from the previous step, with its origin located at the center point of the free curve feature region. The X-axis extends along the central generatrix, and the Y-axis is perpendicular to the central generatrix. To express the longitudinal (Y-direction) boundary of this feature region, a mathematical function needs to be constructed to describe the upper and lower boundary values ​​corresponding to each point on the X-axis. This function is the height function, which physically represents the change in the vertical expansion range of the region caused by changes in position along the generatrix. The height function is defined as a continuous curve that varies with the horizontal coordinate (local X-axis), possessing differentiability and adjustability, and is used to determine the morphological distribution of the feature in the longitudinal direction.

[0024] Based on implicit modeling, the height function is defined as a threshold expression for boundary determination. Specifically, a mathematical expression is constructed such that the longitudinal boundary of the free curve feature region is defined by the conditional relationship of whether a point is inside the region. That is, in the local coordinate system, all points whose ordinate value is less than the output of a certain height function are considered to belong to the feature region. This expression does not directly draw the boundary curve, but instead defines the determination rule for the "existence" of the region through a continuous function. This implicit expression has good scalability and computational stability, and is suitable for subsequent optimization differentiation operations and structural continuity control. In structural optimization iterations, this determination rule can serve as a fundamental building block for the objective function and constraint functions, supporting key optimization operations such as gradient calculation and sensitivity analysis.

[0025] A height function model with adjustable parameters is constructed and used as a core component of the variable design variables in the optimization process. To enhance the expressive power of this function for structural geometry, it needs to be constructed as a function with high degrees of freedom, such as through piecewise interpolation, spline functions, or basis function weighting to construct a height function model with local control characteristics and global continuity. In particular, in a preferred embodiment of the invention, a B-spline function is used to express the height function, distributing several control points on the local X-axis and assigning each control point a vertical value to define the maximum boundary value at that location. By adjusting the position and weight of these control points, the shape of the height function curve can be arbitrarily adjusted, thereby generating predictable and controllable boundary changes in the longitudinal direction of the entire free curve feature region. This method does not rely on fixed primitives or preset geometric templates and can naturally adapt to the needs of structural morphology complexity and detail changes, making it particularly suitable for structural optimization tasks sensitive to longitudinal contours, such as capillary channels, branch channels, and load-bearing members.

[0026] The constructed height function parameterized expression is integrated with the structural optimization design process and associated with subsequent modules such as width function, Boolean modeling, and boundary control. In the structural optimization problem, the height function control point parameters of each free curve feature are uniformly included in the design variable set, forming an overall optimization model with the structural performance objective function (such as minimum compliance, volume constraints, etc.). Through continuous iteration of these parameter values ​​by the optimization algorithm, the boundary contour of the feature in the vertical direction will also evolve accordingly, realizing dynamic changes from straight structures to bent structures, and from narrow-domain structures to wide-domain structures. At the same time, thanks to the implicit expression and continuous differentiability of the height function, boundary sensitivity analysis and gradient calculation can be easily performed in each round of optimization, improving optimization efficiency. Ultimately, the obtained free curve feature will present a clear, continuous, and manufacturable boundary effect in the vertical morphology, solving the technical shortcomings of traditional feature-driven methods that cannot adapt to structural functional requirements due to shape rigidity, and meeting the dual goals of improving structural performance and manufacturing adaptability.

[0027] In summary, this step constructs a height function expression mechanism based on a local coordinate system, which implicitly describes the boundary change process of free curve features in the longitudinal direction. It has high expressive power, high control accuracy and good scalability, and provides key support for subsequent feature contour generation, optimized variable definition and structural performance improvement, demonstrating strong innovation and engineering practical value.

[0028] S3, construct a width function to describe the lateral boundary, and perform a Boolean intersection operation between the width function and the height function through a continuously differentiable KS function to generate a two-dimensional free curve feature profile with a closed boundary; This step involves constructing a lateral boundary based on a width function and generating a two-dimensional free curve feature profile through continuous and differentiable Boolean intersection with a height function. The aim is to achieve a closed, smooth, and deformable expression of structural boundaries, overcoming problems such as discontinuous structural connections and rigid boundary splicing in traditional geometric construction, and providing strong geometric support for high-precision optimization of complex structures.

[0029] A lateral boundary description framework is established in the local coordinate system of the free-form curve feature, and a width function is defined for the structural boundary changes in the lateral direction. This local coordinate system has the center of the feature region as the origin, the generatrix direction as the X-axis, and the direction perpendicular to the generatrix as the Y-axis. In contrast to the height function, which describes the changes in the upper and lower boundaries in the Y-direction, the width function defines the lateral boundary constraints of the structure at each local X-coordinate position. Specifically, the width function describes the criterion for determining whether a local Y-coordinate falls within the feature region, i.e., how the lateral boundary contracts, expands, or fluctuates with the position in the X-direction. Similar to the height function, the width function is also constructed using implicit modeling, enabling flexible expression of various lateral boundary deformation requirements, such as symmetrical and asymmetrical, linear and nonlinear deformations. It is particularly suitable for representing typical geometric forms such as channel-like structures, symmetrical structures, and edge transition zones.

[0030] The width function constructed above is combined with the established height function to construct a complete closed boundary for the two-dimensional free curve feature region. In traditional explicit geometric modeling methods, boundaries in two directions are often joined using graphical Boolean operations, such as intersection or union, to obtain the target region. However, such methods have limitations of being discontinuous and non-differentiable, making it difficult to meet the requirements for continuous and differentiable boundaries in structural optimization, thus limiting the accuracy and stability of sensitivity analysis. To address this, this implementation adopts a continuous and differentiable functional Boolean combination strategy—introducing the KS function (Kreisselmeier-Steinhauser function) to achieve an "intersection-style" fusion of the height and width functions. As a differentiable and smoothly transitioning mathematical tool, the KS function can combine multiple implicit functions into a single region expression, ensuring that the generated boundary has both logical combination relationships and the ability to perform numerical differentiation and deformation propagation. This design completely solves the differentiation difficulties and structural instability problems caused by boundary discontinuities in structural optimization from a geometric expression perspective.

[0031] An implicit representation model for the feature contour of a two-dimensional free curve is defined by combining the height and width functions of the KS function, and this model is used for the precise definition of feature unit boundaries. Specifically, this contour model no longer relies on the explicit trajectory drawing of the contour line, but instead uses a conditional function value judgment rule to determine whether any point within the region belongs to the feature structure region. This implicit representation not only has a compact mathematical structure but can also be directly embedded into the objective function and constraints of structural optimization algorithms, forming a tight coupling between structural performance and geometric morphology. Furthermore, the continuity and smoothness of the KS function enable natural transitions between multiple features. In applications such as multi-feature fusion and multi-branch structure construction, no additional processing of connection methods is required to achieve geometric continuity and engineering consistency. This is difficult to achieve in traditional optimization models, demonstrating the high degree of innovation of this method at the structural construction expression level.

[0032] The obtained closed two-dimensional feature contour model is embedded into the structural optimization modeling process as a basic component for generating unit-level structural morphology, participating in the topology optimization evolution process of the overall structure. In specific applications, the shape of this closed contour model is controlled by width and height functions, which are themselves defined by adjustable control parameters. This means that the feature contour can be freely adjusted as design variables evolve during the optimization process. In each iteration, the structural optimization algorithm updates these design variables based on the current structural performance objectives (such as maximum stiffness and minimum compliance) and constraints such as volume and stress, thereby causing synchronous changes in the feature boundary in the X and Y directions. Since the contour boundary is closed and continuously differentiable, the entire structure maintains geometric integrity and manufacturability during the evolution process, greatly improving the solution stability and design practicality of the optimization model. Ultimately, the resulting optimized structure not only has clear boundaries and a reasonable shape, but also supports post-processing modeling and precision manufacturing, avoiding technical problems such as structural fracture, boundary ambiguity, and manufacturing difficulties in traditional methods. This fully demonstrates the engineering value and technological advancement of this method in the field of structural modeling and optimization.

[0033] In summary, this step, by constructing a Boolean combination mechanism between a continuously adjustable width function and a height function, achieves a complete geometric contour expression of free curve features while maintaining the closure and differentiability of the structural boundary. This provides core geometric support for achieving high-quality structural topology optimization and has broad application prospects and promotional value.

[0034] S4 expresses the height and width functions using B-spline interpolation, and constructs a continuous smooth curve through a set of adjustable control points, thereby achieving flexible and variable modeling of the free curve feature contour boundary. This step applies the B-spline interpolation method to the modeling schemes of the height and width functions, aiming to construct continuous, smooth, and flexibly deformable free-form curve boundaries through a set of controllable geometric parameters. This achieves a structural feature modeling process with high degrees of freedom, strong adaptability, and good manufacturability. Based on the local adjustability and global continuity of spline curves, this method introduces interpolation control points and node vectors to construct spline functions, providing an adjustable design variable space for feature-driven structural optimization.

[0035] In the local coordinate system of the feature contour, the distribution area of ​​interpolation points and the number of interpolation nodes for the height and width functions are determined. Taking the height function as an example, it is defined in the X-axis direction of the free curve feature local coordinate system, representing the maximum range of the longitudinal boundary corresponding to each horizontal coordinate position. For B-spline interpolation modeling, several control point positions need to be set on the X-axis as the node support area of ​​the spline function. The horizontal coordinate positions of these control points can be evenly distributed or non-uniformly arranged according to the structural stress or boundary change trend; the vertical coordinate is the function value at that position, controlling the height of the boundary at that point. Similarly, the width function is defined in the Y-axis direction to describe the lateral boundary extension range of each longitudinal position, and also uses a set of control points to complete the interpolation support. The number of nodes will directly affect the complexity and degree of freedom of the interpolation curve. The more nodes, the stronger the curve's expressive power; the fewer nodes, the simpler the expression and the fewer control variables, suitable for simple structural regions. The number of control points can be set according to the feature complexity and the granularity of structural optimization design.

[0036] A B-spline function is constructed and interpolated to generate height and width function curves with good mathematical properties. In this embodiment, cubic B-splines are preferably used as the basic interpolation function, as they have second-order continuous derivatives, ensuring smooth transitions at all connection points. Specifically, the height function can be represented as a linear combination of several basis functions, where each basis function is associated with a control point, forming the shape support of the curve in the corresponding segment. The interpolation expression is: the height function is a weighted sum of a set of B-spline basis functions, where the ordinate values ​​of the control points are used as weights. The width function is expressed similarly, except that its direction of action is along the Y-axis. By controlling the longitudinal (or transverse) values ​​of each control point, the boundary shape can be customized throughout the feature region. This modeling method avoids the parameter rigidity and adjustment difficulties brought about by traditional function forms, greatly improving the modeling flexibility.

[0037] Besides B-spline functions, height and width functions can also be expressed through interpolation using quadratic and cubic splines. Quadratic splines possess first-order derivative continuity, making them suitable for structural modeling scenarios with relatively gentle boundary changes. They offer high computational efficiency and are easy to implement. Cubic splines possess second-order derivative continuity, better preserving the smoothness of the boundary and the natural transition of the structural curve, making them suitable for topology optimization problems with high requirements for geometric continuity. These spline methods can all flexibly adjust the structural boundary by setting interpolation nodes and control points, meeting the basic requirements of controllability and differentiability for free-form curve feature modeling.

[0038] By constructing node vectors and performing normalization transformations, the B-spline function is expressed within a unified parameter range, enhancing the model's controllability and stability. To ensure a unified parameter domain for the spline curves during computation, this implementation transforms the interpolation process into a normalized parameter space. By mapping the horizontal (or vertical) coordinates to a standard interval, such as [0,1], node vectors are constructed, and the spline order is defined, completing the initialization and normalization of the spline basis functions. These node vectors can be non-uniform vectors; dense placement of nodes in the initial, middle, or final segments improves the curve's representation resolution, thus adapting to regions sensitive to boundary changes. Based on this, the entire B-spline function is defined as a function of standardized parameters, facilitating the unified handling of boundary control logic for various features in subsequent structural optimization algorithms. Furthermore, the node vectors, as one of the input parameters, can be dynamically adjusted according to actual optimization accuracy requirements, exhibiting excellent scalability.

[0039] This invention incorporates the height and width functions expressed by B-spline interpolation into the structural optimization design variable system. By moving control points, continuous morphological adjustments to the feature boundaries are achieved, supporting geometric changes in the structure during topological evolution. In this invention, the position of each control point is an optimization design variable. In each iteration, the optimization algorithm adjusts the control point values ​​according to the structural performance target, thereby guiding local or global changes in the B-spline curve morphology, allowing the feature boundaries to naturally extend, contract, bend, or deform. In structural topology optimization, this free boundary control mechanism provides a highly free design space for optimization, enabling the optimization results to better match the actual structural performance requirements while maintaining boundary continuity and geometric differentiability, thus enhancing the structure's manufacturing adaptability and engineering practicality. Ultimately, the free curve features generated through the above modeling strategy not only possess mathematical completeness and physical expressiveness but also exhibit good adaptability, manufacturability, and structural integrity in practical engineering, providing a unified geometric modeling platform for multiphysics structural optimization.

[0040] In summary, this step, by employing B-spline interpolation to model the height and width functions respectively, constructs highly expressive boundary curves based on control points, basis functions, and node vectors. This not only enhances the modeling freedom of free curve features but also demonstrates significant advantages in optimized control, geometric continuity, and manufacturing friendliness. It is a key step in achieving geometric control and performance improvement of complex structures.

[0041] S5, set the control parameters of each function in the free curve feature as structural optimization design variables, establish a feature-driven topology optimization model based on the structural performance objective function and constraints, and use the design variables as optimization independent variables;

[0042] This step combines the structural performance objectives with geometric controllability by efficiently parameterizing the structural boundary morphology, constructing a structural topology optimization system with a small number of variables, strong expressive power, and high computational efficiency. This method breaks through the traditional modeling approach of density methods or level set methods that rely on all-field variables. By controlling the overall structural morphology changes with a small number of design variables, it maximizes structural performance while ensuring clear boundaries and manufacturability.

[0043] All control parameters in the free-curve feature model are extracted and explicitly defined as design variables in structural optimization. This free-curve feature consists of two main functions: a height function and a width function, which control the longitudinal and lateral boundary variations of the structure, respectively. In previous embodiments, these two functions were expressed using B-splines, with each function defined by several control points and node vectors. The positional parameters of the control points, especially their ordinate or abscissa values, directly determine the shape of the feature boundary curve. In the optimization modeling stage, this invention explicitly defines the relevant values ​​of each control point as design variables, forming a set of design variable vectors for structural optimization. For example, in a structural layout containing four free-curve features, if each feature's height and width functions are represented by five control points, the total number of design variables is forty. Compared to the traditional density method, which often requires tens of thousands of element density values ​​as variables, this method significantly reduces the dimensionality of variables while retaining the rich expressive power of the structural morphology, significantly improving the efficiency of optimization modeling and solution.

[0044] A topology optimization problem is constructed with structural performance as the objective function, and the aforementioned design variables are embedded in the model as optimization independent variables. The performance objective function can be flexibly defined according to different design requirements, such as minimizing compliance (i.e., maximizing structural stiffness), maximizing stress uniformity, maximizing thermal diffusion efficiency, and minimizing fluid channel resistance. In this preferred embodiment, "minimizing structural compliance" is the optimization objective, and the optimization region is a two-dimensional rectangular region, including several adjustable free curve features. The response of the entire region is simulated by a finite element solver. The objective function is typically expressed as an integral of displacement and load, and the structural compliance response is obtained through the structural finite element model. The design variables, i.e., the geometric control parameters of the free curves, will generate sensitivity information through this objective function and participate in the optimization iteration.

[0045] In structural optimization, geometric, physical, and manufacturing constraints are set to ensure a balance between structural performance and engineering feasibility. Common constraints in typical topology optimization problems include volume constraints, boundary integrity constraints, geometric manufacturability constraints, and minimum feature size constraints. For example, in this invention, the upper limit of the total material volume of the structure is set to 50% of the design area, ensuring that the final generated structure meets performance requirements without excessive redundancy. Furthermore, minimum spacing limits can be set between features to prevent features from being too close together, leading to manufacturing difficulties. Upper and lower limits are also set for the control point positions of the width and height function interpolation curves to avoid excessive curve fluctuations that could cause boundary discontinuities or structural unfeasibility. These constraints can be input into the optimization algorithm along with the objective function during the optimization modeling process, forming a multi-objective constraint solution system.

[0046] A suitable structural optimization algorithm is selected, and the design variables are used as independent variables for iterative updates to achieve the evolution and optimization of the structural topology. In this embodiment, the Moving Asymptote Method (MMA) is used as the optimization algorithm, and the objective function, constraint function, and their corresponding sensitivities are input into the solver. Since the height and width functions are implicitly expressed continuous curves, their derivatives with respect to the design variables can be efficiently calculated using the chain rule, thus the overall model has good differentiability and numerical stability. In each iteration, the optimization algorithm updates the control point parameters based on the current structural performance feedback, thereby driving the adjustment of the free curve features in terms of geometry, realizing the movement, deformation, contraction, and even splitting and combination of the structure. The optimization process continues until the convergence condition is met, and the final output structure has clear boundaries, high performance indicators, and manufacturability, while using fewer design variables and significantly reducing computational overhead.

[0047] In summary, this implementation method sets all geometric control parameters in the free curve features as structural optimization design variables, and constructs an efficient and compact topology optimization model in conjunction with the objective function and constraints. This breaks through the bottlenecks of traditional methods in terms of variable dimension, boundary continuity, and manufacturing adaptability, and provides a new high-performance, low-complexity modeling paradigm for multi-physics, multi-objective structural optimization problems. It has significant theoretical innovation and engineering application value.

[0048] S6 employs an iterative optimization algorithm to update design variables. By gradually adjusting the geometric control parameters of the free curve features, it achieves the splitting, combination, and movement of feature regions during the structural optimization process, resulting in a structural layout with clear boundaries and optimized performance. Initialize all design variables for the free curve feature, namely the control points in the height and width functions of the feature region. Each free curve feature is defined by multiple control points, whose coordinates are design variables. Before optimization begins, the position of each control point is initially set; these initial values ​​can be determined by uniform distribution or according to a preset design scheme. Establish the objective function. In structural optimization problems, a common objective is to minimize the compliance of the structure and maximize its stiffness. Compliance is represented by the relationship between displacement and force, and its calculation expression is as follows: In the formula, Structural compliance, or the objective function, refers to the degree of structural deformation, typically used to represent structural stiffness. A lower compliance value indicates less deformation and greater stiffness. Since compliance is the objective function, the optimization algorithm aims to minimize this value. In the optimization process, minimizing compliance means maximizing structural stiffness, thereby improving structural performance. It is a displacement vector, a displacement vector This represents the displacement of each node in the structure under the action of external forces. For each node, It is a vector containing its displacements in three directions (X, Y, Z). For a two-dimensional problem, Typically a two-dimensional vector, representing the planar displacement of the structure. The stiffness matrix is ​​a matrix that represents the stiffness of a structure. It describes the relationship between the deformation and force of a structure under external forces. The stiffness matrix is ​​the core parameter describing structural stiffness; a larger value indicates that the structure is more difficult to deform, meaning it has greater stiffness and lower compliance. During optimization, the stiffness matrix is ​​immutable; it depends on the geometry and material properties of the structure. It is the transpose of the displacement vector, representing the displacement vector. The transpose of means converting the displacement vector from a column vector to a row vector. Matrix multiplication requires a combination of row and column vectors. It is an optimized region. It is a volume element, representing a small volume element used in integration calculations. For two-dimensional structures, It is an area element; for three-dimensional structures, it is a volume element. In the integral of compliance, It is used to calculate the weighted sum of displacement and stiffness over the entire volume or area of ​​the structure; In this objective function, minimizing compliance means the optimization goal is to reduce structural deformation, thereby increasing its stiffness. In this step, the initial values ​​of all design variables (i.e., control point coordinates) are input into the objective function and become the basis for subsequent optimization calculations.

[0049] In each iteration of optimization, an optimization algorithm (such as Moving Asymptote Method, MMA) is used to update the design variables. For each free curve feature, the coordinates of its control points are adjusted by the optimization algorithm. The updated control points will drive changes in the structural morphology and gradually optimize the objective function. The optimization algorithm adjusts the design variables by minimizing the objective function. For each design variable, the update formula is as follows: In the formula, and These are the updated control point coordinates, representing the new positions adjusted by the optimization algorithm. Through iterative optimization, the algorithm adjusts the design variables based on the derivative information of the objective function. and The location ultimately yields new design variables. and This makes the structure behave more optimally in the objective function. and These are the coordinates of the control points for the current free curve feature, within the height function. Indicates the first The ordinates of the control points, and in the width function... Indicates the first The x-coordinates of the control points The learning rate controls the step size in each optimization update. It determines the speed or magnitude of control point updates. In optimization algorithms, the learning rate controls the magnitude of changes in the design variables (control point coordinates) in each iteration. If the learning rate is too large, it may cause oscillations during the optimization process, preventing convergence to the optimal solution; if the learning rate is too small, the convergence speed is too slow, resulting in low optimization efficiency. Therefore, choosing a reasonable learning rate is crucial to ensuring the success of the optimization algorithm. and It is the objective function For design variables and The partial derivatives of the objective function represent the partial derivatives of the objective function. Sensitivity to the coordinates of each control point: These gradients tell us how sensitive the objective function is to changes in each design variable (i.e., the coordinates of each control point). Specifically, partial derivatives reflect the rate of change of the objective function with respect to the control point location, i.e., how adjusting the design variables can reduce (or increase) the value of the objective function, thereby improving structural performance. For example, Refers to the objective function For the ordinate of the control points Sensitivity to change Refers to the objective function x-coordinate of control points Sensitivity to change. The optimization algorithm uses these gradient values ​​to determine the direction and magnitude of adjusting the control points; In this step, the updated design variables (i.e., the new control point coordinates) and and This will further influence the evolution of characteristic morphology in the next step.

[0050] After each optimization iteration, based on the updated design variables (i.e., the new coordinates of the control points) and The boundaries of the free curve features of the structure will change accordingly. These changes are manifested as the splitting, combination or movement of the feature regions, depending on the adjustment of the design variables. The updated control points will affect the boundary morphology of the feature region. For example, when the control points are adjusted to new positions, the feature boundary will change, thus affecting the overall layout of the structure. Based on the updated free curve feature, the new structural compliance is calculated. It then compares the result with that of the previous iteration.

[0051] Through multiple iterations, the optimization algorithm gradually brings the feature region towards its optimal shape, satisfying structural performance and geometric constraints. During the optimization process, the structure's shape undergoes multiple iterations, with each optimization update based on new design variable values, thereby achieving optimization and adjustment of the structural topology.

[0052] This invention effectively solves the problem that fixed-form features cannot express complex structural morphologies in traditional optimization methods by introducing a design optimization method based on free-form curve features. By constructing a free-form curve feature region containing a central generatrix and employing two-dimensional coordinate transformation, the free-form curve feature is transferred from the global coordinate system to a local coordinate system with the generatrix as the X-axis, enabling flexible control of the structure's curvature in any direction. Combining implicit modeling of height and width functions with B-spline interpolation, this invention achieves precise control of the structural boundary, clearly expressing complex shapes such as curved elongated structures and micro-channels. Furthermore, by adjusting the parameters of the control points, flexible boundary variations are achieved, enhancing the geometric freedom and flexibility of the design.

[0053] This invention also establishes a feature-driven topology optimization model based on structural performance as the objective function and constraints by setting the control parameters in the free curve features as structural optimization design variables. During the optimization process, an iterative optimization algorithm is used to gradually adjust the control parameters, enabling the structure to spontaneously split, merge, and move during topology optimization, thereby obtaining a structural layout with optimized performance and clear boundaries. This method avoids the occurrence of redundant regions and inefficient structures, effectively improves the design freedom and performance optimization level of the structure, and ensures that the optimization results achieve the best balance between performance and manufacturing feasibility.

[0054] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings, but the present invention is not limited to the described embodiments. For those skilled in the art, various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention, and these variations still fall within the protection scope of the present invention.

Claims

1. A method for designing and optimizing free curve features, characterized in that, Includes the following steps: S1. Construct a free curve feature region containing the central generatrix. By setting the direction and position of the central generatrix, use a two-dimensional coordinate transformation formula to transform the free curve feature region from the global coordinate system to a local coordinate system with the central generatrix as the X-axis, thereby realizing the bending control of the structure in any direction. S2 uses a height function to describe the change of the vertical boundary, uses an implicit modeling method to construct the boundary of the region with a ordinate smaller than the height function value, and achieves the adjustability of the vertical shape of the feature region by controlling the height function parameter; S3, construct a width function to describe the lateral boundary, and perform a Boolean intersection operation between the width function and the height function through a continuously differentiable KS function to generate a two-dimensional free curve feature profile with a closed boundary; S4 expresses the height and width functions using B-spline interpolation, and constructs a continuous smooth curve through a set of adjustable control points, thereby achieving flexible and variable modeling of the free curve feature contour boundary. S5, set the control parameters of each function in the free curve feature as structural optimization design variables, establish a feature-driven topology optimization model based on the structural performance objective function and constraints, and use the design variables as optimization independent variables; S6 employs an iterative optimization algorithm to update design variables. By gradually adjusting the geometric control parameters of the free curve features, it achieves the splitting, combination, and movement of feature regions during the structural optimization process, resulting in a structural layout with clear boundaries and optimized performance.

2. The design optimization method for a free curve feature according to claim 1, characterized in that, Step S1 includes: Determine the center generatrix parameters of the free curve feature region. The generatrix is ​​a straight line passing through the center point of the region, and set the direction angle of the generatrix. Establish a local coordinate system with the generatrix as the X-axis, and use a two-dimensional coordinate transformation formula to convert points in the global coordinate system into points in the local coordinate system; Construct the initial shape region of the free curve feature in the local coordinate system, and define the longitudinal and lateral boundaries of the region through the height function and the width function; Based on the local coordinate system and the definition of the central generatrix, the feature region is registered in the structural modeling system, and the generatrix orientation angle, center position coordinates, and boundary control parameters are set as optimization variables.

3. The design optimization method for free curve features according to claim 1, characterized in that, Step S2 includes: Construct a height function in a local coordinate system with the central generatrix as the X-axis to define the longitudinal boundary; An implicit modeling approach is used to define regions with ordinates less than the height function value as the internal regions of the feature. The height function is expressed by interpolation using B-spline functions, and a height curve with continuous derivatives is formed using multiple control points; The ordinates of the control points are incorporated into the structural optimization model as design variables, and the dynamic changes of the longitudinal boundaries of the feature region are achieved through iterative updates using optimization algorithms.

4. The design optimization method for a free curve feature according to claim 1, characterized in that, Step S3 includes: A width function to describe the lateral boundary is constructed in the local coordinate system, and the range of boundary variation in the lateral direction is defined by implicit modeling. The width function and the height function are intersected by a Boolean operation using a continuously differentiable KS function to generate a closed two-dimensional free curve feature profile. The two-dimensional free curve feature profile is used as a unit component in the structural optimization model, and the synchronous change of the profile boundary in the horizontal and vertical directions is controlled by iterative updating of design variables.

5. The design optimization method for a free curve feature according to claim 1, characterized in that, Step S4 includes: B-spline interpolation is used to model the height function and the width function respectively, and a continuous smooth boundary representation composed of control points, node vectors and B-spline basis functions is constructed. The position of each control point is set as an optimization design variable, and the control point values ​​are updated during the iteration process through an optimization algorithm, so as to achieve adjustable modeling and morphological evolution control of the free curve feature contour boundary.

6. The design optimization method for a free curve feature according to claim 1, characterized in that, Step S5 includes: Extract the control point parameters of the height and width functions from the features of the free curve, and use the position of each control point as a structural optimization design variable; A topology optimization model is constructed with the objective function of minimizing structural compliance, and constraints such as upper limit of volume, minimum spacing, and upper and lower limits of control points are set. By inputting control points as optimization variables into the optimization algorithm, the optimization evolution of the structural topology is achieved through sensitivity calculation and iterative updates.

7. The design optimization method for a free curve feature according to claim 1, characterized in that, Initialize all design variables for the free curve feature. Each free curve feature is defined by multiple control points. Before starting optimization, the position of each control point is initially set, and an objective function is established. Compliance is represented as the relationship between displacement and force, and the calculation expression is as follows: In the formula, It refers to structural compliance, i.e., the objective function. It is a displacement vector. It is the stiffness matrix. It is the transpose of the displacement vector. It is an optimized region. It is a volume element; In each iteration of optimization, the design variables are updated using an optimization algorithm. For each free curve feature, the coordinates of its control points are adjusted through the optimization algorithm. The updated control points will drive changes in the structural morphology and gradually optimize the objective function. The update formula for each design variable is as follows: In the formula, and These are all updated control point coordinates. and These are the coordinates of the control points for the current free curve feature, within the height function. Indicates the first The ordinates of the control points, and in the width function... Indicates the first The x-coordinates of the control points It's the learning rate. and It is the objective function For design variables and The partial derivatives; After each optimization iteration, the boundary of the free curve feature of the structure will change accordingly based on the updated design variables; The updated control points will affect the boundary morphology of the feature region. Based on the updated free curve features, the new structural compliance will be calculated. It then compares the result with that of the previous iteration.

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