Method for constructing topology description function
By constructing a continuously differentiable topology description function, the problem of discontinuous derivatives in thickness-varying spline components is solved, achieving efficient topology optimization that is suitable for complex engineering designs.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-12-31
- Publication Date
- 2026-04-14
AI Technical Summary
Existing topology description methods suffer from derivative discontinuities in spline components with continuously varying thicknesses, leading to low efficiency in optimization algorithms and difficulty in handling complex engineering problems.
A continuously differentiable topological description function is constructed. By defining the centerline and thickness function of the spline component and combining it with the inverse world transformation expression, a globally continuously differentiable topological description function is generated to determine the affiliation of material points.
It achieves accurate geometric description of variable thickness spline components and can efficiently and stably calculate derivatives, which promotes its application in gradient optimization and improves optimization efficiency.
Smart Images
Figure CN121859481A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of structural topology optimization technology, and in particular to a method for constructing a topology description function. Background Technology
[0002] In the field of structural topology optimization, explicit methods define design boundaries through parametric geometric models, and the design results can be directly used for manufacturing. These methods have achieved good results in describing structures such as beams and shells.
[0003] However, existing topology description methods face significant challenges when the design object is extended to spline components with continuously varying thickness. Existing techniques typically define thickness variations in a piecewise or discrete manner, resulting in derivative discontinuities in the topology description function near thickness control points.
[0004] This mathematical deficiency severely limits the efficiency of optimization algorithms. Modern topology optimization relies on efficient gradient-based algorithms, but existing describing functions cannot provide stable and usable gradient information, forcing optimization to employ derivative-free algorithms. These algorithms converge slowly with large variable spaces, have high computational costs, and are difficult to handle complex engineering problems.
[0005] Therefore, there is an urgent need in this field for a new method for constructing topological description functions that can solve the problem of low optimization efficiency caused by the non-differentiability of functions. Summary of the Invention
[0006] Based on this, it is necessary to propose a method for constructing topology description functions to address the above problems, aiming to solve the problem that the constructed topology description functions have low effectiveness in optimization.
[0007] In a first aspect, embodiments of the present invention provide a method for constructing a topological description function, the method comprising: Obtain a first function and a second function; wherein, the first function uses parameter coordinates as variables to determine the first coordinates of the point in the world coordinate system corresponding to the parameter coordinates for determining the center line of the spline component of the variable thickness spline component in the preset design domain; the second function uses the parameter coordinates as variables to determine the thickness of the variable thickness spline component at the parameter coordinates; both the first function and the second function are continuously differentiable mathematical operation functions. Based on the first function and the second function, a third function is constructed; wherein, the third function is used to determine the target ordinate of the material point in the local coordinate system based on the second coordinate of the material point in the world coordinate system and the corresponding target parameter coordinate; wherein, the target parameter coordinate is the parameter coordinate corresponding to the projection point of the material point on the center line of the spline component; Based on the third function, construct the topology description function.
[0008] In some embodiments, the above method includes at least one of the following: The expression for the first function is as follows: , ; Where u represents the parameter coordinates, This represents the first function. , Let represent the functions used to determine the x-coordinate and y-coordinate in the first coordinate system, respectively. for B-spline basis functions Let i be the third coordinate of the i-th control point in the world coordinate system. The x and y coordinates in the third coordinate system are respectively represented. The control points are used to determine the center line of the spline component. n+1 is the preset number of control points. The expression for the second function is as follows: ; Where u represents the parameter coordinates, The thickness value corresponding to the parameter coordinates, where m represents the preset value. Represents the predefined polynomial coefficients; The expression for the third function is as follows: ; in, This represents the target ordinate of the material point. These represent the x-coordinate and y-coordinate of the second coordinate system, respectively. Indicates the coordinates of the target parameter. Let x be the derivative component of the x-axis in the world coordinate system, which varies with the target parameter coordinates. This is the derivative component of the y-axis in the world coordinate system, which varies with the target parameter coordinates.
[0009] In some embodiments, the above and The expression is as follows: ; ; in, Let i+1 be the fourth coordinate of the control point in the world coordinate system. This represents the parameter coordinate value corresponding to the (i+k+1)th node of the B-spline curve. This represents the parameter coordinate value corresponding to the (i+1)th node of the B-spline curve.
[0010] In some embodiments, the expression for the above topology description function is as follows: ; in, This represents the topology description function. Let p represent the third function, where p represents a preset even number greater than zero.
[0011] In some embodiments, constructing a topology description function based on the third function includes: constructing a topology description function based on the third function and the fourth function; wherein the fourth function uses the second coordinate as a variable to determine the target parameter coordinates.
[0012] In some embodiments, the above method includes: constructing an inverse world transformation expression; the inverse world transformation expression is as follows: ; ; Where u represents the parameter coordinates, These represent the x-coordinate and y-coordinate of the parameter coordinates in the world coordinate system, respectively. , These represent the coordinates of the parameter coordinates in the world coordinate system, and the x-coordinate and y-coordinate in the local coordinate system, respectively. Represents the scaling matrix. Represents the rotation matrix. Represents the translation matrix; Will , , Substituting the expression into the inverse world transformation expression, we obtain the simplified inverse world transformation expression as follows: ; in, This represents the first function. These are functions used to determine the x-coordinate and y-coordinate corresponding to the first coordinate, respectively. Let x be the derivative component of the x-axis in the world coordinate system, which varies with the parametric coordinates. Let be the derivative component of the y-axis in the world coordinate system, varying with the parametric coordinates; let , and u= , = as well as = This yields the fourth function; These represent the x-coordinate and y-coordinate of the second coordinate system, respectively. The coordinates of the target parameter are represented.
[0013] In some embodiments, the scaling matrix described above is as follows: ; in, Represents the scaling matrix, This represents the second function; The rotation matrix is shown below: ; ; ; in, Represents the rotation matrix, Indicates the rotation angle. Let x be the derivative component of the x-axis in the world coordinate system, which varies with the parametric coordinates. The derivative component of the y-axis in the world coordinate system, which varies with the parametric coordinates; The translation matrix is shown below: ; in, Denotes the translation matrix, This represents the first function. These are functions used to determine the x-coordinate and y-coordinate corresponding to the first coordinate, respectively.
[0014] In some embodiments, the expression for the fourth function described above is as follows: ; ; ; ; ; ; in, These represent the x-coordinate and y-coordinate of the second coordinate system, respectively. The coordinates of the target parameter are represented.
[0015] In some embodiments, the above method includes: defining basic line elements; the basic line elements are line segments located on the y-axis in the local coordinate system, with a preset length in both the positive and negative directions of the y-axis; for each parameter coordinate, determining a scaling matrix corresponding to the parameter coordinate based on the parameter coordinate and the second function, determining a rotation matrix corresponding to the parameter coordinate based on the parameter coordinate, and determining a translation matrix corresponding to the parameter coordinate based on the parameter coordinate and the first function; mapping the basic line elements in the local coordinate system to the world coordinate system through scaling, rotation, and translation operations according to the scaling matrix, rotation matrix, and translation matrix; determining the trajectory swept by the basic line elements during the mapping to the world coordinate system through scaling, rotation, and translation operations as the geometric region of the variable thickness spline component; wherein, the rotation angle in the rotation matrix makes the transformed local coordinate system... The axial direction is consistent with the unit tangent direction at the parameter coordinate on the center line of the spline component in the world coordinate system.
[0016] In some embodiments, after constructing the topology description function as described above, the method includes: obtaining the target coordinates of the target material point in the design domain in the world coordinate system; substituting the target coordinates into the topology description function to obtain a target value calculated by the topology description function; and when the target value is greater than... In the case where the target material point is located inside the variable thickness spline assembly; when the target value is equal to In the case where the target material point is located at the boundary of the variable thickness spline assembly; when the target value is less than In this case, it is determined that the target material point is located outside the variable thickness spline assembly.
[0017] Secondly, embodiments of the present invention provide an apparatus for constructing a topological description function, the apparatus comprising: The function acquisition module is used to acquire a first function and a second function; wherein, the first function uses parameter coordinates as variables to determine the first coordinates of the point in the world coordinate system corresponding to the parameter coordinates for determining the center line of the spline component of the variable thickness spline component in the preset design domain; the second function uses the parameter coordinates as variables to determine the thickness of the variable thickness spline component at the parameter coordinates; both the first function and the second function are continuously differentiable mathematical operation functions. The first construction module is used to construct a third function based on the first function and the second function; wherein, the third function is used to determine the target ordinate of the material point in the local coordinate system based on the second coordinate of the material point in the world coordinate system and the corresponding target parameter coordinate; wherein, the target parameter coordinate is the parameter coordinate corresponding to the projection point of the material point on the center line of the spline component; The second building module is used to build a topology description function based on the third function.
[0018] Thirdly, embodiments of the present invention provide a computer-readable storage medium storing a computer program, which, when executed by a processor, causes the processor to perform the steps of the method described in the first aspect.
[0019] Fourthly, embodiments of the present invention provide a computer device including a memory and a processor, the memory storing a computer program, which, when executed by the processor, causes the processor to perform the steps of the method described in the first aspect.
[0020] The method, apparatus, device, and storage medium for constructing a topology description function according to embodiments of the present invention construct a globally continuously differentiable topology description function by composing a first, second, and third continuously differentiable function. This topology description function can not only accurately describe the geometry of a variable-thickness spline component, but its derivative with respect to design variables can also be analytically derived or conveniently calculated using automatic differentiation techniques. This greatly promotes its application in gradient-based topology optimization, enabling efficient and stable finding of variable-thickness structure design schemes with excellent performance, and improving optimization efficiency. Therefore, the method for constructing the topology description function in this application can improve the effectiveness of topology description function construction. Attached Figure Description
[0021] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.
[0022] in: Figure 1 A flowchart illustrating the method for constructing a topology description function provided in an embodiment of the present invention; Figure 2 This is a schematic diagram of a variable thickness spline assembly provided in an embodiment of the present invention; Figure 3This is a schematic diagram illustrating the process of world transformation of basic line elements provided in an embodiment of the present invention; Figure 4 This is a structural block diagram of an apparatus for constructing a topology description function provided in an embodiment of the present invention; Figure 5 A structural block diagram of a computer device provided in an embodiment of the present invention. Detailed Implementation
[0023] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0024] refer to Figure 1 , Figure 1 This is a flowchart illustrating the method for constructing a topology description function provided in an embodiment of this application. Specifically, it includes the following steps S1-S3: Step S1: Obtain the first function and the second function.
[0025] The first function p(u) mentioned above uses the parameter coordinates as variables to determine the first coordinates of the point corresponding to the parameter coordinates on the center line of the spline component of the variable thickness spline component in the world coordinate system. The first coordinates mentioned above are two-dimensional coordinates.
[0026] The aforementioned parameter coordinates u∈[u0, u1] are normalized one-dimensional variables defined within a preset design domain. u0 and u1 can be set according to requirements, and [u0, u1] is a preset range. For the first function, multiple specific parameter coordinates determined from the preset range can be given, thereby the first function determines the first coordinates corresponding to the multiple parameter coordinates. The points corresponding to the multiple first coordinates can be used to determine the centerline of the spline component. Therefore, the first function uses parameter coordinates as variables to determine the first coordinates of the points in the preset design domain that correspond to the parameter coordinates and are used to determine the centerline of the spline component for the variable thickness spline component, in the world coordinate system. The first function p(u) can be a continuously differentiable function with u as the independent variable.
[0027] The second function described above uses the parameter coordinates as variables to determine the thickness of the variable thickness spline component within a preset design domain. The second function W(u) can be a continuously differentiable function with the parameter coordinate u as its independent variable, used to determine the thickness W(u) of the variable thickness spline component at the parameter coordinate u within the design domain.
[0028] Since the second function is used to determine the thickness of the variable-thickness spline component within the preset design domain, the parameter coordinates can be taken from the parameter coordinates corresponding to points on the center line of the variable-thickness spline component to determine the thickness (width) at those points. The thickness at a point is the length in the orthogonal direction to that point, which is perpendicular to the tangential direction of that point. Therefore, the second function can use the parameter coordinates corresponding to points on the center line of the spline component as variables to determine the thickness of the variable-thickness spline component at those points.
[0029] Both the first and second functions mentioned above are continuously differentiable mathematical operation functions.
[0030] The first function mentioned above can be a quasi-uniform B-spline curve function, or it can be described by a non-uniform B-spline curve (NURBS), Bézier curve, polynomial curve, or other parametric curve function. The key is that the curve function has a parametric form and is differentiable.
[0031] The second function mentioned above In addition to polynomial functions, B-spline functions, piecewise linear functions, trigonometric functions, or other parameterized functions can also be used to describe the function, as long as it can be defined by a set of control parameters and is continuously differentiable.
[0032] Step S2: Construct the third function based on the first and second functions.
[0033] Among them, the third function mentioned above Used to determine the second coordinate of the material point Q in the world coordinate system based on the design domain. ), and the corresponding target parameter coordinates. Determine the target ordinate of the material point Q in the local coordinate system. .
[0034] The target parameter coordinates mentioned above are the parameter coordinates of the projection point of the material point on the center line of the spline assembly. That is, the target parameter coordinates are defined as the parameter coordinates of the orthogonal projection point of the material point on the center line P(u) of the spline assembly. .
[0035] Step S3: Construct the topology description function based on the third function.
[0036] The aforementioned topological description function can be used to determine the target value corresponding to a material point based on its target ordinate. The target value is used to determine whether the material point is located within the geometric region of the variable thickness spline component.
[0037] In this embodiment, a topology optimization method based on a topology description function can be used for topology optimization.
[0038] In this embodiment, a globally continuously differentiable topological description function is constructed by composing continuously differentiable functions (a first function, a second function, and a third function). This topological description function can not only accurately describe the geometry of the variable-thickness spline component, but its derivatives with respect to design variables (such as P(u) and w(u)) can also be analytically derived or conveniently calculated using automatic differentiation techniques. This greatly promotes its application in gradient-based topology optimization, enabling efficient and stable search for variable-thickness structure design schemes with excellent performance.
[0039] In some embodiments, the expression for the first function described above can be as follows: , ; in, Represents the first function, , Let these represent the functions used to determine the x-coordinate and y-coordinate in the first coordinate system, respectively. for B-spline basis functions Let i be the third coordinate of the i-th control point in the world coordinate system. These represent the x-coordinate and y-coordinate in the third coordinate system, respectively. The control points are used to determine the centerline of the spline component, and n+1 is the preset number of control points.
[0040] In some embodiments, the expression for the second function described above is as follows: ; Where u represents the parameter coordinates, This represents the thickness value corresponding to the parameter coordinates, where m represents the preset value. Represents the predefined polynomial coefficients.
[0041] The pre-defined polynomial coefficients can be obtained through the following steps: Obtain the coordinates of the preset m+1 parameters from the preset range.
[0042] For the f-th parameter coordinate among m+1 parameter coordinates f Substituting the coordinates of the f-th parameter into the initial second function, we obtain the thickness expression corresponding to the coordinates of the f-th parameter. The polynomial coefficients in the initial second function Not yet determined. The value of f ranges from 1 to m+1, and f is a positive integer.
[0043] Solve the polynomial coefficients of the linear equations for the Vandermonde matrix using the following coefficient matrix. : .
[0044] In some embodiments, the expression for the third function described above can be as follows: ; in, The target ordinate of the material point. These represent the x-coordinate and y-coordinate of the second coordinate system, respectively. Indicates the coordinates of the target parameter. Let x be the derivative component of the x-axis in the world coordinate system, which varies with the target parameter coordinates. This is the derivative component of the y-axis in the world coordinate system, which varies with the target parameter coordinates.
[0045] In some embodiments, constructing the third function based on the first and second functions can be achieved by constructing an initial third function based on the first function, the second function, and the derivative vector components of the x and y axes of the world coordinate system that vary with the parameter coordinates, and letting u = ... x= and y= Determine the third function.
[0046] The initial third function is as follows: ; in,( () represents the coordinates in the world coordinate system. , represents the first function, Indicates the second function, The derivative component of the x-axis in the world coordinate system, which varies with the parametric coordinates. This represents the derivative component of the y-axis in the world coordinate system, which varies with the parametric coordinates.
[0047] In some embodiments, the above and The expression can be as follows: ; ; in, It represents the fourth coordinate of the (i+1)th control point in the world coordinate system.
[0048] In some embodiments, the expression for the above topology description function can be as follows: ; in, Represents the topology description function. Let p represent the third function, and p represent a pre-defined even number greater than zero.
[0049] This is used to control the slope of the topological description function near the boundary of the spline component. The assignment of this topological description function to the material point Q can be defined as follows: like Point Q is located inside the component; like Point Q is located at the component boundary; like Point Q is located outside the component.
[0050] Since the selection of point Q is arbitrary, the above topological description function can determine the affiliation of any material point in the design domain, and can therefore be used for the topological description of variable thickness spline components.
[0051] In some embodiments, the above-described construction of the topology description function based on the third function may include, but is not limited to, the following steps: Construct a topology description function based on the third and fourth functions.
[0052] The fourth function uses the second coordinate as a variable to determine the target parameter coordinates.
[0053] In some embodiments, the method may further include the following steps: Construct the inverse transformation expression of the world transformation.
[0054] The inverse world transformation expression is as follows: ; ; Where u represents the parameter coordinates, Represents the coordinates in the world coordinate system, i.e. These represent the x and y coordinates of the parameter coordinates in the world coordinate system, respectively. , () represents the coordinates in the local coordinate system. , These represent the coordinates of the parameter coordinates in the world coordinate system, and the x and y coordinates in the local coordinate system, respectively. Represents the scaling matrix. Represents the rotation matrix. This represents the translation matrix.
[0055] Will , , Substituting the expression into the inverse world transformation expression, we obtain the simplified inverse world transformation expression as follows: ; in, Represents the first function, These are functions used to determine the x-coordinate and y-coordinate corresponding to the first coordinate, respectively. Let x be the derivative component of the x-axis in the world coordinate system, which varies with the parametric coordinates. This represents the derivative component of the y-axis in the world coordinate system, which varies with the parameter coordinates.
[0056] make , and u= , = as well as = This leads to the fourth function. These represent the x-coordinate and y-coordinate of the second coordinate system, respectively. Indicates the coordinates of the target parameter.
[0057] In some embodiments, the scaling matrix described above can be as follows: ; in, Represents the scaling matrix. This represents the second function.
[0058] In some embodiments, the rotation matrix described above can be as follows: ; ; ; in, Represents the rotation matrix. Indicates the rotation angle. Let x be the derivative component of the x-axis in the world coordinate system, which varies with the parametric coordinates. This represents the derivative component of the y-axis in the world coordinate system, which varies with the parameter coordinates.
[0059] In some embodiments, the translation matrix described above can be as follows: ; in, Represents the translation matrix. Represents the first function, These are functions used to determine the x-coordinate and y-coordinate corresponding to the first coordinate, respectively.
[0060] In some embodiments, the expression for the fourth function is as follows: ; ; ; ; ; ; in, These represent the x-coordinate and y-coordinate of the second coordinate system, respectively. Indicates the coordinates of the target parameter.
[0061] The expression for the above topology description function can be shown below:
[0062] ; ; ; ; ; .
[0063] In some embodiments, before constructing the topology description function based on the third and fourth functions, the following steps may also be included: Obtain the inverse world transformation expression, which is shown below: ; ; These represent the x-coordinate and y-coordinate of the parameter coordinates in the world coordinate system, respectively. , These represent the coordinates mapped from the world coordinate system to the corresponding baseline elements in the local coordinate system. , These represent the scaling matrix, rotation matrix, and translation matrix, respectively. The geometric region of the aforementioned variable thickness spline component can be determined by the trajectory swept by the basic line elements in the local coordinate system during the process of mapping to the world coordinate system through scaling, rotation, and translation operations. The basic line elements are line segments in the local coordinate system that are located on the y-axis and have a preset length in both the positive and negative directions of the y-axis.
[0064] Will , Substituting the expression into the inverse world transformation expression, the inverse world transformation expression can be simplified as follows: ; in, ; ; ; ; ; ; ; in, This represents the second function, where u represents the coordinates of the parameters; Indicates the rotation angle. Let i be the third coordinate of the i-th control point in the world coordinate system. Let i+1 be the fourth coordinate of the control point in the world coordinate system; , The first function is used to determine the x-coordinate and y-coordinate corresponding to the first coordinate.
[0065] make =0 and u= , = as well as = ,get .
[0066] The first function and Expanding the B-spline basis functions in the equation into polynomials yields the following: ; .
[0067] Based on the B-spline basis functions expanded into polynomials, the first function and... as follows: ; ; in, ; .
[0068] Will , Substitution and expand on the topic From the univariate higher-order equation, we obtain the fourth function. Among them, This represents the parameter coordinate value corresponding to the (i+k+1)th node of the B-spline curve. This represents the parameter coordinate value corresponding to the (i+1)th node of the B-spline curve. , , Represents the polynomial coefficients of the B-spline basis functions.
[0069] In some embodiments, before constructing the third function based on the first and second functions, the following steps may be included, but are not limited to: Define the basic line element.
[0070] The aforementioned basic line element is a line segment located on the y-axis in the local coordinate system, with a preset length in both the positive and negative directions of the y-axis.
[0071] For each parameter coordinate, the scaling matrix corresponding to the parameter coordinate is determined based on the parameter coordinate and the second function; the rotation matrix corresponding to the parameter coordinate is determined based on the parameter coordinate; and the translation matrix corresponding to the parameter coordinate is determined based on the parameter coordinate and the first function. Based on the scaling matrix, rotation matrix, and translation matrix, the basic line elements in the local coordinate system are mapped to the world coordinate system through scaling, rotation, and translation operations. The trajectories swept by the basic line elements during the mapping to the world coordinate system through scaling, rotation, and translation operations are determined as the geometric region of the variable thickness spline component.
[0072] Wherein, the rotation angle in the rotation matrix makes the transformed local coordinate system The axial direction is consistent with the unit tangent direction at the parameter coordinates on the center line of the spline component in the world coordinate system.
[0073] In some embodiments, after constructing the topology description function, the method may further include the following steps: Obtain the target coordinates of the target material point in the design domain in the world coordinate system.
[0074] Substitute the target coordinates into the topological description function to obtain the target value calculated by the topological description function.
[0075] When the target value is greater than In this case, the target material point is determined to be located inside the variable thickness spline assembly. When the target value equals... In the case of a target material point located at the boundary of a variable-thickness spline assembly, the target material point is determined to be within this boundary. When the target value is less than... In this case, the target material point is determined to be located outside the variable thickness spline assembly.
[0076] The above method of substituting the target coordinates into the topological description function to obtain the target value calculated by the topological description function can be achieved by substituting the target coordinates into the fourth function to obtain the target parameter coordinates corresponding to the target coordinates, and then substituting the target coordinates and the corresponding target parameter coordinates into the topological description function. The target value is obtained.
[0077] Since the selection of target material points is arbitrary, the above topology description function can determine the affiliation of any material point in the design domain, and can therefore be used for the topology description of variable thickness spline components.
[0078] To better understand the above method, the embodiments of this application also provide the following description: Structural topology optimization is an advanced structural design method that aims to find the optimal distribution of materials within the design domain, thereby maximizing certain structural properties (such as stiffness and strength) while satisfying various constraints (such as volume and displacement). Explicit topology optimization is an important class of methods, characterized by the use of parametric geometric components to explicitly describe the structural boundaries. This avoids problems such as checkerboard patterns and mesh dependencies common in traditional implicit methods (such as variable density methods), and is naturally compatible with computer-aided design systems.
[0079] In explicit topology optimization frameworks, the moving deformable component method is one of the mainstream techniques. In this method, the optimized structure is composed of a series of components with explicit geometric parameters combined through Boolean operations. These components typically use basic geometric elements such as straight lines, arcs, or simple polynomial curves as the central skeleton, supplemented by constant or piecewise simple thickness functions to describe changes in cross-sectional dimensions. While these components achieve explicit boundary description, the central skeleton has a simple shape, inflexible thickness variations, and difficulty in accurately representing complex, smooth force flow paths. In recent years, various components based on spline curve functions have effectively solved these problems. These methods utilize the powerful geometric description capabilities of B-spline curves, enabling the generation of smooth, complex structural boundaries with fewer design variables, significantly improving the degrees of freedom in optimization and the manufacturability of the results.
[0080] However, existing methods still face significant challenges in constructing variable-thickness spline components. First, most methods rely on the normal distance between nodes and the spline curve to determine node affiliation (whether it lies within the widened spline curve). However, calculating the normal distance efficiently and accurately in practice is difficult, especially for high-curvature or self-intersecting curves. Second, to circumvent the computational difficulty of the normal distance, some methods employ a shortest distance approximation. However, this approximation introduces strong discontinuities, leading to instability in sensitivity analysis and easy divergence in iteration during optimization. Furthermore, in cases of complex curve deformation (such as self-intersection or overlap), the component may exhibit geometric distortion, affecting the structural rationality and manufacturability. These problems severely limit the reliability and applicability of existing methods in complex topology design, necessitating a more robust and efficient method for constructing variable-thickness spline components.
[0081] The core of this invention lies in providing a geometrically accurate and mathematically sound model of a movable deformable component, laying the foundation for subsequent explicit topology optimization. The method includes the following steps: Step 1: Define the centerline and thickness function of the variable thickness spline component (the centerline function is the first function mentioned above, and the thickness function is the second function mentioned above).
[0082] Adopt one The quasi-uniform B-spline curve is used as the centerline of the component. This centerline is controlled by B-spline curve control points. The uniquely determined mathematical expression of its (first function) is:
[0083] (1).
[0084] in, for The second-order B-spline basis function, its mathematical form is about polynomials; These are the parametric coordinates of the curve; and These are the x and y coordinates of the control point, respectively. and These are the x and y coordinates of the first coordinate of the B-spline curve as it changes with the parametric coordinates. Control points. It can be preset, ( This represents the third coordinate.
[0085] use A polynomial describes the thickness function of the component as a function of the parametric coordinates: (2); in, The coefficients are polynomial coefficients, defined by a set of specified parameter coordinates. Controlled thickness value The Vandermonde matrix is uniquely determined and solved using the following system of linear equations with coefficient matrices: (3); The variable thickness spline component defined using the above method is as follows: Figure 2 As shown, Figure 2 This is a schematic diagram of a variable thickness spline component provided in an embodiment of this application.
[0086] Step 2: World transformation drives basic line element sweep, defining the component geometry region.
[0087] The basic line element is defined as a local coordinate system. A parallel line A line segment with a length of 1 along both the y-axis and the negative y-axis (i.e., in the local coordinate system, a line segment located on the y-axis with a length of 1 in both the positive and negative directions of the y-axis, such as...). Figure 3 As shown, Figure 3This is a schematic diagram of the process of performing world transformation on the basic line element provided in the embodiments of this application. Through a world transformation matrix determined by the geometric characteristics of the center line, the basic line element is continuously mapped from the local coordinate system to the world (global) coordinate system, and its sweep trajectory constitutes the geometric region of the variable thickness spline component.
[0088] The world transformation is an affine transformation that combines scaling, rotation, and translation operations, achieved through matrices. Implementation. For example... Figure 3 As shown, this transformation will change the points in the local coordinate system Mapped to world coordinate system : (4); In the formula, , and These are the scaling matrix, rotation matrix, and translation matrix, respectively.
[0089] The scaling matrix is used to scale the basic line elements. Directional scaling This multiplies the component thickness, as shown in the following expression: (5); The purpose of the rotation matrix is to rotate the scaled line elements in... In-plane counterclockwise rotation angle The expression is as follows: (6); Depend on Figure 3 It can be seen that the rotation angle Make the transformed local coordinate system The axis direction and the spline component centerline in the world coordinate system are in the current parameter coordinates unit tangent direction Consistency, that is:
[0090] (7); Based on the fundamental theory of B-spline functions, the unit tangent direction... Calculated using the following expression:
[0091] (8); The translation matrix is used to translate the center point of the scaled and rotated line element to the corresponding point on the center line of the spline component in the world coordinate system. The expression is as follows: (9); Step 3: Construct the analytical expression of the component topology description function based on the inverse world transformation.
[0092] This step is used to determine the coordinates of any point Q in the world coordinate system, assuming its coordinates are... That is, whether the second coordinate mentioned above is located inside the spline component. The determination method is as follows: use the inverse world transformation to map point Q onto the baseline element of the corresponding projected local coordinate system (the line connecting point Q and the center line of the spline component is perpendicular to the tangent vector of the center line of the spline component), and then pass through the baseline element. The conditional solution point Q corresponds to the projection point parameter of the spline component centerline. (parameter coordinates of the projection point) That is, the coordinates of the above target parameters ), and use the mapped Construct a topological description function for analytical judgment.
[0093] Step 3.1: Establish the inverse transformation expression for the world transformation.
[0094] Multiply both sides of equation (4) on the left We obtain the inverse transformation expression for the world transformation: (10); in, (11); Substituting equations (5)-(7) and equation (9) into the equations, the inverse transformation expression of the world transformation can be simplified as follows: (12); For point Q: (13); This equation mathematically represents the inverse world transformation that maps point Q to its projection point on the component centerline (i.e., the midpoint of the baseline element).
[0095] Step 3.2: Solve for the projection parameters of point Q corresponding to the centerline of the spline component. .
[0096] For the baseline element Therefore, the first expression in equation (13) can be rewritten as: (14); Based on the second and third equations of equation (8), the above equation can be further rewritten as: (15); In the above formula , , and All are about The piecewise polynomial (the piecewise intervals are the same as those of the B-spline curve) is described in the following way.
[0097] Expand the B-spline basis functions in equations (1) and (8) into polynomials:
[0098] (16); in, Represents the polynomial coefficients of the B-spline basis functions; Therefore, the first equation of equations (1) and (8) can be rewritten as:
[0099] (17); in,
[0100] (18); Substituting equation (18) into equation (15), we expand it to: A higher-order equation of one variable:
[0101] (19); in,
[0102]
[0103]
[0104] (20); Equation (19) can be solved directly using Matlab's built-in function `roots`. .
[0105] Step 3.3: Establish the topology description function.
[0106] According to the second equation of equation (13), relative coordinates (The above target ordinate) is calculated as follows: (twenty one); Its physical meaning is the ratio of the projected distance between point Q and the center line of the spline component to the thickness of the spline component at the projection point, i.e., the relative thickness. This is the third function mentioned above.
[0107] A topological description function in analytical form Determine the affiliation of point Q. A preferred implementation of this function is as follows: (twenty two); in, For an even number greater than zero (such as...) This function controls the slope of the topological description function near the boundaries of the spline component. The assignment of point Q is defined as follows: like Point Q is located inside the component; like Point Q is located at the component boundary; like Point Q is located outside the component.
[0108] Since the selection of point Q is arbitrary, the above topological description function can determine the affiliation of any material point in the design domain, and can therefore be used for the topological description of variable thickness spline components.
[0109] The beneficial effects of this step: Due to the world change and its inverse transformation All coordinates are derived from the control points of the B-spline curve. and control thickness value That is, the design variables are constructed through continuously differentiable mathematical operations, and therefore the topological description function constructed from this is... The design variables are also continuously differentiable. This excellent mathematical property means that the function can be directly used for analytical sensitivity analysis, and its partial derivatives can be calculated accurately and efficiently, and are also parameter-dependent near the boundaries of the spline component. Control. This provides a solid foundation for embedding the variable thickness spline component of the present invention into the gradient optimization algorithm, completely avoiding the problems of calculation error, instability and convergence difficulty that may be caused by numerical difference method or geometric approximation method.
[0110] In summary, based on the design variables in step 1, this invention provides a method for constructing variable-thickness spline components by driving basic line element sweeping and generating analytical topology description functions through steps 2 and 3. The component model constructed by this method has precise geometric definition and excellent mathematical properties, and is particularly suitable for explicit topology optimization processes with high requirements for computational stability and convergence efficiency.
[0111] Explanation of alternative solutions for each step of the above implementation method: The core of the technical solution claimed in this invention lies in the construction method of the variable thickness spline component, and the above steps provide a preferred method for implementing this method. However, those skilled in the art should understand that, without departing from the core idea of this invention, there are various alternative solutions to the above steps, and these solutions can also achieve the purpose of this invention.
[0112] Regarding step 1 (defining the centerline and thickness function): Centerline curve type: In addition to quasi-uniform B-spline curves, the centerline can also be described using non-uniform B-spline curves (NURBS), Bézier curves, polynomial curves, or other parametric curves. As long as the curve has a parametric form and is differentiable, it can be applied to this framework.
[0113] Thickness function form: thickness function In addition to using polynomial functions, B-spline functions, piecewise linear functions, trigonometric functions, or other parameterized functions can also be used to describe the function, as long as it can be defined by a set of control parameters and is continuously differentiable.
[0114] Regarding step 2 (world transformation-driven basic line element sweep): Basic line element shape: In addition to line segments, the basic line elements used for sweeping can also be other simple geometric shapes such as circles and ellipses. Their size can be a unit of 1 or other fixed values, which can be adjusted through a scaling matrix.
[0115] Transformation order and form: The order of scaling, rotation, and translation transformations can be adjusted according to different local coordinate system definitions. The mathematical expression of world transformation is the core, and its specific matrix form can be equivalently transformed according to the coordinate system used (such as whether it includes homogeneous coordinates).
[0116] Regarding step 3 (constructing the topology description function): Projection point parameters Method for determining: Determine the parameters corresponding to point Q In addition to solving for the vertical projection point (minimum distance point), other methods such as chord length parameterization approximation and nearest control point parameters can also be used to determine the associated parameters in certain application scenarios. .
[0117] Projection point parameters Solution methods: In addition to using the roots function in MATLAB, equation (19) can also be solved by using the eigenvalue method of the adjoint matrix, the iterative method, the real root isolation method (such as Sturm's theorem), intelligent algorithms (such as genetic algorithms), or dedicated root-finding tools (such as MPSolve). These alternative algorithms can solve univariate high-order equations and are all within the scope of protection of this invention.
[0118] The specific form of the topological description function: The topological description function given by equation (22) is a preferred continuously differentiable implementation. Other function forms with similar properties can also be used, such as those based on exponential functions. Or, or other continuously differentiable functions whose values are greater than 0 inside the component, equal to 0 at the boundary, and less than 0 outside. Power exponent. Other positive values can be taken; even numbers ensure the function value is symmetric, but they are not the only choice.
[0119] Attribution determination method: after obtaining local coordinates In addition to using continuous topological description functions, judgment inequalities can also be used in scenarios where continuous sensitivity is not required. We can directly determine the discrete attribution (internal / external) by checking if the condition is true or false. However, this would negate the advantage of analytic differentiation.
[0120] The beneficial effect of this invention is that it fundamentally improves the mathematical completeness and computational stability of explicit topology optimization. Specifically, this invention abandons traditional methods that rely on numerical discretization or geometric approximation, and instead uses a rigorous coordinate world transformation of the baseline segment (basic segment) to map any node in the design domain to a local coordinate system based on spline curves, thereby constructing an analytical form of the topology description function. This function can continuously and differentiably characterize the component geometry, achieving a precise mathematical description of the node affiliation relationships.
[0121] The core advantage of this improvement lies in the continuous differentiability of the topological description function itself, which provides a solid mathematical foundation for sensitivity analysis in the subsequent optimization process. Since the derivative of this function with respect to component geometric parameters (such as control point coordinates and thickness parameters) can be analytically calculated, sensitivity calculations are accurate and efficient, completely avoiding instability issues that may arise from numerical difference or approximation methods. Consequently, the convergence and robustness of the optimization iteration process are significantly enhanced.
[0122] Furthermore, the analytical construction method based on world transformation can naturally handle complex deformations of spline curves (such as high curvature and self-intersection), effectively preventing component geometry distortion and misclassification, and ensuring the rationality and manufacturability of the optimization results. In addition, the structural boundaries generated by this method have good smoothness, reducing post-processing requirements and improving integration efficiency with CAD systems.
[0123] To better implement the above method, embodiments of this application provide an apparatus for constructing a topological description function, referring to... Figure 4 , Figure 4 A structural block diagram of an apparatus for constructing a topology description function is provided in an embodiment of this application, as shown below. Figure 4 As shown, the apparatus 400 for constructing the topology description function may specifically include the following: The function acquisition module 401 is used to acquire a first function and a second function; wherein, the first function uses parameter coordinates as variables to determine the first coordinates of the point in the world coordinate system corresponding to the parameter coordinates for determining the center line of the spline component of the variable thickness spline component in the preset design domain; the second function uses parameter coordinates as variables to determine the thickness of the variable thickness spline component at the parameter coordinates; both the first function and the second function are continuously differentiable mathematical operation functions. The first construction module 402 is used to construct a third function based on the first function and the second function; wherein, the third function is used to determine the target ordinate of the material point in the local coordinate system based on the second coordinate of the material point in the world coordinate system in the design domain and the corresponding target parameter coordinate; wherein, the target parameter coordinate is the parameter coordinate corresponding to the projection point of the material point on the center line of the spline component.
[0124] The second building module 403 is used to build a topology description function based on the third function.
[0125] The apparatus 400 for constructing topology description functions provided in this application embodiment can execute the technical solution shown in the above method embodiment. Its implementation principle and beneficial effects are similar, and will not be repeated here.
[0126] Figure 5 An internal structural diagram of a computer device in one embodiment is shown. This computer device can specifically be a terminal or a server. Figure 5 As shown, the computer device includes a processor, memory, and a network interface connected via a system bus. The memory includes a non-volatile storage medium and internal memory. The non-volatile storage medium stores an operating system and may also store a computer program that, when executed by the processor, enables the processor to implement an age recognition method. The internal memory may also store a computer program that, when executed by the processor, enables the processor to implement the age recognition method. Those skilled in the art will understand that... Figure 5 The structure shown is merely a block diagram of a portion of the structure related to the present application and does not constitute a limitation on the computer device to which the present application is applied. Specific computer devices may include more or fewer components than those shown in the figure, or combine certain components, or have different component arrangements.
[0127] In one embodiment, a computer device is provided, including a memory and a processor, the memory storing a computer program that, when executed by the processor, causes the processor to perform the following steps: Obtain a first function and a second function; wherein, the first function uses parameter coordinates as variables to determine the first coordinates of the point in the world coordinate system corresponding to the parameter coordinates for determining the center line of the spline component of the variable thickness spline component in the preset design domain; the second function uses parameter coordinates as variables to determine the thickness of the variable thickness spline component at the parameter coordinates; and both the first function and the second function are continuously differentiable mathematical operation functions. Based on the first function and the second function, a third function is constructed; wherein, the third function is used to determine the target ordinate of the material point in the local coordinate system based on the second coordinate of the material point in the world coordinate system and the corresponding target parameter coordinate; wherein, the target parameter coordinate is the parameter coordinate corresponding to the projection point of the material point on the center line of the spline component; Based on the third function, construct the topology description function.
[0128] In one embodiment, a computer-readable storage medium is provided storing a computer program that, when executed by a processor, causes the processor to perform the following steps: Obtain a first function and a second function; wherein, the first function uses parameter coordinates as variables to determine the first coordinates of the point in the world coordinate system corresponding to the parameter coordinates for determining the center line of the spline component of the variable thickness spline component in the preset design domain; the second function uses parameter coordinates as variables to determine the thickness of the variable thickness spline component at the parameter coordinates; and both the first function and the second function are continuously differentiable mathematical operation functions. Based on the first function and the second function, a third function is constructed; wherein, the third function is used to determine the target ordinate of the material point in the local coordinate system based on the second coordinate of the material point in the world coordinate system and the corresponding target parameter coordinate; wherein, the target parameter coordinate is the parameter coordinate corresponding to the projection point of the material point on the center line of the spline component; Based on the third function, construct the topology description function.
[0129] Those skilled in the art will understand that all or part of the processes in the above embodiments can be implemented by a computer program instructing related hardware. The program can be stored in a non-volatile computer-readable storage medium, and when executed, it can include the processes of the embodiments described above. Any references to memory, storage, databases, or other media used in the embodiments provided in this application can include non-volatile and / or volatile memory. Non-volatile memory can include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory can include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in various forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), dual data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), RAMbus direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and RAMbus dynamic RAM (RDRAM), etc.
[0130] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.
[0131] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of this patent application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims. Please enter the specific implementation details.
Claims
1. A method for constructing a topological description function, characterized in that, The method includes: Obtain a first function and a second function; wherein, the first function uses parameter coordinates as variables to determine the first coordinates of the point in the world coordinate system corresponding to the parameter coordinates for determining the center line of the spline component of the variable thickness spline component in the preset design domain; the second function uses the parameter coordinates as variables to determine the thickness of the variable thickness spline component at the parameter coordinates; both the first function and the second function are continuously differentiable mathematical operation functions. Based on the first function and the second function, a third function is constructed; wherein, the third function is used to determine the target ordinate of the material point in the local coordinate system based on the second coordinate of the material point in the world coordinate system and the corresponding target parameter coordinate; wherein, the target parameter coordinate is the parameter coordinate corresponding to the projection point of the material point on the center line of the spline component; Based on the third function, construct the topology description function.
2. The method according to claim 1, characterized in that, The method includes at least one of the following: The expression for the first function is as follows: ， ； Where u represents the parameter coordinates, This represents the first function. , Let represent the functions used to determine the x-coordinate and y-coordinate in the first coordinate system, respectively. for B-spline basis functions Let i be the third coordinate of the i-th control point in the world coordinate system. The x and y coordinates in the third coordinate system are respectively represented. The control points are used to determine the center line of the spline component. n+1 is the preset number of control points. The expression for the second function is as follows: ； Where u represents the parameter coordinates, This represents the thickness value corresponding to the parameter coordinates, where m represents a preset value. Represents the predefined polynomial coefficients; The expression for the third function is as follows: ； in, This represents the target ordinate of the material point. These represent the x-coordinate and y-coordinate of the second coordinate system, respectively. Indicates the coordinates of the target parameter. Let x be the derivative component of the x-axis in the world coordinate system, which varies with the target parameter coordinates. This is the derivative component of the y-axis in the world coordinate system, which varies with the target parameter coordinates.
3. The method according to claim 2, characterized in that, The and The expression is as follows: ； ； in, Let i+1 be the fourth coordinate of the control point in the world coordinate system. This represents the parameter coordinate value corresponding to the (i+k+1)th node of the B-spline curve. This represents the parameter coordinate value corresponding to the (i+1)th node of the B-spline curve.
4. The method according to claim 1, characterized in that, The expression for the topology description function is as follows: ； in, This represents the topology description function. Let p represent the third function, where p represents a preset even number greater than zero.
5. The method according to claim 1, characterized in that, The step of constructing the topology description function based on the third function includes: Based on the third and fourth functions, a topology description function is constructed; wherein, the fourth function uses the second coordinate as a variable to determine the target parameter coordinates.
6. The method according to claim 5, characterized in that, The method includes: Construct the inverse world transformation expression; the inverse world transformation expression is as follows: ； ； Where u represents the parameter coordinates, These represent the x-coordinate and y-coordinate of the parameter coordinates in the world coordinate system, respectively. , These represent the coordinates of the parameter coordinates in the world coordinate system, and the x-coordinate and y-coordinate in the local coordinate system, respectively. Represents the scaling matrix. Represents the rotation matrix. Represents the translation matrix; Will , , Substituting the expression into the inverse world transformation expression, we obtain the simplified inverse world transformation expression as follows: ； in, This represents the first function. These are functions used to determine the x-coordinate and y-coordinate corresponding to the first coordinate, respectively. Let x be the derivative component of the x-axis in the world coordinate system, which varies with the parametric coordinates. The derivative component of the y-axis in the world coordinate system, which varies with the parametric coordinates; make , and u= , = as well as = This yields the fourth function; These represent the x-coordinate and y-coordinate of the second coordinate system, respectively. The coordinates of the target parameter are represented.
7. The method according to claim 6, characterized in that, The scaling matrix is shown below: ； in, Represents the scaling matrix, This represents the second function; The rotation matrix is shown below: ； ； ； in, Represents the rotation matrix, Indicates the rotation angle. Let x be the derivative component of the x-axis in the world coordinate system, which varies with the parametric coordinates. The derivative component of the y-axis in the world coordinate system, which varies with the parametric coordinates; The translation matrix is shown below: ； in, Denotes the translation matrix, This represents the first function. These are functions used to determine the x-coordinate and y-coordinate corresponding to the first coordinate, respectively.
8. The method according to claim 5, characterized in that, The expression for the fourth function is as follows: ； ； ； ； ； ； in, These represent the x-coordinate and y-coordinate of the second coordinate system, respectively. The coordinates of the target parameter are represented.
9. The method according to claim 1, characterized in that, The method includes: Define a basic line element; the basic line element is a line segment located on the y-axis in the local coordinate system, and whose length is a preset length in both the positive and negative directions of the y-axis; For each parameter coordinate, a scaling matrix corresponding to the parameter coordinate is determined based on the parameter coordinate and the second function; a rotation matrix corresponding to the parameter coordinate is determined based on the parameter coordinate; and a translation matrix corresponding to the parameter coordinate is determined based on the parameter coordinate and the first function. Based on the scaling matrix, rotation matrix, and translation matrix, the basic line elements in the local coordinate system are mapped to the world coordinate system through scaling, rotation, and translation operations. The trajectories swept by the basic line elements during the mapping process are determined as the geometric region of the variable thickness spline component. The rotation angle in the rotation matrix makes the transformed local coordinate system... The axial direction is consistent with the unit tangent direction at the parameter coordinate on the center line of the spline component in the world coordinate system.
10. The method according to claim 1, characterized in that, After constructing the topology description function, the method includes: Obtain the target coordinates of the target material point in the design domain in the world coordinate system; Substitute the target coordinates into the topological description function to obtain the target value calculated by the topological description function; When the target value is greater than In the case where the target material point is located inside the variable thickness spline assembly; when the target value is equal to In the case where the target material point is located at the boundary of the variable thickness spline assembly; when the target value is less than In this case, it is determined that the target material point is located outside the variable thickness spline assembly.