Isogeometric non-probabilistic reliability topological optimization and geometric post-processing integration method

Abstract

The invention discloses an isogeometric non-probabilistic reliability topological optimization and geometric post-processing integration method, and belongs to the technical field of structural topological optimization design. The method comprises the following steps: constructing a geometric model by using the same group of NURBS primary functions, analyzing the model and parameterizing a level set function, and realizing unification of CAD and CAE models; for material attributes and loads with interval uncertainty, non-probability displacement reliability constraints are established based on a vertex combination method and an optimized feature distance; taking the minimization of the structural volume as a target, performing sensitivity analysis in combination with a shape derivative and an adjoint method, and driving boundary evolution of a level set to obtain a topological configuration meeting a reliability requirement; and automatically outputting the explicit geometric model through zero level set boundary extraction and parameterized projection-based boundary repair. According to the method, integration from reliability topological optimization to manufacturable geometric model is realized, and the safety and design efficiency of the structure in an uncertain environment are improved.

Description

Technical Field

[0001] This invention belongs to the field of structural topology optimization design technology, specifically involving an integrated method for equal geometric nonprobabilistic reliability topology optimization and geometric post-processing. Background Technology

[0002] In modern engineering structural design, topology optimization, as a powerful tool for determining the optimal material distribution during the conceptual design phase, has been widely used in aerospace, mechanical manufacturing, and other fields. Traditional topology optimization methods are mostly based on finite element analysis (FEA), but their accuracy and computational efficiency heavily depend on the quality of mesh generation, and there is a data gap between the geometric model and the analysis model, leading to cumbersome design processes and compromised geometric accuracy.

[0003] In recent years, isogeometric analysis (IGA) has provided a new approach to solving the aforementioned problems. IGA uses non-uniform rational B-spline (NURBS) basis functions from computer-aided design (CAD) as the analysis shape functions, achieving a unity of geometric description and mechanical analysis in mathematical expression. This avoids model transformation errors and ensures high-order geometric continuity. Combining IGA with topology optimization, especially with level set methods that can clearly describe boundaries, has become a research hotspot in high-precision structural optimization design.

[0004] However, most existing topology optimization methods based on isogeometry are built on deterministic assumptions, neglecting uncertainties such as material property fluctuations and load dispersion that are common in engineering practice. If these uncertainties are not considered, the optimized structure may fail prematurely during actual service, leading to safety hazards. To handle these uncertainties, probabilistic reliability methods require a large amount of sample data to construct an accurate probability distribution, which is often difficult to achieve in the early stages of engineering projects where data is scarce. Non-probabilistic interval models, on the other hand, only require upper and lower bound information on parameters and are more suitable for small sample situations, but research combining them with isogeometry topology optimization is currently insufficient.

[0005] On the other hand, the results obtained by topology optimization methods such as level sets are implicit functional expressions that cannot be directly recognized and manufactured by CAD / CAM systems. Existing research focuses on the optimization process itself, while the crucial post-processing step of transforming implicit topological configurations, which may have numerical defects, into explicit, smooth geometric models that meet manufacturing requirements often relies on manual intervention, resulting in low automation and becoming a bottleneck in the integrated design and manufacturing process.

[0006] Therefore, there is an urgent need for an integrated solution that can take into account the reliability requirements brought about by uncertainties within the high-precision framework of isogeometric analysis, and automatically complete the conversion from implicit topological results to manufacturable geometric models, thereby truly achieving a seamless connection from "design" to "manufacturing". Summary of the Invention

[0007] To address the aforementioned technical problems, this invention provides an integrated method for equal geometric nonprobabilistic reliability topology optimization and geometric post-processing, which eliminates the conversion error between design and analysis models, improves the safety of structural service in small sample environments, and realizes the transformation from implicit optimization configuration to a manufacturable, explicit geometric model.

[0008] To achieve the above objectives, the present invention adopts the following technical solution:

[0009] An integrated method for isogeometric nonprobabilistic reliability topology optimization and geometric post-processing is proposed, and its implementation steps are as follows:

[0010] Step 1: Construct a nonprobabilistic reliability topology optimization model based on isogeometric analysis, where the expansion coefficient corresponding to the NURBS control point is used as the design variable, the minimization of the structural volume fraction is used as the optimization objective, and the displacement reliability meets the predetermined requirements as the constraint condition.

[0011] Step 2: Construct a geometric model based on NURBS basis functions. Given the structural design domain, load and displacement boundary conditions, calculate the information of control points, Gaussian points and other geometric elements. Define holes in the structural design domain to initialize the level set function. Use Greville points and NURBS basis functions to parameterize the level set function and transform the Hamilton-Jacobi partial differential equation into an ordinary differential equation.

[0012] Step 3: Calculate the stiffness matrix of each element based on the material distribution corresponding to the current parameterized level set function, assemble the overall stiffness matrix, and solve for the displacement response using the equilibrium equation and isogeometric analysis method.

[0013] Step 4: For material properties and external loads with bounded uncertainties, calculate the upper and lower bounds of the displacement response using the interval parameter vertex combination method; based on nonprobabilistic reliability theory, calculate the optimized characteristic distance as a displacement reliability metric.

[0014] Step 5: Calculate the sensitivity information of the objective function and reliability constraints based on the principle of shape derivative, thereby calculating the velocity field driving the evolution of the level set function; update the expansion coefficients using the first-order Euler forward method, thereby driving the evolution of the structural configuration; determine whether the current structural configuration meets the convergence condition. If it meets the convergence condition, perform subsequent geometric post-processing; if it does not meet the condition, continue iterating and repeating steps 3 to 5 until a topology-optimized configuration defined by the final level set function that meets the reliability requirements is obtained.

[0015] Step 6: Extract the zero-level set boundary of the final level set scalar field constructed using NURBS basis functions and expansion coefficients; use a three-step strategy based on parametric projection to repair the extracted boundary and obtain the complete structural boundary.

[0016] Step 7: Based on the preset geometric feature threshold, perform geometric feature recognition and active control on the extracted boundary, including measures such as numerical artifact removal, small hole expansion, thickening of ultra-fine components and smoothing of sharp corners, and output an explicit geometric model represented by point sets.

[0017] In the above steps, step one establishes a topology optimization model based on isogeometric analysis and optimized feature distance constraints; step two uses Greville collocation method and NURBS basis functions, more commonly used in isogeometric analysis, to parameterize the level set function, transforming the Hamilton-Jacobi partial differential equation into a system of ordinary differential equations that are easier to solve; step three solves the structural response based on isogeometric analysis, ensuring mathematical consistency between the geometric model and the analysis model; step four introduces material elastic modulus and load dispersion, and performs a reliability assessment of the current displacement response based on an interval model; step five achieves rapid convergence of structural topology optimization based on the parameterized level set method; step six accurately extracts the structural boundary and ensures the integrity of the boundary and the watertightness of the structure through a repair strategy, and the obtained structural boundary is represented using a point set; step seven realizes the identification and control of different geometric features such as numerical artifacts, micro-holes, finer components, and sharp corners, improving the manufacturability of the structure. Specifically:

[0018] Step two includes: calculating the parametric coordinates of Greville points based on the node vectors of NURBS, wherein each Greville point corresponds one-to-one with a NURBS control point; constructing a basis function matrix using the function values ​​of the NURBS basis functions at Greville points; and representing the level set function as the product of this basis function matrix and a vector composed of expansion coefficients.

[0019] Step three includes: calculating the material distribution at each point in the design domain using the Heaviside projection function based on the parameterized level set function; constructing the displacement field using NURBS basis functions based on the material distribution, and calculating the strain matrix of the isogeometric elements; integrating the strain matrix, material elasticity matrix, and Jacobian determinant of the geometric mapping at Gaussian integration points to calculate the element stiffness matrix; assembling the stiffness matrices of all elements according to their corresponding degrees of freedom to form the overall stiffness matrix; and obtaining the displacement response field of the structure by solving the equilibrium equations consisting of the overall stiffness matrix, load vectors, and boundary conditions.

[0020] In step four, the upper and lower bounds of the displacement response are calculated using the interval parameter vertex combination method. Specifically, the uncertainty parameters of the material properties and external loads are expressed as intervals characterized by their center values ​​and radii. By traversing all vertex combinations formed at the upper and lower limits of the interval parameters, the equilibrium equations are solved to obtain the corresponding displacement responses. The maximum and minimum values ​​among all displacement responses are taken as the upper and lower bounds of the displacement responses, respectively.

[0021] In step four, the optimized feature distance is calculated as a reliability index. Specifically, the interval of the displacement response is compared with the preset permissible displacement interval, and the optimized feature distance is defined as a piecewise function of the relative positional relationship between the two intervals. When the optimized feature distance is less than or equal to zero, it is determined that the critical displacement meets the reliability constraints.

[0022] In step five: based on the principle of shape derivative, the sensitivity of the volume objective function and displacement constraint function to the design variables is calculated using the adjoint method; the sensitivity of the displacement reliability constraint to the design variables is calculated using the chain rule and the partial derivatives of the reliability index with respect to the upper and lower bounds of the displacement; the sensitivity of the volume objective function and the sensitivity of the reliability constraint are weighted and combined to construct the evolution velocity field that drives the boundary of the horizontal set to move along its normal direction.

[0023] The repair of incomplete open boundaries in step six specifically involves: establishing a one-dimensional parametric coordinate system along the outer boundary of the design domain; projecting the start and end points of each open boundary onto the boundary of the design domain, obtaining their parametric coordinates, and sorting them according to their numerical values; for adjacent projection points that do not belong to the same open boundary after sorting, generating a stitching path connecting the two points along the boundary of the design domain to close the corresponding open boundaries.

[0024] In step seven, the geometric features are adjusted and smoothed, including: identifying and deleting boundaries whose feature size is smaller than a first threshold to eliminate numerical artifacts; for hole features whose area enclosed by the boundary is smaller than a second threshold, performing a uniform outward expansion operation based on its geometric center; when the minimum distance between two boundaries is less than a third threshold, shrinking the smaller boundary towards the center of the two boundaries; identifying sharp vertices on the boundary whose included angle between adjacent line segments is less than a fourth threshold, and smoothing the vertex region by cutting new anchor points on its two adjacent sides and using a triangle centroid reconstruction algorithm.

[0025] In a second aspect, the present invention provides an electronic device, comprising: one or more processors; and a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned integrated method for equal geometric nonprobabilistic reliability topology optimization and geometric post-processing.

[0026] Thirdly, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enable the processor to implement the aforementioned integrated method for equal geometric nonprobabilistic reliability topology optimization and geometric post-processing.

[0027] The beneficial effects of this invention are as follows:

[0028] First, by using the same set of NURBS basis functions to uniformly construct geometric models, analysis models, and level set functions, this invention fundamentally eliminates the geometric errors and data interaction costs caused by the conversion between CAD and CAE models in the traditional finite element method, achieving seamless integration of the mathematical languages ​​of design, analysis, and optimization, and ensuring high-precision geometric description and mechanical analysis.

[0029] Second, to address the uncertain parameters that are prevalent in engineering practice but lack sufficient samples, a non-probabilistic interval model and an optimized feature distance reliability index are introduced to construct a topology optimization model under displacement reliability constraints. This method can effectively evaluate the structural safety margin under small sample conditions, drive the generation of topological configurations that still meet performance requirements under extreme loads or material disturbances, and significantly improve the service reliability and safety of structures in real uncertain environments.

[0030] Third, this invention constructs a fully automated and intelligent geometric post-processing workflow. By extracting the boundaries of implicit horizontal fields, repairing open boundaries based on parametric projection, and automatically identifying and actively controlling geometric features such as numerical artifacts, small holes, extremely fine components, and sharp corners, it can directly output smooth, complete, and manufacturing-compliant explicit geometric models (such as point sets or CAD entities), greatly shortening the cycle from optimization results to manufacturable drawings, and powerfully promoting the engineering implementation of topology optimization technology and the integration of design and manufacturing. Attached Figure Description

[0031] Figure 1 This is an overall flowchart of the integrated method for equal geometric nonprobabilistic reliability topology optimization and geometric post-processing of the present invention;

[0032] Figure 2 This is a schematic diagram of the geometric model and boundary conditions of the embodiment;

[0033] Figure 3 These are the iterative curves of volume fraction and reliability during the topology optimization process of the embodiment;

[0034] Figure 4 This is a detailed schematic diagram of the geometric model after geometric post-processing in the embodiment;

[0035] Figure 5 This refers to the geometric model represented by a set of points and the generated entity structure in the embodiment. Detailed Implementation

[0036] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0037] like Figure 1As shown, this invention discloses an integrated method for isogeometric nonprobabilistic reliability topology optimization and geometric post-processing. This method targets continuum structures affected by uncertainties. First, it constructs a geometric model, analytical model, and level set functions using unified NURBS basis functions, and solves the structural response based on isogeometric analysis. Second, it introduces a nonprobabilistic interval model to quantify material and load fluctuations, establishes a displacement reliability assessment index based on optimized feature distances, and performs sensitivity analysis of the reliability index using the adjoint method and chain rule, constructing a suitable evolution velocity field to drive the update of the expansion coefficient, thereby realizing topological configuration evolution under reliability constraints. Then, it utilizes the analytical properties of NURBS basis functions combined with the MarchingSquares algorithm to extract initial boundaries from the implicit level set scalar field, and employs a repair strategy based on parametric projection to achieve automated stitching of open boundaries. Finally, by controlling the coordinate points of the boundaries, it achieves accurate identification and active control of geometric features such as numerical artifacts, small holes, excessively thin rods, and sharp corners, outputting an explicit geometric model with high geometric fidelity. This invention addresses pain points such as language errors in CAD and CAE systems, difficulties in evaluating uncertain parameters in small samples in practical engineering, and the inability to directly manufacture implicit boundaries. Within an isogeometric analysis framework, it implements a topology optimization method integrating reliability analysis and geometric post-processing, promoting integrated design and manufacturing, improving structural safety, and shortening the development cycle. The following section combines... Figure 1 Describe the method in detail, such as Figure 1 As shown, the method includes the following steps:

[0038] Step 1: Using the expansion coefficient corresponding to the control point as the design variable, displacement reliability as the constraint, and minimizing the structural volume fraction as the optimization objective, construct a non-probabilistic reliability topology optimization model.

[0039] Specifically, considering the dual requirements of lightweight structures and high safety in fields such as aerospace, an optimization mathematical model based on isogeometric analysis and incorporating non-probabilistic reliability constraints is constructed:

[0040] ,

[0041] In the formula, find represents finding the design variable, which is the expansion coefficient corresponding to the control point. n and m represent the number of control points of the NURBS basis functions along two different parameter directions, respectively. It is the total number of control points, and the objective function is... This represents the volume fraction of the structure, D denotes the structural design domain, N is the number of isomorphic elements, p+1 and q+1 represent the number of Gaussian integration points along two different parameter directions within each isomorphic element, p and q represent the degrees of the basis functions along the two parameter directions, k, i, and j represent the corresponding subscript indices, and w...i w j Then represents the weight of the corresponding Gaussian integration point, and J represents the Jacobian matrix mapping from the standard Gaussian integration domain to physical space. It is the Heaviside step function, which is a level set function. The projection function, corresponding to the material distribution in the structure, is expressed as follows:

[0042] ,

[0043] st. indicates constraints, including displacement reliability constraints, equilibrium equations, Dirichlet boundary conditions, and Neumann boundary conditions. N represents the reliability constraint function. c Represents the number of reliability constraints. It is an index of reliability constraints. Represents the true displacement field. Represents the displacement permission space Any virtual displacement within, Representatives applied to the Dirichlet border displacement on, It is the elastic tensor of the structural material. Indicates Neumann boundary The corresponding normal direction. The bilinear form of strain energy and the linear form of load in the equilibrium equations are expressed as follows:

[0044] ,

[0045] In the formula, b represents the body force acting on the structure. Represents the traction force at the Neumann boundary.

[0046] Step 2: Construct a geometric model based on NURBS basis functions. Given the structural design domain, load and displacement boundary conditions, calculate the information of control points, Gaussian points, and other geometric elements. Define holes in the structural design domain to initialize the level set function. Use Greville points and NURBS basis functions to parameterize the level set function, and transform the Hamilton-Jacobi partial differential equation into an ordinary differential equation.

[0047] Specifically, the initial control point coordinates are determined based on the geometry of the model, and appropriate basis function orders and node vectors are selected to construct the design domain model S of the structure.

[0048] ,

[0049] In the formula, a biquadratic NURBS surface is constructed using tensor products, where n and m represent the number of control points along different parameter directions, respectively. This is the biquadratic NURBS basis function corresponding to the position of this parameter, where i and j represent the indices of the control point in different parameter directions. This represents control point information. After constructing the geometric model based on NURBS basis functions, the information of control points, Gaussian points, and other geometric elements is calculated. Holes are defined within the structural design domain to initialize the level set function. The level set function is then parameterized using Greville points and NURBS basis functions, transforming the Hamilton-Jacobi partial differential equation into an ordinary differential equation. The parametric coordinates of the Greville points are calculated using the following formula:

[0050] ,

[0051] In the formula, It is an open-node vector The i-th node, p represents the degree of the NURBS basis function, and n represents the number of control points in the NURBS curve, meaning there is a one-to-one correspondence between Greville points and control points. The level set function is obtained by interpolation over all Greville points in the following form:

[0052] ,

[0053] In the formula, Let A be the level set function at all Greville points, and let A be the matrix consisting of the NURBS basis functions corresponding to the Greville points. The basis function represents the j-th Greville point corresponding to the i-th control point. The number of Greville points and control points are both [missing information]. , These are the expansion coefficients corresponding to the Greville points. Through this parameterization process, the level set function is divided into spatial and temporal components. During the optimization process, the spatial term A does not change with the iteration process; that is, only the expansion coefficients remain. The (time term) is updated with pseudo-time t, and the Hamilton-Jacobi partial differential equation is transformed into the following system of ordinary differential equations:

[0054] .

[0055] In the formula, The gradient represents the basis function matrix. It is the gradient magnitude of the level set function. The normal velocity, representing the boundary movement, drives the evolution of the structural boundary, thereby obtaining the desired topological configuration. The gradient magnitude is calculated using the following formula:

[0056] ,

[0057] In the formula, the gradient of the NURBS basis function with respect to the physical coordinates can be calculated using the Jacobian matrix.

[0058] Step 3: Calculate the stiffness matrix of each element based on the material distribution corresponding to the current level set function, assemble the overall stiffness matrix, and solve for the structural response using the isogeometric analysis method.

[0059] Specifically, the displacement field is first expressed using NURBS basis functions as follows:

[0060] ,

[0061] In the formula, The displacement represents the corresponding control point. The analytical expression for the displacement field is consistent with the geometric parameterized expression. Expanding the above equation in vector form yields:

[0062] ,

[0063] In the formula, It is the displacement vector corresponding to the control point. R represents the number of control points. i It is the NURBS basis function corresponding to the i-th control point. , Let T represent the displacement components of the i-th control point along the two physical directions, and T denote the transpose. According to the principle of minimum potential energy, the element stiffness matrix is ​​obtained by integration in the physical space. Isogeometric analysis establishes the physical space using control point information, and the parameter space is mapped to the physical space using NURBS basis functions. Simultaneously, the integration domain of each element in the parameter space can be transformed into a standard Gaussian integration domain. Therefore, the integral term of the element stiffness matrix in the physical space can be transformed into the Gaussian integration domain, and further into the sum of Gaussian integration points:

[0064] ,

[0065] In the formula, w i w j These are the weights of the Gaussian integration points along the two parameter directions, p+1 and q+1 represent the number of Gaussian integration points along each parameter direction within each isogeometric unit, and J represents the Jacobian matrix mapping from the standard Gaussian integration domain to physical space.

[0066] ,

[0067] B is the strain-displacement matrix corresponding to the isogeometric element, which is calculated using the basis function matrix R:

[0068] .

[0069] After assembling the element stiffness matrices, the global stiffness matrix K is obtained. Then, the global displacement field is calculated using the equilibrium equation Ku=F, where F represents the overall load column vector.

[0070] Step 4: For material properties and external loads with bounded uncertainties, calculate the upper and lower bounds of the displacement response using the interval parameter vertex combination method; based on nonprobabilistic reliability theory, calculate the optimized characteristic distance as a displacement reliability metric.

[0071] Specifically, using interval form Represents the uncertainty parameter. and Let represent the center value and radius of the interval, respectively. Considering the interval fluctuations in the material's elastic modulus and the load amplitude, the equilibrium equations are affected by uncertainties:

[0072] ,

[0073] That is, the displacement response of the structure has upper and lower bounds, and the displacement response interval at the constraint point. The following interval parameter vertex method is used to obtain:

[0074] ,

[0075] In the formula, M represents the number of uncertain parameters, and r represents the different combinations of vertices of the interval parameter vertices. When considering uncertainty, and Let represent the actual displacement response range and the permissible displacement range, respectively. Then, at the target reliability of ... At that time, the first The optimal feature distance corresponding to each displacement reliability constraint is defined as the following piecewise function:

[0076] .

[0077] When conducting subsequent reliability assessments, the current optimized feature distance is calculated based on the above formula. When the optimized feature distance d≤0, the reliability constraint is considered to be satisfied.

[0078] Step 5: Calculate the sensitivity information of the objective function and reliability constraints based on the principle of shape derivative, thereby calculating the velocity field that drives the evolution of the level set function; update the expansion coefficients using the first-order Euler forward method, thereby driving the evolution of the structural configuration; determine whether the current structural configuration meets the convergence condition. If the convergence condition is met, perform subsequent geometric post-processing; if not, continue iterating until a topology-optimized configuration that meets the reliability requirements is obtained.

[0079] Specifically, the derivative of the reliability index with respect to pseudo-time t is decomposed into partial derivatives with respect to the upper and lower bounds of the displacement using the chain rule:

[0080] ,

[0081] In the formula, and The calculation is as follows:

[0082] ,

[0083] ,

[0084] To calculate the time derivative of the displacement, Lagrange multipliers are introduced. Construct a Lagrangian function under deterministic displacement constraints using the virtual displacement field w, and transform it into an unconstrained optimization problem as follows:

[0085] ,

[0086] Calculate the derivative of the Lagrangian function with respect to pseudo-time t, and introduce the adjoint field. Implicit terms can be eliminated by solving the adjoint equation of the following form:

[0087] ,

[0088] The formula for calculating the sensitivity information corresponding to the deterministic displacement constraint and the volume objective function is as follows:

[0089] ,

[0090] Combining the sensitivity information of the volume function and the sensitivity information of the reliability constraint, the evolution velocity field of the level set function in the nonprobabilistic reliability topology optimization process is constructed as follows:

[0091] ,

[0092] For the above system of ordinary differential equations, the first-order Euler forward method is used for solution, and the update rule for the expansion coefficients is as follows:

[0093] .

[0094] The convergence conditions for determining whether the current structural configuration satisfies the convergence criteria include:

[0095] ,

[0096] In the formula, iter represents the current iteration step, and step represents the interval step. and These represent the optimized feature distance and structure volume fraction corresponding to the current iteration step, respectively. and These represent the convergence tolerance for reliability constraints and volume fraction variations, respectively.

[0097] Step 6: Extract the zero level set boundary of the level set scalar field constructed using NURBS basis functions and expansion coefficients; use a three-step strategy based on parametric projection to repair the extracted boundary and obtain the complete structural boundary.

[0098] Specifically, the level set function value at any point in the design domain can be obtained using the optimized NURBS basis functions and expansion coefficients:

[0099] ,

[0100] The Marching Squares algorithm is used to extract high-precision zero-level set boundaries as the initial boundaries of the structure. Closed boundaries and open boundaries are distinguished based on whether the beginning and end of the boundary are connected. Closed boundaries are directly retained, while open boundaries need to be processed as follows before they can be used as part of the structure boundary.

[0101] Establish a one-dimensional parametric coordinate system in the counterclockwise direction along the boundary of the design domain. Establish a mapping from physical space points to parameter space. Project the start and end endpoints of all open boundaries onto the design domain boundary to obtain the projection point set. Calculate the parametric coordinates of each projection point. Sort by z-value in ascending order; if adjacent sorting points If they do not belong to the same open boundary, a stitching path is generated along the design domain boundary to close the open curve. The final boundary set includes closed boundaries, open boundaries, and stitching paths.

[0102] Step 7: Based on the preset geometric feature threshold, perform geometric feature recognition and active control on the extracted boundary, including measures such as numerical artifact removal, small hole expansion, thickening of ultra-fine components and smoothing of sharp corners, and output an explicit geometric model represented by point sets.

[0103] Specifically, the feature dimensions of each boundary are calculated, and the diagonal length of the largest bounding box is taken as the reference dimension. Threshold coefficients are set for various geometric features, which are respectively the artifact thresholds. Minimum size threshold Minimum gap threshold Sharp corner threshold ;

[0104] For feature size smaller The boundaries can be considered as isolated islands or tiny pores caused by numerical artifacts and removed.

[0105] for For smaller hole features, the coordinates of all points on the boundary are modified according to the following formula:

[0106] ,

[0107] In the formula, It is a boundary The number of all points is calculated based on the uniform outward expansion of the boundary from the center, preserving the original shape of the hole to the greatest extent possible. Represents the initial coordinates of points on the boundary. This represents its expanded coordinates.

[0108] For finer gaps, if the minimum distance between the two boundaries Less than For smaller boundaries Perform the following shrink operation:

[0109] ,

[0110] In the formula, It is a boundary Center to Boundary The minimum distance.

[0111] For the sharp angle feature, calculate each discrete point rotation angle at the location :

[0112] ,

[0113] When the rotation angle At that time, according to the adjacent side Different processing methods are chosen based on the length of the adjacent edge. Length less than the specified cut-off length If the adjacent edges are long enough, define two anchor points at a distance of on the two adjacent edges. and The centroid of the triangle formed by the anchor point and the current discrete point is taken as the new point. The position. This process is repeated until the angle constraint is met, and finally a sequence is used. Replace the origin point .

[0114] Example:

[0115] To more fully illustrate the applicability of this invention in practical engineering, this invention addresses, for example... Figure 2The topology optimization design of the rib web corresponding to the NASASC (2)-0714 airfoil shown is performed under displacement reliability constraints. The NASA SC (2)-0714 airfoil is a supercritical airfoil with a maximum thickness of only 14% of the chord length. This airfoil exhibits unique rear load characteristics under high-speed conditions, and its geometry and stress are relatively complex. To ensure the integrity of the upper and lower edge strips, the space within 5% of the thickness of the upper and lower edge strips is reserved as a non-design domain, and the remaining portion within the web is defined as the design domain. The length of the design domain is 0.8m, divided into 80×20 IGA elements, and the NURBS basis function order is 3. A non-uniform distributed load q = 0.05 + 2x is applied above the rib. 3 The material has a nominal elastic modulus of 1 MPa and a Poisson's ratio of 0.3, with a strength of N / m and fixed supports at both ends. Assume that both the material's elastic modulus and the load exhibit a 5% divergence. To more objectively reflect the overall stiffness performance under distributed loads, the displacement reliability constraint is applied to the average displacement rather than the maximum displacement on the upper surface of the rib. The average displacement constraint range is set to [180, 200] μm, and the target reliability is 0.95.

[0116] Figure 3 The iterative history curves of volume fraction and displacement reliability during the rib topology optimization process are shown, and the convergence results of the topology optimization are presented. From Figure 3 The iterative process shows that, due to the highly nonlinear nature of this optimization problem, the objective function and constraint function fluctuate wildly in the early stages of iteration. However, as the iteration progresses, the topology gradually stabilizes, and both the objective function and constraint function converge in the later stages. The current displacement reliability of the obtained topology is 0.9533, which satisfies the constraint conditions. The rib volume fraction has become 65.73% of the original, resulting in a significant weight reduction.

[0117] The geometric post-processing parameter settings are shown in Table 1. The converged results are then subjected to geometric post-processing and adjustment according to these parameters. Figure 4 This shows the details of the geometric model obtained after geometric post-processing, where the blue curve represents the original topological configuration boundary and the red curve represents the post-processed configuration boundary. Figure 4 It can be observed that under these geometric parameter settings, the micro-hole 1 was expanded, the thinner rod at point 2 was thickened, and the numerical artifact 3 was removed, effectively controlling various geometric features. At this point, the volume fraction of the geometric model is 66.39%, indicating that post-processing has minimal impact on the volume fraction, only fine-tuning local features and preserving the characteristics of the original configuration to a large extent.

[0118] Table 1

[0119]

[0120] Figure 5It demonstrates the geometric model represented by a set of points and the solid structure generated in CATIA after complete optimization algorithms and geometric post-processing.

[0121] In summary, this invention proposes an integrated method for topology optimization and geometric post-processing based on non-probabilistic geometric properties. First, a geometric model, analytical model, and level set function are constructed using NURBS basis functions, achieving a unified mathematical language. Then, considering the uncertainties of material elastic modulus and external loads, a non-probabilistic interval model is introduced for reliability analysis, and an optimization feature distance reliability index is established. Sensitivity analysis of the reliability index is performed based on the adjoint method and chain rule, thereby constructing a suitable level set evolution rate to drive boundary changes, and updating design variables using the first-order Euler forward method. Finally, boundary extraction is performed on the obtained configuration, and its geometric features are identified and controlled, resulting in an explicit and smooth geometric model that meets manufacturing requirements.

[0122] In a second aspect, the present invention provides an electronic device, comprising: one or more processors; and a memory for storing one or more programs; wherein, when the one or more programs are executed by the one or more processors, the one or more processors implement the aforementioned integrated method for equal geometric nonprobabilistic reliability topology optimization and geometric post-processing.

[0123] Thirdly, the present invention provides a computer-readable storage medium having executable instructions stored thereon, which, when executed by a processor, enable the processor to implement the aforementioned integrated method for equal geometric nonprobabilistic reliability topology optimization and geometric post-processing.

[0124] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above descriptions are merely specific embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An integrated method for isogeometric nonprobabilistic reliability topology optimization and geometric post-processing, characterized in that, include: Step 1: Construct a nonprobabilistic reliability topology optimization mathematical model based on isogeometric analysis, where the expansion coefficient corresponding to the NURBS control point is used as the design variable, the minimization of the structural volume fraction is used as the optimization objective, and the reliability of the key displacement meets the predetermined requirements as the constraint condition. Step 2: Based on NURBS basis functions, parameterize the level set functions defined in the design domain using Greville points; Step 3: Based on the material distribution defined by the current parametrically represented level set function, calculate the displacement response of the structure under load using the isogeometric analysis method; Step 4: Considering the interval uncertainty of material properties and external loads, calculate the upper and lower bounds of the displacement response using the interval parameter vertex combination method; based on the non-probabilistic reliability theory, calculate the optimized characteristic distance as the reliability index of the key displacement. Step 5: Based on the reliability index and displacement response, calculate the sensitivity of the objective function and constraints to the design variables, construct the level set evolution velocity field, update the expansion coefficients to drive the evolution of the level set function, and thus update the topology optimization model; Repeat steps three through five until the convergence condition is met, and obtain the topological configuration defined by the final level set function. Step 6: Extract the zero level set curve from the final level set function as the initial structural boundary, and repair the incomplete open boundary to obtain the closed boundary profile. Step 7: Perform geometric feature recognition on the boundary contour, and adjust and smooth the recognized geometric features according to manufacturing requirements to output an explicit geometric model that can be directly used for manufacturing.

2. The integrated method for isogeometric nonprobabilistic reliability topology optimization and geometric post-processing according to claim 1, characterized in that, Step two includes: calculating the parametric coordinates of Greville points based on the node vectors of NURBS, wherein each Greville point corresponds one-to-one with a NURBS control point; constructing a basis function matrix using the function values ​​of the NURBS basis functions at Greville points; and representing the level set function as the product of this basis function matrix and a vector composed of expansion coefficients.

3. The integrated method for isogeometric nonprobabilistic reliability topology optimization and geometric post-processing according to claim 1, characterized in that, Step three includes: calculating the material distribution at each point in the design domain using the Heaviside projection function based on the parameterized level set function; constructing the displacement field using NURBS basis functions based on the material distribution, and calculating the strain matrix of the isogeometric elements; integrating the strain matrix, material elasticity matrix, and Jacobian determinant of the geometric mapping at Gaussian integration points to calculate the element stiffness matrix; assembling the stiffness matrices of all elements according to their corresponding degrees of freedom to form the overall stiffness matrix; and obtaining the displacement response field of the structure by solving the equilibrium equations consisting of the overall stiffness matrix, load vectors, and boundary conditions.

4. The integrated method for isogeometric nonprobabilistic reliability topology optimization and geometric post-processing according to claim 1, characterized in that, In step four, the upper and lower bounds of the displacement response are calculated using the interval parameter vertex combination method. Specifically, the uncertainty parameters of the material properties and external loads are expressed as intervals characterized by their center values ​​and radii. By traversing all vertex combinations formed at the upper and lower limits of the interval parameters, the equilibrium equations are solved to obtain the corresponding displacement responses. The maximum and minimum values ​​among all displacement responses are taken as the upper and lower bounds of the displacement responses, respectively.

5. The integrated method for isogeometric nonprobabilistic reliability topology optimization and geometric post-processing according to claim 1, characterized in that, In step four, the optimized feature distance is calculated as a reliability index. Specifically, the interval of the displacement response is compared with the preset permissible displacement interval, and the optimized feature distance is defined as a piecewise function of the relative positional relationship between the two intervals. When the optimized feature distance is less than or equal to zero, it is determined that the critical displacement meets the reliability constraints.

6. The integrated method for isogeometric nonprobabilistic reliability topology optimization and geometric post-processing according to claim 1, characterized in that, In step five: based on the principle of shape derivative, the sensitivity of the volume objective function and displacement constraint function to the design variables is calculated using the adjoint method; the sensitivity of the displacement reliability constraint to the design variables is calculated using the chain rule and the partial derivatives of the reliability index with respect to the upper and lower bounds of the displacement; the sensitivity of the volume objective function and the sensitivity of the reliability constraint are weighted and combined to construct the evolution velocity field that drives the boundary of the horizontal set to move along its normal direction.

7. The integrated method for isogeometric nonprobabilistic reliability topology optimization and geometric post-processing according to claim 1, characterized in that, The repair of incomplete open boundaries in step six specifically involves: establishing a one-dimensional parametric coordinate system along the outer boundary of the design domain; projecting the start and end points of each open boundary onto the boundary of the design domain, obtaining their parametric coordinates, and sorting them according to their numerical values; for adjacent projection points that do not belong to the same open boundary after sorting, generating a stitching path connecting the two points along the boundary of the design domain to close the corresponding open boundaries.

8. The integrated method for isogeometric nonprobabilistic reliability topology optimization and geometric post-processing according to claim 1, characterized in that, In step seven, the geometric features are adjusted and smoothed, including: identifying and deleting boundaries whose feature size is smaller than a first threshold to eliminate numerical artifacts; for hole features whose area enclosed by the boundary is smaller than a second threshold, performing a uniform outward expansion operation based on its geometric center; when the minimum distance between two boundaries is less than a third threshold, shrinking the smaller boundary towards the center of the two boundaries; identifying sharp vertices on the boundary whose included angle between adjacent line segments is less than a fourth threshold, and smoothing the vertex region by cutting new anchor points on its two adjacent sides and using a triangle centroid reconstruction algorithm.

9. An electronic device, characterized in that, include: One or more processors; Memory, used to store one or more programs; Wherein, when one or more programs are executed by the one or more processors, the one or more processors implement the integrated method for equal geometric nonprobabilistic reliability topology optimization and geometric post-processing as described in any one of claims 1-8.

10. A computer-readable storage medium, characterized in that, It stores executable instructions that, when executed by a processor, enable the processor to implement the integrated method for equal geometric nonprobabilistic reliability topology optimization and geometric post-processing as described in any one of claims 1-8.

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