A method, apparatus, device, and storage medium for topology optimization of irregularly shaped mirrors.

Abstract

This application discloses a method, apparatus, device, and storage medium for topology optimization of irregularly shaped mirrors. The method establishes a finite element analysis model by constructing parameters, determines the design domain and non-design domain based on the finite element analysis model, then establishes a topology optimization model based on the design domain and non-design domain, determines the optimization formula in the topology optimization model, then constructs the preprocessing basis function of the irregularly shaped mirror based on the mirror node information in the non-design domain, and updates the topology optimization model in combination with the optimization formula, thereby obtaining the optimal topology structure of the irregularly shaped mirror that takes into account low RMS value under the condition of weight allowance.

Description

Technical Field

[0001] This application relates to the field of optical technology, and in particular to a method, apparatus, device and storage medium for optimizing the topology of irregularly shaped mirrors. Background Technology

[0002] In high-performance optical systems, such as those used in space remote sensing and airborne reconnaissance, mirrors are core components for beam control and imaging. Their design currently faces two stringent constraints based on practical needs: first, to meet the platform's dynamic performance and launch costs, the structure must be extremely lightweight; second, to ensure final image quality, the mirror surface must maintain nanometer-level surface accuracy under various complex operating conditions, characterized by the mirror's RMS (Root Mean Square) value. Especially with the improvement of optical system performance, to meet the demands of large field of view, high resolution, multi-band multiplexing, and extremely compact optical path layouts, "irregular" mirrors with non-circular or asymmetrical apertures are increasingly used, making their lightweight and conformal design even more complex and prominent.

[0003] Topology optimization, as an advanced design method for achieving extreme lightweighting, traditionally aims to minimize structural compliance or maximize fundamental frequency, with volume fraction and fundamental frequency as constraints. However, this method has fundamental flaws: ① The mechanical performance optimization objective is severely disconnected from the system's optical performance indicators (mirror RMS value). That is, the optical surface deformation of a high-stiffness structure may precisely couple with low-order aberrations that degrade imaging quality, leading to excessive mirror RMS values. For irregularly shaped mirrors, the relationship between structural deformation and optical aberrations is complex due to the irregular aperture, making empirical adjustment difficult and exacerbating this problem. ② Traditional surface fitting and evaluation methods (such as standard Zernike polynomials) are only orthogonal within the unit circle domain. For irregularly shaped mirrors such as rectangular, hexagonal, elliptical, and ring-shaped mirrors, there are boundary adaptability issues with irregular apertures, resulting in insufficient accuracy in the mirror RMS value calculation itself, failing to provide reliable feedback for optimization. Summary of the Invention

[0004] This application provides a method, apparatus, device, and storage medium for topology optimization of irregularly shaped mirrors, solving the technical problem of inaccurate RMS value calculation for irregular apertures in the prior art. By employing an irregular aperture basis function reconstruction algorithm, preprocessed basis functions are customized for irregular apertures to ensure the accuracy of mirror RMS value calculation. Then, RMS constraints are directly introduced into the topology optimization model, so that the material distribution evolution is directly controlled by optical indicators, realizing optically dominated design. Finally, the sensitivity formula is used to analyze the unit density through RMS, achieving efficient and accurate sensitivity analysis, ensuring rapid convergence of the optimization process, and reducing the mirror RMS value of irregularly shaped mirrors under working conditions.

[0005] In a first aspect, embodiments of this application provide a method for topology optimization of irregularly shaped mirrors, including: A finite element analysis model is established based on the structural parameters of the irregularly shaped mirror, and the design domain and non-design domain are determined based on the finite element analysis model. A topology optimization model is established based on the design domain and the non-design domain, and the optimization formula of the topology optimization model is determined. Obtain the mirror node information in the finite element analysis model, and construct the Zernike basis function for the irregular aperture based on the mirror node information in the finite element analysis model; The irregular aperture Zernike basis function is decoupled in the discrete domain by using the irregular aperture basis function reconstruction algorithm to obtain the preprocessed basis function. By combining the optimized formula and the preprocessing basis function, an iterative optimization step is performed to update the topology optimization model, resulting in an updated irregular mirror topology optimization model.

[0006] In conjunction with the first aspect, in one possible implementation, the construction of the Zernike basis function for the irregular aperture based on the mirror node information in the finite element analysis model includes: A local coordinate system is established with the mirror normal of the non-design domain in the finite element analysis model as the Z-axis, and the coordinate information of the mirror node is obtained as the mirror node information. The reference radius is calculated based on the mirror node information, and the Zernike polynomial parameterization expansion is performed on the mirror node information based on the reference radius to construct the Zernike basis function for the irregular aperture.

[0007] In conjunction with the first aspect, in one possible implementation, the irregular aperture Zernike basis function is subjected to discrete-domain basis vector decoupling processing using an irregular aperture basis function reconstruction algorithm to obtain preprocessed basis functions, including: Calculate the inner product matrix of the Zernike basis functions of the irregular aperture in the non-design domain, and process the inner product matrix using a matrix transformation formula to obtain the transformation matrix; The Zernike basis functions of the irregular aperture are decoupled in the discrete domain using the transformation matrix to obtain preprocessed basis functions.

[0008] In conjunction with the first aspect, in one possible implementation, the iterative optimization step, which combines the optimized formulation and the preprocessed basis function to update the topology optimization model and obtain an updated irregular mirror topology optimization model, includes: The iterative optimization steps include: The normal displacement vector of the irregular-shaped reflector is calculated using the finite element analysis model. The Zernike coefficients for the irregular aperture are obtained by fitting the mirror normal displacement vector of the irregularly shaped mirror based on the preprocessing basis function, and by subtracting the preprocessing basis function from the mirror normal displacement vector of the irregularly shaped mirror. The mirror information obtained by fitting the Zernike coefficients of the irregular aperture is used to obtain the Zernike fitting residual of the irregular aperture. The root mean square of the Zernike fitting residuals of the irregular aperture is calculated to update the RMS value of the irregular mirror surface. Based on the finite element analysis model, the objective function and constraint function of the optimization formula in the current iterative optimization step are calculated to obtain the objective function value and constraint function value. Sensitivity analysis is performed on the objective function and constraint function in the optimized formula to obtain the sensitivity analysis results; Based on the sensitivity analysis results, the element density in the topology optimization model is iteratively updated, and the updated element density is rearranged according to the element density index order to generate the updated element density. The topology optimization model is adjusted based on the updated unit density, and it is determined whether the adjusted topology optimization model has reached the preset convergence condition based on the objective function value, the constraint function value, and the updated mirror RMS value, or whether the maximum number of iterations has been reached based on the current iteration step number; wherein, the preset convergence condition includes the lower limit of the structural fundamental frequency of the irregular mirror, the upper limit of the structural proportion constraint, and the upper limit of the RMS constraint. If the adjusted topology optimization model fails to meet the preset convergence condition and the current iteration step does not reach the maximum iteration step, the iterative optimization steps are repeated. If the adjusted topology optimization model reaches the preset convergence condition or the current iteration step reaches the maximum iteration step, then the iterative optimization step is stopped.

[0009] In conjunction with the first aspect, in one possible implementation, the formula for calculating the RMS value of the irregularly shaped reflector is:

[0010] In the formula, This indicates the RMS value of the irregularly shaped mirror. This represents the normal displacement vector of the irregularly shaped mirror obtained after calculation using the finite element analysis model. Indicates based on the previous Preprocessing basis functions and the first The mirror normal displacement vector obtained by multiplying the Zernike coefficients of the irregular aperture is... This represents the total number of mirror nodes. The transpose symbol for a matrix. This indicates the order of the preprocessing basis functions.

[0011] In conjunction with the first aspect, in one possible implementation, the sensitivity analysis of the objective function and constraint function in the optimization formula includes: Sensitivity analysis was performed on the structural compliance in the objective function, and the analysis formula is as follows:

[0012] In the formula, In the topology optimization model, the first... The cell density of each cell, Let K represent the structural compliance of the irregularly shaped mirror, K represent the structural stiffness matrix of the irregularly shaped mirror, and U represent the structural nodal displacements of the irregularly shaped mirror. The symbol for the transpose of a matrix; Sensitivity analysis is performed on the natural frequencies in the constraint function, and the analysis formula is as follows:

[0013] In the formula, In the topology optimization model, the first... The cell density of each cell, The structure of the irregularly shaped mirror is shown in section 1. The natural frequency is given by K, where K represents the structural stiffness matrix of the irregularly shaped mirror, and M represents the structural mass matrix of the irregularly shaped mirror. This represents the structural mode shape vector of the irregularly shaped mirror; Sensitivity analysis was performed on the RMS value in the constraint function, and the analysis formula is as follows:

[0014] In the formula, In the topology optimization model, the first... The element density of each element, where K represents the structural stiffness matrix of the irregularly shaped mirror. The displacement of the structural nodes of the irregularly shaped mirror is represented by , and RMS represents the RMS value of the mirror surface. This represents the mirror normal displacement vector obtained after the structure is calculated using the finite element analysis model. Indicates based on the previous Preprocessing basis functions and the first The mirror normal displacement vector obtained by multiplying the Zernike coefficients of the irregular aperture is... This represents the total number of mirror nodes. This indicates the order of the preprocessing basis functions.

[0015] In conjunction with the first aspect, in one possible implementation, the iterative update of the cell density in the topology optimization model based on the sensitivity analysis results includes: The sensitivity analysis results are filtered and smoothed, and the element density in the topology optimization model is updated using the moving asymptote method; the calculation formula for filtering the sensitivity analysis results is as follows:

[0016] In the formula, This indicates the sensitivity after filtering. and This represents the element density of different elements in the topology optimization model. Indicated by unit Centered on the minimum filtration radius The set of all elements within a circular or spherical region of radius . Indicates the minimum filtration radius. Represents the weighting function. Indicates the original sensitivity. This represents the objective function or constraint function; where,

[0017] In the formula, Indicates the minimum filtration radius. Represents the element in the topology optimization model Center point and unit The geometric distance between the center points.

[0018] Secondly, embodiments of this application provide a topology optimization device for irregularly shaped reflectors, comprising: The finite element analysis model establishment module is used to establish a finite element analysis model based on the structural parameters of the irregularly shaped mirror, and to determine the design domain and non-design domain based on the finite element analysis model. A topology optimization model building module is used to build a topology optimization model based on the design domain and the non-design domain, and to determine the optimization formula of the topology optimization model; The irregular aperture Zernike basis function construction module is used to obtain the mirror node information in the finite element analysis model and construct the irregular aperture Zernike basis function based on the mirror node information in the finite element analysis model. The preprocessing basis function construction module is used to perform basis vector decoupling processing on the Zernike basis function of the irregular aperture in the discrete domain using the irregular aperture basis function reconstruction algorithm to obtain the preprocessed basis function. The iterative optimization processing module is used to combine the optimization formula and the preprocessing basis function to perform iterative optimization steps, update the topology optimization model, and obtain the updated irregular mirror topology optimization model.

[0019] Thirdly, embodiments of this application provide an apparatus comprising: a processor; a memory for storing processor-executable instructions; wherein, when the processor executes the executable instructions, it implements the method as described in the first aspect or any possible implementation of the first aspect.

[0020] Fourthly, embodiments of this application provide a non-volatile computer-readable storage medium, the non-volatile computer-readable storage medium including storage for storing a computer program or instructions that, when executed, cause the method described in the first aspect or any possible implementation of the first aspect to be implemented.

[0021] One or more technical solutions provided in the embodiments of this application have at least the following technical effects or advantages: This application embodiment establishes a finite element analysis model by constructing parameters, and determines the design domain and non-design domain based on the finite element analysis model; establishes a topology optimization model based on the design domain and non-design domain, and determines the optimization formula based on the topology optimization model; constructs the preprocessing basis function of the irregular-shaped mirror based on the mirror node information in the non-design domain, and performs iterative optimization steps in combination with the optimization formula to iteratively optimize the element density in the design domain; optimizes the topology optimization model based on the iteratively optimized element density to obtain the optimized topology optimization model of the irregular-shaped mirror, thereby improving the accuracy of RMS value calculation under irregular aperture. Attached Figure Description

[0022] To more clearly illustrate the technical solutions of the embodiments of this application, the drawings used in the description of the embodiments of this application or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0023] Figure 1 A flowchart illustrating a topology optimization method for irregularly shaped mirrors provided in this application embodiment; Figure 2 A geometric configuration of an irregularly shaped reflector provided in an embodiment of this application; Figure 3 A finite element mesh model of an irregularly shaped reflector provided in this application embodiment; Figure 4 The optimization iteration curves of the objective function and constraint function provided in the embodiments of this application; Figure 5 This is an optimized result of the irregularly shaped reflector provided in the embodiments of this application; Figure 6 Another optimization result of the irregularly shaped reflector provided in the embodiments of this application; Figure 7 This is a schematic diagram of an irregularly shaped reflector topology optimization device provided in an embodiment of this application. Detailed Implementation

[0024] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.

[0025] The following description of some technologies involved in the embodiments of this application is provided to aid understanding and should be considered merely exemplary. Therefore, those skilled in the art should recognize that various changes and modifications can be made to the embodiments described herein without departing from the scope and spirit of this application. Similarly, for clarity and brevity, some descriptions of well-known functions and structures are omitted in the following description.

[0026] Figure 1 This is a flowchart illustrating a topology optimization method for irregularly shaped mirrors provided in an embodiment of this application. Figure 1 This is merely one execution order shown in the embodiments of this application and does not represent the only execution order for a method of optimizing the topology of irregularly shaped mirrors. Where the final result can be achieved, Figure 1 The steps shown can be performed in parallel or in reverse order.

[0027] A finite element analysis model is established based on the structural parameters of the irregularly shaped mirror, and the design domain and non-design domain are determined based on the finite element analysis model.

[0028] Specifically, firstly, the structural parameters of the irregularly shaped mirror are determined based on the requirements of the optical system (such as important parameters of optoelectronic products like astronomical telescopes and airborne scanning mirrors, including but not limited to field of view, resolution, system size, and operating environment). These structural parameters include, but are not limited to, the aperture (e.g., rectangular, hexagonal), initial thickness, support type (back support, side support), and material properties. An initial structural model is then constructed based on these parameters. Next, a finite element analysis model is established based on the initial structural model. In this embodiment, the finite element analysis model is... ,in, This represents the structural load vector of the irregularly shaped mirror. This represents the structural stiffness matrix of the irregularly shaped mirror. This indicates the displacement of the structural nodes of the irregularly shaped mirror.

[0029] After establishing the finite element analysis model, material information (including but not limited to elastic modulus, Poisson's ratio, and density) is set. Then, based on the installation and stress requirements of the irregularly shaped mirror in the optical system, the boundary conditions of the irregularly shaped mirror are set. Finally, the mirror surface and connection area are selected as the non-design domain, and other areas are selected as the design domain (i.e., the back rib area).

[0030] A topology optimization model is established based on the design domain and non-design domain, and the optimization formula of the topology optimization model is determined. Then, manufacturing process constraints (including but not limited to draft constraints, symmetry constraints and minimum size control) are applied to the design domain to obtain a more reasonable and easier-to-manufacture topology optimization model.

[0031] The optimized formulation of the topology optimization model is as follows: Searching for:

[0032] Minimize:

[0033] constraint:

[0034]

[0035]

[0036]

[0037]

[0038] In the formula, In the topology optimization model, the first... The cell density of each cell, This indicates the structural flexibility of the irregularly shaped mirror. This represents the structural load vector of the irregularly shaped mirror. This represents the structural stiffness matrix of the irregularly shaped mirror. This represents the displacement of structural nodes in the irregularly shaped mirror. This indicates the structural fundamental frequency constraint of the irregularly shaped mirror. The structure of the irregularly shaped mirror is shown in section 1. First natural frequency, This indicates the lower limit of the structural fundamental frequency of the irregularly shaped mirror. This indicates the structural volume of the irregularly shaped mirror. This indicates the upper limit of the structural proportion constraint for irregularly shaped mirrors. This represents the RMS value of the irregularly shaped mirror. The RMS value is derived from the mirror node information of the irregularly shaped mirror through Zernike polynomial parametric expansion to obtain the Zernike basis functions for the irregular aperture. Then, an irregular aperture basis function reconstruction algorithm is used to decouple the basis vectors in the discrete domain to obtain preprocessed basis functions. Finally, the preprocessed basis functions are subtracted based on the mirror normal displacement vector of the irregularly shaped mirror. The fitting residuals were obtained by fitting the mirror surface information (which was obtained by fitting the Zernike coefficients of the irregular aperture), and finally the root mean square value of the fitting residuals was calculated to obtain the RMS value. This indicates the corresponding upper limit of the RMS constraint.

[0039] After obtaining the optimized formulation of the topology optimization model, it is also necessary to obtain the mirror node information in the finite element analysis model, and construct the Zernike basis function for the irregular aperture based on the mirror node information in the finite element analysis model.

[0040] Specifically, first, determine the number of terms in the Zernike basis functions to be constructed and the number of fitting subtraction terms (such as translation, tilt, and defocus); then, establish a local coordinate system with the mirror normal of the non-design domain as the Z-axis to obtain the coordinate information of the mirror nodes. And the ID information of the mirror nodes, and based on the coordinate information of the mirror nodes. Calculate the reference radius, and then perform Zernike polynomial parameterization expansion on the coordinate information of the mirror nodes based on the reference radius to construct the Zernike basis function for the irregular aperture.

[0041] The formula for calculating the reference radius is as follows:

[0042] In the formula, Indicates the reference radius. The x-coordinate represents the coordinate information of the mirror node. The vertical coordinate represents the coordinate information of the mirror node.

[0043] In this embodiment, the reference radius is a parameter that needs to be normalized when performing Zernike polynomial parameterization expansion. For example, if the reference radius is 10 and the coordinates are (8, 9), then the normalized coordinates are (0.8, 0.9).

[0044] Since the mirror surface is irregularly shaped and does not satisfy the standard Zernike orthogonality, this embodiment, after constructing the irregular aperture Zernike basis function, also needs to use the irregular aperture basis function reconstruction algorithm to decouple the basis vectors in the discrete domain to obtain the preprocessed basis function. Specifically, the Zernike basis function of the irregular aperture is calculated in the non-design domain mirror region. Inner product matrix Inner product matrix This reflects the degree of mutual coupling of standard Zernike polynomials after they lose orthogonality in the irregular aperture region.

[0045] Wherein, the inner product matrix for:

[0046] In the formula, Indicates the first i Item and the j The degree of coupling between the Xiangzernick polynomials. It is calculated using the following formula:

[0047] In the formula, , Indicates the first i Item and the j Xiang Zernik polynomial, Indicates non-design domain mirror area The area.

[0048] After obtaining the inner product matrix Then, the inner product matrix is ​​transformed using the matrix transformation formula. The transformation matrix is ​​obtained through processing. .

[0049] The matrix transformation formula in this embodiment is:

[0050] In the formula, Represents the transformation matrix The transpose of the matrix, The symbol for transpose of a matrix.

[0051] By performing basis vector decoupling on the Zernike basis functions of the irregular aperture in the discrete domain using the transformation matrix, preprocessed basis functions are obtained.

[0052] To facilitate numerical calculations of the preprocessing basis functions, this embodiment organizes the values ​​of the preprocessing basis functions at discrete points into column vectors, generating a basis function column vector, specifically as follows:

[0053] In the formula, W represents the column vector of basis functions. Indicates the first Orthogonal Zernike polynomials for irregular apertures of order 1 Represents the transformation matrix. Represents the elements in the transformation matrix. Indicates the first Standard Zernikic polynomials, Let represent a column vector consisting of standard Zernike polynomials. This represents the number of terms in a standard Zernike polynomial.

[0054] After obtaining the preprocessed basis functions, the iterative optimization steps are performed by combining the optimization formula and the preprocessed basis functions to update the topology optimization model, resulting in the updated topology optimization model of the irregular mirror.

[0055] The above iterative optimization steps specifically include: 1) The normal displacement vector of the irregular-shaped mirror is calculated by finite element analysis model.

[0056] 2) The Zernike coefficients of the irregular aperture are obtained by fitting the mirror normal displacement vector of the irregularly shaped mirror based on the preprocessed basis functions, and the Zernike coefficients are obtained by subtracting the preprocessed Zernike coefficients from the mirror normal displacement vector of the irregularly shaped mirror. The mirror information obtained by fitting the Zernike coefficients of the irregular aperture is used to obtain the Zernike fitting residuals of the irregular aperture.

[0057] 3) Calculate the root mean square of the Zernike fitting residuals for irregular apertures to update the RMS value of the irregular mirror surface.

[0058] Specifically, the mirror normal displacement vector of the irregularly shaped mirror is obtained through a finite element analysis model. First, a batch of first-type responses (including but not limited to responses calculating displacement, volume, mass, stress, and fundamental frequency) are established. The mirror normal displacement vector S of the irregularly shaped mirror is then calculated using the finite element analysis model, and this mirror normal displacement vector is... The input third-type response is fitted with Zernike polynomials to calculate the RMS value of the irregularly shaped mirror. Then, sensitivity analysis is performed on the objective function and constraint functions in the topology optimization model to obtain the RMS sensitivity. It is important to note that the ID order of the mirror nodes input into the third-type response must be consistent with the ID order of the extracted mirror nodes, forming a correspondence.

[0059] After obtaining the normal displacement vector of the mirror, the least squares method is used to fit the mirror deformation based on the preprocessed basis function and the normal displacement vector of the mirror, and the RMS value of the irregular mirror is calculated.

[0060] Specifically, the least squares method is first used to fit the previous data. Zelenek polynomials, yielding the first Fit coefficients of order (i.e., the former) Zernickel coefficient of irregular aperture), and then through the front Preprocessing basis functions and the first Fit coefficients of order Multiplying them yields the mirror normal displacement vector. . In the formula, Indicates the preceding Order of preprocessing basis functions, Indicates the preceding The fitting coefficients of order.

[0061] Among them, the fitting coefficient It is obtained through the least squares method, and the specific calculation formula is as follows:

[0062] Finally, the RMS value of the irregularly shaped mirror is obtained using the formula for calculating the RMS value of the mirror surface.

[0063] The formula for calculating the RMS value of an irregularly shaped mirror is:

[0064] In the formula, This indicates the RMS value of the irregularly shaped mirror. This represents the normal displacement vector of the irregularly shaped mirror obtained after calculation using the finite element analysis model. Indicates based on the previous Preprocessing basis functions and the first The mirror normal displacement vector obtained by multiplying the Zernike coefficients of the irregular aperture is... This represents the total number of mirror nodes. The transpose symbol for a matrix. This indicates the order of the preprocessing basis functions. .

[0065] 4) After updating the RMS value of the irregular-shaped mirror, calculate the objective function and constraint function of the optimization formula in the current iterative optimization step according to the finite element analysis model, and obtain the objective function value and constraint function value.

[0066] 5) Perform sensitivity analysis on the objective function and constraint function in the optimization formula to obtain the sensitivity analysis results.

[0067] Specifically, the sensitivity analysis results in this embodiment include the sensitivity analysis results of the objective function and the sensitivity analysis results of the constraint function.

[0068] Specifically, sensitivity analysis is performed on the objective function and constraint functions in the topology optimization model, including: Sensitivity analysis was performed on the structural compliance in the objective function, and the analysis formula is as follows:

[0069] In the formula, In the topology optimization model, the first... The cell density of each cell, Let K represent the structural compliance of the irregularly shaped mirror, K represent the structural stiffness matrix of the irregularly shaped mirror, and U represent the structural nodal displacements of the irregularly shaped mirror. The symbol for transpose of a matrix.

[0070] Furthermore, the structural compliance in the objective function can be calculated using the structural compliance calculation formula. The specific formula for calculating structural compliance is as follows:

[0071] In the formula, C K represents the structural compliance matrix, and U represents the structural stiffness matrix. The symbol for transpose of a matrix.

[0072] Sensitivity analysis is performed on the natural frequencies in the constraint function, and the analysis formula is as follows:

[0073] In the formula, In the topology optimization model, the first... The cell density of each cell, The structure of the irregularly shaped mirror is shown in section 1. The natural frequency is given by K, where K represents the structural stiffness matrix of the irregularly shaped mirror, and M represents the structural mass matrix of the irregularly shaped mirror. This represents the structural mode shape vector of the irregularly shaped mirror. The symbol for transpose of a matrix.

[0074] Furthermore, the natural frequencies in the constraint functions can be obtained by performing modal analysis on the finite element model to obtain the structure's first natural frequency. The natural frequency; the volume fraction can be determined based on the design variables (cell density) of each cell in the current iteration step. The volume distribution of all units within the design domain is obtained by statistical topology optimization model design.

[0075] The specific process of modal analysis (i.e., eigenvalue analysis) of the finite element model is as follows: Based on the global stiffness matrix K and global mass matrix M at the current iteration step, the generalized eigenvalue equations are constructed:

[0076] In the formula, This represents the eigenvalues ​​obtained from modal analysis of the finite element model.

[0077] Solve the equation using numerical algorithms (such as the Lanczos method or the iterative subspace method) to obtain the eigenvalues ​​of the corresponding order. Then, through the formula... Extracting the first part of the irregular mirror structure The first natural frequency.

[0078] Sensitivity analysis was performed on the RMS value in the constraint function, and the analysis formula is as follows:

[0079] In the formula, In the topology optimization model, the first... The element density of each element, where K represents the structural stiffness matrix of the irregularly shaped mirror. The displacement of the structural nodes of the irregularly shaped mirror is represented by , and RMS represents the RMS value of the mirror surface. This represents the mirror normal displacement vector obtained after the structure is calculated using the finite element analysis model. Indicates based on the previous Preprocessing basis functions and the first The mirror normal displacement vector obtained by multiplying the Zernike coefficients of the irregular aperture is... This represents the total number of mirror nodes. This indicates the order of the preprocessing basis functions. .

[0080] 6) Iteratively update the element density in the topology optimization model based on the sensitivity analysis results, and rearrange the updated element density according to the element density index order to generate the updated element density.

[0081] Specifically, the sensitivity analysis results are filtered according to the sensitivity filtering calculation formula, and then the sensitivity of the unit is smoothed by weighted averaging of the sensitivity of all units in the neighboring unit. Finally, the mathematical programming subproblem is established and solved by the Method of Moving Asymptotes (MMA), which transforms the complex global nonlinear optimization problem into a series of approximate subproblems. The numerical update and truncation of the design variables are carried out, and the unit density in the topology optimization model is updated according to the solution results of the subproblems.

[0082] In this embodiment, the MMA is built into the Optistruct solver.

[0083] The sensitivity filtering calculation formula is as follows:

[0084] In the formula, This indicates the sensitivity after filtering. and This represents the element density of different elements in the topology optimization model. Indicated by unit Centered on the minimum filtration radius The set of all elements within a circular or spherical region of radius . Indicates the minimum filtration radius. Represents the weighting function. Indicates the original sensitivity. This represents the objective function or constraint function; where,

[0085] In the formula, Indicates the minimum filtration radius. Represents the element in the topology optimization model Center point and unit The geometric distance between the center points.

[0086] 7) Adjust the topology optimization model based on the updated element density (i.e., design variables), and determine whether the adjusted topology optimization model has reached the preset convergence condition based on the objective function value, constraint function value, and updated mirror RMS value, or determine whether the maximum number of iterations has been reached based on the current iteration step number. The preset convergence condition includes the lower limit of the structural fundamental frequency of the irregularly shaped mirror, the upper limit of the structural proportion constraint, and the upper limit of the RMS constraint.

[0087] Specifically, in this embodiment, the element density is used as a design variable for the topology optimization model. It is a value between 0 and 1 (or 0 / 1), indicating whether a certain element in the topology optimization model has material "present" or "absent" (or an intermediate transitional state).

[0088] 8) If the adjusted topology optimization model does not meet the preset convergence condition and the current iteration step does not reach the maximum iteration step, then repeat the iterative optimization steps; 9) If the adjusted topology optimization model reaches the preset convergence condition or the current iteration step reaches the maximum iteration step, then stop executing the iterative optimization steps.

[0089] In this embodiment, the finite element analysis model, acting as a solver for the physical field, needs to know the material properties (such as the elastic modulus E) of each element in the topology optimization model in order to establish the structural stiffness matrix K and solve for the structural nodal displacements U. The topology optimization model, acting as a "decision-maker," determines, based on the results calculated by the finite element analysis model (such as strain energy, stress, and sensitivity), whether the element density of each element in the topology model should increase or decrease in the next iteration during the iterative optimization step.

[0090] like Figure 2The diagram illustrates the geometric configuration of an irregularly shaped reflector. The reflector's optical working surface (mirror surface) is derived from an ellipse with a major radius of 140 mm and a minor radius of 135 mm. This ellipse is intersected by two parallel straight lines, forming a stadium-like (or track-shaped) profile with semi-elliptical ends connected by parallel straight edges in the middle. A 120 mm diameter circular through-hole is located at the center of the reflector. The inner wall of this hole forms a ring-shaped support interface for connection and fixation to the rear structure.

[0091] Figure 3 This is the finite element mesh model of the aforementioned irregularly shaped reflector. The model discretizes the geometric configuration and uses hexahedral elements for calculation. The boundary conditions of the model are set to constrain the degrees of freedom of the entire sidewall (i.e., the annular surface) of the central circular interface in all directions, simulating the fixed connection with the rigid support structure at this point.

[0092] The load case under consideration is as follows: a gravitational acceleration field is applied in the coordinate system containing the mirror, at a 45° angle to the normal direction of the mirror. This case simulates the self-weight load borne by the mirror under a specific installation posture, and its directional decomposition will cause asymmetric deformation of the structure.

[0093] The reflector is made of SiC, and its material parameter is elastic modulus. Poisson's ratio ,density .

[0094] The optimized formula used in this embodiment is as follows: Find:

[0095] min:

[0096] st

[0097]

[0098]

[0099]

[0100]

[0101] The topology optimization model for this reflector uses minimizing structural compliance as the objective function, while also imposing a lower bound on the structural fundamental frequency constraint. Upper limit of structural proportion constraints Upper limit of RMS (surface accuracy) constraint The iterative curves after convergence of the optimization process for the objective function and constraint functions are shown in the figure below. Figure 4As shown, the final compliance converged to 1.667E-5mJ, the fundamental frequency to 4885.69Hz, the RMS to 3.45694nm, and the volume fraction to 0.24551. This indicates that both the objective function and the constraint functions met the convergence requirements. The iteration curves show that the objective function and all constraint functions gradually stabilized and reached convergence during the optimization process, demonstrating that the optimized design maximized structural stiffness while satisfying the constraints of given frequency, lightweight design, and surface accuracy.

[0102] The optimization results of the reflector are as follows: Figure 5 and Figure 6 As shown.

[0103] Figure 5 The elliptical, flat area at the top is shown as the non-design domain (mirror area), which was fully preserved during the optimization process. The complex reinforcing rib structure below the mirror evolved from the design domain, its contours perfectly adapting to the boundary constraints of the irregular aperture. From this perspective, it can be observed that the material forms a support system with specific topological logic on the back of the mirror. This integrated configuration provides uniform stiffness support for the mirror, macroscopically ensuring the structural stability of the reflector under complex load conditions.

[0104] Figure 6 The image showcases the back support structure of the reflector after removing the non-designed area of ​​the mirror. The material is distributed radially from the central annular support interface to the edges, evolving into multiple sets of reinforcing ribs with variable cross-sections. These ribs thicken significantly near the central support, gradually branching and thinning towards the edges, forming a lightweight structure resembling a biomimetic skeleton. The continuous material distribution, clear main force transmission paths, and relatively clear contour boundaries, without obvious gray areas, are beneficial for subsequent geometric reconstruction and engineering manufacturing.

[0105] This application embodiment introduces an irregular aperture basis function reconstruction algorithm to generate customized preprocessed basis functions for irregular apertures (such as runway-shaped, polygonal, etc.). Compared with the traditional technique of directly applying standard circular domain Zernike polynomials, which leads to the loss of orthogonality, this application embodiment eliminates the coupling interference between fitting coefficients from the mathematical source. This ensures that under irregular boundaries, the calculation of surface accuracy (RMS) is no longer affected by aperture shape distortion, and that the optimization algorithm always iterates based on real and accurate optical evaluation feedback.

[0106] Traditional techniques typically use minimizing structural compliance (i.e., maximizing stiffness) as the optimization objective. This is an indirect mechanical proxy and often cannot avoid surface errors that are highly correlated with imaging quality. The embodiments of this application establish a topology optimization formula with the mirror RMS value as a direct constraint, allowing the evolution of material distribution to be directly controlled by nanoscale optical parameters.

[0107] The sensitivity calculation in this application not only avoids the risk of spurious convergence caused by numerical noise, but also greatly reduces the computational overhead of differentiating ultra-large-scale design variables, ensuring the fast and stable convergence of the optimization process in high-dimensional parameter space.

[0108] While this application provides method operation steps as shown in the embodiments or flowcharts, more or fewer operation steps may be included based on conventional or non-inventive labor. The order of steps listed in this embodiment is merely one possible execution order among many and does not represent the only execution order. In actual device or client product execution, the method can be executed sequentially according to this embodiment or the accompanying drawings, or in parallel (e.g., in a parallel processor or multi-threaded processing environment).

[0109] like Figure 7 As shown in the illustration, this application also provides a topology optimization device for irregularly shaped mirrors. The device includes: The finite element analysis model building module is used to build a finite element analysis model based on the structural parameters of the irregularly shaped mirror, and to determine the design domain and non-design domain based on the finite element analysis model.

[0110] The topology optimization model building module is used to build a topology optimization model based on the design domain and non-design domain, and to determine the optimization formula of the topology optimization model.

[0111] The irregular aperture Zernike basis function construction module is used to obtain the mirror node information in the finite element analysis model and construct the irregular aperture Zernike basis function based on the mirror node information in the finite element analysis model.

[0112] The preprocessing basis function construction module is used to perform basis vector decoupling processing on the Zernike basis function of the irregular aperture in the discrete domain using the irregular aperture basis function reconstruction algorithm to obtain the preprocessed basis function.

[0113] The iterative optimization processing module is used to combine the optimization formula and the preprocessing basis function to perform iterative optimization steps, update the topology optimization model, and obtain the updated topology optimization model of the irregular mirror.

[0114] Some modules in the apparatus described in this application can be described in the general context of computer-executable instructions that are executed by a computer, such as program modules. Generally, program modules include routines, programs, objects, components, data structures, classes, etc., that perform a specific task or implement a specific abstract data type. This application can also be practiced in distributed computing environments where tasks are performed by remote processing devices connected via a communication network. In distributed computing environments, program modules can reside in local and remote computer storage media, including storage devices.

[0115] The apparatus or module described in the above embodiments can be implemented by a computer chip or physical entity, or by a product with a certain function. For ease of description, the above apparatus is described by dividing it into various modules according to their functions. When implementing the embodiments of this application, the functions of each module can be implemented in one or more software and / or hardware. Of course, a module that implements a certain function can also be implemented by combining multiple sub-modules or sub-units.

[0116] The methods, apparatus, or modules described in this application can be implemented in a computer-readable program code manner. The controller can be implemented in any suitable manner, such as a microprocessor or processor and a computer-readable medium storing computer-readable program code (e.g., software or firmware) executable by the (micro)processor, logic gates, switches, application-specific integrated circuits (ASICs), programmable logic controllers, and embedded microcontrollers. Examples of controllers include, but are not limited to, the following microcontrollers: ARC 625D, Atmel AT91SAM, Microchip PIC18F26K20, and Silicon Labs C8051F320. A memory controller can also be implemented as part of the control logic of a memory. Those skilled in the art will also recognize that, in addition to implementing the controller in purely computer-readable program code manner, the same functionality can be achieved by logically programming the method steps to make the controller take the form of logic gates, switches, application-specific integrated circuits, programmable logic controllers, and embedded microcontrollers. Therefore, such a controller can be considered a hardware component, and the means included within it for implementing various functions can also be considered as structures within the hardware component. Alternatively, the device used to implement various functions can be viewed as either a software module that implements the method or a structure within a hardware component.

[0117] This application also provides an apparatus, the apparatus comprising: a processor; a memory for storing processor-executable instructions; wherein, when the processor executes the executable instructions, it implements the method described in this application.

[0118] This application also provides a non-volatile computer-readable storage medium storing a computer program or instructions thereon, which, when executed, enables the method described in this application embodiment to be implemented.

[0119] Furthermore, in the various embodiments of the present invention, each functional module can be integrated into a processing module, or each module can exist independently, or two or more modules can be integrated into a single module.

[0120] The aforementioned storage media include, but are not limited to, Random Access Memory (RAM), Read-Only Memory (ROM), Cache, Hard Disk Drive (HDD), or Memory Card. The memory can be used to store computer program instructions.

[0121] As can be seen from the above description of the embodiments, those skilled in the art can clearly understand that this application can be implemented by means of software plus necessary hardware. Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, can be embodied in the form of a software product, or it can be embodied in the process of data migration. The computer software product can be stored in a storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, mobile terminal, server, or network device, etc.) to execute the methods described in various embodiments or some parts of the embodiments of this application.

[0122] The various embodiments described in this specification are presented in a progressive manner. Similar or identical parts between embodiments can be referred to interchangeably. Each embodiment focuses on its differences from other embodiments. All or part of this application can be used in numerous general-purpose or special-purpose computer system environments or configurations. Examples include: personal computers, server computers, handheld or portable devices, tablet devices, mobile communication terminals, multiprocessor systems, microprocessor-based systems, programmable electronic devices, network PCs, minicomputers, mainframe computers, and distributed computing environments including any of the above systems or devices, etc.

[0123] The above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit this application. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of this application.

Claims

1. A method for topology optimization of irregularly shaped mirrors, characterized in that, include: A finite element analysis model is established based on the structural parameters of the irregularly shaped mirror, and the design domain and non-design domain are determined based on the finite element analysis model. A topology optimization model is established based on the design domain and the non-design domain, and the optimization formula of the topology optimization model is determined. Obtain the mirror node information in the finite element analysis model, and construct the Zernike basis function for the irregular aperture based on the mirror node information in the finite element analysis model; The irregular aperture Zernike basis function is decoupled in the discrete domain by using the irregular aperture basis function reconstruction algorithm to obtain the preprocessed basis function. By combining the optimized formula and the preprocessing basis function, an iterative optimization step is performed to update the topology optimization model, resulting in an updated irregular mirror topology optimization model.

2. The method for topology optimization of irregularly shaped mirrors according to claim 1, characterized in that, The construction of the Zernike basis function for the irregular aperture based on the mirror node information in the finite element analysis model includes: A local coordinate system is established with the mirror normal of the non-design domain in the finite element analysis model as the Z-axis, and the coordinate information of the mirror node is obtained as the mirror node information. The reference radius is calculated based on the mirror node information, and the Zernike polynomial parameterization expansion is performed on the mirror node information based on the reference radius to construct the Zernike basis function for the irregular aperture.

3. The method for topology optimization of irregularly shaped mirrors according to claim 1, characterized in that, The irregular aperture basis function reconstruction algorithm is used to perform basis vector decoupling processing on the irregular aperture Zernike basis function in the discrete domain to obtain preprocessed basis functions, including: Calculate the inner product matrix of the Zernike basis functions of the irregular aperture in the non-design domain, and process the inner product matrix using a matrix transformation formula to obtain the transformation matrix; The Zernike basis functions of the irregular aperture are decoupled in the discrete domain using the transformation matrix to obtain preprocessed basis functions.

4. The topology optimization method for an irregularly shaped mirror according to claim 1, characterized in that, The iterative optimization step, combining the optimized formula and the preprocessed basis function, updates the topology optimization model to obtain an updated irregular mirror topology optimization model, including: The iterative optimization steps include: The normal displacement vector of the irregular-shaped reflector is calculated using the finite element analysis model. The Zernike coefficients for the irregular aperture are obtained by fitting the mirror normal displacement vector of the irregularly shaped mirror based on the preprocessing basis function, and by subtracting the preprocessing basis function from the mirror normal displacement vector of the irregularly shaped mirror. The mirror information obtained by fitting the Zernike coefficients of the irregular aperture is used to obtain the Zernike fitting residual of the irregular aperture. The root mean square of the Zernike fitting residuals of the irregular aperture is calculated to update the RMS value of the irregular mirror surface. Based on the finite element analysis model, the objective function and constraint function of the optimization formula in the current iterative optimization step are calculated to obtain the objective function value and constraint function value. Sensitivity analysis is performed on the objective function and constraint function in the optimized formula to obtain the sensitivity analysis results; Based on the sensitivity analysis results, the element density in the topology optimization model is iteratively updated, and the updated element density is rearranged according to the element density index order to generate the updated element density. The topology optimization model is adjusted based on the updated unit density, and it is determined whether the adjusted topology optimization model has reached the preset convergence condition based on the objective function value, the constraint function value, and the updated mirror RMS value, or whether the maximum number of iterations has been reached based on the current iteration step number; wherein, the preset convergence condition includes the lower limit of the structural fundamental frequency of the irregular mirror, the upper limit of the structural proportion constraint, and the upper limit of the RMS constraint. If the adjusted topology optimization model fails to meet the preset convergence condition and the current iteration step does not reach the maximum iteration step, then the iterative optimization steps are repeated. If the adjusted topology optimization model reaches the preset convergence condition or the current iteration step reaches the maximum iteration step, then the iterative optimization step is stopped.

5. The topology optimization method for an irregularly shaped mirror according to claim 4, characterized in that, The formula for calculating the RMS value of an irregularly shaped mirror is: In the formula, This indicates the RMS value of the irregularly shaped mirror. This represents the normal displacement vector of the irregularly shaped mirror obtained after calculation using the finite element analysis model. Indicates based on the previous Preprocessing basis functions and the first The mirror normal displacement vector obtained by multiplying the Zernike coefficients of the irregular aperture is... This represents the total number of mirror nodes. The transpose symbol for a matrix. This indicates the order of the preprocessing basis functions.

6. The topology optimization method for an irregularly shaped mirror according to claim 4, characterized in that, The sensitivity analysis of the objective function and constraint function in the optimized formula includes: Sensitivity analysis was performed on the structural compliance in the objective function, and the analysis formula is as follows: In the formula, In the topology optimization model, the first... The cell density of each cell, Let K represent the structural compliance of the irregularly shaped mirror, K represent the structural stiffness matrix of the irregularly shaped mirror, and U represent the structural nodal displacements of the irregularly shaped mirror. The symbol for the transpose of a matrix; Sensitivity analysis is performed on the natural frequencies in the constraint function, and the analysis formula is as follows: In the formula, In the topology optimization model, the first... The cell density of each cell, The structure of the irregularly shaped mirror is shown in section 1. The natural frequency is given by K, where K represents the structural stiffness matrix of the irregularly shaped mirror, and M represents the structural mass matrix of the irregularly shaped mirror. This represents the structural mode shape vector of the irregularly shaped mirror. The symbol for the transpose of a matrix; Sensitivity analysis was performed on the RMS value in the constraint function, and the analysis formula is as follows: In the formula, In the topology optimization model, the first... The element density of each element, where K represents the structural stiffness matrix of the irregularly shaped mirror. The displacement of the structural nodes of the irregularly shaped mirror is represented by , and RMS represents the RMS value of the mirror surface. This represents the mirror normal displacement vector obtained after the structure is calculated using the finite element analysis model. Indicates based on the previous Preprocessing basis functions and the first The mirror normal displacement vector obtained by multiplying the Zernike coefficients of the irregular aperture is... This represents the total number of mirror nodes. This indicates the order of the preprocessing basis functions.

7. The method for topology optimization of an irregularly shaped mirror according to claim 5, characterized in that, The iterative update of the cell density in the topology optimization model based on the sensitivity analysis results includes: The sensitivity analysis results are filtered and smoothed, and the element density in the topology optimization model is updated using the moving asymptote method; the calculation formula for filtering the sensitivity analysis results is as follows: In the formula, This indicates the sensitivity after filtering. and This represents the element density of different elements in the topology optimization model. Indicated by unit Centered on the minimum filtration radius The set of all elements within a circular or spherical region of radius . Indicates the minimum filtration radius. Represents the weighting function. Indicates the original sensitivity. This represents the objective function or constraint function; where, In the formula, Indicates the minimum filtration radius. Represents the element in the topology optimization model Center point and unit The geometric distance between the center points.

8. A topology optimization device for irregularly shaped mirrors, characterized in that, include: The finite element analysis model building module is used to build a finite element analysis model based on the structural parameters of the irregularly shaped mirror, and to determine the design domain and non-design domain based on the finite element analysis model. A topology optimization model building module is used to build a topology optimization model based on the design domain and the non-design domain, and to determine the optimization formula of the topology optimization model; The irregular aperture Zernike basis function construction module is used to obtain the mirror node information in the finite element analysis model and construct the irregular aperture Zernike basis function based on the mirror node information in the finite element analysis model. The preprocessing basis function construction module is used to perform basis vector decoupling processing on the Zernike basis function of the irregular aperture in the discrete domain using the irregular aperture basis function reconstruction algorithm to obtain the preprocessed basis function. The iterative optimization processing module is used to combine the optimization formula and the preprocessing basis function to perform iterative optimization steps, update the topology optimization model, and obtain the updated irregular mirror topology optimization model.

9. An apparatus for performing a topology optimization method for irregularly shaped mirrors, characterized in that, include: processor; Memory used to store processor-executable instructions; When the processor executes the executable instructions, it implements the method as described in any one of claims 1 to 7.

10. A non-volatile computer-readable storage medium, characterized in that, Includes storage of computer programs or instructions that, when executed, cause the method as described in any one of claims 1 to 7 to be implemented.

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