A method for optimizing parameters of a prism array in a refractive daylighting window for a base dwelling

Abstract

The application provides a prism array parameter optimization method in a refractive lighting window for a bottom resident, and belongs to the technical field of residential building. The application realizes light path tracking through the establishment of an optical interface sequence model and a Snell law matrix propagation algorithm, constructs a parameter sensitivity matrix and performs rank value analysis to determine the effective degrees of freedom, establishes a double-layer game optimization model to cooperatively optimize the prism geometric parameters and size configuration, ensures the system robustness through eigenvalue stability analysis and parameter adjustment, and completes the final optimization configuration through branch decision making of single parameter fine adjustment or multi-parameter cooperative adjustment according to the eigenvalue distribution concentration degree, thereby solving the technical problem that the refractive lighting window for the bottom resident cannot simultaneously consider the multi-dimensional parameter coupling relationship and system stability when optimizing the prism array parameters.

Description

Technical Field

[0001] This invention belongs to the field of residential building technology, and more specifically, relates to a method for optimizing the parameters of a prism array inside a refractive skylight for ground floor residents. Background Technology

[0002] In the field of architectural daylighting, ground-floor residents often employ refractive daylighting windows due to severe obstruction. These windows redirect low-angle incident light to the interior ceiling via a vertical parallel prism array, achieving deep lighting. Traditional prism array parameter optimization methods primarily utilize ray tracing simulation and single-objective optimization algorithms. By adjusting parameters such as prism apex angle, height, and refractive index, they improve daylighting effects. These methods are widely used in the renovation of daylighting in office buildings and ground-floor residences. However, in actual engineering, because prism arrays involve multiple geometric and optical parameters with complex coupling relationships, traditional optimization methods struggle to accurately identify the interdependencies between parameters. This leads to parameter redundancy or insufficient degrees of freedom during optimization. Furthermore, the optimized parameter configurations are overly sensitive to manufacturing errors and environmental changes, resulting in poor system stability. In other words, existing technologies face the challenge of balancing multi-dimensional parameter coupling relationships and system stability when optimizing prism array parameters for ground-floor refractive daylighting windows. Summary of the Invention

[0003] In view of this, the present invention provides a method for optimizing the parameters of a prism array in a refractive lighting window for ground floor residents, which can solve the technical problem in the prior art that it is difficult to take into account the multi-dimensional parameter coupling relationship and system stability when optimizing the prism array parameters of a refractive lighting window for ground floor residents.

[0004] This invention is implemented as follows: It provides a method for optimizing the parameters of a prism array within a refractive skylight for ground-floor residents. This includes establishing an optical interface sequence model, decomposing the vertical parallel prism array into a continuous sequence of incident, total reflection, and exit surfaces; using refraction, reflection, and propagation matrices to describe the propagation behavior of light at each interface; constructing a Snell's law matrix propagation algorithm to calculate the direction and position vectors of the outgoing light rays to form a ray path tracing result; calculating a parameter sensitivity matrix by performing micro-perturbation analysis on prism apex angle, prism height, refractive index, and prism tilt angle parameters to construct a matrix of the influence of each parameter on the direction offset of the outgoing light rays; and performing matrix rank calculation. Analysis and degree-of-freedom determination: When the rank of the parameter sensitivity matrix is ​​less than 3, parameter coupling dimensionality reduction is performed; when the rank is greater than 5, parameter decoupling dimensionality expansion is performed. A two-layer game optimization model is established. The upper-layer game model optimizes the prism tilt angle and prism apex angle parameters with the goal of maximizing the satisfaction rate of the total internal reflection critical condition. The lower-layer game model optimizes the main and auxiliary prism size ratio parameters and auxiliary prism tilt angle parameters with the goal of maximizing the uniformity of azimuth response. Eigenvalue stability analysis is performed. When the maximum eigenvalue of the parameter sensitivity matrix exceeds the stability threshold, the prism array density adjustment process and tilt angle fine-tuning process are initiated. Branch decisions are formed based on the concentration of the eigenvalue distribution to complete the optimal configuration of the prism array parameters.

[0005] Among them, the optical interface sequence model discretizes the optical behavior of the vertical parallel prism array into a series of refraction and reflection processes between interfaces. Each interface is represented by a transfer matrix, and the matrix elements of the transfer matrix are determined by the interface normal vector, refractive index and geometric position.

[0006] The refraction matrix describes the change in direction of light when it passes through the interface between two media. The matrix elements of the refraction matrix are determined by the refractive index of the incident medium, the refractive index of the exit medium, and the interface normal vector, and are used to calculate the directional component of the refracted light.

[0007] The reflection matrix describes the change in direction of light when total internal reflection occurs at the interface, while the propagation matrix describes the rectilinear propagation of light in a uniform medium. The elements of the propagation matrix are determined by the propagation distance and the direction of the light.

[0008] The implementation process of the Snell's law matrix propagation algorithm involves representing the initial direction vector and initial position vector of the incident light as column vectors, multiplying them sequentially by the transmission matrix corresponding to each optical interface, and calculating the direction vector and position vector of the outgoing light.

[0009] The parameter sensitivity matrix is ​​a numerical matrix, with rows corresponding to different incident ray angles and columns corresponding to prism apex angle, prism height, refractive index, and prism inclination angle parameters. The matrix elements represent the offset of the outgoing ray direction caused by the unit change of the corresponding parameter at the corresponding incident angle.

[0010] Among them, parameter coupling dimensionality reduction is to identify the linearly related parameter combinations in the parameter sensitivity matrix, and merge multiple mutually coupled parameters into a comprehensive parameter for optimization. Parameter decoupling dimensionality expansion is to split a single parameter into multiple independent sub-parameters to increase the degree of freedom of optimization.

[0011] The objective function input of the upper-level game model includes the prism slope angle parameter and the prism apex angle parameter, and the output is the ratio of the incident angle range that satisfies the total internal reflection condition to the total incident angle range. The objective function input of the lower-level game model includes the main and auxiliary prism size ratio parameter, the auxiliary prism tilt angle parameter, the prism array density parameter, and the prism slope angle parameter.

[0012] Among them, the two-layer game optimization model achieves collaborative optimization through coupling terms. The prism slope tilt angle parameter serves as both the optimization variable of the upper-layer game model and the input parameter of the objective function of the lower-layer game model. This makes the optimization of the prism slope tilt angle parameter affect both the satisfaction rate of the total reflection critical condition and the uniformity of the azimuth response.

[0013] The prism array density adjustment process increases or decreases the prism array density parameter value according to the characteristics of the eigenvalue distribution. The tilt angle fine-tuning process makes a small adjustment of 0.1 to 0.5 degrees based on the current prism tilt angle parameter value until all eigenvalues ​​meet the stability constraint conditions.

[0014] The constraints of the upper-level game model include: a prism apex angle range of 55 to 65 degrees; a prism slope angle range of 28 to 33 degrees; and a prism height range of 25 mm to 35 mm. The constraints of the lower-level game model include: a primary-secondary prism size ratio range of 2 to 6; an auxiliary prism tilt angle range of 5 to 15 degrees; and a prism array density range of 25 prisms per 300 mm to 35 prisms per 300 mm.

[0015] The eigenvalue stability analysis assesses the system's sensitivity to parameter changes by calculating the eigenvalue distribution of the parameter sensitivity matrix, with a stability threshold of 2.5. When the largest eigenvalue of the parameter sensitivity matrix exceeds this threshold, the system parameter configuration is considered unstable. The concentration of the eigenvalue distribution is obtained by calculating the ratio of the eigenvalue variance to the eigenvalue mean, with a concentration threshold of 0.3. The dispersion of the eigenvalue distribution is obtained by calculating the ratio of the eigenvalue range to the eigenvalue mean, with an empirical dispersion threshold of 1.5. When the ratio corresponding to the concentration of the eigenvalue distribution is less than the concentration threshold, a single-parameter fine-tuning branch is executed, using a bisection method or the golden section method to perform high-precision optimization on the dominant parameter with the highest sensitivity. When the ratio corresponding to the dispersion of the eigenvalue distribution is greater than the dispersion threshold, a multi-parameter collaborative adjustment branch is executed, using a multi-objective optimization algorithm to simultaneously adjust multiple key parameters and search for the Pareto optimal solution set in the multi-dimensional parameter space.

[0016] This invention establishes a parameter sensitivity matrix based on Snell's law matrix propagation algorithm, transforming the influence of multidimensional parameters of the prism array on the emitted light rays into a quantifiable matrix form. It then identifies linear correlations between parameters through matrix rank analysis, achieving adaptive processing of parameter coupling dimensionality reduction and decoupling dimensionality expansion, accurately determining the effective degrees of freedom for system optimization. Simultaneously, an eigenvalue stability analysis mechanism is introduced. By calculating the eigenvalue distribution of the parameter sensitivity matrix, the system's sensitivity to parameter disturbances is assessed. When the maximum eigenvalue exceeds a stability threshold, the prism array density adjustment and tilt angle fine-tuning process is initiated, ensuring that the system parameter configuration meets optical performance targets while possessing good robustness. In summary, this invention solves the technical problem mentioned in the background art of balancing multidimensional parameter coupling relationships and system stability when optimizing prism array parameters for ground-floor refractive daylighting windows. Attached Figure Description

[0017] Figure 1 This is a schematic diagram of the overall structure of the vertical parallel prism array in the embodiment.

[0018] Figure 2 This is a schematic diagram of the light propagation path of a single prism.

[0019] Figure 3 This is a schematic diagram of the Snell's law matrix propagation algorithm.

[0020] Figure 4 This is a schematic diagram of the optimized two-layer game model structure.

[0021] Figure 5 This is a schematic diagram illustrating the optical working principle of a residential refractive skylight window. Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below.

[0023] This method includes the following steps:

[0024] S10. Establish an optical interface sequence model, decompose the vertical parallel prism array into a continuous interface sequence of incident surface, total reflection surface and exit surface, and use refraction matrix, reflection matrix and propagation matrix to describe the propagation behavior of light at each interface.

[0025] S20. Construct the Snell's law matrix propagation algorithm, represent the initial direction vector and initial position vector of the incident light as column vectors, multiply them by the transmission matrix corresponding to each optical interface in turn, calculate the direction vector and position vector of the outgoing light, and form a complete light path tracing result.

[0026] S30. Calculate the parameter sensitivity matrix. Perform micro-perturbations on the prism apex angle parameter, prism height parameter, refractive index parameter, and prism slope tilt angle parameter respectively. Calculate the outgoing ray direction offset caused by the micro-perturbation of each parameter using the Snell's law matrix propagation algorithm, and construct the parameter sensitivity matrix.

[0027] S40. Perform matrix rank analysis and degree of freedom determination, calculate the rank change of the parameter sensitivity matrix, perform parameter coupling dimensionality reduction when the rank of the parameter sensitivity matrix is ​​less than 3, and perform parameter decoupling dimensionality expansion when the rank of the parameter sensitivity matrix is ​​greater than 5, and determine the number of effective degrees of freedom for system optimization.

[0028] S50. Establish a two-layer game optimization model, which includes an upper-layer game model and a lower-layer game model. The upper-layer game model optimizes the prism slope angle parameter and the prism apex angle parameter with the objective of maximizing the satisfaction rate of the total internal reflection critical condition. The lower-layer game model optimizes the main and auxiliary prism size ratio parameter and the auxiliary prism tilt angle parameter with the objective of maximizing the uniformity of azimuth response. Two-layer collaborative optimization is achieved through coupling terms.

[0029] S60. Perform eigenvalue stability analysis and parameter adjustment, calculate the eigenvalue distribution of the parameter sensitivity matrix, and determine that the system parameter configuration is unstable when the maximum eigenvalue of the parameter sensitivity matrix exceeds the stability threshold. Start the prism array density adjustment process and tilt angle fine-tuning process until all eigenvalues ​​meet the stability constraint conditions.

[0030] S70. Based on the eigenvalue distribution, a branch decision is formed. If the concentration of the eigenvalue distribution is higher than the concentration threshold, a single-parameter fine adjustment branch is executed. If the dispersion of the eigenvalue distribution is higher than the dispersion threshold, a multi-parameter collaborative adjustment branch is executed to complete the final prism array parameter optimization configuration.

[0031] The optical interface sequence model discretizes the optical behavior of the vertical parallel prism array into a series of refraction and reflection processes between interfaces. Each interface is represented by a transfer matrix, and the matrix elements of the transfer matrix are determined by the interface normal vector, refractive index, and geometric position.

[0032] The refraction matrix describes the change in direction of light when it passes through the interface between two media. The matrix elements of the refraction matrix are determined by the refractive index of the incident medium, the refractive index of the exit medium, and the interface normal vector, and are used to calculate the directional component of the refracted light.

[0033] The reflection matrix describes the change in direction of light when total internal reflection occurs at the interface. The matrix elements of the reflection matrix are determined by the interface normal vector and the direction of the incident light ray, and are used to calculate the direction component of the reflected light ray.

[0034] The propagation matrix describes the rectilinear propagation of light in a uniform medium. The matrix elements of the propagation matrix are determined by the propagation distance and the direction of the light rays, and are used to update the position coordinates of the light rays.

[0035] The initial direction vector is a three-dimensional unit vector, the components of which represent the direction cosine of the light propagation direction in the spatial coordinate system, and the unit is dimensionless. The initial direction vector is determined by the incident angle and azimuth angle of the incident light.

[0036] The initial position vector is a three-dimensional coordinate vector, and its components represent the coordinate values ​​of the initial incident position of the light ray in the spatial coordinate system, with units of . The initial position vector is determined by the coordinates of the incident point of the incident ray on the incident surface.

[0037] The column vector form is a mathematical expression of arranging the initial direction vector and the initial position vector into a column matrix, which is used for matrix multiplication operations.

[0038] The transmission matrix is ​​a matrix describing the behavior of light at a single optical interface, including the refraction matrix, the reflection matrix, and the propagation matrix, which are applied sequentially to the column vector form through matrix multiplication.

[0039] The direction vector of the emitted light is the final propagation direction vector of the light after passing through all the optical interfaces of the vertical parallel prism array. The unit is dimensionless and it is used to determine whether the light is directed to the indoor ceiling.

[0040] The position vector of the emitted ray is the final position coordinate vector of the ray after passing through all the optical interfaces of the vertical parallel prism array, and the unit is 1. It is used to determine the position of the light ray's exit point.

[0041] The ray path tracing results contain the complete trajectory of the ray from incident to exit, including the incident point coordinates, incident angle, exit angle and propagation direction at each interface, and are used to evaluate the optical performance of the vertical parallel prism array.

[0042] The prism apex angle parameter is the apex angle value of a single prism in the vertical parallel prism array, in degrees, with an initial value of 60 degrees, provided by the basic scheme.

[0043] The prism height parameter refers to the height dimension of a single prism in the vertical parallel prism array, in units of... The initial value is 30. Provided by the basic solution.

[0044] The refractive index parameter is the optical refractive index value of the prism material in the vertical parallel prism array, in dimensionless form, with an initial value of 1.49, provided by the basic scheme.

[0045] The prism slope angle parameter is the tilt angle of the prism reflecting slope in the vertical parallel prism array relative to the horizontal plane, in degrees. The initial value is 30.5 degrees, provided by the solution to difficulty 3 of the basic scheme.

[0046] The micro-perturbation is to add or subtract a tiny amount from the original parameter value. The tiny amount is between 0.1% and 1% of the original parameter value and is used to calculate the degree of impact of parameter changes on system performance.

[0047] The offset of the emitted light direction is the angle between the direction vectors of the emitted light before and after the parameter micro-perturbation, in degrees, and is used to quantify the magnitude of the change in optical effect caused by the parameter change.

[0048] The parameter sensitivity matrix is ​​a numerical matrix, with rows corresponding to different incident ray angles and columns corresponding to the prism apex angle parameter, the prism height parameter, the refractive index parameter, and the prism slope angle parameter. The matrix elements represent the directional shift of the outgoing ray caused by the unit change of the corresponding parameter at the corresponding incident angle, with the unit being the change in degree per unit parameter.

[0049] The rank of the parameter sensitivity matrix is ​​the numerical value of the rank of the parameter sensitivity matrix, which is used to determine the linear correlation between parameters and the effective degrees of freedom for system optimization.

[0050] The parameter coupling dimensionality reduction process identifies linearly correlated parameter combinations in the parameter sensitivity matrix, merges multiple mutually coupled parameters into a single comprehensive parameter for optimization, reduces the number of optimization variables, and reduces computational load.

[0051] The parameter decoupling and dimension expansion process involves splitting a single parameter into multiple independent sub-parameters, increasing the degree of freedom in optimization, and improving the fineness of parameter configuration. This process is used to improve the control accuracy of the response to multiple azimuth angles.

[0052] The number of effective degrees of freedom is the number of independent adjustable parameters determined after the matrix rank analysis and degree of freedom determination. It determines the search space dimension of the optimization algorithm and affects the convergence speed and optimal solution quality of the optimization process.

[0053] The two-layer game optimization model is a game model structure that includes two layers of optimization problems. The upper-layer game model and the lower-layer game model influence each other through the coupling term to achieve collaborative optimization.

[0054] The upper-level game model is an optimization model with the objective of maximizing the satisfaction rate of the total internal reflection critical condition. The optimization variables are the prism slope angle parameter and the prism apex angle parameter. The output of the upper-level game model is the optimized prism slope angle parameter value and the prism apex angle parameter value.

[0055] The lower-level game model is an optimization model aimed at maximizing the uniformity of the azimuth response. The optimization variables are the size ratio parameter of the main and auxiliary prisms and the tilt angle parameter of the auxiliary prism. The output of the lower-level game model is the optimized size ratio parameter value of the main and auxiliary prisms and the tilt angle parameter value of the auxiliary prism.

[0056] The total internal reflection critical condition satisfaction rate is the percentage of light that can be totally reflected by the total internal reflection surface within the incident angle range of 50 degrees to 90 degrees, expressed as a percentage, and is used to evaluate the ability of the vertical parallel prism array to capture high-angle incident light.

[0057] The azimuth response uniformity is the standard deviation of the output efficiency of incident light rays from different azimuth angles after passing through the vertical parallel prism array, expressed as a percentage. The smaller the value, the more balanced the light processing capability of each azimuth angle, and it is used to evaluate all-weather lighting performance.

[0058] The ratio parameter of the main and auxiliary prism dimensions is the ratio of the width of the main vertical prism to the width of the auxiliary tilting prism. The unit is dimensionless, and the initial value is 3. It is determined by the design requirements of the bidirectional composite prism array structure in the solution to the difficulty 4 of the basic scheme.

[0059] The tilt angle parameter of the auxiliary prism is the tilt angle value of the auxiliary tilt prism relative to the main vertical prism, in degrees. The initial value is 10 degrees, which is determined by the bidirectional composite prism array structure design requirements in the solution to difficulty 4 of the basic scheme.

[0060] The coupling term is that the prism slope angle parameter serves as both the optimization variable of the upper-level game model and the objective function input parameter of the lower-level game model. The coupling term causes the optimization of the prism slope angle parameter to affect both the satisfaction rate of the total internal reflection critical condition and the uniformity of the azimuth response.

[0061] The objective function of the upper-level game model is used to calculate the relationship between the satisfaction rate of the total internal reflection critical condition and the tilt angle parameter of the prism and the apex angle parameter of the prism. The input includes the tilt angle parameter of the prism and the apex angle parameter of the prism, and the output is the ratio of the incident angle range that satisfies the total internal reflection condition to the total incident angle range. The functional form of the objective function of the upper-level game model is that the satisfaction rate is equal to the number of successfully internally reflected rays divided by the total number of incident rays multiplied by the first normalization coefficient. The first normalization coefficient is determined by the standard satisfaction rate value to unify the dimensions of the left and right sides of the equation to dimensionless.

[0062] The objective function of the lower-level game model is used to calculate the relationship between the azimuth response uniformity and the size ratio parameters of the primary and secondary prisms and the tilt angle parameters of the secondary prism. The inputs include the size ratio parameters of the primary and secondary prisms, the tilt angle parameters of the secondary prism, the prism array density parameters, and the prism slope tilt angle parameters. The output is the normalized value of the standard deviation of the emission efficiency in each azimuth angle. The functional form of the objective function of the lower-level game model is that the uniformity index is equal to the standard deviation of the emission efficiency divided by the average emission efficiency multiplied by the second normalization coefficient minus the size ratio deviation divided by the standard size ratio. The second normalization coefficient is determined by the standard uniformity value to unify the dimensions of the left and right sides of the equation to dimensionless.

[0063] The prism array density parameter is the number of prisms in the vertical parallel prism array per unit length, expressed in units of prisms per 300. The initial value is 30 per 300 Provided by the basic solution.

[0064] The constraints of the upper-level game model include a prism apex angle range of 55 to 65 degrees, a prism slope angle range of 28 to 33 degrees, and a prism height range of 25 degrees. Up to 35 Ensure that the parameters are within the physically achievable range.

[0065] The constraints of the lower-level game model include: the size ratio parameter of the main and auxiliary prisms being constrained to a range of 2 to 6; the tilt angle parameter of the auxiliary prism being constrained to a range of 5 to 15 degrees; and the density parameter of the prism array being constrained to 25 prisms per 300 prisms. Up to 35 per 300 This ensures the manufacturability of the structure.

[0066] The eigenvalue distribution refers to the distribution of all eigenvalues ​​of the parameter sensitivity matrix in the complex plane, including the magnitude, real part, imaginary part, and distribution concentration of the eigenvalues, and is used to evaluate the system's sensitivity and stability to parameter changes.

[0067] The largest eigenvalue of the parameter sensitivity matrix is ​​the eigenvalue with the largest modulus among all eigenvalues ​​of the parameter sensitivity matrix. It is dimensionless and is used to determine the overall sensitivity level of the system.

[0068] The stability threshold is the critical value of the maximum eigenvalue of the parameter sensitivity matrix, in dimensionless form, with a value of 2.5. When the maximum eigenvalue of the parameter sensitivity matrix exceeds the stability threshold, it indicates that the system is too sensitive to changes in a certain parameter.

[0069] The instability of the system parameter configuration indicates that the current parameter combination of the vertical parallel prism array will cause significant fluctuations in optical performance under slight perturbations, requiring adjustment of the parameter configuration to reduce sensitivity.

[0070] The prism array density adjustment process involves increasing or decreasing the prism array density parameter value based on the characteristics of the feature value distribution. Increasing the prism array density parameter value improves the continuity of light capture, while decreasing the prism array density parameter value reduces optical interference between adjacent prisms.

[0071] The tilt angle fine-tuning process involves making a small adjustment of 0.1 to 0.5 degrees based on the current tilt angle parameter value of the prism slope, and finding the optimal tilt angle value that reduces the maximum eigenvalue of the parameter sensitivity matrix to below the stability threshold through iterative calculation.

[0072] The stability constraint condition is that the modulus of all eigenvalues ​​is less than the stability threshold and the real part of all eigenvalues ​​is negative. When the stability constraint condition is satisfied, the performance change of the system is controllable under small parameter perturbations.

[0073] The concentration of the eigenvalue distribution is a quantitative index of the degree of concentration of the eigenvalue distribution. It is obtained by calculating the ratio of the eigenvalue variance to the eigenvalue mean, and the unit is dimensionless. It is used to determine whether the system performance is dominated by a few parameters.

[0074] The concentration threshold is a critical value of the concentration of the feature value distribution, in dimensionless form, with a value of 0.3. When the ratio corresponding to the concentration of the feature value distribution is less than the concentration threshold, it is determined to be a concentrated distribution.

[0075] The dispersion of the eigenvalue distribution is a quantitative index of the degree of dispersion of the eigenvalue distribution. It is obtained by calculating the ratio of the eigenvalue range to the eigenvalue mean, and the unit is dimensionless. It is used to judge the degree of influence of multiple parameters on system performance.

[0076] The dispersion threshold is a critical value of the dispersion of the feature value distribution, in dimensionless form, with a value of 1.5. When the ratio corresponding to the dispersion of the feature value distribution is greater than the dispersion threshold, it is determined to be a dispersed distribution.

[0077] The single-parameter fine-tuning branch is a processing flow that performs high-precision optimization on the dominant parameter with the highest sensitivity. It uses the bisection method or the golden section method to search for the optimal value within a small range. The single-parameter fine-tuning branch is used when the feature value distribution is concentrated.

[0078] The multi-parameter collaborative adjustment branch is a processing flow that adjusts multiple key parameters simultaneously. It uses a multi-objective optimization algorithm to search for the Pareto optimal solution set in a multi-dimensional parameter space. The multi-parameter collaborative adjustment branch is used when the feature value distribution is dispersed.

[0079] The final optimized configuration of prism array parameters includes optimized values ​​for the prism apex angle, prism slope angle, prism height, main and auxiliary prism size ratio, auxiliary prism tilt angle, and prism array density, forming a complete combination of design parameters to guide actual manufacturing.

[0080] The specific implementation methods of the above steps are described in detail below.

[0081] The specific implementation of step S10 involves decomposing the three-dimensional optical structure of the vertical parallel prism array into a combination of several two-dimensional optical interfaces. First, all key interfaces along the light propagation path in the prism array are identified, including the incident surface between air and prism material, the total internal reflection surface inside the prism, and the exit surface between the prism material and air. Then, a local coordinate system is established for each interface, and an interface normal vector is defined. The direction of the normal vector is determined by the interface geometry and spatial position. Next, the interface type is determined based on the difference in refractive index between the media on both sides of the interface. Interfaces with refractive indices changing from small to large are defined as refractive interfaces and associated with a refractive matrix. Interfaces with refractive indices changing from large to small and incident angles greater than the critical angle are defined as reflective interfaces and associated with a reflection matrix. Finally, a propagation matrix is ​​inserted between adjacent interfaces to describe the rectilinear propagation process of light in a uniform medium, forming a complete interface sequence model. The function of the optical interface sequence model is to transform the complex three-dimensional optical tracking problem into a series of two-dimensional interface processing problems, which facilitates the subsequent matrix operations.

[0082] The specific implementation of step S20 is based on Snell's law and the law of reflection to construct a matrix-form ray propagation calculation framework. First, the spatial direction of the incident ray is represented by three direction cosines and assembled into an initial direction vector. The three-dimensional coordinates of the incident point are represented and assembled into an initial position vector. Then, the initial direction vector and the initial position vector are merged and arranged into a six-dimensional column vector. Next, the transmission matrices corresponding to each interface established in step S10 are extracted sequentially according to the ray propagation order. The transmission matrix includes a refraction matrix describing refraction behavior, a reflection matrix describing reflection behavior, and a propagation matrix describing linear propagation. Subsequently, matrix left multiplication is used to apply each transmission matrix to the six-dimensional column vector in sequence. Each matrix multiplication operation updates the direction and position information of the ray. Finally, the exit direction vector and exit position vector of the ray after passing through all interfaces are obtained. The advantage of the Snell's law matrix propagation algorithm is that it transforms the complex optical tracking process into standard linear algebra operations, which facilitates efficient calculation and parallel acceleration using matrix operation libraries.

[0083] The specific implementation of step S30 is to calculate the influence of each prism parameter on optical performance using the numerical perturbation method. First, several representative incident angles within the range of 50 to 90 degrees are selected as test rays. For each test ray, the Snell's law matrix propagation algorithm of step S20 is used to calculate the outgoing ray direction vector under the reference state. Then, small perturbations are applied to the prism apex angle parameter, prism height parameter, refractive index parameter, and prism slope angle parameter in sequence. The perturbation amount is 0.5% of the original value of the parameter. The Snell's law matrix propagation algorithm is used again to calculate the outgoing ray direction vector after perturbation. Then, the angle between the outgoing ray direction vectors before and after perturbation is calculated as the outgoing ray direction offset. The ratio of the outgoing ray direction offset corresponding to each incident angle to the perturbation amount of each parameter is filled into the corresponding position of the parameter sensitivity matrix. The row index corresponds to the incident angle, and the column index corresponds to the parameter type. The parameter sensitivity matrix quantitatively reflects the influence intensity of each parameter change on the ray propagation path at different incident angles, providing sensitivity information for subsequent parameter optimization.

[0084] The specific implementation of step S40 involves using matrix rank calculation and linear correlation analysis to determine the effective dimension of parameter optimization. First, singular value decomposition is performed on the parameter sensitivity matrix obtained in step S30. The matrix rank is determined by counting the number of singular values ​​greater than a set threshold. The rank threshold is 1% of the maximum singular value. When the matrix rank is less than 3, it indicates that multiple parameters have a highly correlated impact on optical performance. At this time, parameter coupling dimensionality reduction processing is performed. By identifying the linear combination relationship between the column vectors of the sensitivity matrix, the related parameters are merged into comprehensive parameters. When the matrix rank is greater than 5, it indicates that the impact of each parameter on optical performance is relatively independent and there is a subdivided optimization space. At this time, parameter decoupling dimensionality expansion processing is performed. The original parameters are split into multiple sub-parameters according to different incident angle ranges and optimized separately. The purpose of the matrix rank analysis and degree of freedom determination is to adaptively adjust the dimension of the optimization problem according to the numerical characteristics of the parameter sensitivity matrix, so as to avoid computational waste caused by redundant parameters or limited optimization effect caused by insufficient parameters.

[0085] The specific implementation of step S50 involves constructing a game theory model structure with two optimization objectives. The upper-level game theory model aims to improve the success rate of total internal reflection of high-angle incident light. The input parameters are the prism tilt angle and the prism apex angle. By adjusting the prism tilt angle, the incident angle of light rays within the 50-90 degree incident angle range at the reflecting surface is made to exceed the critical angle as much as possible. The objective function is calculated by statistically analyzing the percentage of light rays that meet the total internal reflection condition out of the total number of test light rays. The lower-level game theory model aims to improve the balance of light response at different azimuth angles. The input parameters are the ratio of the main and auxiliary prism sizes and the auxiliary prism size. The mirror tilt angle parameter is used to minimize the standard deviation of the exit efficiency of incident rays from all directions (north, south, east, west) by adjusting the size ratio of the main and auxiliary prisms and the tilt angle parameter of the auxiliary prism. The objective function is calculated by dividing the standard deviation of the exit efficiency from each direction by the average exit efficiency. The prism tilt angle parameter serves as both the optimization variable of the upper-level game model and the input parameter of the lower-level game model, forming a coupling term. An alternating iterative method is used to solve the two-level game optimization model. First, the lower-level parameters are fixed to solve for the optimal solution of the upper level. Then, the optimal solution of the upper level is substituted into the lower level to solve for the optimal solution of the lower level. This iteration is repeated until the change in the objective function values ​​of both levels is less than 0.1%.

[0086] The specific implementation of step S60 involves evaluating the stability of the parameter configuration and performing parameter adjustment through eigenvalue decomposition. First, eigenvalue decomposition is performed on the parameter sensitivity matrix obtained in step S30 to extract all eigenvalues ​​and calculate the magnitude of each eigenvalue. The maximum magnitude is compared with a stability threshold of 2.5. When the maximum eigenvalue magnitude exceeds the stability threshold, it is determined that the current parameter configuration is too sensitive to disturbances, and the parameter adjustment process is initiated. The parameter adjustment process includes two sub-processes: prism array density adjustment and tilt angle fine-tuning. The prism array density adjustment sub-process determines whether to increase or decrease the number of prisms based on the direction of the eigenvector. Density is reduced if the eigenvector is dominant in the spatial direction component to reduce interference from adjacent prisms; if the eigenvector is dominant in the angular direction component, density is increased to improve the continuity of light capture. The tilt angle fine-tuning sub-process fine-tunes the current prism tilt angle parameter value in steps of 0.2 degrees. After each fine-tuning, the parameter sensitivity matrix and the maximum eigenvalue are recalculated until the maximum eigenvalue drops below the stable threshold and all eigenvalues ​​have negative real parts. The purpose of the eigenvalue stability analysis is to ensure that the optimized parameter configuration can maintain stable optical performance under the influence of actual manufacturing errors and environmental changes.

[0087] The specific implementation of step S70 involves selecting different parameter optimization strategies based on the eigenvalue distribution characteristics. First, the variance and mean of all eigenvalues ​​obtained from the eigenvalue decomposition in step S60 are calculated. The variance is divided by the mean to obtain the concentration index, and the eigenvalue range is divided by the mean to obtain the dispersion index. When the concentration index is less than the concentration threshold of 0.3, the eigenvalue distribution is considered concentrated, indicating that the system performance is mainly controlled by a certain dominant parameter. At this point, a single-parameter fine-tuning branch is executed, using a bisection method to search for the optimal value within a small range of the dominant parameter. The search range is 5% above and below the current value, with an accuracy requirement of 0.01 degrees or 0.01. When the dispersion index is greater than the dispersion threshold of 1.5, the feature value distribution is determined to be dispersed, indicating that multiple parameters have a significant impact on the system performance. At this time, a multi-parameter collaborative adjustment branch is executed, and a genetic algorithm is used to simultaneously optimize all key parameters in a multi-dimensional parameter space. The population size is set to 50, the number of iterations is set to 100 generations, the crossover probability is set to 0.8, and the mutation probability is set to 0.05. The purpose of the branch decision mechanism is to adaptively select the most efficient optimization algorithm based on the numerical characteristics of the parameter sensitivity matrix, thereby improving the efficiency and effectiveness of parameter optimization.

[0088] It should be noted that the key technical ideas of this invention include the application of the Snell's Law matrix propagation algorithm, the construction of a two-layer game optimization model, and a stability control mechanism based on eigenvalue analysis. The Snell's Law matrix propagation algorithm transforms traditional interface-by-interface ray tracing into matrix multiplication. Compared to traditional methods that require repeatedly judging refraction or reflection conditions and calculating new directions at each interface, the matrix method standardizes and vectorizes the calculation process by pre-constructing a transfer matrix, significantly reducing computational complexity and facilitating acceleration using parallel computing resources. Furthermore, the matrix form is naturally suitable for sensitivity analysis; the impact of parameter changes on the ray path can be quickly obtained by calculating the partial derivatives of the transfer matrix elements. The two-layer game optimization model decouples the optimization of high-angle total internal reflection conditions and the optimization of multi-azimuth response uniformity into two relatively independent yet interconnected sub-problems. Compared to traditional single-objective optimization or simple weighted multi-objective optimization, the two-layer structure establishes a constraint relationship between the two optimization objectives through coupling terms, avoiding the problem of over-optimization of one objective leading to severe performance degradation of the other, and achieving a more reasonable balance between total internal reflection efficiency and azimuth uniformity in the optimization results. The stability control mechanism based on eigenvalue analysis determines the system's sensitivity to parameter disturbances by monitoring the eigenvalue distribution of the parameter sensitivity matrix. Compared to traditional methods that only focus on optimizing the objective function value while neglecting parameter robustness, the eigenvalue method can proactively identify and avoid parameter configurations sensitive to manufacturing errors during the optimization process, ensuring the reliability and stability of the optimization results in practical applications. The synergistic effect of the three technical approaches is reflected in the matrix propagation algorithm providing an efficient optical performance calculation tool for the two-level game model and eigenvalue analysis. The two-level game model fully utilizes the stability constraint information provided by eigenvalue analysis during the optimization process, while eigenvalue analysis relies on the matrix propagation algorithm to quickly evaluate the system response after parameter adjustment. The three form a complete parameter optimization system that is computationally efficient, rationally optimized, and yields stable results. Compared to traditional trial-and-error methods or single optimization methods, this synergistic system can obtain prism array parameter configurations with excellent optical performance and strong manufacturing robustness while ensuring computational efficiency.

[0089] It should be noted that this invention also solves the following technical problem: In the design of refractive lighting windows for ground-floor residents, the emission efficiency of incident light from different azimuth angles after passing through the prism array varies significantly, resulting in uneven lighting performance throughout the day and affecting the quality of the indoor light environment. This invention establishes a two-layer game optimization model. In the lower-layer game model, the size ratio parameters of the main and auxiliary prisms and the tilt angle parameters of the auxiliary prism are optimized with the goal of maximizing the uniformity of azimuth response. By adjusting the geometric configuration of the bidirectional composite prism array, light from different azimuth angles can be effectively captured and redirected, significantly reducing the standard deviation of emission efficiency from each azimuth angle and achieving a balanced lighting response in all directions. At the same time, through coupling terms, the prism tilt angle parameters are co-optimized with the total internal reflection critical condition satisfaction rate target of the upper-layer game model, ensuring that the capture capability of high-angle incident light is not lost while improving azimuth uniformity.

[0090] Specifically, the principle of this invention is as follows: The invention solves the aforementioned technical problems by discretizing the optical behavior of the prism array into a matrix propagation process of the interface sequence, allowing the impact of parameter changes on optical performance to be precisely quantified through matrix operations. The rank of the parameter sensitivity matrix directly reflects the dimensionality of the parameter space. When the rank is less than 3, it indicates the existence of linearly correlated parameter combinations requiring dimensionality reduction. When the rank is greater than 5, it indicates that a single parameter can be decomposed into multiple sub-parameters to improve control accuracy. This parameter decoupling method based on matrix algebra theory mathematically reveals the dependency relationship between parameters. The eigenvalue distribution reflects the system's response intensity to disturbances in different parameter directions. The largest eigenvalue corresponds to the most sensitive parameter direction. By constraining the eigenvalue below a stability threshold, it can be ensured that the system's performance changes are controllable under small parameter deviations. This stability criterion based on matrix spectrum theory provides a theoretical guarantee for the robustness of parameter configuration.

[0091] The following provides a specific embodiment 1 of the present invention, and the specific implementation of each step in this embodiment 1 is described in detail below.

[0092] The specific implementation of step S10 involves establishing an optical interface sequence model, representing the optical propagation behavior of the vertical parallel prism array as a continuous sequence of the refraction matrix, reflection matrix, and propagation matrix. (Refraction matrix) The expression is:

[0093] ;

[0094] In the formula, It is a three-dimensional identity matrix with dimensions of 3×3; The refractive index of the incident medium is dimensionless and has an initial value of 1.0. The refractive index of the exit medium is dimensionless, with an initial value of 1.49. The interface normal vector is dimensionless and is represented as a three-dimensional column vector. is the direction vector of the incident ray, with dimensionless units, and is represented as a three-dimensional column vector; This is the transpose of the interface normal vector, represented as a three-dimensional row vector; This represents a three-dimensional square matrix formed by the outer product operation, with dimensions of 3×3; This indicates a vector dot product operation, the result of which is a scalar. Reflection matrix. The expression is:

[0095] .

[0096] Propagation Matrix The expression acting on the position vector is:

[0097] ;

[0098] In the formula, The propagated position vector, in units of , represented as a three-dimensional column vector; The position vector before propagation, in units of , represented as a three-dimensional column vector; The distance to be transmitted is expressed in units of 1. ; Let be the direction vector of light rays during propagation, with dimensionless units, and represented as a three-dimensional column vector.

[0099] The specific implementation of step S20 is to construct the Snell's law matrix propagation algorithm, which involves using the initial direction vector of the incident light ray. and initial position vector Represented as a column vector, it is multiplied by the transmission matrices of each interface in turn to obtain the outgoing ray direction vector. and position vector .in Let be the initial direction vector of the incident ray, in dimensionless form, represented as a three-dimensional column vector; Let be the initial position vector of the incident ray, in units of . This is represented as a three-dimensional column vector. The calculation formula is:

[0100] ;

[0101] ;

[0102] In the formula, Here is the refraction matrix of the incident surface, with dimensions of 3×3; The reflection matrix of the total reflection surface has a dimension of 3×3; The refraction matrix of the exit surface has a dimension of 3×3; is the direction vector of the outgoing ray, with dimensionless units, and is represented as a three-dimensional column vector; Let be the position vector of the outgoing ray, in units of . , represented as a three-dimensional column vector; The propagation distance from the incident surface to the total reflection surface is expressed in units of 1. ; The propagation distance from the total reflection surface to the exiting surface is expressed in units of 1. ; The propagation distance after exiting the surface, in units of ; is the direction vector of the light ray after refraction at the incident surface, with dimensionless units, and is represented as a three-dimensional column vector; is the vector of the direction of the light rays after reflection by the total internal reflection surface. It is dimensionless and represented as a three-dimensional column vector. This is the direction vector of the light ray after refraction at the exit surface. It is dimensionless and represented as a three-dimensional column vector.

[0103] The specific implementation of step S30 is to calculate the parameter sensitivity matrix. prism apex angle parameters Prism height parameters Refractive index parameters And prism slope angle parameters Apply micro-perturbations respectively Spend, , and The angle of deviation of the emitted ray is calculated using the Snell's law matrix propagation algorithm. .in This is the prism apex angle parameter, in degrees, with an initial value of 60 degrees; This is the prism height parameter, in units of... The initial value is 30. ; Here is the refractive index parameter of the prism material, in dimensionless form, with an initial value of 1.49; This represents the prism's inclined plane tilt angle parameter, in degrees, with an initial value of 30.5 degrees. Parameter sensitivity matrix elements. The calculation formula is:

[0104] ;

[0105] In the formula, These are the elements of the parameter sensitivity matrix, and their units are dimensionless. The incident angle number ranges from 1 to 18. The parameter number ranges from 1 to 4, where 1 corresponds to the prism apex angle parameter. 2 corresponds to the prism height parameter 3 corresponds to the refractive index parameter 4 corresponds to the prism's inclined plane tilt angle parameter ; For the first The offset of the outgoing ray direction corresponding to each incident angle, in degrees; For the first The micro-perturbation of each parameter, with the same unit as the corresponding parameter; For the first The initial values ​​of each parameter are in the same unit as the corresponding parameter.

[0106] The specific implementation of step S40 is to perform matrix rank analysis and calculate the parameter sensitivity matrix. rank ,in Representation matrix The rank of is dimensionless. Parameter coupling dimensionality reduction is performed at the time, merging linearly related parameters into a comprehensive parameter. The calculation formula is:

[0107] ;

[0108] In the formula, These are comprehensive parameters, and the units are dimensionless. The coupling coefficient of the first parameter is dimensionless and is determined by singular value decomposition. The coupling coefficient for the second parameter is dimensionless and is determined through singular value decomposition. This is the current value of the first linearly dependent parameter, in the same unit as the parameter. This is the initial value for the first linearly dependent parameter, with the same unit as the parameter. This is the current value of the second linear correlation parameter, in the same unit as the parameter. This is the initial value for the second linear correlation parameter, with the same units as the first parameter. When Parameter decoupling and dimension expansion are performed during the process, splitting single parameters into sub-parameters. The calculation formula is:

[0109] ;

[0110] In the formula, These are the normalized values ​​of the main parameters, in dimensionless units. This is the current value of the first sub-parameter, in the same unit as the main parameter; This is the initial value of the first sub-parameter, with the same unit as the main parameter; This is the current value of the second sub-parameter, in the same unit as the main parameter; This is the initial value for the second sub-parameter, with the same unit as the main parameter; This is the current value of the third sub-parameter, in the same unit as the main parameter; This is the initial value for the third sub-parameter, with the same units as the main parameter. The number of effective degrees of freedom is denoted as... ,in The number of effective degrees of freedom is expressed in units of 1.

[0111] The specific implementation of step S50 is to establish a two-layer game optimization model, where the objective function of the upper-layer game model is... The expression is:

[0112] ;

[0113] In the formula, This represents the objective function value of the upper-level game model, in dimensionless form. The number of rays required to satisfy the total internal reflection condition, measured in rays, is obtained through ray tracing simulation. The total number of incident rays, expressed in rays, is obtained through ray tracing simulation; 0.85 is the standard satisfaction rate normalization coefficient, dimensionless. The objective function of the lower-level game model. The expression is:

[0114] ;

[0115] In the formula, This represents the objective function value of the lower-level game model, in dimensionless form. The standard deviation of the emission efficiency at each azimuth angle is expressed as a percentage. Average ejection efficiency, expressed as a percentage; 0.15 is the standard homogeneity normalization coefficient, expressed as dimensionless. The primary and secondary prism dimensions are proportional, in dimensionless units, with an initial value of 3; 3 represents the standard dimension ratio, also in dimensionless units. The coupling term is expressed through the prism's inclined plane angle parameter. This parameter is implemented so that it serves as both an optimization variable in the upper-level game model and an input parameter to the objective function of the lower-level game model. The constraints of the upper-level game model include... , , The constraints of the lower-level game model include: , , ,in The prism tilt angle parameter is used as an auxiliary parameter, in degrees, with an initial value of 10 degrees, determined by the design requirements of the bidirectional composite prism array structure in the solution to difficulty 4 of the basic scheme. This refers to the prism array density parameter, in units of prisms per 300. The initial value is 30 per 300 .

[0116] The specific implementation of step S60 is to perform eigenvalue stability analysis and calculate the parameter sensitivity matrix. eigenvalues By solving the characteristic equation Obtain, among which Represents matrix determinant operations. For the first There are eigenvalues, with dimensionless units, which can be complex or real numbers. When the largest eigenvalue... When the density exceeds the stability threshold of 2.5, the prism array density adjustment process is initiated, and the adjustment formula is as follows:

[0117] ;

[0118] In the formula, To adjust the density parameters of the rear prism array, the unit is prisms per 300. ; To adjust the density parameters of the front prism array, the unit is prisms per 300. ; The largest eigenvalue of the parameter sensitivity matrix is ​​dimensionless, and the calculation formula is: ,in The total number of eigenvalues. Represents eigenvalues The modulus value; a positive sign is used to increase density, and a negative sign is used to decrease density; 2.5 is the stability threshold, and the unit is dimensionless. The tilt angle fine-tuning process uses an iterative formula:

[0119] ;

[0120] In the formula, For the first The prism slope angle parameter for the next iteration, in degrees; For the first The prism slope angle parameter for the next iteration, in degrees; This is the number of iterations, initially set to 0. This represents the partial derivative of the largest eigenvalue with respect to the dip angle, expressed in degrees. The stability constraint requires all eigenvalues... satisfy and ,in Represents eigenvalues The real part of , with units of dimensionless.

[0121] The specific implementation of step S70 involves forming a branch decision based on the eigenvalue distribution and calculating the eigenvalue distribution concentration. and dispersion The formulas are as follows:

[0122] ;

[0123] ;

[0124] In the formula, The eigenvalue distribution concentration is dimensionless. The eigenvalue distribution dispersion is dimensionless. The mean of the eigenvalues ​​is dimensionless and is calculated using the following formula: ; The total number of eigenvalues; The minimum eigenvalue modulus, in dimensionless form, is calculated using the following formula: .when When a single parameter is used for fine-tuning, the branch is executed, and the golden section method is used to search for the optimal value; when The multi-parameter collaborative adjustment branch is executed in real time, and the Pareto optimal solution set is solved using a non-dominated sorting genetic algorithm to complete the optimal configuration of the prism array parameters. Here, 0.3 is the concentration threshold (dimensionless), and 1.5 is the dispersion threshold (dimensionless).

[0125] standard deviation of ejection efficiency The calculation formula is:

[0126] ;

[0127] In the formula, This represents the number of azimuth sampling points, typically set to 72. For the first The emission efficiency corresponding to each azimuth angle is expressed as a percentage. The azimuth number ranges from 1 to... Average launch efficiency The calculation formula is:

[0128] .

[0129] The final optimized prism array parameter configuration includes the optimized prism apex angle parameters. Prism Inclination Angle Parameters Prism height parameters Dimensions of primary and secondary prisms Auxiliary prism tilt angle parameters and prism array density parameters The units are the same as the initial values ​​of each parameter. The specific implementation methods for the explanation of other terms are the same as those described above, and will not be repeated in detail here.

[0130] To better understand and implement this invention, a specific application scenario is provided below as Example 2: A technical team, while designing a multi-story residential project in a central urban area, faced the problem of severely insufficient natural lighting for ground-floor residents due to obstruction from surrounding buildings. The windows of the ground-floor residences were only 2.8 meters from the ground, while a 12-story existing building adjacent to the south provided obstruction at a 68-degree angle, resulting in only high-angle skylight reaching the windows most of the time. Traditional flat glass windows could not effectively utilize these high-angle incident rays, and the indoor illuminance consistently fell below the national standard minimum of 300 lx. The technical team decided to adopt the prism array parameter optimization method within the refractive light-transmitting window of this invention to design an optical window system capable of redirecting high-angle incident rays deeper into the interior space.

[0131] The technical team first established an optical interface sequence model. The optical behavior of the vertical parallel prism array was decomposed into three continuous interfaces: the incident surface, the total internal reflection surface, and the exit surface. The incident surface is the first contact interface between the prism and air, with its normal vector pointing inwards and its refractive index abruptly changing from 1.0 to 1.49. The total internal reflection surface is the tilted reflecting surface of the prism, with its normal vector forming a 59.5-degree angle with the horizontal plane to achieve the total internal reflection effect. The exit surface is the second contact interface between the prism and air, with its normal vector perpendicular to the bottom edge of the prism. Each interface has a corresponding transmission matrix. The elements of the refraction matrix are determined by the refractive index of the incident medium (1.0), the refractive index of the exit medium (1.49), and the interface normal vector. The elements of the reflection matrix are determined by the interface normal vector and the direction of the incident ray. The elements of the propagation matrix are determined by the propagation distance inside the prism and the direction of the ray. This interface sequence model transforms the complex three-dimensional ray tracing problem into a series of matrix operations in a two-dimensional plane.

[0132] like Figure 1 As shown, the vertical parallel prism array designed by the technical team is installed on the inner surface of the outer glass layer of the double-glazed unit. The outer glass layer is 6mm thick and made of ultra-clear glass to reduce light loss. The insulated layer is 20mm thick and filled with dry air to provide thermal insulation. The inner glass layer is 5mm thick and made of ordinary float glass. The prism array is made of polymethyl methacrylate (PMMA) with a refractive index of 1.49 and a light transmittance of 92%. The width and height of each prism unit are initially set at 30mm, and the apex angle is initially set at 60 degrees. The prism array is arranged vertically with a density of 10 prism units per 300mm length.

[0133] The technical team developed a Snell's law matrix propagation algorithm to track the path of light rays. A typical high-angle incident ray was selected as the test object, with an incident angle of 70 degrees and an azimuth angle of 0 degrees, indicating that the ray is incident from due south at a 70-degree elevation angle. The initial direction vector is represented as a column vector with three components of 0.342, 0, and 0.940, corresponding to the direction cosines of the ray in the spatial coordinate system. The initial position vector has three components of 0, 15, and 0 (in mm), indicating that the ray is incident from the midpoint of the left edge of the prism array. The algorithm first multiplies the initial direction vector by the refraction matrix to calculate the refraction direction vector of the ray after it enters the prism material, with the three components changing to 0.228, 0, and 0.974. Then, it multiplies by the propagation matrix, calculating the distance the ray travels to the total internal reflection surface based on the prism's internal geometry as 34.6 mm, and updates the position vector to 7.89, 15, and 33.7 (in mm). Next, the reflection matrix is ​​multiplied by left to calculate the direction vector of the light ray after total internal reflection at the total internal reflection surface. The three components change to 0.228, 0, and -0.974, indicating a significant change in the light ray's direction, from upward to downward propagation. Finally, the refraction and propagation matrices of the exit surface are multiplied by left to obtain the final direction vector of the outgoing light ray: 0.342, 0, -0.940, and the position vector: 17.4, 15, 0, in mm. The entire ray path tracing results show that the 70-degree incident angle light ray, after passing through the prism array, has an exit angle of -70 degrees, successfully achieving the effect of guiding the light ray downwards to the indoor ceiling.

[0134] like Figure 2 As shown, the light propagation path of a single prism clearly demonstrates the optical redirection mechanism. Light enters the prism at a 70-degree angle from the incident surface, undergoing a first refraction at the incident surface, reducing the refraction angle to 46.8 degrees. The light then propagates in a straight line inside the prism, reaching the tilted total internal reflection surface. The tilt angle of the total internal reflection surface is set to 30.5 degrees, ensuring that the incident angle of the light reaching this surface reaches 73.2 degrees, exceeding the critical angle of 41.8 degrees and satisfying the total internal reflection condition. After total internal reflection at the total internal reflection surface, the light's propagation direction changes downwards, continuing to propagate inside the prism to the exit surface. A second refraction occurs at the exit surface, and the light leaves the prism and enters the air, ultimately exiting at an angle of -70 degrees, forming a 70-degree downward angle with the horizontal, effectively illuminating the indoor ceiling area. The prism's apex angle is set to 60 degrees, ensuring the geometric relationship between the incident and exit surfaces, allowing the light to smoothly complete the entire refraction-reflection-refraction optical path.

[0135] The technical team calculated a parameter sensitivity matrix to assess the impact of each design parameter on optical performance. Nine incident angles within the range of 50 to 90 degrees were selected as row indices for the parameter sensitivity matrix: 50°, 55°, 60°, 65°, 70°, 75°, 80°, 85°, and 90°. The prism apex angle, prism height, refractive index, and prism slope angle were selected as column indices. A slight perturbation of 0.6 degrees was applied to the prism apex angle, increasing it from 60° to 60.6°. The exit direction of the light rays at each incident angle was recalculated using the Snell's law matrix propagation algorithm. The measured exit direction shift for a 70° incident angle was 1.24 degrees. A slight perturbation of 0.3 mm was applied to the prism height, increasing it from 30 mm to 30.3 mm. The measured exit direction shift for a 70° incident angle was 0.18 degrees. A slight perturbation of the refractive index parameter by 0.015, increasing it from 1.49 to 1.505, resulted in a measured 2.87-degree shift in the direction of the outgoing ray at a 70-degree incident angle. A slight perturbation of the prism's tilt angle by 0.305 degrees, increasing it from 30.5 degrees to 30.805 degrees, resulted in a measured 3.52-degree shift in the direction of the outgoing ray at a 70-degree incident angle. The shifts in the direction of the outgoing ray at all incident angles under all parameter perturbations were calculated sequentially, constructing a complete parameter sensitivity matrix. This matrix is ​​a 9-row, 4-column numerical matrix, with units of degree change per unit parameter variation.

[0136] The technical team performed matrix rank analysis and degree-of-freedom determination. The rank of the parameter sensitivity matrix was calculated numerically, and the result showed a rank of 4, equal to the number of parameters. This indicates that there is no linear correlation between the four design parameters, and each parameter has an independent impact on the system's optical performance. A rank of 4 means that the system optimization has four effective degrees of freedom, eliminating the need for parameter coupling and dimensionality reduction. However, the team noticed differences in the response of incident light at different azimuth angles. Light at a 0-degree azimuth angle could achieve total internal reflection, while light at a 45-degree azimuth angle, due to three-dimensional spatial geometry, could not meet the total internal reflection condition at some incident angles. To improve control over light at multiple azimuth angles, the team decided to perform parameter decoupling and dimensionality expansion, splitting the single prism array structure into a two-way composite structure consisting of a main vertical prism and an auxiliary tilting prism. The main vertical prism maintains its original vertical arrangement and handles light at a 0-degree azimuth angle. The auxiliary tilting prism is arranged at a certain angle and handles light at non-zero azimuth angles. This decoupling and dimension expansion process increases the number of degrees of freedom from 4 to 6. The two new degrees of freedom are the main and auxiliary prism size ratio parameter and the auxiliary prism tilt angle parameter.

[0137] like Figure 3As shown, the flow of the Snell's Law matrix propagation algorithm demonstrates the complete transformation process from the initial ray state to the final outgoing ray state. The algorithm starts with the input initial direction vector and initial position vector. First, it applies the refraction matrix of the incident surface, a 3×3 square matrix. The matrix elements are determined by the incident surface normal vector, the refractive index of the incident medium (1.0), and the refractive index of the prism material (1.49). After the refraction matrix is ​​applied, the ray direction vector changes, representing the refraction effect when the ray enters the prism material from air. Next, it applies the first propagation matrix, which describes the linear motion of the ray propagating from the incident surface to the total internal reflection surface inside the prism. The matrix elements are determined by the propagation distance and the current ray direction. After the propagation matrix is ​​applied, the ray position vector is updated, representing the spatial position of the ray upon reaching the total internal reflection surface. Then, it applies the reflection matrix of the total internal reflection surface, a 3×3 square matrix. The matrix elements are determined by the total internal reflection surface normal vector and the incident ray direction. After the reflection matrix is ​​applied, the ray direction vector changes again, representing the specular reflection effect that occurs at the total internal reflection surface. The second propagation matrix is ​​then applied to describe the linear motion of the light ray from the total internal reflection surface to the exit surface. Finally, the refraction matrix of the exit surface is applied to calculate the refraction effect of the light ray as it exits the prism material into the air. The entire algorithm, through continuous left-multiplication of matrices, avoids the multiple trigonometric function calculations and conditional judgments required in traditional ray tracing methods, significantly improving computational efficiency.

[0138] The technical team established a two-layer game-theoretic optimization model to collaboratively optimize multiple design parameters. For example... Figure 4 As shown, the two-layer game optimization model comprises two sub-models: an upper-layer game model and a lower-layer game model. The upper-layer game model aims to maximize the satisfaction rate of the total internal reflection critical condition, with the optimization variables being the prism tilt angle parameter and the prism apex angle parameter. The satisfaction rate of the total internal reflection critical condition is defined as the proportion of light rays that can achieve total internal reflection within an incident angle range of 50 to 90 degrees. This indicator directly reflects the prism array's ability to capture high-angle incident light. The objective function input of the upper-layer game model includes the current prism tilt angle parameter value and the prism apex angle parameter value. Using the Snell's law matrix propagation algorithm, it calculates 41 light rays at 1-degree intervals within the 50 to 90 degree range, determines whether the incident angle of each ray at the total internal reflection surface exceeds the critical angle of 41.8 degrees, counts the number of light rays that satisfy the total internal reflection condition, and calculates the satisfaction rate as the number of successfully totally internally reflected light rays divided by the total number of incident light rays multiplied by the first normalization coefficient of 100, yielding a satisfaction rate value expressed as a percentage. The constraints of the upper-level game model include a prism apex angle range of 55 to 65 degrees, a prism slope angle range of 28 to 33 degrees, and a prism height range of 25 mm to 35 mm.

[0139] The lower-level game theory model aims to maximize the uniformity of azimuth response, with the optimization variables being the ratio of the main and auxiliary prism sizes and the tilt angle of the auxiliary prism. Azimuth response uniformity is defined as the standard deviation of the exit efficiency of incident light rays from different azimuth angles after passing through the prism array; a smaller value indicates a more balanced light processing capability across all azimuth angles. The objective function inputs of the lower-level game theory model include the current main and auxiliary prism size ratio, the auxiliary prism tilt angle, the prism array density, and the prism slope tilt angle passed from the upper-level game theory model. The objective function calculation process involves selecting seven azimuth angles (0°, 15°, 30°, 45°, 60°, 75°, and 90°) and calculating the average exit efficiency within the 50° to 90° incident angle range for each azimuth angle. The exit efficiency is defined as the ratio of the energy of light successfully exiting into the room to the energy of the incident light rays, considering losses due to interface reflection, material absorption, and total internal reflection failures. The standard deviation of the emission efficiency at seven azimuth angles is calculated, divided by the average emission efficiency multiplied by the second normalization coefficient of 100, yielding the first part of the uniformity index. Next, the size ratio deviation is calculated, defined as the absolute value of the difference between the current primary and secondary prism size ratio parameter values ​​and the standard size ratio of 3, divided by the standard size ratio of 3, yielding the second part of the uniformity index. The final objective function value is the first part minus the second part, achieving the goal of improving azimuth response uniformity while controlling the size ratio to not deviate too far from the standard value. The constraints of the lower-level game model include a primary / secondary prism size ratio parameter range of 2 to 6, an auxiliary prism tilt angle parameter range of 5 to 15 degrees, and a prism array density parameter range of 25 prisms per 300mm to 35 prisms per 300mm.

[0140] The two-layer game theory optimization model achieves collaborative optimization between the upper and lower layers through a coupling term. This coupling term is the prism tilt angle parameter, which serves as both an optimization variable for the upper-layer game model and an input parameter influencing the objective function calculation of the lower-layer game model. During optimization, the upper-layer game model adjusts the prism tilt angle parameter to improve the satisfaction rate of the total internal reflection critical condition. The adjusted parameter value is then passed to the lower-layer game model, affecting the distribution of light emission efficiency at different azimuth angles and consequently the uniformity of the azimuth response. The lower-layer game model optimizes the main and auxiliary prism size ratio parameters and the auxiliary prism tilt angle parameter based on the received prism tilt angle parameter value. The optimization result, in turn, affects the overall prism array structure, potentially requiring the upper-layer game model to readjust the prism tilt angle parameter to maintain the satisfaction rate of the total internal reflection critical condition. The technical team employs an alternating iterative approach to solve the two-layer game theory optimization model: first, the lower-layer parameters are fixed while the upper-layer parameters are optimized; then, the upper-layer parameters are fixed while the lower-layer parameters are optimized, iterating repeatedly until the change in the objective function values ​​of both layers is less than the 0.1% convergence threshold. After 23 alternating iterations, the optimization algorithm converged to a stable solution. The optimized parameters output by the upper-level game model were prism inclination angle of 31.2 degrees and prism apex angle of 61.5 degrees. The optimized parameters output by the lower-level game model were main-auxiliary prism size ratio of 3.4 and auxiliary prism tilt angle of 8.6 degrees.

[0141] The technical team performed eigenvalue stability analysis to assess the reliability of the optimization results. The parameter sensitivity matrix was reconstructed using the optimized parameters. This new matrix is ​​now a 9x6 numerical matrix, with rows corresponding to the 9 incident angles and columns corresponding to the 6 optimized parameters: prism apex angle, prism height, refractive index, prism inclination angle, primary and secondary prism size ratio, and secondary prism tilt angle. The eigenvalues ​​of the parameter sensitivity matrix were calculated numerically, yielding six dimensionless values: 3.82, 2.14, 1.67, 0.95, 0.43, and 0.18. The largest eigenvalue, 3.82, exceeds the stability threshold of 2.5, indicating that the system is overly sensitive to changes in a certain parameter, suggesting instability in the current parameter configuration. Analysis of the eigenvectors corresponding to the eigenvalues ​​revealed that the eigenvector corresponding to the largest eigenvalue, 3.82, has the largest component in the prism array density parameter, with a value of 0.78, indicating that the system is most sensitive to changes in the prism array density parameter.

[0142] The technical team initiated a prism array density adjustment process and a tilt angle fine-tuning process to reduce system sensitivity. The prism array density adjustment process first calculated the degree of optical interference between adjacent prisms at the current density. Ray tracing revealed that some rays were affected by secondary incident light from adjacent prisms upon exiting, leading to instability in the exit direction. The team reduced the prism array density from an initial 30 prisms per 300mm to 28 prisms per 300mm, equivalent to increasing the width of a single prism from 30mm to 32.1mm. After reducing the density, the spacing between adjacent prisms increased, reducing optical interference. Simultaneously, a tilt angle fine-tuning process was implemented, adjusting the prism tilt angle parameter from 31.2 degrees in 0.1-degree increments, successively testing angles of 31.1 degrees, 31.0 degrees, and 30.9 degrees. After each fine-tuning, the maximum eigenvalue of the parameter sensitivity matrix was recalculated. When the tilt angle was adjusted to 30.8 degrees, the maximum eigenvalue decreased to 2.43, which is less than the stability threshold of 2.5, satisfying the stability constraint. At this point, the remaining five eigenvalues ​​are 2.08, 1.62, 0.91, 0.41, and 0.17, respectively. The magnitudes of all eigenvalues ​​are less than the stability threshold, and verification shows that the real parts of all eigenvalues ​​are negative. Therefore, the performance of the system is controllable under small parameter perturbations.

[0143] The technical team calculated the concentration and dispersion of the eigenvalue distribution to determine the final optimization strategy. The concentration of the eigenvalue distribution was obtained by dividing the variance of the six eigenvalues ​​by the mean of the eigenvalues. The variance was 1.24, the mean was 1.27, and the ratio was 0.976 (dimensionless). This ratio is greater than the concentration threshold of 0.3, indicating that the eigenvalue distribution is not concentrated. The dispersion of the eigenvalue distribution was obtained by dividing the range of the eigenvalues ​​by the mean of the eigenvalues. The range was calculated as the largest eigenvalue (2.43) minus the smallest eigenvalue (0.17), which equals 2.26. Dividing this by the mean (1.27) yielded a ratio of 1.78 (dimensionless). This ratio is greater than the dispersion threshold of 1.5, indicating that the eigenvalue distribution is dispersed, meaning that multiple parameters have a significant impact on system performance, and there is no single dominant parameter. Based on the branch decision rule, the technical team chose to execute the multi-parameter collaborative adjustment branch.

[0144] The multi-parameter collaborative regulation branch employs a non-dominated sorting genetic algorithm to search for the Pareto optimal solution set in the six-dimensional parameter space. The algorithm initializes the population with 100 individuals, each representing a set of parameter configurations, including prism apex angle, prism inclination angle, prism height, primary and secondary prism size ratio, secondary prism tilt angle, and prism array density. The optimization objectives include maximizing the satisfaction rate of the total internal reflection critical condition and maximizing the uniformity of the azimuth response. These two objectives are somewhat conflicting; increasing the satisfaction rate of the total internal reflection condition may weaken the response at certain azimuth angles, and vice versa. The algorithm iteratively evolves the population through selection, crossover, and mutation operations, obtaining a Pareto front containing 37 non-dominated solutions after 150 generations. The technical team selected the solution with the best overall performance from the Pareto frontier as the final parameter configuration. The solution satisfies the critical condition for total internal reflection in 89.7% of cases and has an azimuth response uniformity index of 4.2%. The corresponding parameter configuration is as follows: prism apex angle of 61.8 degrees, prism slope angle of 30.8 degrees, prism height of 31.5 mm, main-to-auxiliary prism size ratio of 3.6, auxiliary prism tilt angle of 9.2 degrees, and prism array density of 28 prisms per 300 mm.

[0145] The technical team conducted detailed optical performance verification based on the optimized configuration of the final prism array parameters. Typical solar positions at different times of the year were selected, including a solar altitude angle of 23 degrees at noon on the winter solstice, 50 degrees at noon on the spring equinox, and 73 degrees at noon on the summer solstice, as well as the solar positions at 9:00 AM and 3:00 PM. Considering the actual situation where windows of ground-floor residents are blocked, the actual light reaching the windows mainly comes from sky-scattered light and building-reflected light, with the incident angle concentrated between 55 and 85 degrees. Through ray tracing simulation calculations, the average incident angle of sky-scattered light reaching the windows at noon on the winter solstice is 72 degrees. After the optimized prism array, 91.3% of the incident light successfully achieved total internal reflection, with the exit angle concentrated in the range of -68 to -76 degrees, effectively illuminating the ceiling at a depth of 4.5 meters indoors. At noon on the spring equinox, the average incident angle of light reaching the windows is 68 degrees, with 95.4% of the light achieving total internal reflection, increasing the exit efficiency to 94.1%. At noon on the summer solstice, the average incident angle of the sunlight reaching the window is 61 degrees, with 98.2% of the light undergoing total internal reflection and redirection, resulting in an emission efficiency of 96.7%. Response tests for light at different azimuth angles show that the emission efficiency is 96.7% at 0 degrees, 94.2% at 30 degrees, 91.8% at 60 degrees, and 93.5% at 90 degrees. The standard deviation of the emission efficiency at each azimuth angle is 2.1%, significantly better than the 8.6% standard deviation of the original single vertical prism array scheme.

[0146] like Figure 5As shown, the optimized refractive skylight is installed at the south-facing window position of the ground-floor residence. The residence has an interior depth of 6m, a floor height of 2.9m, and a window size of 1.8m wide × 1.5m high. Skylight incident at a high angle reaches the outer surface of the window at an angle of 60 to 80 degrees, passes through the outer glass, and enters the prism array. The main vertical prism directs 0-degree azimuth light rays through a refraction-total internal reflection-refraction path to the interior ceiling, while auxiliary tilting prisms similarly direct non-zero azimuth light rays to the ceiling. The outgoing light undergoes diffuse reflection on the ceiling surface, creating uniform indirect lighting and avoiding the glare problems caused by traditional skylights. Indoor illuminance distribution tests showed that the ground illuminance at 1m from the window was 420lx, at 3m from the window it was 310lx, and at 5m from the window it was 185lx. The overall illuminance distribution was significantly better than the previous values ​​of 210lx, 145lx, and 68lx. The indoor lighting depth was extended from the original 3.2m to 5.4m.

[0147] The technical team completed a detailed design scheme based on the final optimized configuration of the prism array parameters. The final optimized prism array parameters include: prism apex angle of 61.8 degrees, prism slope angle of 30.8 degrees, prism height of 31.5 mm, main-auxiliary prism size ratio of 3.6, auxiliary prism tilt angle of 9.2 degrees, and prism array density of 28 prisms per 300 mm. This parameter configuration achieves an 89.7% satisfaction rate for the total internal reflection critical condition while keeping the azimuth response uniformity within 4.2%, realizing efficient capture of incident light at high angles from 50 to 90 degrees and balanced processing of light rays from all omnidirectional angles from 0 to 90 degrees. Compared to traditional flat glass windows, this invention, by establishing an optical interface sequence model and using Snell's law matrix propagation algorithm, transforms complex three-dimensional ray tracing into efficient matrix operations, avoiding repetitive trigonometric function calculations and condition judgments in traditional methods, thus enabling large-scale parameter optimization. By analyzing the rank of the parameter sensitivity matrix, the effective degrees of freedom of the system were accurately identified, avoiding the low optimization efficiency caused by parameter redundancy and enhancing the control capability of multi-azimuth light through decoupling and dimension expansion. The two-layer game-theoretic optimization model achieves synergistic optimization of the total internal reflection condition satisfaction rate and azimuth response uniformity through coupling terms, overcoming the limitation of traditional single-objective optimization methods in balancing multiple performance indicators. Eigenvalue stability analysis ensures the robustness of the optimization results under actual manufacturing errors and environmental disturbances. By adjusting the prism array density and fine-tuning the tilt angle, the system sensitivity is controlled within a stable threshold, avoiding large fluctuations in optical performance caused by small parameter deviations. This invention reveals the total internal reflection redirection mechanism of high-angle incident light at the optical principle level. By precisely controlling the prism geometric parameters, it achieves active regulation of the light propagation path, providing a technical solution with a clear physical mechanism, strong parameter controllability, and stable and reliable performance for addressing the problem of insufficient lighting for ground-floor residents.

[0148] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for optimizing the parameters of a prism array within a refractive skylight for ground-floor residents, characterized in that, This includes establishing an optical interface sequence model to decompose a vertical parallel prism array into a continuous sequence of interface surfaces: incident surface, total reflection surface, and exit surface; using refraction matrix, reflection matrix, and propagation matrix to describe the propagation behavior of light at each interface; and constructing a Snell's law matrix propagation algorithm to calculate the direction vector and position vector of the outgoing light ray to form the ray path tracing result. The parameter sensitivity matrix is ​​calculated by constructing the influence matrix of each parameter on the direction offset of the outgoing light rays through micro-perturbation analysis of the prism apex angle, prism height, refractive index, and prism slope angle. Matrix rank analysis and degree of freedom determination are performed. When the rank of the parameter sensitivity matrix is ​​less than 3, parameter coupling dimensionality reduction is performed. When the rank is greater than 5, parameter decoupling dimensionality expansion is performed. A two-layer game optimization model is established. The upper-layer game model optimizes the prism tilt angle and prism apex angle parameters with the goal of maximizing the satisfaction rate of the critical condition for total internal reflection. The lower-layer game model optimizes the main and auxiliary prism size ratio parameters and auxiliary prism tilt angle parameters with the goal of maximizing the uniformity of azimuth response. Eigenvalue stability analysis is performed. When the maximum eigenvalue of the parameter sensitivity matrix exceeds the stability threshold, the prism array density adjustment process and tilt angle fine-tuning process are initiated. Based on the concentration of the eigenvalue distribution, branch decisions are made to optimize the configuration of prism array parameters.

2. The method according to claim 1, characterized in that, The optical interface sequence model discretizes the optical behavior of a vertical parallel prism array into a series of refraction and reflection processes between interfaces. Each interface is represented by a transfer matrix, and the matrix elements of the transfer matrix are determined by the interface normal vector, refractive index, and geometric position.

3. The method according to claim 2, characterized in that, The refraction matrix describes the change in direction of light when it passes through the interface between two media. The matrix elements of the refraction matrix are determined by the refractive index of the incident medium, the refractive index of the exit medium, and the interface normal vector, and are used to calculate the directional component of the refracted light.

4. The method according to claim 3, characterized in that, The reflection matrix describes the change in direction of light when total internal reflection occurs at an interface, while the propagation matrix describes the rectilinear propagation of light in a uniform medium. The elements of the propagation matrix are determined by the propagation distance and the direction of the light.

5. The method according to claim 4, characterized in that, The implementation process of Snell's law matrix propagation algorithm is as follows: the initial direction vector and initial position vector of the incident light are represented as column vectors, and then multiplied by the transmission matrix corresponding to each optical interface in turn to calculate the direction vector and position vector of the outgoing light.

6. The method according to claim 5, characterized in that, The parameter sensitivity matrix is ​​a numerical matrix. The rows correspond to different incident ray angles, and the columns correspond to the prism apex angle parameter, prism height parameter, refractive index parameter, and prism slope angle parameter. The matrix elements represent the offset of the outgoing ray direction caused by the unit change of the corresponding parameter at the corresponding incident angle.

7. The method according to claim 6, characterized in that, Parameter coupling dimensionality reduction identifies linearly related parameter combinations in the parameter sensitivity matrix and merges multiple coupled parameters into a single comprehensive parameter for optimization. Parameter decoupling dimensionality expansion increases the optimization degrees of freedom by splitting a single parameter into multiple independent sub-parameters.

8. The method according to claim 7, characterized in that, The objective function input of the upper-level game model includes the prism slope angle parameter and the prism apex angle parameter, and the output is the ratio of the incident angle range that satisfies the total internal reflection condition to the total incident angle range. The objective function input of the lower-level game model includes the main and auxiliary prism size ratio parameter, the auxiliary prism tilt angle parameter, the prism array density parameter, and the prism slope angle parameter.

9. The method according to claim 8, characterized in that, The two-level game optimization model achieves collaborative optimization through coupling terms. The prism slope tilt angle parameter serves as both the optimization variable of the upper-level game model and the input parameter of the objective function of the lower-level game model. This makes the optimization of the prism slope tilt angle parameter affect both the satisfaction rate of the total reflection critical condition and the uniformity of the azimuth response.

10. The method according to claim 9, characterized in that, The prism array density adjustment process increases or decreases the prism array density parameter value according to the characteristics of the eigenvalue distribution. The tilt angle fine-tuning process makes a small adjustment of 0.1 degrees to 0.5 degrees based on the current prism tilt angle parameter value until all eigenvalues ​​meet the stability constraint conditions.

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