A structure grid radiation simulation method based on optical path ordering

Abstract

The application discloses a structured grid radiation simulation method based on optical path ordering, and belongs to the field of radiation transmission simulation. The method reads and analyzes a structured grid file, constructs a structured data set, discretizes a solid angle, calculates the projection value of the geometric center of all units on the unit vector of each discrete direction, sorts the units according to the projection value, and obtains the unit calculation sequence along the actual propagation path of the light. The radiation transfer equation in each unit is solved according to the sequence, the calculation result of the upstream unit is transmitted to the downstream unit through the connecting surface as the incident boundary condition, and the global input radiation is converged through iterative calculation. The application ensures that the calculation sequence is consistent with the real propagation path of the radiation energy by optical path physical ordering, eliminates the non-physical reverse radiation transmission, improves the calculation accuracy and convergence, and can directly process general engineering grid data, thereby being convenient for integrated application.

Description

Technical Field

[0001] This invention belongs to the field of radiation transfer simulation technology, specifically relating to a structured grid radiation simulation method, and more specifically to a radiation simulation method based on optical path ordering. Background Technology

[0002] Radiation transfer simulation technology has been systematically applied in several cutting-edge engineering fields, including thermal radiation analysis, aircraft infrared detection, remote sensing monitoring, and space target characteristic analysis. In the field of thermal radiation analysis, Chinese patent application number 202311416357.3, "Method and Device for Optimizing the Calculation Grid of Solid Rocket Engine Exhaust Flame Radiation," constructs a structured external flow field grid by rearranging the grid points and employing a preset densification strategy, thereby improving the calculation accuracy and efficiency of the exhaust flame radiation field. In infrared radiation detection, Chinese patent application number 202510795349.7, "A Target Infrared Radiation Characteristics Calculation Method Based on Neural Networks," proposes a dual BP neural network collaborative architecture, realizing a "conditional calculation" mechanism that improves analysis speed while ensuring spectral accuracy.

[0003] Despite significant advancements in existing technologies, core challenges remain in practical engineering applications. Radiation, inherently propagating in the form of rays, exhibits clear directionality and path dependence. When numerically solving radiation transfer equations, the transfer relationship of radiative energy between grid cells must be properly handled. Traditional methods, when discretizing computational domains, often neglect the physical characteristics of radiative transfer, directly employing grid ordering based on geometric proximity. This leads to non-physical reverse radiation transfer phenomena during computation, where the radiative energy of downstream grids incorrectly influences upstream grids, contradicting the actual physical process. In complex structures or multidimensional problems, such as radiative transfer under high-speed flight conditions involving multi-field coupling of aerodynamics, heat, structure, and radiation, traditional grid ordering methods struggle to accurately capture the radiative energy transfer sequence. Solving radiation transfer equations requires that grid cells be ordered according to the upstream and downstream relationships of energy beam propagation to ensure that the radiation intensity calculation of each grid node correctly depends on upstream information and provides input to the downstream. The lack of such optical path-guided sorting mechanism not only leads to the loss of accuracy in numerical solutions, but also causes convergence problems. Especially when dealing with optically thick media or strong gradient radiation fields, the calculation error will be significantly amplified, becoming a key bottleneck restricting the improvement of the accuracy and efficiency of radiation transfer simulation. Summary of the Invention

[0004] The purpose of this invention is to solve the problems of non-physical reverse radiation transmission, low calculation accuracy and poor convergence in existing radiation simulations caused by the grid order not following the physical path of radiation propagation. Therefore, this invention proposes a radiation simulation method that physically orders the structural grid cells according to the optical path direction.

[0005] To achieve the above objectives, the technical solution provided by this invention is:

[0006] A method for simulating radiation in structured meshes based on optical path ordering is provided, including the following steps:

[0007] Step 1, Data Input and Parsing: Read the structured mesh data file that describes the geometry and topology of the computational domain, parse and extract the node, face, cell and boundary condition information, and construct a structured dataset that can be used in subsequent computation steps;

[0008] Step 2, Direction Discretization and Path Sorting: Discretize the solid angle representing the radiation transmission direction space according to the defined angle domain to obtain multiple discrete radiation transmission directions; for each discrete direction, calculate the projection value of the geometric center of all elements on the unit vector of that discrete direction; and sort all elements according to the order of the projection values ​​to obtain the element calculation sequence that conforms to the radiation along the ray propagation path, wherein the boundary elements that serve as the radiation incident starting point are sorted first.

[0009] Step 3, Sequential Solution and Information Transmission: For each discrete direction, based on the element calculation sequence determined in Step 2, solve the radiation transfer equation of each element one by one to obtain the radiation intensity distribution inside the element; during the solution process, the calculation result of the current element is transmitted to its downstream adjacent element through the connection surface as the incident boundary condition of the downstream element.

[0010] Step 4, Global Iteration and Convergence Judgment: Based on the radiation intensity distribution in all discrete directions calculated in Step 3, calculate the global input radiation; determine whether the relative error between the input radiation in the current iteration step and the input radiation in the previous iteration step meets the preset convergence criterion; if not, return the current result as a new radiation source term to Step 3 for the next iteration; if it meets the criterion, output the final radiation intensity field as the simulation result.

[0011] Furthermore, in step 1, the structured mesh data file is an ASCII code format mesh file exported using Fluent software.

[0012] Furthermore, in step 2, the discretization of the solid angle specifically involves uniformly dividing the spherical coordinate system into a specified number of solid angles, each solid angle representing a discrete radiation transmission direction.

[0013] Furthermore, in step 3, before solving the radiation transfer equation of the element, each structured element is discretized by internal interpolation to generate an internal node set for accurate calculation.

[0014] Furthermore, in step 3, the angle discretization method used to solve the radiation transfer equation is the discrete coordinate method.

[0015] Furthermore, in step 3, solving the radiation transfer equation for each element specifically includes:

[0016] Step 3.1: Assign absorption coefficient, scattering coefficient, refractive index and temperature to the medium within the unit cell, and assign emissivity, reflectivity and temperature to the discrete points on the unit cell boundary;

[0017] Step 3.2: Based on temperature, calculate the blackbody radiation intensity of the medium and the wall according to Planck's law, and use it as the source term of the radiation transfer equation;

[0018] Step 3.3: Based on the parameters, source terms, and incident conditions from the upstream unit, solve the radiation transfer equation after discrete coordinate method and spatial discretization. The equation includes angular redistribution terms caused by gradient refractive index to obtain the radiation intensity at each discrete point in the unit.

[0019] Furthermore, in the equation solving step 3.3, the radiation intensity is treated as an unknown quantity, and a system of linear equations is formed within the element for solving; for the boundary discrete points with known radiation intensity, they are represented in the system of equations as fixed values.

[0020] Furthermore, in step 4, the global radiation energy field is the input radiation, which is obtained by integrating the radiation intensity over all discrete directions; the convergence criterion is that the relative error of the input radiation is less than [a certain value]. .

[0021] The advantages of this invention are:

[0022] The structural mesh radiation simulation method based on optical path ordering provided by this invention ensures that the calculation order strictly follows the actual physical propagation path of radiant energy by physically ordering all elements according to the projection value of the element's geometric center in the direction of light propagation. This core improvement fundamentally eliminates the non-physical reverse radiation transfer phenomenon caused by incorrect ordering, making the simulation process consistent with the physical essence. Furthermore, since the calculation of each element strictly follows the optical path order and correctly depends on the radiation information of its upstream elements, the accumulation of numerical errors caused by information transmission errors is significantly reduced, improving the accuracy of the overall radiation intensity calculation. Simultaneously, the correct physical ordering provides a stable and clear information flow direction for the iterative solution process, effectively avoiding iterative oscillations or non-convergence problems caused by chaotic dependencies, and greatly improving the convergence of the calculation. In addition, this method constructs a clear process from structural mesh data analysis, optical path ordering, sequential solution to iterative convergence, enabling its convenient integration into existing engineering simulation systems. It provides a solution for practical problems involving high-temperature radiation transmission that combines physical accuracy, numerical stability, and engineering practicality. Attached Figure Description

[0023] The above and / or other features and advantages of the present invention will become more readily understood from the following description with reference to the accompanying drawings, in which:

[0024] Figure 1 This is a flowchart illustrating the overall implementation steps of the structured mesh radiation simulation method based on optical path ordering according to an embodiment of the present invention.

[0025] Figure 2 This is a schematic diagram illustrating a specific implementation of the structured grid radiation simulation method based on optical path ordering in this invention.

[0026] Figure 3 This is a schematic diagram illustrating the sorting of structural grid cells along the optical path direction in an embodiment of the present invention;

[0027] Figure 4 This is a schematic diagram illustrating the information transfer method between units when performing calculations based on the unit sorting order in an embodiment of the present invention;

[0028] Figure 5 This is an example diagram of the simulation results of thermal radiation transmission calculation in a nozzle model using the method of an embodiment of the present invention. Detailed Implementation

[0029] The present invention will now be described in detail with reference to the accompanying drawings and exemplary embodiments thereof. It should be noted that the following detailed description of the present invention is for illustrative purposes only and is not intended to limit the scope of the invention.

[0030] This invention aims to provide a radiation transfer simulation method with accurate physical processes and high numerical stability. Its core lies in sorting the computational grid according to the optical path to overcome the non-physical reverse radiation transfer problem caused by improper sorting in traditional methods, thereby improving computational accuracy and convergence.

[0031] This invention provides a structural mesh radiation simulation method based on optical path ordering, the overall implementation process of which is as follows: Figure 1 As shown. Figure 2 As shown, this method begins with a pre-generated structured mesh data file describing the geometry and topology of the computational domain, such as an ASCII code format file exported using Fluent software and conforming to its conventions. Through a series of steps, it ultimately outputs a high-precision three-dimensional radiation intensity field. Each step will be described in detail below.

[0032] Step S1: Data Input and Parsing

[0033] First, the system reads the target structure mesh data file. It parses the file line by line, identifying and extracting the key information necessary to define the computational domain. This information mainly includes four categories: nodes (containing point numbers and their 3D coordinates), faces (containing face numbers, the sequence of nodes constituting the face, the elements connected to the face, and the face's normal vector), elements (i.e., volume elements, containing element numbers, the sequence of nodes constituting the element, and the faces contained within the element), and boundary conditions (containing boundary names, types, and the numbers of the faces they contain). The system reorganizes and stores this information, constructing a structured dataset that can be efficiently accessed in all subsequent computational steps. This step is fundamental to all subsequent precise calculations; it transforms the external mesh file into data objects with clear logical relationships that can be directly manipulated within the program, preparing for physical sorting and numerical solving.

[0034] Step S2: Directional Discretization and Path Sorting

[0035] The purpose of this step is to determine a physically sound unit calculation order for each radiation propagation direction. First, based on the angle domain defined by the simulation accuracy requirements (i.e., the required number and range of solid angles), the solid angles representing the space of the radiation transmission direction are discretized. Specifically, the spherical coordinate system is uniformly divided into a specified number of solid angles, each solid angle representing a discrete radiation transmission direction. Then, for each discrete direction, the core optical path sorting operation is performed, the principle and process of which are as follows: Figure 3 As shown.

[0036] The sorting is based on the projection value of the element's geometric center onto the discrete-direction unit vector. Mathematically, this projection value is the scalar product of the element's centroid coordinate vector and the optical path direction unit vector. Specifically, if the optical path direction unit vector is... ,unit The coordinates of the barycenter are Then its projection value The calculation formula is: The magnitude of this projection value physically represents the distance of the center of the cell along the current direction of light propagation.

[0037] By sorting all elements in ascending order of this projection value, the resulting element sequence strictly follows the actual, unidirectional propagation path of radiated energy along this direction. This mechanism fundamentally ensures that the calculation process proceeds along the optical path, with upstream elements calculated before downstream elements, thus completely avoiding the physically unacceptable "reverse radiation transfer" problem that may occur with traditional sorting based on geometric numbering or proximity.

[0038] Figure 3This process is clearly illustrated: First, the projection values ​​of the boundary cells (shown in blue in the diagram) are calculated and sorted, for example, the projection values ​​of cells 1, 2, and 3 are calculated. , , According to the calculation results The calculation order is determined to be 1, 2, 3; after all boundary elements are sorted, the internal elements (shown in red in the figure) are then sorted, for example, elements 5 and 6, based on their projection values. The order is determined to be 5 and 6. This "boundary element priority sorting" strategy ensures that the boundary incident conditions, which are the source of radiant energy, are handled correctly first, providing the correct physical starting point for the entire computational sequence. It is this sorting step that allows the entire numerical solution process to be based on correct physical causal relationships, laying a crucial foundation for improving computational accuracy and promoting iterative convergence.

[0039] Step S3: Sequential Solving and Information Transfer

[0040] After obtaining the element calculation sequence for each discrete direction, the sequential solution stage begins. To improve computational accuracy, further internal interpolation discretization can be performed on each structured hexahedral element before solving the radiation transfer equations within the element. For example... Figure 4 As shown, an original cell (represented by its 8 black vertices) is uniformly divided into a finer mesh (e.g., 3×3×3), generating a series of internal discrete points (gray dots in the figure). These newly added discrete points, together with the original vertices, constitute the set of computational nodes inside the cell used for accurate solution.

[0041] For each cell in the current discrete direction and sorting sequence, the solution for its radiation intensity is based on the numerical solution of the radiation transfer equation, and the specific process is as follows:

[0042] (1) Parameter assignment: Assign radiation characteristic parameters to the current element and its contained discrete points, specifically including: assigning absorption coefficients to the medium within the current element. Scattering coefficient Refractive index With temperature And assign emissivity, reflectivity and temperature to the boundary points;

[0043] (2) Radiation source term calculation: The blackbody radiation intensity is calculated based on the temperature of the medium or wall. The emission from the medium itself and the wall follows Planck's blackbody radiation law. The blackbody radiation intensity is calculated based on the wavelength range of the radiation transfer simulation.

[0044]

[0045] In the formula, For wavelength, wavelength The blackbody radiation intensity of the spectrum below, , and These are all constants, namely Planck's constant, the speed of light, and Boltzmann's constant; in practical applications, these constants are combined into the first radiation constant. Second radiation constant In this embodiment, , ;

[0046] If we simulate thermal radiation, its radiant energy is obtained by integrating over the entire wavelength range, that is, by integrating Planck's law over the entire wavelength range, resulting in the blackbody radiant energy:

[0047]

[0048] In the formula, For Stefan-Boltzmann constant, in this embodiment, we take... ;

[0049] (3) Radiative Transfer Equation (RTE): The emission, absorption, and scattering processes of radiative transfer can be represented by the RTE. The radiation intensity can be described as a function of angle and space, specifically:

[0050]

[0051] In the formula, Indicates spatial location , angle direction ,time ,wavelength The spectral radiance at that point; This is the spectral attenuation coefficient, which is numerically equivalent to the spectral absorption coefficient. and spectral scattering coefficient sum; Indicates the direction of the shot towards the target micro-element; Let be the scattering phase function, representing the radiation from Direction of fire The probability distribution of ; c is the speed of light, that is, the spectral radiation intensity propagates at the speed of light; The refractive index, as a factor, determines the intensity of radiation as it changes along the path of radiative transmission. According to Fermat's principle, the radiation intensity will propagate along a curve, and the refractive index... The gradient in each direction can be expressed by the following ray equation:

[0052]

[0053] Combining the ray equation and the differential term of the angle variable with respect to the optical path, in RTE Regarding optical path The differential operator can be expressed as:

[0054]

[0055] In the formula, Let be the direction cosine, where and These are the zenith angle and circumference angle of the radiation intensity, respectively. , , Refractive index about , , The derivative of a direction is defined as follows:

[0056]

[0057] (3) Simplification and Discretization of the governing equations: After neglecting the time-dependent changes in radiation intensity, the RTE can be simplified to:

[0058]

[0059] In the formula, Ω is represented as:

[0060]

[0061] left end The angular redistribution term caused by the gradient refractive index, which includes the partial derivative in the angular direction, is expressed as:

[0062]

[0063] The right end For the radiation source term, it is represented as:

[0064]

[0065] The radiation transfer equation is determined based on the spectral wavelength range used in the physics problem. In the calculation, only the parameter states within this wavelength range are considered, and the relevant parameters are omitted from the description. The discrete coordinate method is used to discretize the angle variables of the RTE, and the solid angle is discretized into... one direction and corresponding weights and In a certain discrete direction The radiation transfer equation is expressed as follows:

[0066]

[0067] In the formula, the angular redistribution term for:

[0068]

[0069] Source Item Represented as:

[0070]

[0071] In the formula, Indicates the direction of angles, zenith angle and inscribed angle number. This represents the scattering phase function between discrete directions. Indicates direction Radiation intensity on and These represent the recursive formulas for the zenith angle and the circumference angle, respectively, during the process of discretizing the radiation intensity into various directions using the discrete coordinate method.

[0072] (4) Spatial discretization and element solution: Spatial element discretization is performed using Lagrange interpolation basis functions. Discretize the first derivative of the radiation intensity with respect to spatial coordinates as follows:

[0073]

[0074] In the formula, The natural coordinates within the isoparametric element depend on Changes; isoparametric natural coordinate system To spatial coordinate system Jacobian matrix, Let be the radiation intensity at each node within the isoparametric element, which is the unknown quantity. The number of all nodes in the isoparametric element;

[0075] Within each element, based on the element discrete points in step S2, the discrete points within a single element are divided into boundary points and points to be calculated. Boundary points are assigned corresponding radiation intensity values ​​(from upstream elements or physical boundary conditions). Points to be calculated are balanced according to the discrete radiation transfer equations, and radiation intensity is used as the unknown quantity to construct a system of linear equations:

[0076]

[0077] In the above formula, This is a coefficient matrix, representing the coefficients of discrete points within a specific cell at a specific angle. At boundary points, the corresponding element has a value of 1. It is a set matrix of source terms and boundary values, where the element value at the boundary point is the known radiation intensity value at its discrete point;

[0078] (5) Information transmission: such as Figure 4 As shown, the solution process strictly follows the element calculation sequence determined in step S2. After the current element is calculated, its result is immediately passed to its direct downstream neighboring element in the sequence through the connection surface, and is used as the known incident boundary condition when the downstream element solves the linear equation system, substituted into its right-hand vector. This mechanism, which strictly relies on the upstream and downstream relationships determined by the optical program sequence and transmits physical information through the connecting surfaces, ensures that the coupling and transmission of radiated energy between grids completely simulates the real propagation process, making the entire solution chain numerically stable and information-coherent.

[0079] Step S4: Global Iteration and Convergence Judgment

[0080] After all discrete directions have undergone the above calculations, the global radiation intensity distribution for the current iteration step is obtained. To determine whether the calculation has reached radiation equilibrium, a global radiation energy field needs to be calculated as a criterion. Here, incident radiation is used. This involves integrating the radiation intensity in all directions at every point in space, representing the total radiant energy flux received at that point. The system calculates the input radiation field for the current iteration step and compares it with the result of the previous iteration step to evaluate its maximum relative error. If this error does not meet the preset convergence criterion, in this embodiment, the convergence criterion is an error less than... If the error is not met, it indicates that the radiation field has not yet reached equilibrium. At this point, the currently calculated radiation intensity field is used as a new, more realistic radiation source term and fed back to step S3, initiating the next iteration to gradually approximate the steady-state solution. If the error meets the convergence criterion, it indicates that the calculation has converged, and the system outputs the final full-field radiation intensity distribution as the simulation result. This iterative-judgment-feedback closed-loop process enables the method to handle complex coupling effects such as scattering and multiple reflections, ultimately obtaining a physically self-consistent and numerically stable high-precision solution.

[0081] To verify the feasibility and effectiveness of the method proposed in this invention, it was applied to a typical engineering scenario. For example... Figure 5As shown, a thermal radiation transfer simulation was performed on a nozzle model flying at an altitude of 6 km and a speed of Ma=1.2. In the computational domain, the temperature of the nozzle's outer skin ranged from 308 K to 312 K, the nozzle exit temperature was 999 K, and the external air temperature was 249.15 K. The outer skin wall of the nozzle was set as a directional diffuse reflective wall, with its emissivity varying with angle, a maximum value of 0.3, and a minimum value of 0.1. The nozzle exit section was set as a blackbody boundary. The remaining four walls of the computational domain were set as blackbody walls at the air temperature (249.15 K), meaning they completely absorbed all incident radiation and emitted it outwards as blackbody radiation at that temperature. Using the method of this invention, the structured mesh file (a Fluent-readable structured mesh in ASCII code format) was read, and the discrete numbers for the circumferential angle and zenith angle were set to 8×4 (a total of 32 discrete directions). After iterative calculation, the radiation intensity distribution in multiple directions in space was successfully obtained.

[0082] The figure extracts four directions to illustrate the spatial distribution of radiation intensity. The results clearly show that the radiation intensity is low in the lower-temperature skin region (blue area in the figure), while the high-temperature nozzle exit region exhibits extremely strong radiation (red area in the figure), which is completely consistent with physical expectations. This embodiment fully demonstrates that the method of the present invention can effectively handle real engineering structure meshes, accurately set complex physical boundary conditions, and precisely simulate the transmission and distribution of radiation energy according to the optical path direction, possessing high engineering practical value.

[0083] Finally, it should be noted that the features mentioned and / or shown in the above description of exemplary embodiments of the present invention can be combined in the same or similar manner with one or more other embodiments, combined with or substituted for corresponding features in other embodiments. These combined or substituted technical solutions should also be considered to be included within the scope of protection of the present invention.

Claims

1. A structural mesh radiation simulation method based on optical path ordering, characterized in that, Includes the following steps: Step 1, Data Input and Parsing: Read the structured mesh data file that describes the geometry and topology of the computational domain, parse and extract the node, face, cell and boundary condition information, and construct a structured dataset that can be used in subsequent computation steps; Step 2, Direction Discretization and Path Sorting: Discretize the solid angles representing the radiation transmission direction space according to the defined angle domain to obtain multiple discrete radiation transmission directions; for each discrete direction, calculate the projection value of the geometric center of all elements onto the unit vector of that discrete direction. All elements are sorted according to the size of the projection values ​​to obtain a calculation sequence of elements that conforms to the path of radiation propagation along the light ray. Among them, the boundary elements that serve as the starting point of radiation incidence are sorted first. Step 3, Sequential Solution and Information Transmission: For each discrete direction, based on the element calculation sequence determined in Step 2, solve the radiation transfer equation of each element one by one to obtain the radiation intensity distribution inside the element; during the solution process, the calculation result of the current element is transmitted to its downstream adjacent element through the connection surface as the incident boundary condition of the downstream element. Step 4, Global Iteration and Convergence Judgment: Based on the radiation intensity distribution in all discrete directions calculated in Step 3, calculate the global input radiation; determine whether the relative error between the input radiation in the current iteration step and the input radiation in the previous iteration step meets the preset convergence criterion; if not, return the current result as a new radiation source term to Step 3 for the next iteration; if it meets the criterion, output the final radiation intensity field as the simulation result.

2. The structural mesh radiation simulation method according to claim 1, characterized in that, In step 1, the structured mesh data file is an ASCII code format mesh file exported using Fluent software.

3. The structural mesh radiation simulation method according to claim 1, characterized in that, In step 2, the discretization of the solid angles specifically involves uniformly dividing the spherical coordinate system into a specified number of solid angles, with each solid angle representing a discrete radiation transmission direction.

4. The structural mesh radiation simulation method according to claim 1, characterized in that, In step 3, before solving the radiation transfer equation of the element, each structured element is discretized by internal interpolation to generate an internal node set for accurate calculation.

5. The structural mesh radiation simulation method according to claim 4, characterized in that, In step 3, the angle discretization method used to solve the radiation transfer equation is the discrete coordinate method.

6. The structural mesh radiation simulation method according to claim 5, characterized in that, Step 3, solving the radiation transfer equation for each element specifically includes: Step 3.1: Assign absorption coefficient, scattering coefficient, refractive index and temperature to the medium within the unit, and assign emissivity, reflectivity and temperature to discrete points on the unit boundary; Step 3.2: Based on temperature, calculate the blackbody radiation intensity of the medium and the wall according to Planck's law, and use it as the source term of the radiation transfer equation; Step 3.3: Based on the parameters, source terms, and incident conditions from the upstream unit, solve the radiation transfer equation after discrete coordinate method and spatial discretization. The equation includes an angular redistribution term caused by gradient refractive index to obtain the radiation intensity at each discrete point in the unit.

7. The structural mesh radiation simulation method according to claim 1 or 6, characterized in that, In the equation solving step 3.3, the radiation intensity is treated as an unknown quantity, and a system of linear equations is formed within the unit for solving; for the boundary discrete points with known radiation intensity, they are represented in the form of fixed values ​​in the system of equations.

8. The structural mesh radiation simulation method according to claim 1, characterized in that, In step 4, the global radiation energy field is the input radiation, which is obtained by integrating the radiation intensity over all discrete directions; the convergence criterion is that the relative error of the input radiation is less than [a certain value]. .

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