A method for solving inverse problems of thermal process based on improved cuckoo search algorithm

Abstract

The application provides a heat process inverse problem solving method based on an improved cuckoo algorithm, and relates to the technical field of heat transfer inverse problems. The application collects temperature field information to be inverted through experiments; determines the type of the inverse problem, sets a target function, and determines the constraint conditions corresponding to the inversion parameters; writes a forward calculation program to calculate the fitness value of the target function; and uses the improved cuckoo algorithm to invert the parameters to be inverted to obtain an inversion solution. The application improves the defects of the original cuckoo algorithm, such as weak global search ability, low convergence precision, and easy local convergence, caused by the too simple parameter setting of the original cuckoo algorithm, which cannot balance the global search and local search, and the low utilization rate of population information, improves the convergence speed while improving the solution quality, and has wide practical significance in solving various heat process inverse problems.

Description

Technical Field

[0001] This invention relates to the field of inverse problems in heat transfer technology, and in particular to a method for solving inverse problems of thermal processes based on an improved cuckoo algorithm. Background Technology

[0002] In mathematics, a forward problem is the process of calculating the results of a system given its parameters. In heat transfer, the process of solving the thermal conductivity differential equation to obtain the temperature distribution of an object based on its thermal properties and boundary conditions is called a forward thermal problem. Conversely, the process of solving for the parameters of a physical system based on known data is called an inverse problem. Inverse thermal problems involve inferring thermal properties and boundary conditions from the internal temperature distribution of an object. Inverse problems are categorized based on the specific problem being solved: parameter identification inverse problems (solving unknown thermal properties from a known temperature field or gradient), boundary identification inverse problems (solving boundary conditions), initial field identification inverse problems (solving initial temperature), and source term identification inverse problems (solving the location and magnitude of source terms). Solving these types of problems has significant practical applications in metallurgy, aerospace, biomedicine, and power engineering.

[0003] Early research on the inverse problem of thermal processes primarily employed regularization methods and gradient-based algorithms. The Tikhonov regularization method, in particular, reduces ill-posedness by introducing a regularization parameter. Therefore, the selection of this parameter significantly impacts algorithm performance: an excessively large parameter can lead to overly weighted regularization terms, failing to yield a solution that closely matches the original problem; conversely, an excessively small parameter cannot prevent overfitting, reducing the reliability of the solution. In practice, the optimal regularization parameter varies for each problem, limiting the method's effectiveness. Gradient-based optimization algorithms include conjugate gradient methods, least squares methods, and Newton's method. These methods converge quickly, but the inverse solution is highly dependent on initial values, and gradient information is often difficult to obtain. When the model is complex, iterative calculations require significant time and computational resources.

[0004] Currently, intelligent optimization algorithms, such as the Cuckoo Algorithm, are being introduced into solving inverse problems of hot processes. In the Cuckoo Algorithm, each valid solution to the problem is initialized as a nest. By mimicking the cuckoo's habit of using other nests to lay its eggs, the algorithm continuously updates the position of feasible solutions through levy flight and greedy selection, eventually obtaining a satisfactory result. However, due to the overly simple parameter settings of the original algorithm, which fail to balance global and local search and have low population information utilization, it often suffers from slow search speed, low computational accuracy, and a tendency to get trapped in local convergence during the solution of inverse problems of hot processes, making it impossible to obtain an accurate inverse solution. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a method for solving the inverse problem of thermal processes based on an improved cuckoo algorithm.

[0006] A method for solving the inverse problem of hot processes based on an improved cuckoo algorithm includes the following steps:

[0007] Step 1: Collect the temperature field information to be inverted, i.e. the temperature distribution of the object, through experiments using the preset physical property parameters, initial conditions and boundary conditions;

[0008] The experiment involves taking a sample to be tested, first measuring the thermal conductivity of the sample, then heating the sample, and finally obtaining the temperature distribution of the sample through a thermocouple after the sample temperature stabilizes.

[0009] Step 2: Determine the inverse problem type, set the fitness function, and determine the constraints corresponding to the inverse parameters, that is, the upper and lower limits of the predicted values ​​of the inverse parameters as constraints;

[0010] The inverse problem types include parameter identification inverse problems and boundary identification inverse problems. Boundary identification includes first-type boundary conditions and second-type boundary conditions, namely boundary temperature and heat flux. The parameters to be inverted are thermal conductivity, boundary temperature and heat flux density, respectively, and the inversion parameters are regarded as unknown conditions.

[0011] The fitness function is set as follows: Where M is the number of measurement points. The discrete temperature field obtained in step 1, The discrete temperature field estimated by the algorithm;

[0012] Step 3: Write a forward calculation program to calculate the estimated discrete temperature field.

[0013] The function of the forward calculation program is to obtain a workable solution by inputting preset physical property parameters, known conditions in the initial and boundary conditions, and the cuckoo population, and then solving the heat conduction differential equation through forward calculation.

[0014] Step 4: Use the improved Cuckoo algorithm to invert the parameters to be inverted and obtain the inversion solution;

[0015] Step 4.1: Initialize the population; each position in the initialized population represents a feasible solution;

[0016] Step 4.2: Calculate according to the forward calculation program written in Step 3. Substitute the fitness function set in step 2 into the fitness function to calculate the fitness value of each cuckoo;

[0017] Step 4.3: Update the location of the bird's nest using the improved global search operator;

[0018] The improved global search operator is as follows:

[0019]

[0020] in, It is the position of the i-th bird's nest in the t-th iteration; The optimal bird's nest position in the current iteration; Levy(β) represents Levy flight; α is the improved adaptive step size factor.

[0021]

[0022] Among them, I iter This refers to the current iteration number, I. max It is the maximum number of iterations, I min It is the minimum number of iterations, and its value is 1; α min The minimum step size scaling factor is set to 0.01; α max The maximum step size scaling factor is set to 1;

[0023] ξ i The newly introduced traction factor is defined as follows:

[0024]

[0025] in, It is the fitness function value of the current best individual in the population; It is the fitness function value of an individual in the current population;

[0026] Step 4.4: Use the greedy selection operator to retain cuckoo offspring with smaller fitness function values;

[0027] The greedy selection operator is as follows:

[0028]

[0029] Step 4.5: Update the location of the bird's nest using an improved local search operator;

[0030] The improved local search operator formula is as follows:

[0031]

[0032] Where r is a random number vector following a normal distribution from 0 to 1; ε is a random number vector following a uniform distribution from 0 to 1; and represents the positions of Bird's Nest j and Bird's Nest k in the current iteration number, randomly selected; Heaviside(x) is the step function; Pa is the improved adaptive discard probability, defined as:

[0033]

[0034] Among them, Pa max To maximize the discard probability, set it to 0.5; Pa min To minimize the discard probability, it is set to 0.01;

[0035] Step 4.6: Use the greedy selection operator to retain the cuckoo offspring with smaller fitness function values;

[0036] Step 4.7: Evaluate whether the termination condition is met, i.e. whether the preset maximum iteration step size has been reached. If the termination condition is met, the algorithm ends and the final inversion solution is obtained; otherwise, return to step 4.2.

[0037] The beneficial effects of adopting the above technical solution are as follows:

[0038] This invention provides a method for solving the inverse problem of hot processes based on an improved cuckoo algorithm, which has the following beneficial effects:

[0039] 1. The improved method applied in this invention introduces the concept of adaptive parameters and applies them to the adaptive step size scaling factor α and the adaptive discard probability Pa. α and Pa are initialized to large values ​​in the initial search phase. Large step sizes and a high discard probability maintain population diversity, enabling rapid global search. As iterations proceed, α and Pa gradually decrease. Controlling the randomness of the step size and the discard probability allows the search focus to gradually shift from global to local, achieving higher search accuracy and convergence speed.

[0040] 2. An improved method applied in this invention creates the concept of a traction factor. The traction factor ξ i This represents the traction force exerted by the optimal individual on other individuals in the population, and its value is related to the fitness function. Weaker individuals experience a greater traction force, accelerating their approach to feasible solutions and reducing the blindness of the search; better individuals are less affected by the optimal individual, allowing them to fully preserve their individual information and explore the nearby solution space more thoroughly, which helps generate new feasible solutions. Therefore, the addition of the traction factor effectively reconciles the contradiction between individual information and optimal value information, greatly improving information exchange among cuckoo populations, enhancing optimization efficiency, accelerating convergence, and ensuring population diversity.

[0041] 3. This invention is based on an improved Cuckoo algorithm for solving inverse problems of thermal processes. It overcomes the shortcomings of existing classical algorithms, such as slow convergence speed, easy getting stuck in local convergence, and low utilization of population information. While improving the convergence speed of the algorithm, it also improves the quality of the inverse solution. It can be applied to various inverse problems of thermal conductivity, boundary temperature, and heat flux density. It has broad practical significance for guiding the solution of inverse problems and the engineering application of inverse problems. Attached Figure Description

[0042] Figure 1 This is a flowchart of the inverse problem-solving method for hot processes based on the improved cuckoo algorithm in an embodiment of the present invention;

[0043] Figure (a) is the overall flowchart, and Figure (b) is the detailed flowchart.

[0044] Figure 2 This is an iterative diagram of the adaptive step size scaling factor α and the adaptive discard probability Pa in an embodiment of the present invention.

[0045] Figure (a) is the iterative graph of the adaptive step size scaling factor α, and Figure (b) is the iterative graph of the adaptive discard probability Pa.

[0046] Figure 3 This is a convection model of a square cavity pipe in an embodiment of the present invention;

[0047] Figure 4 The following are convergence graphs of the original and improved Cuckoo algorithm for the CEC2017-f9 test function in this embodiment of the invention.

[0048] Figure 5 The following are convergence graphs of the original and improved cuckoo algorithms in embodiments of the present invention;

[0049] Figure (a) shows the convergence graphs of the original and improved cuckoo algorithms for the inverse problem of thermal conductivity.

[0050] Figure (b) - Convergence diagrams of the original and improved Cuckoo algorithm for the boundary temperature inverse problem;

[0051] Figure (c) - Convergence graphs of the original and improved Cuckoo algorithm for the inverse heat flux density problem. Detailed Implementation

[0052] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0053] A method for solving the inverse problem of hot processes based on an improved cuckoo algorithm, such as... Figure 1 As shown in (a) and (b), the steps include:

[0054] Step 1: Collect the temperature field information to be inverted, i.e. the temperature distribution of the object, through experiments using the preset physical property parameters, initial conditions and boundary conditions;

[0055] The experiment involves taking a sample to be tested, first measuring the thermal conductivity of the sample, then heating the sample, and finally obtaining the temperature distribution of the sample through a thermocouple after the sample temperature stabilizes.

[0056] Step 2: Determine the inverse problem type, set the fitness function, and determine the constraints corresponding to the inverse parameters, that is, the upper and lower limits of the predicted values ​​of the inverse parameters as constraints;

[0057] The inverse problem types include parameter identification inverse problems and boundary identification inverse problems. Boundary identification includes first-type boundary conditions and second-type boundary conditions, namely boundary temperature and heat flux. The parameters to be inverted are thermal conductivity, boundary temperature and heat flux density, respectively, and the inversion parameters are regarded as unknown conditions.

[0058] The fitness function is set as follows: Where M is the number of measurement points. The discrete temperature field obtained in step 1, The discrete temperature field estimated by the algorithm;

[0059] Step 3: Write a forward calculation program to calculate the estimated discrete temperature field.

[0060] The function of the forward calculation program is to obtain a workable solution by inputting preset physical property parameters, known conditions in the initial and boundary conditions, and the cuckoo population, and then solving the heat conduction differential equation through forward calculation.

[0061] The numerical method used for the forward calculation is the spectral method, and the locus of points is Chebyshev-Gauss-Lobattopoints.

[0062] Step 4: Use the improved Cuckoo algorithm to invert the parameters to be inverted and obtain the inversion solution.

[0063] Step 4.1: Initialize the population; each position in the initialized population represents a feasible solution;

[0064] Step 4.2: Calculate the fitness value of each cuckoo according to the fitness function;

[0065] Step 4.3: Update the location of the bird's nest using the improved global search operator;

[0066] The improved global search operator is as follows:

[0067]

[0068] in, It is the position of the i-th bird's nest in the t-th iteration; The optimal bird's nest position in the current iteration; Levy(β) represents Levy flight; α is the improved adaptive step size factor.

[0069]

[0070] Among them, I iter This refers to the current iteration number, I. max It is the maximum number of iterations, I min It is the minimum number of iterations, and its value is 1; α min The minimum step size scaling factor is set to 0.01; α max This is the maximum step size scaling factor, set to 1.

[0071] ξ i The newly introduced traction factor is defined as follows:

[0072]

[0073] in, It is the fitness function value of the current best individual in the population; It is the fitness function value of an individual in the current population;

[0074] Step 4.4: Use the greedy selection operator to retain cuckoo offspring with smaller fitness function values;

[0075] The greedy selection operator is as follows:

[0076]

[0077] Step 4.5: Update the location of the bird's nest using an improved local search operator;

[0078] The improved local search operator formula is as follows:

[0079]

[0080] Where r is a random number vector following a normal distribution from 0 to 1; ε is a random number vector following a uniform distribution from 0 to 1; and represents the positions of Bird's Nest j and Bird's Nest k in the current iteration number, randomly selected; Heaviside(x) is the step function; Pa is the improved adaptive discard probability, defined as:

[0081]

[0082] Among them, Pa max To maximize the discard probability, set it to 0.5; Pa min To minimize the discard probability, it is set to 0.01. The iteration result is shown in the figure below. Figure 2 As shown

[0083] Step 4.6: Use the greedy selection operator to retain the cuckoo offspring with smaller fitness function values;

[0084] Step 4.7: Evaluate whether the termination condition is met, i.e. whether the preset maximum iteration step size has been reached. If the termination condition is met, the algorithm ends and the final inversion solution is obtained; otherwise, return to step 4.2.

[0085] In step 4 of this embodiment, the improved Cuckoo algorithm is used to solve the inverse problems of one-dimensional thermal conductivity, boundary temperature of a circular region, and heat flux density of a square region, respectively. The inverse heat flux density problem model is as follows: Figure 3 As shown, hot air flows in from the left and out from the right, with the temperatures at the left and right boundaries being respectively... and The upper wall is insulated, while the heat flux density of the lower wall is unknown.

[0086] The error takes different forms depending on the type of inverse problem being solved. The error formula for the thermal conductivity inverse problem is as follows:

[0087] Where, λ exact For accurate thermal conductivity, it is set to 1, λ estimate The thermal conductivity is obtained from the inversion calculation.

[0088] The error formula for the boundary temperature inverse problem is:

[0089] Where N is the number of discrete boundary points, T exact For precise boundary temperature, set to

[0090] The error formula for the inverse problem of heat flux density is:

[0091] Where Z is the total number of time layers, q exact (t) is the exact solution of the heat flux density at time t, which is set as a function of time, q estimate (t) is the inversion solution of heat flux density at time t.

[0092] When testing the inverse problem solving method for hot processes based on the improved cuckoo algorithm (SECSA) and the original cuckoo algorithm (CSA) in this embodiment, Figure 4 The fitness value variation curve for optimization problem f9 in the CEC2017 test set shows a significant improvement in both convergence speed and accuracy, verifying the superiority of the improved algorithm. The inversion results of the thermal process inverse problem solution method based on the improved Cuckoo algorithm are compared with those of the original algorithm. Each algorithm underwent 30 independent experiments, and the results are shown in the figures and tables. Table 1 shows the error analysis for the thermal conductivity inverse problem, Table 2 shows the error analysis for the boundary temperature inverse problem, and Table 3 shows the error analysis for the heat flux density inverse problem. Figure 5 (a) is the convergence graph of the original and improved cuckoo algorithm for the inverse problem of thermal conductivity;

[0093] Figure 5 (b) Convergence graphs of the original and improved Cuckoo algorithm for the boundary temperature inverse problem; Figure 5 (c) Convergence graphs of the original and improved Cuckoo Algorithms for the inverse heat flux density problem. Referring to the table above, it is found that in the one-dimensional constant thermal conductivity problem, the CSA accuracy can reach 1.1092 × 10⁻⁶. -14 The accuracy of SECSA can reach 1.6283×10⁻⁶. -16 This represents an improvement of two orders of magnitude. In the inverse problem of boundary temperature for circular regions, the two algorithms achieve similar accuracies, reaching 10-1. -5 For the inversion of heat flux density, CSA falls into local convergence for the two-dimensional square cavity convection problem, with an accuracy of 2.1000 × 10⁻⁶. -3 SECSA overcomes this problem with its superior search capabilities, achieving significant improvements in convergence speed and computational accuracy, with an accuracy of 3.1546 × 10⁻⁶. -5 This represents an improvement of two orders of magnitude. A comparison of the convergence plots reveals that SECSA's convergence speed and search accuracy are both superior to CSA.

[0094] Table 1

[0095]

[0096] Table 2

[0097]

[0098] Table 3

[0099]

[0100] In summary, this invention provides valuable guidance for solving inverse problems of thermal processes and for their practical applications.

[0101] The improved Cuckoo algorithm, when used to invert unknown parameters such as physical properties, boundary temperatures, and heat flux densities based on known physical property parameters, boundary conditions, and temperature distributions measured at measuring points, exhibits fast inversion speed, high convergence accuracy, and is less prone to local convergence. It has strong guiding significance for solving inverse problems and other engineering optimization problems.

[0102] The above description is merely a preferred embodiment of this disclosure and an explanation of the technical principles employed. Those skilled in the art should understand that the scope of the invention involved in the embodiments of this disclosure is not limited to technical solutions formed by specific combinations of the above-described technical features, but should also cover other technical solutions formed by arbitrary combinations of the above-described technical features or their equivalents without departing from the above-described inventive concept. For example, technical solutions formed by substituting the above-described features with (but not limited to) technical features with similar functions disclosed in the embodiments of this disclosure.

Claims

1. A method for solving the inverse problem of thermal processes based on an improved cuckoo algorithm, characterized in that, Includes the following steps: Step 1: Collect the temperature field information to be inverted, i.e. the temperature distribution of the object, through experiments using the preset physical property parameters, initial conditions and boundary conditions; Step 2: Determine the inverse problem type, set the fitness function, and determine the constraints corresponding to the inverse parameters, that is, the upper and lower limits of the predicted values ​​of the inverse parameters as constraints; Step 3: Write a forward calculation program to calculate the estimated discrete temperature field. ; Step 4: Use the improved Cuckoo algorithm to invert the parameters to be inverted and obtain the inversion solution; Step 4.1: Initialize the population; each position in the initialized population represents a feasible solution; Step 4.2: Calculate according to the forward calculation program written in Step 3. Substitute the fitness function set in step 2 into the fitness function to calculate the fitness value of each cuckoo; Step 4.3: Update the location of the bird's nest using the improved global search operator; The improved global search operator is as follows: ； in, It is the position of the i-th bird's nest in the t-th iteration; This represents the optimal location for the Bird's Nest in the current iteration. Flying for Levy; For the improved adaptive step size factor: ； in, This refers to the current iteration number. It is the maximum number of iterations. It is the minimum number of iterations, and its value is 1. The minimum step size scaling factor is set to 0.01; The maximum step size scaling factor is set to 1; The newly introduced traction factor is defined as follows: ； in, It is the fitness function value of the current best individual in the population; It is the fitness function value of an individual in the current population; Step 4.4: Use the greedy selection operator to retain cuckoo offspring with smaller fitness function values; The greedy selection operator is as follows: ； Step 4.5: Update the location of the bird's nest using an improved local search operator; Step 4.6: Use the greedy selection operator to retain the cuckoo offspring with smaller fitness function values; Step 4.7: Evaluate whether the termination condition is met, i.e. whether the preset maximum iteration step size has been reached. If the termination condition is met, the algorithm ends and the final inversion solution is obtained; otherwise, return to step 4.

2.

2. The method for solving the inverse problem of thermal processes based on the improved cuckoo algorithm according to claim 1, characterized in that, The experiment described in step 1 involves taking the sample to be tested, first measuring the thermal conductivity of the sample, then heating the sample, and finally obtaining the temperature distribution of the sample through a thermocouple after the sample temperature stabilizes.

3. The method for solving the inverse problem of thermal processes based on the improved cuckoo algorithm according to claim 1, characterized in that, Step 1 specifically includes the following steps: The inverse problem types in step 2 include parameter identification inverse problems and boundary identification inverse problems. Boundary identification includes first-type boundary conditions and second-type boundary conditions, namely boundary temperature and heat flux. The parameters to be inverted are thermal conductivity, boundary temperature and heat flux density, respectively. The inversion parameters are regarded as unknown conditions. The fitness function is set as follows: Where M is the number of measurement points, The discrete temperature field obtained in step 1, This is the discrete temperature field estimated through forward calculation.

4. The method for solving the inverse problem of thermal processes based on the improved cuckoo algorithm according to claim 1, characterized in that, The forward calculation program described in step 3 serves to obtain a workable solution by inputting the preset physical property parameters, known conditions in the initial and boundary conditions, and the cuckoo population. This solution is obtained through forward calculation of the heat conduction differential equation. .

5. The method for solving the inverse problem of thermal processes based on the improved cuckoo algorithm according to claim 1, characterized in that, The improved local search operator formula described in step 4.5 is as follows: ； in, It is a vector of random numbers that follows a normal distribution from 0 to 1; It is a vector of random numbers that follows a uniform distribution from 0 to 1; and The Bird's Nest is randomly selected from the current iteration number. And Bird's Nest Location; It is a step function; The improved adaptive discard probability is defined as: ； in, To maximize the discard probability, set it to 0.5; To minimize the discard probability, it is set to 0.01.

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