CN113375828B - Temperature field reconstruction method based on ultrasonic waves - Google Patents
Temperature field reconstruction method based on ultrasonic waves Download PDFInfo
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Abstract
The invention discloses a temperature field reconstruction method based on ultrasonic waves. And then, in the second stage, the region to be measured is divided into a plurality of fine grids with smaller areas, and the reconstruction resolution and the reconstruction precision of the temperature field are improved by utilizing the strong generalization capability of the Gaussian process regression algorithm. The two-stage reconstruction algorithm obtained by combining the advantages of the two algorithms can solve the problem of measurement data information shortage, simultaneously realizes the balance between the number of grid divisions and the reconstruction resolution, improves the reconstruction precision and has better use performance.
Description
Technical Field
The invention relates to the technical field of temperature field reconstruction, in particular to a temperature field reconstruction method based on ultrasonic waves.
Background
Currently, the commonly used temperature measurement methods can be classified into contact type and non-contact type. The contact type temperature measuring method comprises a thermocouple thermometer, an expansion type thermometer, a pressure gauge thermometer, a blackbody cavity thermometer, an optical fiber thermometry method and the like, the measuring method can measure only by directly contacting a temperature measuring element with a measured object, and in most cases, the temperature measuring element and the measured object can be measured only when reaching a thermal equilibrium state, so that a measured temperature field is destroyed, therefore, the contact type thermometry method can be limited in use in some occasions and only can be used in a laboratory or a region with low temperature, and the measurement of the temperature field in the whole region is difficult to realize. Contact temperature measurement is taken as a traditional temperature measurement technology, is limited by the high temperature resistance of a heated element material, can only be used for short-time measurement, and has the problems of large field operation amount, incapability of realizing real-time online detection, incapability of providing accurate temperature field distribution parameters and the like.
The measuring instrument and elements of the non-contact measuring method are not in direct contact with the measured medium, and the temperature measurement of targets such as high temperature, strong corrosion and the like can be realized. The non-contact temperature measuring method mainly comprises a radiation method, an optical method and an acoustic method. The acoustic method has the advantages of wide temperature measurement range, high measurement precision, high real-time performance and high adaptability to temperature measurement environments, and is recognized as a technology with the greatest development prospect in the aspect of real-time online detection of temperature fields.
At present, ultrasonic temperature measurement is widely applied to the fields of monitoring of a boiler hearth temperature field, measuring of a submarine hydrothermal port temperature, measuring of a stored grain temperature field and the like. The ultrasonic temperature measurement is mainly based on the correlation between the sound velocity and the temperature of ultrasonic waves in the medium transmission process, and the temperature information of the measured object is deduced by measuring the sound velocity change. Besides the characteristic of non-contact temperature measurement, the ultrasonic temperature measurement technology also has the advantages of wide temperature measurement range, strong environmental adaptability, real-time continuity and the like, and can realize the measurement of spatial temperature distribution.
The key of ultrasonic temperature measurement is selection of a reconstruction algorithm, and the current temperature field reconstruction algorithm is mainly divided into a non-iterative reconstruction algorithm and an iterative reconstruction algorithm. The non-iterative reconstruction algorithm mainly comprises a least square method and a truncated singular value decomposition method; the iterative algorithm comprises an algebraic reconstruction method and a Landweber iteration method.
The least square method is simple and convenient to calculate, but the inverse operation exists in the operation process, so that the number of divided grids in the temperature field reconstructed by the method is smaller than the number of ultrasonic measurement paths. This results in too few temperature field sampling points and large reconstruction errors.
The reconstruction speed of the truncated singular value decomposition is high, the singular value decomposition is firstly carried out on a coefficient matrix formed by an ultrasonic path, then, the items of which the singular values approach zero are abandoned, and the amplification effect of small singular values on the flight time measurement errors is reduced. However, the singular value of the coefficient matrix is often in a continuous descending state, so that it is difficult to select a proper singular value for truncation, and the reconstruction effect of the truncated singular value decomposition is poor.
The algebraic reconstruction method firstly gives an initial value of the reconstructed temperature field, then calculates the error between the flight time based on the initial value and the measured flight time, and then uses the error for correcting the initial value. By iterating continuously until the required accuracy is reached. The algebraic reconstruction method is easily affected by measurement noise, and the reconstruction speed and the reconstruction result are greatly affected by initial values.
The Landweber iteration method has good stability and noise resistance, but has the problems that the numerical solution is excessively smooth, the prior information of a reconstructed object cannot be utilized and the like.
Therefore, a temperature field reconstruction method is needed to solve the problem of lack of measured data information, achieve balance between the number of grid divisions and reconstruction resolution, improve reconstruction accuracy, and improve usability.
Disclosure of Invention
In order to solve the problems in the prior art, the invention provides a temperature field reconstruction method based on ultrasonic waves, which solves the problems mentioned in the background technology.
In order to achieve the purpose, the invention provides the following technical scheme: a temperature field reconstruction method based on ultrasonic waves comprises the steps of dividing a region to be measured into a certain number of coarse grids, and performing optimization solution on an objective function by using a wolf optimization algorithm to quickly obtain temperature values of the coarse grids; and dividing the region to be measured into a plurality of fine grids, obtaining the temperature distribution condition of the global fine grid by utilizing the strong generalization capability of the Gaussian process regression algorithm, drawing and displaying a temperature field reconstruction image, and finally completing the reconstruction of the temperature field.
Preferably, the method for reconstructing the temperature field comprises the following specific steps:
1. dividing the region to be measured into N coarse grids according to the self geometric characteristics of the region to be measured and the number M of effective ultrasonic paths;
2. setting the length value of the ith ultrasonic path in the jth grid as Ai, j; according to the determined number M of the effective ultrasonic paths and the number N of the divided coarse grids, calculating the length value of each ultrasonic path on the coarse grids to form a coefficient matrix A
3. By switching the receiving and transmitting states of the acoustic wave transducer, the corresponding ultrasonic wave flight time under each path is measured by an ultrasonic time delay estimation algorithm to form a flight time matrix b
4. Constructing an L2 norm-based objective function
min{||Ax-b|| 2 }；
In the formula, A is a distance coefficient matrix; x is an acoustic velocity reciprocal vector, and b is an ultrasonic wave flight time data vector;
5. solving x of the target function in the fourth step through a GWO algorithm, defining the position of the wolf pack in the GWO algorithm as a solution of x, calculating the advantages and disadvantages of the wolf pack position through the target function, and obtaining the optimal solution x of x through iterative solution of the GWO algorithm;
6. calculating to obtain corresponding coarse grid temperature vector value Tc through the optimal solution x
7. Taking the center coordinate Cc of the coarse grid and the temperature vector value Tc of the coarse grid as a training sample of the GPR;
8. training samples and covariance functions from GPR
Functional expression of a negative log-likelihood function that constitutes a conditional probability for a training sample>
In which M' is present in>
Is an unknown in the formula, θ represents a set of three unknowns +>
The deviation of theta is calculated and calculated,
wherein, the first and the second end of the pipe are connected with each other,
K n is a calculated covariance matrix that is a function of the measured signal,
based on the partial derivative of the negative log-likelihood function, carrying out minimum operation on the negative log-likelihood function by using a conjugate gradient method, thereby obtaining an optimal solution of the hyperparameter theta;
9. further dividing the area to be measured into fine grids, setting the number of the fine grids to be 10-20 times of the number of the coarse grids, and setting the number of the fine grids to be P to obtain a fine grid center coordinate matrix C f
10. Carrying out Gaussian process regression prediction by using the obtained optimal solution of the hyperparameter theta, and then obtaining a fine grid temperature vector value T f
11. According to the fine grid central coordinate matrix C f And fine grid temperature vector value T f Drawing and displaying a temperature field reconstruction image;
12. and repeating the third step to the eleventh step in a circulating manner, and rebuilding the temperature field for continuous time to obtain the dynamic change condition of the temperature field.
Preferably, in the first step, the number M of the effective ultrasonic paths is calculated according to the actual size of the region to be measured, the complexity of the temperature field to be measured, and the layout and number of the acoustic wave transducers.
Preferably, in the first step, the number N of coarse grids is less than or equal to the number M of effective ultrasonic paths.
Preferably, in step six, the specific process of calculating the corresponding coarse grid temperature vector value Tc is as follows:substituting the optimal solution x into a formula
Wherein L represents the path length, v represents the ultrasound propagation velocity, L and B are fixed constants, and x is ^ based on>
The temperature vector value Tc of the coarse grid can be obtained, and the temperature distribution of the measurement area under the coarse grid can be obtained.
Preferably, in step ten, the fine grid temperature vector value T is obtained f The specific process comprises the following steps: determining the specific form of the covariance function K (X, X) by using the optimal solution of the hyper-parameter theta, and bringing the determined covariance function K (X, X) into
And
in the method, the final prediction mean vector m can be obtained by solving * And its covariance cov (f) * ) The temperature vector value T of the fine grid can be obtained f And obtaining the temperature distribution of the measurement area under the fine grid.
The invention has the beneficial effects that: 1) The invention provides a two-stage temperature field reconstruction algorithm fusing a gray wolf optimization algorithm (GWOO) and a Gaussian Process Regression (GPR). In the first stage, the target function is optimized and solved by using a wolf optimization algorithm, and the temperature distribution information of the coarse grid is obtained by minimizing the flight time error. And in the second stage, on the basis of introducing prior information, performing temperature reconstruction on the fine grid by using Gaussian process regression to obtain the temperature distribution condition of the global fine grid. The gray wolf optimization algorithm belongs to a meta-heuristic algorithm, and on one hand, the probability that the solving process is trapped in local optimization is reduced, the global search performance is improved, and the reconstruction precision of the temperature field is effectively improved. On the other hand, in the conventional algorithm, the target function gradient or the Hessian matrix is usually adopted to determine the search direction, and the GWO algorithm does not need to calculate the gradient or the Hessian matrix of the target function, so that the difficulty of calculating a derivative in the reconstruction process is overcome, and the practical application of the algorithm is facilitated.
In the temperature field reconstruction process, the Gaussian process regression algorithm can reasonably introduce prior information, and model selection is carried out by a standard Bayesian method, so that the model performance is improved, and the calculation complexity is reduced. The method also has the advantages of easy numerical value realization, parameter self-adaption, strong generalization capability and the like, and is beneficial to improving the reconstruction precision of the temperature distribution.
2) The invention optimizes the balance between the division quantity of the area grids and the reconstruction resolution, and the division of the area to be measured into a plurality of small grids according to a certain mode is an important work in the reconstruction process of the temperature field. In general, the small grids are divided according to the principle of equal area. However, the number of the grid divisions does not have a certain criterion, the number of the divisions is large, the reconstruction accuracy is higher, but the reconstruction speed is lower, and more acoustic wave transducers are needed; the number of divisions is small, the reconstruction speed is faster, but the spatial resolution of the coarse grid temperature is lower. In the use process of the temperature field reconstruction algorithm, the area to be measured is divided into a small number of coarse grids in the first stage, and the grey wolf optimization algorithm is used for quickly obtaining the temperature value of the coarse grids. And then, in the second stage, the region to be measured is divided into a plurality of fine grids with smaller areas, and the reconstruction resolution and the reconstruction precision of the temperature field are improved by utilizing the strong generalization capability of the Gaussian process regression algorithm.
The two-stage reconstruction algorithm obtained by combining the advantages of the two algorithms solves the problem of lack of information amount of measured data, realizes the balance between the number of grid divisions and the reconstruction resolution, and improves the reconstruction precision of the temperature field.
Drawings
FIG. 1 is a schematic diagram of an arrangement of an acoustic wave transducer according to embodiment 1 of the present invention;
FIG. 2 is a schematic view of the flight path of the ultrasonic wave according to embodiment 1 of the present invention;
FIG. 3 is a schematic flow chart of the gray wolf optimization algorithm according to embodiment 1 of the present invention;
FIG. 4 is a schematic flowchart of a temperature field reconstruction method according to embodiment 1 of the present invention;
FIG. 5 is a schematic diagram of an iterative process of the gray wolf optimization algorithm in embodiment 1 of the present invention;
fig. 6 is a thermodynamic schematic of the reconstruction of the temperature field in example 1 of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be obtained by a person skilled in the art without making any creative effort based on the embodiments in the present invention, belong to the protection scope of the present invention.
Example 1
The general steps of acoustic temperature field reconstruction can be briefly summarized as follows:
(1) Considering the temperature complexity and the specific application requirements in the actual environment, determining the layout mode of the acoustic wave transducer, and selecting an effective ultrasonic wave propagation path;
(2) Determining a grid division mode of the region to be detected by combining the self geometric characteristics of the region to be detected and the like according to the determined layout of the transducer, the selected effective ultrasonic propagation path and the characteristics of the adopted reconstruction algorithm, and averagely dividing the region to be detected into a plurality of small grids;
(3) Controlling an acoustic wave transducer to transmit and receive ultrasonic signals, analyzing and processing the signals, and obtaining ultrasonic wave flight time TOF corresponding to each ultrasonic wave propagation path by an ultrasonic wave time delay estimation algorithm;
(4) And substituting a group of ultrasonic wave flying time obtained in each detection period and the related parameter information of the area to be detected into a temperature field reconstruction algorithm, calculating the temperature information of the area to be detected, drawing and displaying a temperature field visual image, and storing or transmitting temperature field data according to needs.
Taking an example that 4 transceiver transducers are arranged on each side of a region to be measured, and 16 transducers are used in total, the acoustic transducers are arranged as shown in fig. 1, the region to be measured is divided into 49 (7 × 7) grids, a schematic diagram of an ultrasonic flight path is shown in fig. 2, and 96 effective ultrasonic propagation paths are formed in the figure.
Fundamental principle of ultrasonic temperature measurement
Three basic equations in fluids according to ultrasound: the continuity equation, the state equation and the motion equation are deduced to obtain the functional relation between the sound velocity and the temperature
Where c is the propagation velocity of the ultrasonic wave, T is the absolute temperature, γ =1.402, R = 8.31J/K.mol for air, μ =29 × 10 -3 kg/mol is the gas constant.
It follows that the sound velocity and the absolute temperature T of the medium at equilibrium at a silent disturbance 0 Is proportional to the square root of (c).
Further rewriting formula (1) to
Wherein for a specific gas or a defined gas mixture
Can be considered as a fixed, known constant (in air, B =20.045 is usually taken). It follows that the propagation velocity of ultrasound in a gaseous environment will depend only on the gas temperature, that is to say the propagation velocity of ultrasound is a single-valued function of the gas temperature.
As can be seen from the schematic diagram of the flight path of the ultrasonic wave in FIG. 2, the length L of the flight path is a fixed constant that can be determined in advance, and therefore, the time t of flight of the ultrasonic wave on the flight path is measured L The average temperature T on the single path is obtained according to equation (2) L ：
In the formula v L Is the ultrasonic velocity. Based on the principle of temperature measurement on a single path, the actual acoustic thermometry model can be expressed as: the measurement area is first divided into n discrete grids (n = p × q, p denoting the number of rows and q denoting the number of columns) and the temperature values within each grid are considered to be the same. Considering that the distance of an ultrasonic wave propagation path passing through a certain grid j (j =1, \8230;, n) is a certain known quantity, the TOF data of the ultrasonic wave corresponding to the propagation path i (i =1, \8230;, n) is a measured value, a slowness value (namely a reciprocal value of the sound velocity) under the grid can be obtained through calculation, and then a corresponding temperature value can be obtained according to a functional relation constructed between the temperature and the sound velocity, so that the acoustic temperature measurement process is realized. In summary, the matrix form of the acoustic temperature measurement model is:
Ax＝y (4)
wherein A is an m multiplied by n dimensional distance coefficient matrix; x is an n x 1 dimensional reciprocal sound velocity vector, also called slowness vector; y is an mx 1-dimensional ultrasonic TOF data vector; m is the number of ultrasonic propagation paths; n is the total number of the divided grids.
Considering the noise error in the actual measurement, equation (4) is rewritten as:
Ax＝y+r (5)
where r is the measurement noise vector.
The temperature field reconstruction is essentially to rapidly and effectively solve x under the condition of knowing a matrix A and a vector y, and then a functional relation between sound velocity and temperature is utilized to obtain a temperature value of each grid; and the obtained temperature is used as the temperature of the geometric central point of the grid space, and finally, the detailed temperature distribution is obtained by adopting mathematical methods such as interpolation and the like.
Principle of grey wolf optimization algorithm
The acoustic temperature field reconstruction is essentially to solve a ill-conditioned inverse problem through a small amount of projection data, and the number of unknown variables is often more than that of the projection data; mathematically, there is uncertainty in the solution of the equation sought. Therefore, the solution to the equation should be such that the reconstructed temperature field error should be as small as possible, i.e. solving for x when the minimum of equation (5) is satisfied:
min{‖Ax-y‖ 2 } (6)
the project adopts a Gray Wolf Optimization algorithm (GWOL Optimization) to solve the inverse problem. The GWOOD algorithm flow chart is shown in FIG. 3, and the starting point of the Grey wolf optimization algorithm is to simulate the hunting behaviors of the Grey wolf population surrounding, hunting, searching and attacking prey, and the population position is continuously and iteratively optimized to obtain the optimal solution. The GWO algorithm optimization process comprises the steps of social level layering, tracking, surrounding and attacking prey of the wolf, and the specific steps are as follows.
(1) Social ranking
When designing a GWO algorithm, a mathematical model of a Grey wolf social grade needs to be simulated, an optimal solution, a suboptimal solution and a suboptimal solution are respectively defined as alpha, beta and delta, and the rest solutions are defined as omega. The hunting process (optimization) of the GWO algorithm is mainly led by α, β and δ, and ω obeys the above three wolves in hunting. The optimization process of GWO is mainly guided by the best three solutions (i.e., α, β, and δ) in each generation of population.
(2) Process for enclosing prey
In practice, the grayish wolf population implements a hunting process by surrounding the prey. Wherein the mathematical model surrounding the prey and the optimizing population locations is represented by the following equation:
wherein t represents the current iteration number;
representing a coefficient vector; />
A position vector representing a prey; />
Representing the location vector of the gray wolf.
Wherein the coefficient vector
The calculation formula of (c) is as follows:
in the formula
Represents a linear drop from 2 to 0 in an iterative process; />
Represents a random vector, <' > is selected>
(3) Hunting process
The grayish wolf population has the ability to identify and surround prey. In the hunting process, alpha plays a leading role, and beta and delta can also participate in a matching way; but in an abstract search space, the wolf population cannot acquire accurate position information of a prey (optimal solution). In order to construct the hunting process of the wolf population using the mathematical model, it is assumed that α, β, and δ have a stronger ability to acquire the potential locations of the prey, so in the iteration, it is necessary to save the three optimal solutions (i.e., α, β, and δ) and update the locations of the remaining search agents (ω) according to their locations. The mathematical formula for this process is expressed as:
in the formula
Respectively representing direction vectors among alpha, beta, delta and omega; />
Representing the direction vectors of the next step determined by alpha, beta and delta respectively. C1, C2, and C3 are random coefficients obtained by the formula (10), and A1, A2, and A3 are random coefficients obtained by the formula (9).
(4) Attack hunting process
To build a mathematical model of an attack prey, it can be found out, reduced, with reference to equation (9)
Is greater than or equal to>
The value of (c) also fluctuates. Thus, the algorithm recognizes when>
In the range of [ -1,1]In the interval, the wolf population attacks the game, and the search agent location should be anywhere between the current wolf and game at the next time.
(5) Search hunting procedure
The wolfsbane population mainly realizes prey search according to positions of alpha, beta and delta, appoints that prey is mutually dispersed when the prey is searched, and realizes the prey attack process after gathering. When searching for prey, the constructed dispersion model is assumed to be
When the absolute value of the search result is larger than 1, searching the prey by the search agent, wherein the search mode ensures that the algorithm can carry out global search; in addition, it can be found from formula (10) that>
Is located at [0,2 ]]Random number of interval, i.e. coefficient->
Provides random weighting of the prey to facilitate random incrementing>
Or decrease>
The distance defined in the formula (11) is beneficial to the GWOO algorithm to have more randomness in the optimization process, and the probability of the algorithm falling into the local optimal solution is reduced; and the coefficient is not linearly reduced, so that the possibility that the algorithm jumps out of a local solution is improved.
Regression principle of Gaussian process
A Gaussian Process Regression (GPR) method is a commonly used supervised learning method in machine learning, and the basic Process for Regression problem can be briefly described as follows: firstly, supposing the prior probability of a training sample obeying a Gaussian process, then calculating and solving the corresponding posterior probability under a Bayes framework, then adopting a maximum likelihood method to solve the optimal hyperparameter of the corresponding kernel function, and finally realizing the needed prediction by using the obtained model. The specific steps of the GPR process are shown below.
Given a training sample D = { (x) i ,y i ) | i =1, \8230 |, n }, where x i ∈R d Representing coordinates of a center point of a discrete grid in the measurement area; d represents an input dimension; n represents the number of training samples; y is i E R represents the temperature value of the corresponding grid. In a given finite set of training samples D, f (x) 1 ),f(x 2 ),…,f(x n ) Can be combined withThe set of machine variables, and satisfying the joint Gaussian distribution, all statistical features of the Gaussian Process (GP) can be constructed by a mean function m (x) and a covariance function k (x, x'):
consider that the noise ε contained in the observed target value y is white Gaussian noise (mean 0, variance 0) independent of f (x)
) Since f (x) also satisfies the gaussian distribution, and y also satisfies the gaussian distribution, the set of joint distributions of the observed target values y also forms a gaussian process.
Wherein m (x) represents a mean function; delta. For the preparation of a coating ij Represents the Kronecker delta function, δ when i = j ij ＝1。
The prior distribution of the observed target value y can therefore be expressed as:
according to Bayesian principles, the Gaussian process establishes a prior function within a given set of training samples D, at n * A given test specimen
Down-conversion to posterior distribution, observing target value vector y and test data output vector f * The joint prior distribution between is:
in the formula, K (X, X) * )＝K(X * ,X) T Representing a test sample input X * An n × 1 dimensional covariance matrix with the training sample input X; k (X) * ,X * ) Representing a test sample input X * Its own covariance.
From this, the prediction value f can be calculated * The posterior distribution of (a) is:
f * ∣X,y,X * ～N(m * ,cov(f * )) (18)
in the formula m * Representing a test sample input X * Predicted mean vector of (2): cov (f) * ) Representing test sample input X * The prediction covariance of (2). Wherein the predicted mean vector m * I.e. test data output vector f * The predicted value of (2).
In formulae (19) and (20), f * Outputting vector for test data, y is observation target value vector, K (X, X) * )＝K(X * ,X) T Representing test sample input X * Covariance matrix with training sample input X, K (X, X) denotes covariance of X itself, K (X) * ,X * ) Represents X * The covariance of the self-body,
representing the variance of the noise, I n Is an identity matrix.
Since the covariance function in the GPR method is a symmetric function that satisfies the Mercer condition, and thus the covariance function is equivalent to the kernel function, equation (19) can be rewritten as:
in the formula
Predicted mean vector m * Is a covariance function K (x) i ,x j ) The process changes the complex nonlinear problem into a solution of the linear problem.
According to the gaussian process theory of correlation analysis, a covariance function K (X, X) is used to measure the correlation between the training samples and the prediction samples. Therefore, the selection of the covariance function K (X, X) may affect the prediction result to some extent. Typical covariance functions include square exponential covariance functions, mercer covariance functions, rational quadratic covariance functions, neural network covariance functions, linear covariance functions, and the like. The covariance function can be selected at will in the Gaussian process, and the squared exponential covariance function is selected as the project. The function expression is
Wherein M = diag (l) -2 ) A symmetric matrix representing a hyper-parameter; l represents a relevance measure hyperparameter which represents the relevance of the distance between two data;
a signal variance representing a covariance function; />
Representing the variance of the noise; delta ij Representing a Kronecker operator; α represents a shape parameter of the covariance function. Since three hyper-parameters M>
Is unknown in equation (22), these hyperparameters are generally considered to be a collection { } { (R) } and>
and (6) processing.
The invention solves the hyper-parameter set theta in the covariance function by using a maximum likelihood method: firstly, a negative log-likelihood function of the conditional probability of the training sample is constructed, and then a conjugate gradient method is used for carrying out minimization operation on the negative log-likelihood function, so that the optimal solution of the hyper-parameter is obtained. Wherein, the negative log-likelihood function can be expressed as:
to make a partial derivative of theta, there are
Wherein the content of the first and second substances,
K n is a covariance matrix calculated by the formula (22), and this matrix includes a variable (hyper-parameter θ).
On the basis of the partial derivative obtained by the formula (24), the θ parameter at which the minimum value is obtained by the formula (23) can be obtained by using the conjugate gradient method. The specific form of the covariance function K (X, X) is further determined by equation (22). Finally, the determined covariance function K (X, X) is substituted into a formula (19) and a formula (20), and the final prediction mean vector m can be obtained by solving * And its covariance cov (f) * )。
The flow of the temperature field reconstruction method is shown in fig. 4, and the specific implementation steps are as follows:
1) According to the actual size of the region to be measured of the material object system and the complexity of the temperature field to be measured, a certain number of acoustic wave transducers are arranged, the layout mode of the transducers is determined, and the number of effective ultrasonic wave paths is calculated.
2) The region to be measured is divided into a certain number of coarse grids, and in order to balance the algorithm efficiency and the reconstruction accuracy, the number of the coarse grids is the same as or similar to the number of the effective ultrasonic paths, or the number of the coarse grids is smaller than the number of the effective ultrasonic paths.
3) And calculating the length value of each ultrasonic path on the coarse grid (the length of the ith ultrasonic path on the jth grid is Ai, j) according to the determined number (M) of the effective ultrasonic paths and the number (N) of the divided coarse grids to form a coefficient matrix A.
4) And measuring the corresponding ultrasonic wave flight time under each path by an ultrasonic time delay estimation algorithm by switching the receiving and transmitting states of the acoustic wave transducer to form a flight time matrix b.
5) Target function to be constructed based on L2 norm
min{||Ax-b|| 2 } (25)
Wherein A is a distance coefficient matrix; x is the reciprocal vector of the speed of sound, also called the slowness vector; b is an ultrasonic wave flight time data vector;
6) Applying the GWO algorithm to solve for x of equation (25): the position of the wolf pack in the GWO algorithm is defined as the solution of x, and the advantages and the disadvantages of the wolf pack position are calculated through a formula (25). An optimal solution x is obtained through iterative solution of the GWO algorithm, and the solution process of the GWO algorithm is shown in fig. 5.
7) And (4) calculating the optimal solution x solved by the GWO algorithm through a formula (3) to obtain a corresponding coarse grid temperature value Tc.
8) And calculating a coarse grid center coordinate matrix Cc. The coarse grid center coordinates Cc and the coarse grid temperature value Tc are used together as a training sample of the GPR.
9) And (3) calculating a covariance matrix and a partial derivative matrix of the central coordinates of the coarse grid according to the GPR training sample and a formula (22), thereby forming a function expression of a negative log-likelihood function of the conditional probability of the training sample and the gradient thereof. For unknown hyper-parameters in the negative log-likelihood function, the method uses a conjugate gradient method to solve, and when the negative log-likelihood function obtains a minimum value, the hyper-parameter value at the moment is taken as an optimal value. Solving a hyper-parameter set
After the optimal value is obtained, the next step can be carried out.
10 The area to be measured is further divided into fine grids with smaller areas, and in order to improve the reconstruction accuracy of the temperature field, the fine grids can be divided into fine grids which are sufficiently fine and are more than 10 times of the number of the coarse grids, and are generally divided into 10-20 times of the number of the coarse grids. Supposing that the number of the divided fine grids is P, calculating a fine grid center coordinate matrix C f 。
11 Solving to obtain a hyper-parameter set
After the optimal value of (a), the expression (i.e., the calculation formula of the covariance matrix) of the formula (22) becomes known (the corresponding covariance value can be obtained only by substituting the value (xi, xj)). Substituting the covariance function K (X, X) into equations (19) and (20), equation (19) may also calculate the result (corresponding to the predicted mean vector m) * ). Similarly, since the formula (20) is also composed of a series of covariance matrices, the formula (20) can also calculate the result (corresponding to the predicted covariance cov (f) * ) Calculated prediction mean vector T) f I.e. the fine grid temperature value.
12 Based on a fine-grid center coordinate matrix C f And corresponding fine mesh temperature value T f And drawing and displaying the temperature field reconstruction image. And completing the reconstruction of the temperature field.
13 Continuous-time temperature field reconstruction is carried out if necessary, and the dynamic change condition of the temperature field is obtained. The steps 4) to 12) can be circularly operated. After the algorithm is run once, the current temperature field image can be drawn at the step 12), as shown in fig. 6, the left graph is the reconstructed three-dimensional temperature field thermodynamic diagram, and the right graph is the reconstructed two-dimensional temperature field thermodynamic diagram.
The invention provides a two-stage temperature field reconstruction algorithm fusing a gray wolf optimization algorithm (GWOO) and a Gaussian Process Regression (GPR). In the second stage, on the basis of introducing prior information, the temperature reconstruction of the fine grid is carried out by using Gaussian process regression, and the temperature distribution condition of the global fine grid is obtained. The gray wolf optimization algorithm belongs to a meta-heuristic algorithm, and on one hand, the probability that the solving process is trapped in local optimization is reduced, the global search performance is improved, and the reconstruction precision of the temperature field is effectively improved. On the other hand, in the conventional algorithm, the target function gradient or the Hessian matrix is usually adopted to determine the searching direction, and the GWO algorithm does not need to calculate the gradient or the Hessian matrix of the target function, so that the difficulty of calculating a derivative in the reconstruction process is overcome, and the practical application of the algorithm is facilitated.
In the temperature field reconstruction process, the Gaussian process regression algorithm can reasonably introduce prior information, and model selection is carried out by a standard Bayesian method, so that the model performance is improved, and the calculation complexity is reduced. The method also has the advantages of easy numerical value realization, parameter self-adaption, strong generalization capability and the like, and is beneficial to improving the reconstruction precision of the temperature distribution.
The invention optimizes the balance between the division quantity of the area grids and the reconstruction resolution, and the division of the area to be measured into a plurality of small grids according to a certain mode is an important work in the reconstruction process of the temperature field. In general, the small grids are divided according to the principle of equal area. However, the number of the grid divisions does not have a certain criterion, the number of the divisions is large, the reconstruction accuracy is often higher, but the reconstruction speed is lower, and more acoustic wave transducers are needed; the number of divisions is small, the reconstruction speed is faster, but the spatial resolution of the coarse grid temperature is lower. In the use process of the temperature field reconstruction algorithm, the region to be measured is divided into a small number of coarse grids in the first stage, and the temperature value of the coarse grids is quickly obtained by using the gray wolf optimization algorithm. And then, in the second stage, the region to be measured is divided into a plurality of fine grids with smaller areas, and the reconstruction resolution and the reconstruction precision of the temperature field are improved by utilizing the strong generalization capability of the Gaussian process regression algorithm.
The problem of lack of information quantity of measured data is solved through a two-stage reconstruction algorithm obtained by fusing the advantages of the two algorithms, the balance between the grid division quantity and the reconstruction resolution is realized, and the reconstruction precision of the temperature field is improved.
Although the present invention has been described in detail with reference to the foregoing embodiments, it will be apparent to those skilled in the art that modifications may be made to the embodiments described in the foregoing embodiments, or equivalents may be substituted for elements thereof.
Claims (5)
1. A temperature field reconstruction method based on ultrasonic waves is characterized by comprising the steps of dividing a region to be measured into a certain number of coarse grids, and performing optimization solution on an objective function by using a wolf optimization algorithm to quickly obtain a temperature value of the coarse grids; dividing the region to be measured into a plurality of fine grids, obtaining the temperature distribution condition of the global fine grid by utilizing the strong generalization capability of a Gaussian process regression algorithm, drawing and displaying a temperature field reconstruction image, and finally completing the reconstruction of the temperature field;
the temperature field reconstruction method comprises the following specific steps:
1. dividing the region to be measured into n coarse grids according to the self geometric characteristics of the region to be measured and the number m of effective ultrasonic paths;
2. setting the length value of the ith ultrasonic path in the jth grid as Ai, j; according to the number m of the determined effective ultrasonic wave paths and the number n of the divided coarse grids, the length value of each ultrasonic wave path on the coarse grids is calculated to form a coefficient matrix A
3. By switching the receiving and transmitting states of the acoustic wave transducer, the corresponding ultrasonic wave flight time under each path is measured by an ultrasonic time delay estimation algorithm to form a flight time matrix b
4. Constructing an L2 norm-based objective function
min{||Ax-b|| 2 }；
In the formula, A is a distance coefficient matrix; x is an acoustic velocity reciprocal vector, and b is an ultrasonic wave flight time data vector;
5. solving x of the target function in the fourth step through a GWO algorithm, defining the position of the wolf pack in the GWO algorithm as a solution of x, calculating the advantages and disadvantages of the wolf pack position through the target function, and obtaining the optimal solution x of x through iterative solution of the GWO algorithm;
6. calculating to obtain corresponding coarse grid temperature vector value Tc through the optimal solution x
7. Taking the center coordinate Cc of the coarse grid and the temperature vector value Tc of the coarse grid as a training sample of the GPR;
8. training samples and covariance function based on GPR
Functional expression of a negative log-likelihood function constituting conditional probabilities of training samples
/>
In the formula, M is a group of,
is an unknown in the formula, θ represents a set of three unknowns @>
M＝diag(l -2 ) A symmetric matrix representing the hyperparameter; l represents a relevance measure hyperparameter and represents the relevance of the distance between two data; />
A signal variance representing a covariance function; />
Representing the noise variance; delta ij Representing a Kronecker operator;
the deviation of theta is calculated and calculated,
wherein the content of the first and second substances,
K n is the calculated covariance matrix and,
based on the partial derivative of the obtained negative log-likelihood function, carrying out minimization operation on the negative log-likelihood function by using a conjugate gradient method so as to obtain an optimal solution of the hyperparameter theta;
9. further dividing the area to be measured into fine grids, wherein the number of the fine grids is 10-20 times of that of the coarse grids, and the number of the fine grids is P, so as to obtain a fine grid central coordinate matrix C f
10. Carrying out Gaussian process regression prediction by using the obtained optimal solution of the hyperparameter theta, and then obtaining a fine grid temperature vector value T f
11. According to the fine grid central coordinate matrix C f And fine grid temperature vector value T f Drawing and displaying a temperature field reconstruction image;
12. and repeating the third step to the eleventh step in a circulating manner, and rebuilding the temperature field for continuous time to obtain the dynamic change condition of the temperature field.
2. The temperature field reconstruction method according to claim 1, characterized in that: in the first step, the number m of the effective ultrasonic paths is calculated according to the actual size of the region to be measured, the complexity of the temperature field to be measured, and the layout mode and the number of the acoustic wave transducers.
3. The temperature field reconstruction method according to claim 1, characterized in that: in the first step, the number n of the coarse grids is less than or equal to the number m of the effective ultrasonic paths.
4. The temperature field reconstruction method according to claim 1, characterized in that: in step six, the specific process of calculating to obtain the corresponding coarse grid temperature vector value Tc is as follows: substituting the optimal solution x into a formula
In the formula, L represents a path length, T L Is the average temperature, t, over the path of the sound wave L Is the flight time, v, of the sound wave in the propagation path L Is the average propagation speed of the sound wave on the propagation path, L and B are fixed constants, x is ^ in ^ x ^>
The temperature vector value Tc of the coarse grid can be obtained, and the temperature distribution of the measurement area under the coarse grid can be obtained.
5. The temperature field reconstruction method according to claim 1, characterized in that: in step ten, obtaining a fine grid temperature vector value T f The specific process comprises the following steps: determining the specific form of the covariance function K (X, X) by using the optimal solution of the hyperparameter theta, and substituting the determined covariance function K (X, X) into the optimal solution
And
in the method, the final prediction mean vector m can be obtained by solving * And its covariance cov (f) * ) The temperature vector value T of the fine grid can be obtained f So as to obtain the temperature distribution of the measurement area under the fine grid;
in the formula (f) * Outputting vector for test data, y is observation target value vector, K (X, X) * )＝K(X * ,X) T Representing test sample input X * Covariance matrix with training sample input X, K (X, X) denotes the covariance of X itself, K (X) * ,X * ) Represents X * The covariance of the self-body,
which represents the variance of the noise, is,I n is an identity matrix. />
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