CN114048616A - LM correction method for identifying nonlinear thermal conductivity based on radial integral boundary element method - Google Patents
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Abstract
The invention discloses a correction LM method for identifying nonlinear thermal conductivity based on a radial integral boundary element method. The method comprises the following steps: acquiring initial data through a measuring means; establishing a fine boundary element model, dividing a boundary element grid, and establishing a boundary element format of a transient nonlinear heat conduction equation; calculating an optimization objective function of an inverse problem of the transient nonlinear heat conduction equation; judging whether the target function meets a convergence criterion; and outputting the final identification parameter value, the iteration step number and the optimized objective function value. According to the invention, the complex variable derivation method is introduced into the traditional LM method, so that each coefficient of the sensitivity matrix is accurately and efficiently calculated, and the radial integral boundary element method is introduced into the inverse heat conduction problem, so that the numerical simulation of the positive problem only needs to disperse units on the boundary, the dimensionality of the problem is reduced, the calculation precision of the inverse heat conduction problem is improved, and the application of the method in the identification of the thermophysical parameters of the ultrathin structure is expanded.
Description
Technical Field
The invention relates to a thermal conductivity identification technology in the fields of aerospace, steel industry, chemical industry and material preparation and characterization, in particular to a thermal conductivity identification method for a novel material/structure changing along with temperature.
Background
At present, high-Mach number aircrafts are competitively developed in various countries, which puts new requirements on thermal protection systems, and new heat-proof and heat-insulating materials need to be developed for the new requirements. Such materials generally have nonlinear thermophysical parameters, that is, the thermophysical parameters change with temperature, such as thermal conductivity, specific heat, etc. with temperature. These complex physical parameters of a material are difficult to measure directly by experimental means, especially when measured over a large temperature range. For example, within 0-1200 ℃, every 100 ℃ thermal conductivity is measured, and at this time, not only the reliability of the measuring instrument is significantly affected by the high temperature condition, but also the number to be measured makes the experiment increasingly difficult to perform.
The Inverse problem of heat conduction (Inverse heat conduction) provides a new idea for identifying the thermal conductivity of the material/structure along with the temperature change, and the unknown thermal conductivity can be reversely identified through the temperature measurement value of the measuring point. Aiming at the inverse problem of heat conduction, many scholars propose various solving methods, wherein the key is an optimization algorithm, and at present, the optimization algorithm is mainly divided into two types: random methods and gradient methods. A random algorithm such as a genetic algorithm can search for global optima, but the calculation amount is large, and the calculation cost is high; in contrast, the gradient method is highly accurate and computationally efficient, with the Levenberg-Marquardt (LM) method being favored by scholars. However, the efficiency and accuracy of the gradient method depends to a large extent on the sensitivity matrix coefficients, which represent the partial derivatives of the physical field to the thermal conductivity to be identified. The traditional method calculates the sensitivity by using a difference method: and applying a small change to the identification parameters, and calculating the change of the physical quantity of the measuring point at the moment, wherein the ratio of the latter to the former is the sensitivity coefficient. However, for the non-linearity problem, it is difficult to obtain an accurate value, since the difference method depends heavily on the calculation step size, and there is a cancellation error, and a high calculation cost is required. For this reason, it is necessary to establish a method capable of accurately and efficiently calculating the sensitivity coefficient in the inverse problem.
In addition, the numerical method is also an important part of the inverse problem of thermal conduction, and solving the physical field by using the traditional finite element method makes the inverse problem difficult to be applied to identifying the thermophysical parameters of ultra-thin and ultra-thin structures, such as coatings, bonding layers, honeycomb walls and other structures in a thermal protection system. Since the number of elements required for finite element analysis of ultra-thin and ultra-thin structures is huge, which is hard to bear by general computer resources, the boundary element method maintains the advantage of dividing the elements only on the boundary, and therefore, it is necessary to solve the inverse problem of thermal conduction based on the high-precision boundary element method.
Disclosure of Invention
1. Problems to be solved
Aiming at the problems in the prior art, the invention mainly solves the problems that the traditional LM method lacks a method for stably and accurately calculating the sensitivity coefficient in the process of identifying the nonlinear thermal conductivity and is difficult to be applied to an ultrathin superfine structure, and provides a modified LM method for identifying the nonlinear thermal conductivity based on a radial integral boundary element method, so that the method has higher precision and stability when identifying the thermal conductivity changing along with the temperature.
2. Technical scheme
In order to solve the above problems, the present invention adopts the following technical solutions.
A modified LM method for identifying nonlinear thermal conductivity based on a radial integral boundary element method is provided, wherein thermal conductivity which generally changes along with temperature does not have a functional form, so that thermal conductivity at a specified temperature is identified, and values between temperatures are represented by high-order interpolation values, and the method specifically comprises the following steps:
firstly, acquiring initial data through a measuring means, wherein the data comprises physical magnitude of a measuring point, geometric dimension of a structure, initial condition, boundary condition and initial guess value of a parameter to be identified.
And secondly, establishing a fine boundary element model according to the initial data acquired in the first step, dividing a boundary element grid, dividing the grid only at the boundary, and establishing a boundary element format of a transient nonlinear heat conduction equation.
The transient nonlinear heat conduction equation is as follows:
in formula (I), k is the thermal conductivity as a function of temperature, xiIs a Cartesian space coordinate, p and c are density and specific heat, respectively, t0At the initial time, T is the temperature value and Ω is the calculation domain.
Solving the transient nonlinear heat conduction equation by adopting a boundary element method, introducing a weight function, and expressing the formula (I) as a boundary element weak form:
where q (x) is the heat flow density, Γ is the boundary of the computational domain Ω, and G is the weight function, taking the basic solution of the Green's function.
And (3) integrating the last two terms of the radial basis function expression (VII) by combining a radial integration method:
where R is the distance between the source point and the field point, R is the distance between the field point and the action point, αA、akAnd a0Is a coefficient to be determined, phiAFor radial basis functions, F is expressed as:
and finally, expressing the formula (VII) into a form only containing boundary integration, solving the heat conduction problem based on a numerical integration method, and obtaining a temperature value at the same position as the measuring point.
Thirdly, calculating an optimization objective function of an inverse problem of the transient nonlinear heat transfer equation:
in formula (II), M is the number of measured point temperature values, and y is (y)1,y2,…,yNIs) the parameter vector to be identified, N is the number of identification parameters, Ti *For measuring temperature values, TiTo calculate a temperature value.
Fourthly, judging whether the optimization objective function meets a convergence criterion formula (III), if so, terminating iteration and turning to the fifth step; otherwise, the LM method of the formula (IV) is adopted to obtain the increment delta y of the identification parameter, then the identification parameter value is updated by the formula (V), new transient nonlinear heat transfer equation optimization target function data is obtained, and the second step is returned.
Sk≤ξor|Sk+1-Sk|≤ξ (Ⅲ)
Δy＝[JTJ+μ·diag(JTJ)]-1JT(Ti *-Ti(y)) (Ⅳ)
Wherein S represents an objective function, k represents the number of iterations, ξ represents the convergence accuracy, p is 1-N, μ is a damping factor, di ag represents a matrix diagonal element, J is a sensitivity matrix, and is represented as follows:
in the formula (VI), M and N are respectively the number of measured data and the number of identification parameters, T is a calculated temperature value, and each coefficient of J is calculated by adopting a complex variable derivation method.
In the complex variable derivation method, a small imaginary value ih is added to the argument x, and the function is expanded into a taylor series:
since h is very small, higher order terms can be ignored, and derivative values are directly obtained:
in formula (XI), Im represents an imaginary part.
The LM method is adopted to replace the conjugate gradient method to obtain the increment of the identification parameter, and the formula is as follows:
in the formula (XII)kTo search for the direction, akFor step size, the following is expressed:
where Δ T represents the difference between the calculated and measured values, J is the sensitivity matrix, gkIs an objective function gradient, betakThe conjugate coefficients are expressed as follows:
and fifthly, outputting the final identification parameter value, iterating the steps and optimizing the objective function value.
3. Advantageous effects
Compared with the prior art, the invention has the beneficial effects that:
(1) according to the invention, the complex variable derivation method is introduced into the traditional LM method, so that each coefficient of the sensitivity matrix is accurately and efficiently calculated, the high-precision identification of the nonlinear thermal conductivity is realized, and the precision and the stability of the traditional LM method in the identification of the thermophysical parameters are improved.
(2) Compared with the traditional finite difference method, the complex variable derivation method introduced by the invention does not depend on the calculation step length when calculating the sensitivity matrix coefficient, has no cancellation error, and obviously reduces the calculation cost.
(3) The radial integral boundary element method is introduced into the inverse heat conduction problem, so that numerical simulation of a positive problem only needs to disperse units on a boundary, dimension of the problem is reduced, calculation accuracy of the inverse heat conduction problem is improved, and application of the inverse heat conduction problem in thermophysical parameter identification of an ultra-thin structure is expanded.
Drawings
The accompanying drawings are included to provide a further understanding of the invention, and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention, and together with the description serve to explain the principles of the invention and not to limit the invention. In the drawings:
FIG. 1 is a flow chart of an implementation of a modified LM method for identifying nonlinear thermal conductivity based on a radial integral boundary element method;
FIG. 2 is a graph showing the results of measuring temperature values at 30s for 9 measuring point positions in the example;
FIG. 3 is a diagram of a boundary element computation domain according to an embodiment;
fig. 4 is a graph of thermal conductivity identification with temperature variation without a functional form in the example.
Detailed Description
The following describes in detail the modified LM method for identifying the non-linear thermal conductivity based on the radial integral boundary element method according to the present invention with reference to the drawings and the embodiments.
Embodiment as shown in fig. 1, a modified LM method for identifying nonlinear thermal conductivity based on a radial integral boundary element method, the method comprising the steps of: acquiring initial data through a measuring means; establishing a fine boundary element model, dividing a boundary element grid, and establishing a boundary element format of a transient nonlinear heat conduction equation; calculating an optimization target function of an inverse problem of the transient nonlinear heat conduction equation; judging whether the target function meets a convergence criterion; and outputting the final identification parameter value, the iteration step number and the optimized target function value.
In the first step, initial data is acquired by measurement means. The example is a 0.1m x 0.1m flat plate structure with 4 0.01 m circular holes, an initial temperature of 100 deg.C, 400 deg.C at the upper boundary, 100 deg.C at the lower boundary, and thermal insulation at other boundaries. The initial guesses for the thermal conductivity at 100 deg.C, 200 deg.C, 300 deg.C, 400 deg.C to be identified are all 45W/m deg.C, with values between the temperatures being represented by Lagrangian interpolations. The physical quantity values of the measuring points are the measured temperature values in 30s of the positions of the 9 measuring points in the embodiment as shown in FIG. 2.
And secondly, establishing a fine boundary element model, as shown in a schematic diagram of a boundary element calculation domain of the embodiment in fig. 3, dividing 208 boundary units in total, and establishing a boundary element format of the transient nonlinear heat conduction equation.
First, the control equation for transient heat conduction is given:
wherein k is a thermal conductivity varying with temperature, and the density ρ and the specific heat c are 2000kg/m, respectively3And 200J/kg. degree.C.
Solving a heat conduction equation by adopting a boundary element method, introducing a weight function, and expressing the formula (I) as a boundary element weak form:
wherein G is a weight function, and a basic solution of the Green function is taken.
Integrating the last two terms of the radial basis function expression (VII) by combining the radial integration method:
where R is the distance between the source point and the field point, R is the distance between the field point and the action point, αA、akAnd a0Is a coefficient to be determined, phiAFor radial basis functions, F is expressed as:
and finally, expressing the formula (VII) into a form only containing boundary integration, solving the heat conduction problem based on a numerical integration method, and obtaining a temperature value at the same position as the measuring point.
Thirdly, calculating an optimization objective function of an inverse problem of the transient nonlinear heat transfer equation:
in formula (II), M is the number of measured point temperature values, and y is (y)1,y2,…,yNIs) the parameter vector to be identified, N is the number of identification parameters, Ti *For measuring temperature values, TiTo calculate a temperature value.
Fourthly, judging whether the optimization objective function meets a convergence criterion formula (III), if so, terminating iteration and turning to the fifth step; otherwise, the LM method of the formula (IV) is adopted to obtain the increment delta y of the identification parameter, then the identification parameter value is updated by the formula (V), new transient nonlinear heat transfer equation optimization target function data is obtained, and the second step is returned.
Sk≤ξor|Sk+1-Sk|≤ξ (Ⅲ)
Δy＝[JTJ+μ·diag(JTJ)]-1JT(Ti *-Ti(y)) (Ⅳ)
Wherein S represents an objective function, k represents iteration times, ξ represents convergence accuracy, and 10 is taken-4Where p is 1 to N, μ is the damping factor initially taken at 0.1, diag stands for the matrix diagonal elements, and J is the sensitivity matrix, expressed as follows:
in the formula (VI), M and N are respectively the number of measured data and the number of identification parameters, T is a calculated temperature value, and each coefficient of J is calculated by adopting a complex variable derivation method.
And fifthly, outputting a final result, as shown in fig. 4, which is a thermal conductivity identification result with temperature variation without a functional form in the embodiment. For comparison, the embodiment is identified by adopting the traditional LM method, the calculation step length is respectively 10% and 5% of the parameter to be identified, and the result shows that the traditional LM method depends on the calculation step length and has slow convergence speed, but the method can lead each parameter to be converged to a true value quickly, and the identified thermal conductivity curve is well matched with the true curve.
Finally, it should be noted that the above embodiments are only used for illustrating the technical solutions of the present invention and are not limited. Although the present invention has been described in detail with reference to the embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted without departing from the spirit and scope of the invention as defined in the appended claims.
Claims (5)
1. A LM correction method for identifying nonlinear thermal conductivity based on a radial integral boundary element method is characterized in that:
the method specifically comprises the following steps:
firstly, acquiring initial data by a measuring means, wherein the initial data comprises a physical quantity value of a measuring point, a geometric dimension of a structure, an initial condition, a boundary condition and an initial guess value of a parameter to be identified;
secondly, establishing a fine boundary element model according to the initial data acquired in the first step, dividing boundary element grids, dividing the grids only at the boundary, and establishing a boundary element format of a transient nonlinear heat conduction equation;
the transient nonlinear heat conduction equation is as follows:
in formula (I), k is the thermal conductivity as a function of temperature, xiIs a Cartesian space coordinate, p and c are density and specific heat, respectively, t0At the initial moment, T is a temperature value, and omega is a calculation domain;
thirdly, calculating an optimization objective function of an inverse problem of the transient nonlinear heat transfer equation:
in formula (II), M is the number of measured point temperature values, and y is (y)1，y2，...，yNIs a vector of parameters to be identified, N is the number of identification parameters, Ti *For measuring temperature values, TiTo calculate a temperature value;
fourthly, judging whether the optimization objective function meets a convergence criterion formula (III), if so, terminating iteration and turning to the fifth step; otherwise, obtaining the increment delta y of the identification parameter by adopting the LM method of the formula (IV), then updating the identification parameter value by the formula (V) to obtain new transient nonlinear heat transfer equation optimization target function data, and returning to the second step;
Sk≤ξor|Sk+1-Sk|≤ξ (III)
Δy＝[JTJ+μ·diag(JTJ)]-1JT(Ti *-Ti(y)) (IV)
wherein S represents an objective function, k represents the number of iterations, ξ represents the convergence accuracy, p is 1 to N, μ is a damping factor, diag represents taking a matrix diagonal element, J is a sensitivity matrix, and is represented as follows:
in the formula (VI), M and N are respectively the number of measured data and the number of identification parameters, T is a calculated temperature value, and each coefficient of J is calculated by adopting a complex variable derivation method;
and fifthly, outputting the final identification parameter value, iterating the steps and optimizing the objective function value.
2. The method for identifying a modified LM method for nonlinear thermal conductivity based on the radial integral boundary element method of claim 1 wherein:
in the second step, the transient nonlinear heat conduction equation is solved by adopting a boundary element method, a weight function is introduced, and the formula (I) is expressed into a boundary element weak form:
where q (x) is the heat flow density, Γ is the boundary of the computational domain Ω, and G is the weight function, taking the basic solution of the Green's function.
3. The method for identifying a modified LM method for nonlinear thermal conductivity based on the radial integral boundary element method of claim 1 wherein:
in the second step, the last two-term domain integration of the radial basis function expression (VII) is adopted, and the radial integration method is combined:
where R is the distance between the source point and the field point, R is the distance between the field point and the action point, αA、akAnd a0Is a coefficient of undetermined value, phiAFor radial basis functions, F is expressed as:
and finally, expressing the formula (VII) into a form only containing boundary integration, and solving the heat conduction problem based on a numerical integration method to obtain a temperature value at the same position as the measuring point.
4. The method for identifying a modified LM method for nonlinear thermal conductivity based on the radial integral boundary element method of claim 1 wherein:
in the fourth step, in the complex variable derivation method, a small imaginary part ih is added to the argument x, and the function is expanded into a taylor series:
since h is very small, higher order terms can be ignored, and derivative values are directly obtained:
in formula (XI), Im represents an imaginary part.
5. The method for identifying a modified LM method for nonlinear thermal conductivity based on the radial integral boundary element method of claim 1 wherein:
in the fourth step, the LM method is replaced by a conjugate gradient method to obtain the increment of the identification parameter, and the formula is as follows:
in the formula (XII) dkTo search for the direction, akFor step size, the following is expressed:
where Δ T represents the difference between the calculated and measured values, J is the sensitivity matrix, gkIs an objective function gradient, betakThe conjugate coefficients are expressed as follows:
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| CN105868465A (en) * | 2016-03-28 | 2016-08-17 | 大连理工大学 | LM correction method for identification of heat conductivity changing along with temperature |
| CN105956344A (en) * | 2016-06-23 | 2016-09-21 | 大连理工大学 | Simple rapid solution method for complex-structure multi-dimensional transient nonlinear thermal conduction reverse problem |
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