CN109408926B - Method for solving complex structure multi-dimensional transient nonlinear heat conduction inverse problem - Google Patents

Abstract

The invention provides a method for solving the inverse problem of complex structure multi-dimensional transient nonlinear heat conduction. The method of the invention can identify the thermophysical parameters, boundary conditions and the like of the material/structure. The present invention combines the complex variable derivation method with commercial software for the first time for the inverse problem, so that the complex variable derivation method does not need to solve the source program code of the direct problem, and then uses the gradient method to solve the heat conduction inverse problem. Accurate sensitivity matrix coefficients can be obtained while solving the direct problem, and then the accurate and fast solution of the multi-dimensional transient nonlinear heat conduction inverse problem with complex structures can be realized. Because the present invention adopts commercial software, it can ensure strong applicability, high precision and high efficiency in solving the direct problem, and can ensure high efficiency and high precision in solving the inverse problem because it adopts the gradient method. The invention is particularly suitable for efficient and high-precision solving of complex structure multi-dimensional transient nonlinear heat conduction inverse problems, and has broad engineering application prospects.

Description

A method for solving the inverse problem of multi-dimensional transient nonlinear heat conduction in complex structures

Technical Field

The present invention relates to the fields of aerospace, steel and chemical engineering, and can be used for identifying the thermal physical property parameters of materials/structures, and can also be used for determining the boundary conditions of materials/structures. Specifically, it is a method for solving the inverse problem of multi-dimensional transient nonlinear heat conduction of complex structures.

Background Art

The problem of heat conduction refers to the determination of the temperature field distribution of an object with known boundary conditions, initial conditions, physical conditions and geometric conditions. However, in some cases, some information that should be known is unknown, such as boundary conditions, and inverse problems are needed to determine these unknown quantities. The inverse problem of heat conduction refers to the reverse deduction of boundary conditions, physical conditions, initial conditions, internal heat sources or geometric parameters based on additional information such as the temperature or heat flux distribution of an object. It has a wide range of application backgrounds in the aerospace field, steel industry and chemical industry, for example, the identification of thermophysical parameters of thermal protection system materials that change with temperature, the determination of heat flux density on the surface of steel billets, the determination of aerodynamic thermal boundary conditions, etc.

The solution of the inverse problem of heat conduction includes two parts, the solution of the direct problem and the solution of the inverse problem. For the direct problem, that is, the heat conduction problem, it is usually solved by manual programming. Commonly used solution methods include finite difference method, finite element method, boundary element method and meshless method. For the inverse problem, domestic and foreign scholars have proposed a variety of solution methods, which can be divided into two categories: gradient method and random method. The advantages of the gradient method are high accuracy and high efficiency, and the disadvantage is local convergence. The advantage of the random method is global convergence, and there is no need to determine the sensitivity matrix coefficients, but this type of method has low accuracy and many iterations, that is, low efficiency.

For solving the direct problem, the manual programming methods mentioned above either have poor adaptability to complex structure multi-dimensional transients or have high theoretical requirements, which are very difficult for engineering and technical personnel and have a high threshold for use. For solving the inverse problem, although the random method can avoid the calculation of the sensitivity matrix coefficients, its accuracy and efficiency are low. In comparison, the gradient method has higher accuracy and efficiency.

For the gradient method, accurate determination of the sensitivity matrix is crucial. Usually, the sensitivity matrix is obtained by using the difference method and the complex variable derivative method. The difference method is effective for simple linear problems. The "Simple and Fast Solution Method for the Inverse Problem of Multidimensional Transient Nonlinear Heat Conduction in Complex Structures" with publication number "CN105956344A" uses the difference method to calculate the coefficients of the sensitivity matrix. However, for some multidimensional nonlinear problems with particularly complex geometric structures, the difference method sometimes has destructive errors, making the sensitivity matrix coefficients zero, making the inversion impossible. The complex variable derivative method that can accurately determine the sensitivity matrix usually requires source code.

Summary of the invention

According to the technical problems raised above, a method for solving the inverse problem of multi-dimensional transient nonlinear heat conduction of complex structures is provided by combining the complex variable derivative method with commercial software, which not only ensures the high precision of solving the direct problem, but also ensures the high efficiency of solving the inverse problem. The innovation of the present invention lies in that the complex variable derivative method is combined with commercial software for the first time for the inverse problem, and two complex units suitable for two-dimensional and three-dimensional transient nonlinear heat conduction problems are constructed, and the sensitivity matrix required for the inverse analysis is accurately provided, so as to solve the inverse problem with high efficiency and high precision.

The technical means adopted by the present invention are as follows:

A method for solving the inverse problem of multi-dimensional transient nonlinear heat conduction of a complex structure comprises the following steps:

S1. Construct a complex unit for a multidimensional transient nonlinear heat conduction problem, wherein the complex unit includes two groups of nodes, namely, virtual and real nodes, specifically represented as a+bi, and the multidimensionality is two-dimensional or three-dimensional;

S2. For the parameters that need to be identified, modeling is performed using finite element software, and the measurement information of the physical quantities at the measuring points, initial conditions, physical parameters or boundary conditions, and the assumed initial values of the identification parameters are input;

的形式，从而获得测点物理量的计算值及所需的灵敏度矩阵

S3. Use finite element software to solve the multi-dimensional transient nonlinear heat conduction problem of complex structures. The complex unit is used in the solution and the complex unit is expressed as a matrix.

In the form of, the calculated value of the physical quantity at the measuring point and the required sensitivity matrix are obtained

S4. Calculate the optimization objective function based on the calculated value of the physical quantity at the measuring point and the sensitivity matrix. When the physical quantity at the measuring point is temperature, the specific function is:

Wherein, M is the number of measured data, _Ti * is the measured temperature value, _Ti represents the calculated temperature value, i represents the parameter to be inverted, i=1~M, wherein the object of the optimization objective function also includes a dimensionless form without a specific physical quantity unit;

S5, check whether it converges,

If the convergence criterion (2) is satisfied

F(x ₁ ,x ₂ ,...,x _N )≤ξ or |F ^K+1 -F ^K |≤ξ (2),

Then the iteration ends and the identified parameter results are output, where ξ is an infinitesimal positive number;

进行灵敏度分析，通过公式(4)更新辨识值后返回步骤S3，If not, then the sensitivity matrix calculated by using the complex unit in step S3 is retrieved.

Perform sensitivity analysis, update the identification value using formula (4) and return to step S3.

The sensitivity matrix is specifically:

[J ^T J + μ diag (J ^T J)] δ = J ^T [T _i ^* -T _i (x ₁ , x ₂ ,..., x _N )] (4)

The updated identification value is as follows:

{x ^k+1 }＝{x ^k }+{δ} (5)

为灵敏度矩阵系数，J^T为J的转置矩阵。In the formula, k is the number of iterations, μ is the damping factor,

is the sensitivity matrix coefficient, and J ^T is the transposed matrix of J.

Furthermore, the relaxation factor w is introduced into formula (5), and formula (5) is expanded to formula (6), which is:

{x ^k+1 }＝{x ^k }+w{δ} (6)

Among them, w=0~1.

Furthermore, when the complex structure is a two-dimensional structure, the construction of the plurality of units comprises the following steps:

S11. Construct the solution equation of the traditional transient heat conduction finite element method, specifically:

Among them, for each step of calculation, the temperature {T} _t at time t, heat capacity matrix [C], stiffness matrix [K], and load matrix {P} _t+Δt are all known quantities, and the temperature {T} t+Δt at time _t+Δt is an unknown quantity;

{T}_t+Δt、

分别用

T、

将式(7)转换为式(8)：S12.

{T} _t+Δt ,

Use

T.

Convert equation (7) to equation (8):

虚节点的为在给定扰动h下温度的变化量为

传统的二维4节点有限元单元总自由度为4，复数形式的2维8节点自定义单元的总自由度为8，将复数单元表示为矩阵

的形式，具体为：S31. For heat transfer problems, each node has only one degree of freedom, and the real temperature value of the real node is

The change in temperature of the virtual node under a given disturbance h is

The traditional 2D 4-node finite element has a total degree of freedom of 4, and the complex 2D 8-node custom element has a total degree of freedom of 8. The complex element is represented as a matrix

The form is:

Among them, Re represents the real value and Im represents the imaginary part.

Furthermore, when the complex structure is a three-dimensional structure, the construction of the plurality of units comprises the following steps:

S11. Construct the solution equation of the traditional transient heat conduction finite element method, specifically:

Among them, for each step of calculation, the temperature {T} _t at time t, heat capacity matrix [C], stiffness matrix [K], and load matrix {P} _t+Δt are all known quantities, and the temperature {T} t+Δt at time _t+Δt is an unknown quantity;

{T}_t+Δt、

分别用

T、

将式(7)转换为式(8)：S12.

{T} _t+Δt ,

Use

T.

Convert equation (7) to equation (8):

虚节点的为在给定扰动h下温度的变化量为

传统的三维8节点有限元单元总自由度为8，复数形式的三维16节点自定义单元的总自由度为16，将复数单元表示为矩阵

的形式，具体为：S31. For heat transfer problems, each node has only one degree of freedom, and the real temperature value of the real node is

The change in temperature of the virtual node under a given disturbance h is

The traditional 3D 8-node finite element has a total degree of freedom of 8, and the complex 3D 16-node custom element has a total degree of freedom of 16. The complex element is represented as a matrix

The form is:

Among them, Re represents the real value and Im represents the imaginary part.

Furthermore, in step S31, when the deflection parameter h to be inverted is given, the calculated value of the measured physical quantity and the required sensitivity matrix coefficients are obtained by equations (9) and (10).

Compared with the prior art, the present invention has the following advantages:

The complex variable derivation method in the prior art can accurately calculate the coefficients of the sensitivity matrix, but it requires the source code for solving the direct problem and cannot be used in combination with finite element commercial software. To this end, the present invention combines the complex variable derivation method with commercial software for the first time to solve the inverse problem, so that the complex variable derivation method does not need the source code for solving the direct problem, and then uses the gradient method to solve the inverse problem of heat conduction. The present invention combines the complex variable derivation method with commercial software, which not only ensures the high precision of solving the direct problem, but also ensures the high efficiency of solving the inverse problem. In addition, it greatly reduces the difficulty of use.

Based on the above reasons, the present invention can be widely promoted in the fields of aerospace, steel industry and chemical industry.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

FIG1 is a flow chart of a method for solving an inverse problem of multi-dimensional transient nonlinear heat conduction in a complex structure according to the present invention.

FIG. 2 is a schematic diagram of a two-dimensional 8-node complex unit according to Embodiment 1 of the present invention.

FIG3 is a schematic diagram of a three-dimensional 16-node complex unit according to Embodiment 2 of the present invention.

FIG4 is a schematic diagram of the geometric dimensions of the model in Example 1 of the present invention.

FIG5 is a schematic diagram of boundary conditions of the model in Example 1 of the present invention.

FIG. 6 is a schematic diagram of the identification results of thermal conductivity in Example 1 of the present invention.

FIG. 7 is a schematic diagram of the geometric shape of the model in Example 2 of the present invention.

FIG8 is a schematic diagram of the identification results of the second type of boundary conditions in Example 2 of the present invention.

FIG. 9 is a schematic diagram of the identification results of the third type of boundary conditions in Example 2 of the present invention.

DETAILED DESCRIPTION

In order to enable those skilled in the art to better understand the scheme of the present invention, the technical scheme in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work should fall within the scope of protection of the present invention.

It should be noted that the terms "first", "second", etc. in the specification and claims of the present invention and the above-mentioned drawings are used to distinguish similar objects, and are not necessarily used to describe a specific order or sequence. It should be understood that the data used in this way can be interchanged where appropriate, so that the embodiments of the present invention described herein can be implemented in an order other than those illustrated or described herein. In addition, the terms "including" and "having" and any variations thereof are intended to cover non-exclusive inclusions, for example, a process, method, system, product or device that includes a series of steps or units is not necessarily limited to those steps or units that are clearly listed, but may include other steps or units that are not clearly listed or inherent to these processes, methods, products or devices.

As shown in FIG1 , the present invention provides a method for solving the inverse problem of multi-dimensional transient nonlinear heat conduction of a complex structure, comprising the following steps:

S1. Construct a complex unit for a multidimensional transient nonlinear heat conduction problem, wherein the complex unit includes two groups of nodes, namely, virtual and real nodes, specifically represented as a+bi, and the multidimensionality is two-dimensional or three-dimensional;

S2. For the parameters that need to be identified, modeling is performed using finite element software, and the measurement information of the physical quantities at the measuring points, initial conditions, physical parameters or boundary conditions, and the assumed initial values of the identification parameters are input;

的形式，从而获得测点物理量的计算值及所需的灵敏度矩阵

S3. Use finite element software to solve the multi-dimensional transient nonlinear heat conduction problem of complex structures. The complex unit is used in the solution and the complex unit is expressed as a matrix.

In the form of, the calculated value of the physical quantity at the measuring point and the required sensitivity matrix are obtained

S4. Calculate the optimization objective function based on the calculated value of the physical quantity at the measuring point and the sensitivity matrix. When the physical quantity at the measuring point is temperature, the specific function is:

Wherein, M is the number of measured data, _Ti * is the measured temperature value, _Ti represents the calculated temperature value, i represents the parameter to be inverted, i=1~M, wherein the object of the optimization objective function also includes a dimensionless form without a specific physical quantity unit;

S5, check whether it converges,

If the convergence criterion (2) is satisfied

F(x ₁ ,x ₂ ,...,x _N )≤ξ or |F ^K+1 -F ^K |≤ξ (2),

Then the iteration ends and the identified parameter results are output, where ξ is an infinitesimal positive number;

进行灵敏度分析，通过公式(4)更新辨识值后返回步骤S3，If not, then the sensitivity matrix calculated by using the complex unit in step S3 is retrieved.

Perform sensitivity analysis, update the identification value using formula (4) and return to step S3.

The sensitivity matrix is specifically:

[J ^T J + μ diag (J ^T J)] δ = J ^T [T _i ^* -T _i (x ₁ , x ₂ ,..., x _N )] (4)

The updated identification value is as follows:

{x ^k+1 }＝{x ^k }+{δ} (5)

为灵敏度矩阵系数，J^T为J的转置矩阵。Where k is the number of iterations, μ is the damping factor,

is the sensitivity matrix coefficient, and J ^T is the transposed matrix of J.

Example 1

As shown in FIG2, when the complex structure is a two-dimensional structure, the total degree of freedom of the complex form of the 2D 8-node custom unit is 8, and its geometric dimensions and boundary conditions are shown in FIG4 and FIG5 respectively. The thermal conductivity of the structure varies with temperature and has no functional form. Therefore, the thermal conductivity at a specified temperature can be identified.

The main steps are shown in S1 to S5, wherein step S1 includes:

S11. Construct the solution equation of the traditional transient heat conduction finite element method, specifically:

Among them, for each step of calculation, the temperature {T} _t at time t, heat capacity matrix [C], stiffness matrix [K], and load matrix {P} _t+Δt are all known quantities, and the temperature {T} t+Δt at time _t+Δt is an unknown quantity;

{T}_t+Δt、

分别用

T、

将式(7)转换为式(8)：S12.

{T} _t+Δt ,

Use

T.

Convert equation (7) to equation (8):

The sensitivity matrix calculation of S3 is as follows:

虚节点的为在给定扰动h下温度的变化量为

传统的三维8节点有限元单元总自由度为8，复数形式的三维16节点自定义单元的总自由度为16，将复数单元表示为矩阵

的形式，具体为：S31. For heat transfer problems, each node has only one degree of freedom, and the real temperature value of the real node is

The change in temperature of the virtual node under a given disturbance h is

The traditional 3D 8-node finite element has a total degree of freedom of 8, and the complex 3D 16-node custom element has a total degree of freedom of 16. The complex element is represented as a matrix

The form is:

Among them, Re represents the real value and Im represents the imaginary part.

When the deflection parameter h to be inverted is given, the calculated value of the measured physical quantity and the required sensitivity matrix coefficients are obtained through equations (9) and (10):

Figure 6 shows the thermal conductivity identification result without a functional form in Example 1 of the present invention. It can be seen that for the transient heat conduction inverse problem of a two-dimensional complex structure, convergence can be achieved after 10 iterations, and the thermal conductivity of the material at a specified temperature can be accurately and quickly identified.

Example 2

As shown in Figure 3, when the complex structure is a three-dimensional structure, the total degree of freedom of the three-dimensional 16-node custom unit is 16, and its geometric dimensions are shown in Figure 7. The second and third types of boundary conditions on its outer surface are identified to identify the magnitude of the heat flux density and the convection heat transfer coefficient respectively.

The main steps are shown in S1 to S5, wherein step S1 includes:

S11. Construct the solution equation of the traditional transient heat conduction finite element method, specifically:

Among them, for each step of calculation, the temperature {T} _t at time t, heat capacity matrix [C], stiffness matrix [K], and load matrix {P} _t+Δt are all known quantities, and the temperature {T} t+Δt at time _t+Δt is an unknown quantity;

{T}_t+Δt、

分别用

T、

将式(7)转换为式(8)：S12.

{T} _t+Δt ,

Use

T.

Convert equation (7) to equation (8):

The sensitivity matrix calculation of S3 is as follows:

虚节点的为在给定扰动h下温度的变化量为

传统的三维8节点有限元单元总自由度为8，复数形式的三维16节点自定义单元的总自由度为16，将复数单元表示为矩阵

的形式，具体为：S31. For heat transfer problems, each node has only one degree of freedom, and the real temperature value of the real node is

The change in temperature of the virtual node under a given disturbance h is

The traditional 3D 8-node finite element has a total degree of freedom of 8, and the complex 3D 16-node custom element has a total degree of freedom of 16. The complex element is represented as a matrix

The form is:

Among them, Re represents the real value and Im represents the imaginary part.

In step S31, when the deflection parameter h to be inverted is given, the calculated value of the measured physical quantity and the required sensitivity matrix coefficients are obtained through equations (9) and (10).

8 and 9 show the identification results of the boundary conditions in Example 2 of the present invention. It can be seen that for the inverse problem of transient heat conduction of a three-dimensional complex structure, convergence can be achieved after 9 iterations, and accurate and rapid identification of the boundary conditions can be achieved.

The serial numbers of the above embodiments of the present invention are only for description and do not represent the advantages or disadvantages of the embodiments.

In the above embodiments of the present invention, the description of each embodiment has its own emphasis. For parts that are not described in detail in a certain embodiment, reference can be made to the relevant descriptions of other embodiments.

Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein by equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims (1)

1. A method for solving the inverse problem of multi-dimensional transient nonlinear heat conduction of complex structures, characterized by comprising the following steps: S1. Construct a complex unit for a multidimensional transient nonlinear heat conduction problem, wherein the complex unit includes two groups of nodes, namely, virtual and real nodes, specifically represented as a+bi, and the multidimensionality is two-dimensional or three-dimensional; S2. For the parameters that need to be identified, modeling is performed using finite element software, and the measurement information of the physical quantities at the measuring points, initial conditions, physical parameters or boundary conditions, and the assumed initial values of the identification parameters are input; S3. Use finite element software to solve the multi-dimensional transient nonlinear heat conduction problem of complex structures. The complex unit is used in the solution and the complex unit is expressed as a matrix.

In the form of, the calculated value of the physical quantity at the measuring point and the required sensitivity matrix are obtained

S4. Calculate the optimization objective function based on the calculated value of the physical quantity at the measuring point and the sensitivity matrix. When the physical quantity at the measuring point is temperature, the specific function is:

Wherein, M is the number of measured data, _Ti * is the measured temperature value, _Ti represents the calculated temperature value, i represents the parameter to be inverted, i=1~M, wherein the object of the optimization objective function also includes a dimensionless form without a specific physical quantity unit; S5, check whether it converges, If the convergence criterion (2) is satisfied F(x ₁ ,x ₂ ,...,x _N )≤ξ or |F ^K+1 -F ^K |≤ξ (2), Then the iteration ends and the identified parameter results are output, where ξ is an infinitesimal positive number; If not, then the sensitivity matrix calculated by using the complex unit in step S3 is retrieved.

Perform sensitivity analysis, update the identification value using formula (4) and return to step S3. The sensitivity matrix is specifically:

[J ^T J + μ diag (J ^T J)] = J ^T [T _i ^* -T _i (x ₁ , x ₂ ,..., x _N )] (4) The updated identification value is as follows: {x ^k+1 }＝{x ^k }+{δ} (5) In the formula, k is the number of iterations, μ is the damping factor,

is the sensitivity matrix coefficient, J ^T is the transposed matrix of J; Introducing the relaxation factor w into formula (5), formula (5) is expanded to formula (6), which is: {x ^k+1 }＝{x ^k }+w{δ} (6) Wherein, w = 0 to 1; When the complex structure is a two-dimensional structure, the construction of the plurality of units comprises the following steps: S11. Construct the solution equation of the traditional transient heat conduction finite element method, specifically:

Among them, for each step of calculation, the temperature {T} _t at time t, heat capacity matrix [C], stiffness matrix [K], and load matrix {P} _t+Δt are all known quantities, and the temperature {T} t+Δt at time _t+Δt is an unknown quantity; S12.

{T} _t+Δt ,

Use

T.

Convert equation (7) to equation (8):

S31. For heat transfer problems, each node has only one degree of freedom, and the real temperature value of the real node is

The change in temperature of the virtual node under a given disturbance h is

The traditional 2D 4-node finite element has a total degree of freedom of 4, and the complex 2D 8-node custom element has a total degree of freedom of 8. The complex element is represented as a matrix

The form is:

Among them, Re represents the real value, and Im represents the imaginary part; When the complex structure is a three-dimensional structure, the construction of the plurality of units comprises the following steps: S11. Construct the solution equation of the traditional transient heat conduction finite element method, specifically:

Among them, for each step of calculation, the temperature {T} _t at time t, heat capacity matrix [C], stiffness matrix [K], and load matrix {P} _t+Δt are all known quantities, and the temperature {T} t+Δt at time _t+Δt is an unknown quantity; S12.

{T} _t+Δt ,

Use

T.

Convert equation (7) to equation (8):

S31. For heat transfer problems, each node has only one degree of freedom, and the real temperature value of the real node is

The change in temperature of the virtual node under a given disturbance h is

The traditional 3D 8-node finite element has a total degree of freedom of 8, and the complex 3D 16-node custom element has a total degree of freedom of 16. The complex element is represented as a matrix

The form is:

Among them, Re represents the real value, and Im represents the imaginary part; In step S31, when the deflection parameter h to be inverted is given, the calculated value of the measured physical quantity and the required sensitivity matrix coefficients are obtained through equations (9) and (10).

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