CN105677993A - Numerical value general solution method for heat conduction heat source position recognition inverse problem - Google Patents

Abstract

The invention relates to a numerical value general solution method for a heat conduction heat source position recognition inverse problem. The numerical value general solution comprises the following steps that a heat conduction source position recognition inverse problem is described, the next step is directly executed if a heat conduction source is a point source, and if the heat conduction source is a non-point source, a conversion algorithm is adopted to convert the non-point source inverse problem into a point source inverse problem, and then the next step is executed; a homogeneous solution and a particular solution are calculated to construct a numerical value general solution; a system of linear equations is solved, and heat source position parameters are obtained. According to the method, the numerical value general solution meeting a heat conduction differential equation and using the heat source parameters as variables is constructed based on the finite element numerical solution, the heat conduction position recognition inverse problem is converted into a multivariate function extremum problem, and the heat source parameters are rapidly obtained through inversion. The method not only can inverse the point heat source position, but also can inverse the positions of heat sources in any shape, thereby being wide in application range, high in adaptability and good in engineering application prospect.

Description

A kind of thermally conductive heat source position identifies the numerical value general solution method of indirect problem

Technical field

A kind of method that the present invention relates to conduction of heat source item location recognition indirect problem, identifies the numerical value general solution method of indirect problem particularly to a kind of thermally conductive heat source position.

Background technology

Heat conduction inverse problem application is the study hotspot in engineering in recent years, and it is the emphasis of thermal conduction study basic research. Heat conduction inverse problem is just showing good application prospect in engineering in recent years, has a wide range of applications background at industrial circles such as power engineering, metallurgical and moulds.

Heat conduction inverse problem refers to for heat conduction problem, it is known that the structure temperature distribution after sometime, the thermal physical property parameter of inverting structure, heat source function or geometry or boundary condition etc., here it is heat conduction inverse problem. It will be apparent that heat conduction inverse problem compares conduction of heat direct problem, its difficulty solved more greatly, also more time-consuming.

Existing many about the patent of solution of inverse problems algorithm at present, application surface is also relatively wider. As Chinese invention patent application number 200710051566 discloses one " research method of Hydraulic and Hydro-Power Engineering hydraulics indirect problem ", propose pulse-spectrum method, Discrete Optimization Method, perturbation method, controlling metho, the component framework of waterpower hydroelectric project hydro science indirect problem. This invention belongs to the method for distributed parameter system indirect problem, but not the method for source item identification indirect problem.

Chinese invention patent application number 201210350077 discloses one " solving the numerical method of the indirect problem of subsonic flow ", carries out the numerical method of the indirect problem of solid wall surface geometry design under given solid wall surface pressure condition for a class. This invention belongs to the method for Geometric Shape Recognition indirect problem, but not the method for source item identification indirect problem.

Chinese invention patent application number 201210366939 discloses one " solving the numerical method of a class indirect problem with the Eulerian equation of Lagrangian Form ", it is proposed to the Two Dimensional Euler Equations of a kind of new Lagrangian Form solves the indirect problem of solid wall surface geometry design. This invention falls within the method for Geometric Shape Recognition indirect problem, but not the method for source item identification indirect problem.

Chinese invention patent application number 201410095196.7 gives " the farmland component temperature retrieval method based on global optimization approach ", adopts global optimization approach simulated annealing that object function is carried out minimization and achieves the inverting of farmland component temperature.

Chinese invention patent application 201410032593.X gives " a kind of numerical value general solution method of lines of thermal conduction source strength identification indirect problem ", and this is to know method for distinguishing about thermally conductive heat source strength parameter.

Summary of the invention

For the above-mentioned problems in the prior art, it is an object of the invention to propose a kind of thermally conductive heat source position and identify the numerical value general solution method of indirect problem. The method is possible not only to inverse point heat source position, and can inverting arbitrary shape heat source position, therefore wide application, strong adaptability, there is good future in engineering applications.

Described a kind of thermally conductive heat source position identifies the numerical value general solution method of indirect problem, it is characterised in that comprise the steps:

1) description of conduction of heat source item location recognition indirect problem

This conduction of heat source item location recognition indirect problem describes as follows: at thermal source q_s(x, y z), seek thermal source q temperature field θ under effect_sParameter, wherein in order to measure, point is upper given measures a temperature θ to given supplementary condition_d;

Thermal conduction under steady state is described as shown in formula (1):

$\begin{array}{cc}\begin{array}{c}\begin{array}{cc}\mathrm{\λ}{\▿}^2\mathrm{\θ}+q_s=0& (x,y,z\∈\mathrm{\Ω})\end{array}\\ \mathrm{\θ}{(x,y,z)}_b={\mathrm{\θ}}_b(x,y,z)\\ -\mathrm{\λ}{\left(\frac{\∂\mathrm{\θ}}{\∂n}\right)}_v=h(\mathrm{\θ}{|}_v-{\mathrm{\θ}}_f)\end{array}& \left(1\right)\\ -\mathrm{\λ}{\left(\frac{\∂\mathrm{\θ}}{\∂n}\right)}_w=f_2& \left(2\right)\end{array}$

In formula: (x, y, z) for temperature for θ; For Laplacian Differential Approach operator; Ω is the definition territory of problem; B is First Boundary Condition; V is third boundary condition; W is second kind boundary condition, and h is surface coefficient of heat transfer or convection transfer rate; θ_fFor heat exchange medium temperature; q_sFor heat source function; λ is heat conductivity, and n is border normal direction, f₂For heat flow density, wherein First Boundary Condition is assigned temperature; Third boundary condition is heat convection; Second kind boundary condition is heat flow density, and heat source function contains intensity and position;

2) if step 1) described in conduction of heat source be point source, then be directly entered step 3), if step 1) described in conduction of heat source be non-point source, then adopt transfer algorithm, non-point source indirect problem be converted into point source indirect problem, enter back into step 3);

3) solve homogeneous solution and particular solution, construct numerical value general solution

Basic theories according to Solutions of Ordinary Differential Equations, formula (1) general solution is made up of homogeneous solution and particular solution, and wherein homogeneous solution is to make q in formula (1)_s=0 solves, and particular solution then makes q_s=1 solves;

Rhetoric question topic (1) has the point source of k diverse location, and thermal source is expressed as:

$q_s=q(x,y,z)=\underset{i=1}{\overset{k}{\Σ}}\left(\mathrm{\δ}\right(x_i,y_i,z_i\left){\mathrm{\η}}_i\right)---\left(3\right)$

In formula: δ (x_i,y_i,z_i) for position function, (x_i,y_i,z_i) it is location parameter; η_iFor the intensive parameter of i-th point source, steady state problem, η_iFor constant; K is the number of point source;

The position of source item, according to the character of problem or engineering significance, is provided its possible position range by a,

I-th source item q in formula (3)_siLocation variable be (x_i,y_i,z_i), this location variable sets its excursion as x_i1≤x_i≤x_i2、y_i1≤y_i≤y_i2、z_i1≤z_i≤z_i2, wherein x_i1,x_i2,y_i1,y_i2,z_i1,z_i2For given value; Introduce dimensionless location parameter variable and contribute to the simplification of follow-up formula: $\begin{array}{ccc}{\mathrm{\α}}_i^1=(x_{i1}-x_i)/(x_{i1}-x_{i2}),& {\mathrm{\α}}_i^2=(y_{i1}-y_i)/(y_{i1}-y_{i2}),& {\mathrm{\α}}_i^3=(z_{i1}-z_i)/(z_{i1}-z_{i2})\end{array},$ And have $0\≤{\mathrm{\α}}_i^k\≤1,k=1,2,3;$

B only considers a variable, such as x_i, calculate particular solution;

Calculating k source strength respectively is η_iFinite Element Numerical Solution, Finite Element Numerical Solution is meant that here: the problem definition territory Ω that formula (1) is provided, utilize computer pass through Finite Element Method to i-th point source calculate at 2 end points (x_i1,x_i2) to have intensity be η_iHeat point source effect give a definition the temperature field in the Ω of territory, be designated as numerical value particular solution

Make q in formula (1)_s=0, solve homogeneous solution θ=θ₁; Adopting dimensionless location parameter variable, being expressed as thus constructing Numerical Temperature general solution θ:

$\mathrm{\θ}={\mathrm{\θ}}_1+\underset{i=1}{\overset{k}{\Σ}}\left(\right(1-{\mathrm{\α}}_i^1){\mathrm{\θ}}_{qi}^1+{\mathrm{\α}}_i^1{\mathrm{\θ}}_{qi}^2)---\left(4\right)$

In formula:For respective coordinates x_iDimensionless location parameter, be unknown quantity to be asked;Particular solution for point source i;

By given supplementary condition, namely some measurement, point considering, numerical value general solution (4) measures temperature θ with given_dError, obtain the residuals squares cost function of problem, as follows:

$g\left({\mathrm{\α}}^1\right)=\underset{j=1}{\overset{m}{\Σ}}{({\mathrm{\θ}}_{1j}+\underset{i=1}{\overset{k}{\Σ}}(\left(1-{\mathrm{\α}}_i^1\right){\mathrm{\θ}}_{qij}^1+{\mathrm{\α}}_i^1{\mathrm{\θ}}_{qij}^2)-{\mathrm{\θ}}_{dj})}^2---\left(5\right)$

Wherein: m is measure dot number;θ_djFor measuring some the measurement temperature of j, θ_1jBe measure some j homogeneous solution,It it is the i-th point source particular solution at j point;

Source item identification indirect problem formula (1) is converted into the extreme-value problem of the function of many variables that represented by formula (5) is variable with heat source position parameter, solves extreme value, tries to achieve the solution of indirect problem;

C is to location variable y_i,z_i, respectively repeat b step, obtain similar (5) formula respectively withFor the expression formula of variable, as (6) formula provides, then go to step 4) solve;

$\begin{array}{c}g\left({\mathrm{\α}}^2\right)=\underset{j=1}{\overset{m}{\Σ}}{({\mathrm{\θ}}_{1j}+\underset{i=1}{\overset{k}{\Σ}}(\left(1-{\mathrm{\α}}_i^2\right){\mathrm{\θ}}_{qij}^1+{\mathrm{\α}}_i^2{\mathrm{\θ}}_{qij}^2)-{\mathrm{\θ}}_{dj})}^2\\ g\left({\mathrm{\α}}^3\right)=\underset{j=1}{\overset{m}{\Σ}}{({\mathrm{\θ}}_{1j}+\underset{i=1}{\overset{k}{\Σ}}(\left(1-{\mathrm{\α}}_i^3\right){\mathrm{\θ}}_{qij}^1+{\mathrm{\α}}_i^3{\mathrm{\θ}}_{qij}^2)-{\mathrm{\θ}}_{dj})}^2\end{array}---\left(6\right)$

4) solve system of linear equations, obtain heat source position parameter

Seek the extreme-value problem of formula (5), formula (6), position parametric variable is differentiated, even:

$\∂g/\∂{\mathrm{\α}}_i^p=0,i=1,2,...,k;p=1,2,3$

In formula: i correspondence i-th unknown position variables parameter, three directions of p respective coordinates, wherein 1 corresponding (5), first expression formula of 2 correspondings (6), second expression formula of 3 correspondings (6), thus obtains calculating the system of linear equations of point source location parameter:

A α=B (7)

$\begin{array}{c}B={\left(\underset{j}{\overset{m}{\Σ}}\right({\mathrm{\θ}}_{dj}-{\mathrm{\θ}}_{1j}-\underset{i}{\overset{k}{\Σ}}{\mathrm{\θ}}_{qij}^1\left)\right({\mathrm{\θ}}_{qpj}^2-{\mathrm{\θ}}_{qpj}^1\left)\right)}_{k\×1}\\ =\underset{j}{\overset{m}{\Σ}}\left[\begin{array}{c}({\mathrm{\θ}}_{dj}-{\mathrm{\θ}}_{1j}-\underset{i}{\overset{k}{\Σ}}{\mathrm{\θ}}_{qij}^1)({\mathrm{\θ}}_{q1j}^2-{\mathrm{\θ}}_{q1j}^1)\\ .\\ .\\ .\\ ({\mathrm{\θ}}_{dj}-{\mathrm{\θ}}_{1j}-\underset{i}{\overset{k}{\Σ}}{\mathrm{\θ}}_{qij}^1)({\mathrm{\θ}}_{qkj}^2-{\mathrm{\θ}}_{qkj}^1)\end{array}\right]\end{array}---\left(9\right)$

Wherein: α is the dimensionless heat source position parameter vector being made up of k point source, solves each time and only correspond to some coordinate direction;

After solving formula (7), obtain the location parameter of thermal source according to the dimensionless α tried to achieve, as follows:

$\begin{array}{c}{x}_i=x_{i1}-(x_{i1}-x_{i2}){\mathrm{\α}}_i^1\\ y_i=y_{i1}-(y_{i1}-y_{i2}){\mathrm{\α}}_i^2\\ z_i=z_{i1}-(z_{i1}-z_{i2}){\mathrm{\α}}_i^3\end{array},(i=1,2,\mathrm{...},k)---\left(10\right).$

Described a kind of thermally conductive heat source position identifies the numerical value general solution method of indirect problem, it is characterised in that the heat flow density f in second kind boundary condition₂It is considered as border thermal source, the inverting of its parameter and thermal source q_sIdentical.

Described a kind of thermally conductive heat source position identifies the numerical value general solution method of indirect problem, it is characterized in that step 2) described non-point source is converted into the transfer algorithm of point source indirect problem, specifically include following steps: non-point source indirect problem is converted it into point source indirect problem by interpolation method, then step 3 is adopted) and step 4), solve.

Described a kind of thermally conductive heat source position identifies the numerical value general solution method of indirect problem, it is characterised in that the thermal source to spatial domain, and the method changing into point source indirect problem is as follows:

Space thermal source is separated into some hexahedrons or the combination of tetrahedron heat source unit, it may be assumed that

$F=\underset{i}{\Σ}{F}_i---\left(11\right)$

Using heat point source as interpolation knot, hexahedron or tetrahedron element are interpolated, with the interpolating function that point source is interpolation knot, space thermal source is described, thus converting it into the inversion problem of heat point source, obtain the location parameter of point source, thus obtaining the position of space thermal source, express by heat source unit unit interpolating function:

$F_i=\underset{j}{\overset{n}{\Σ}}{N}_j(x,y,z)P_j---\left(12\right)$

Wherein: n is the interpolation knot number of heat source unit, hexahedral element generally having 8 nodes, tetrahedron element generally has 4 nodes.

Use the dimension of notably each physical quantity during this method, namely want dimension to unify. As adopted International System of Units SI, basic dimension is: rice m, kilogram kg, second s, then heat source strength be W (watt), material thermal conductivity dimension be W/ (m DEG C), convection transfer rate dimension be W/ (m²DEG C); And the dimension of temperature adopts DEG C (degree Celsius).

By adopting above-mentioned technology, compared with prior art, beneficial effects of the present invention is as follows:

1) present invention is by adopting the building method limited, and according to the character of problem or engineering significance, the position of source item is provided its possible position range, applies thermal source, the homogeneous solution of solving equation (1) and particular solution;

2) a kind of thermally conductive heat source position that the present invention proposes identifies the numerical value general solution method of indirect problem, based on Finite Element Numerical Solution, construct numerical value general solution that meet thermal conduction differential equation, that be variable with Heat-Source Parameters, conduction of heat location recognition indirect problem is converted into Extreme Value Problem of Multi-Variable Functions, and fast inversion goes out Heat-Source Parameters. The method is possible not only to inverse point heat source position, and can inverting arbitrary shape heat source position, therefore wide application, strong adaptability, there is good future in engineering applications.

Accompanying drawing explanation

Fig. 1 solves point of heat transfer source position to identify the flow chart of indirect problem;

Fig. 2 is the x of the invention process case_i1The Finite-Element Solution of application point thermal source;

Fig. 3 is the x of the invention process case_i2The Finite-Element Solution of application point thermal source.

Detailed description of the invention

Below in conjunction with Figure of description and embodiment, the invention will be further described, but protection scope of the present invention is not limited to that:

As it is shown in figure 1, thermally conductive heat source position of the present invention identifies the numerical value general solution method of indirect problem, comprise the steps:

1) description of conduction of heat source item location recognition indirect problem

This conduction of heat source item location recognition indirect problem describes as follows: at thermal source q_s(x, y z), seek thermal source q temperature field θ under effect_sParameter, wherein in order to measure, point is upper given measures a temperature θ to given supplementary condition_d;

Thermal conduction under steady state is described as shown in equation (1):

$\begin{array}{cc}\begin{array}{c}\begin{array}{cc}\mathrm{\λ}{\▿}^2\mathrm{\θ}+q_s=0& (x,y,z\∈\mathrm{\Ω})\end{array}\\ \mathrm{\θ}{(x,y,z)}_b={\mathrm{\θ}}_b(x,y,z)\\ -\mathrm{\λ}{\left(\frac{\∂\mathrm{\θ}}{\∂n}\right)}_v=h(\mathrm{\θ}{|}_v-{\mathrm{\θ}}_f)\end{array}& \left(1\right)\\ -\mathrm{\λ}{\left(\frac{\∂\mathrm{\θ}}{\∂n}\right)}_w=f_2& \left(2\right)\end{array}$

In formula: (x, y, z) for temperature for θ; For Laplacian Differential Approach operator; Ω is the definition territory of problem; B is First Boundary Condition, i.e. assigned temperature; V is third boundary condition, i.e. heat convection; W is second kind boundary condition, i.e. heat flow density, and h is surface coefficient of heat transfer or convection transfer rate; θ_fFor heat exchange medium temperature; q_sFor heat source function (containing intensity and position); λ is heat conductivity, and n is border normal direction, f₂For heat flow density; In the present invention, to the heat flow density f in second kind boundary condition₂, can be considered border thermal source, the inverting of its parameter and thermal source q_sIdentical, it is possible to solve by this method, therefore the available equation (1) of description of Heat Conduction source item location recognition indirect problem is expressed by this method, is left out second kind boundary condition;

2) if step 1) described in conduction of heat source be point source, then be directly entered step 3), if step 1) described in conduction of heat source be non-point source, then adopt transfer algorithm, non-point source indirect problem be converted into point source indirect problem, enter back into step 3);

3) solve homogeneous solution and particular solution, construct numerical value general solution

Basic theories according to Solutions of Ordinary Differential Equations, equation (1) general solution is made up of homogeneous solution and particular solution, and wherein homogeneous solution is to make q in (1) formula_s=0 solve, particular solution then makes q_s=1 solves;

Thermal source q_sForm generally can describe with the thermal source of point, line, surface and body, its midpoint is basic, and the thermal source of other form can be described by the combination of point source certain rule. Therefore below, heat point source is first discussed. Rhetoric question topic (1) has the point source of k diverse location, and thermal source can be expressed as:

$q_s=q(x,y,z)=\underset{i=1}{\overset{k}{\Σ}}\left(\mathrm{\δ}\right(x_i,y_i,z_i\left){\mathrm{\η}}_i\right)---\left(3\right)$

In formula: δ (x_i,y_i,z_i) for position function, (x_i,y_i,z_i) it is location parameter; η_iFor the intensive parameter of i-th point source, steady state problem, η_iFor constant; K is the number of point source.

The core of numerical value general solution method is the homogeneous solution and the particular solution that adopt method of value solving such as finite element method numerical value solving equation (1), thus the numerical value general solution that structure is variable with heat source strength parameter or location parameter. In intensity inversion problem, the location parameter of each point source is known, is therefore easy to when the homogeneous solution of solving equation (1) and particular solution apply thermal source. But the intensive parameter of each point source is known in the inversion problem of position, and position is undetermined, thermal source so cannot be applied, also just cannot the homogeneous solution of solving equation (1) and particular solution.

Therefore it is a feature of the present invention that and be given as building method:

The a character (or engineering significance) according to problem, provides its possible position range to the position of source item.

I-th source item q in formula (3)_siLocation variable be (x_i,y_i,z_i), this location variable can set its excursion as x_i1≤x_i≤x_i2、y_i1≤y_i≤y_i2、z_i1≤z_i≤z_i2, wherein x_i1,x_i2,y_i1,y_i2,z_i1,z_i2For given value.Introduce dimensionless location parameter variable and contribute to the simplification of follow-up formula: $\begin{array}{cc}{\mathrm{\α}}_i^1=(x_{i1}-x_i)/(x_{i1}-x_{i2}),& {\mathrm{\α}}_i^2=(y_{i1}-y_i)/(y_{i1}-y_{i2}),\end{array}$ ${\mathrm{\α}}_i^3=(z_{i1}-z_i)/(z_{i1}-z_{i2}),$ And have $0\≤{\mathrm{\α}}_i^k\≤1,k=1,2,3.$

B only considers a variable, such as x_i, calculate particular solution.

Calculating k source strength respectively is η_iFinite Element Numerical Solution. Here Finite Element Numerical Solution is meant that: the problem definition territory Ω that (1) formula is provided, and utilizes computer to pass through Finite Element Method and calculates i-th point source at 2 end points (x_i1,x_i2) to have intensity be η_iHeat point source effect give a definition the temperature field in the Ω of territory, be designated as numerical value particular solution

Make q in formula (1)_s=0, solve homogeneous solution θ=θ₁; Adopting dimensionless location parameter variable, being expressed as thus constructing Numerical Temperature general solution θ:

$\mathrm{\θ}={\mathrm{\θ}}_1+\underset{i=1}{\overset{k}{\Σ}}\left(\right(1-{\mathrm{\α}}_i^1){\mathrm{\θ}}_{qi}^1+{\mathrm{\α}}_i^1{\mathrm{\θ}}_{qi}^2)---\left(4\right)$

In formula:For respective coordinates x_iDimensionless location parameter, be unknown quantity to be asked;Particular solution for point source i.

By given supplementary condition, namely some measurement, point considering, numerical value general solution (4) measures temperature θ with given_dError, obtain the residuals squares cost function of problem, as follows:

$g\left({\mathrm{\α}}^1\right)=\underset{j=1}{\overset{m}{\Σ}}{({\mathrm{\θ}}_{1j}+\underset{i=1}{\overset{k}{\Σ}}(\left(1-{\mathrm{\α}}_i^1\right){\mathrm{\θ}}_{qij}^1+{\mathrm{\α}}_i^1{\mathrm{\θ}}_{qij}^2)-{\mathrm{\θ}}_{dj})}^2---\left(5\right)$

Wherein: m is measure dot number; θ_djFor measuring some the measurement temperature of j, θ_1jBe measure some j homogeneous solution,It it is the i-th point source particular solution at j point.

So, source item identification indirect problem formula (1) translates into the extreme-value problem of the function of many variables that represented by formula (5) is variable with heat source position parameter, it is easy to solve extreme value, thus trying to achieve the solution of indirect problem.

C is to location variable y_i,z_i, repeat 2.2 steps respectively, obtain similar (5) formula respectively withFor the expression formula of variable, as (6) formula provides, then go to step 4) solve.

$\begin{array}{c}g\left({\mathrm{\α}}^2\right)=\underset{j=1}{\overset{m}{\Σ}}{({\mathrm{\θ}}_{1j}+\underset{i=1}{\overset{k}{\Σ}}(\left(1-{\mathrm{\α}}_i^2\right){\mathrm{\θ}}_{qij}^1+{\mathrm{\α}}_i^2{\mathrm{\θ}}_{qij}^2)-{\mathrm{\θ}}_{dj})}^2\\ g\left({\mathrm{\α}}^3\right)=\underset{j=1}{\overset{m}{\Σ}}{({\mathrm{\θ}}_{1j}+\underset{i=1}{\overset{k}{\Σ}}(\left(1-{\mathrm{\α}}_i^3\right){\mathrm{\θ}}_{qij}^1+{\mathrm{\α}}_i^3{\mathrm{\θ}}_{qij}^2)-{\mathrm{\θ}}_{dj})}^2\end{array}---\left(6\right)$

4) solve system of linear equations, obtain heat source position parameter

Seek the extreme-value problem of (5), (6) formula, position parametric variable can be differentiated, even:

$\∂g/\∂{\mathrm{\α}}_i^p=0,i=1,2,...,k;p=1,2,3$

In formula: i correspondence i-th unknown position variables parameter, three directions of p respective coordinates, 1 corresponding (5) formula, first expression formula of 2 corresponding (6) formulas, second expression formula of 3 corresponding (6) formulas. Thus obtain calculating the system of linear equations of point source location parameter:

A α=B (7)

$\begin{array}{c}B={\left(\underset{j}{\overset{m}{\Σ}}\right({\mathrm{\θ}}_{dj}-{\mathrm{\θ}}_{1j}-\underset{i}{\overset{k}{\Σ}}{\mathrm{\θ}}_{qij}^1\left)\right({\mathrm{\θ}}_{qpj}^2-{\mathrm{\θ}}_{qpj}^1\left)\right)}_{k\×1}\\ =\underset{j}{\overset{m}{\Σ}}\left[\begin{array}{c}({\mathrm{\θ}}_{dj}-{\mathrm{\θ}}_{1j}-\underset{i}{\overset{k}{\Σ}}{\mathrm{\θ}}_{qij}^1)({\mathrm{\θ}}_{q1j}^2-{\mathrm{\θ}}_{q1j}^1)\\ .\\ .\\ .\\ ({\mathrm{\θ}}_{dj}-{\mathrm{\θ}}_{1j}-\underset{i}{\overset{k}{\Σ}}{\mathrm{\θ}}_{qij}^1)({\mathrm{\θ}}_{qkj}^2-{\mathrm{\θ}}_{qkj}^1)\end{array}\right]\end{array}---\left(9\right)$

Wherein: α is the dimensionless heat source position parameter vector (solve each time and only correspond to some coordinate direction) being made up of k point source.

After solving equation (7), the location parameter of thermal source can be obtained according to the dimensionless α tried to achieve, as follows:

$\begin{array}{c}{x}_i=x_{i1}-(x_{i1}-x_{i2}){\mathrm{\α}}_i^1\\ y_i=y_{i1}-(y_{i1}-y_{i2}){\mathrm{\α}}_i^2\\ z_i=z_{i1}-(z_{i1}-z_{i2}){\mathrm{\α}}_i^3\end{array},(i=1,2,\mathrm{...},k)---\left(10\right)$

Conduction of heat location recognition indirect problem for non-point source, it is possible to convert it into point source indirect problem by interpolation method, then adopts above-mentioned steps 2 and 3, solves. Such as the thermal source to spatial domain, the method changing into point source indirect problem is as follows:

Space thermal source is separated into some hexahedrons or the combination of tetrahedron heat source unit, it may be assumed that

$F=\underset{i}{\Σ}{F}_i---\left(11\right)$

Using heat point source as interpolation knot, hexahedron or tetrahedron element are interpolated, express by heat source unit unit interpolating function:

$F_i=\underset{j}{\overset{n}{\Σ}}{N}_j(x,y,z)P_j---\left(12\right)$

Wherein: n is the interpolation knot number of heat source unit, hexahedral element generally having 8 nodes, tetrahedron element generally has 4 nodes.

So with the interpolating function that point source is interpolation knot, space thermal source is described, thus converting it into the inversion problem of heat point source. By above-mentioned steps, the location parameter of point source can be obtained, thus obtaining the position of space thermal source.

Embodiment: for point source

1st step: conduction of heat source item location recognition indirect problem describes

The example of this One-dimensional heat transfer source item location recognition indirect problem can be described as: one-dimensional problem, and long is 1m, midpoint effect 5W heat point source, one end assigned temperature 10 DEG C, and the other end is placed in the air that ambient temperature is 25 DEG C; Material thermal conductivity 20, convection transfer rate 25.Require the position of inverse point thermal source.

The finite element analysis related in this example, it is possible to use finite element analysis software general at present, as Ansys, Cosmos etc. complete;

2nd step: determine the relevant parameter of point source inverting

According to the inversion algorithm that the present invention proposes, it is first determined the relevant parameter of point source inverting is as follows:

1) scope that setpoint source position is possible, namely to x_i1≤x_i≤x_i2, take l/4≤x_i≤ 3l/4;

2) take one in x=1 position and measure point, provide supplementary condition θ according to after accurately solving consideration deviation₁₁=138 (unit is degree Celsius DEG C, lower with, slightly);

3rd step: solve homogeneous solution and particular solution, constructs general solution

Numerical value general solution method according to the present invention, adopts FInite Element to calculate homogeneous solution, x respectively at measurement point_i1、x_i1The particular solution of some effect 5W heat point source, is shown in Fig. 1 and Fig. 2; Extract the temperature value measuring point, have: ${\mathrm{\θ}}_{11}=10,{\mathrm{\θ}}_{q11}^1=72.5,{\mathrm{\θ}}_{q11}^2=\mathrm{197.5.}$

4th step: solving equation, obtains heat source position parameter

According to formula (7)～(9), can invocation point source position parameter be:

$\begin{array}{c}\mathrm{\α}=({\mathrm{\θ}}_{d1}-{\mathrm{\θ}}_{11}-{\mathrm{\θ}}_{q11}^1)/({\mathrm{\θ}}_{q11}^2-{\mathrm{\θ}}_{q11}^1)\\ =(138-10-72.5)/(197.5-72.5)=0.444\end{array}---\left(13\right)$

According to formula (10), it is known that the characteristic of point source location parameter is 0.5, the inversion result that therefore this algorithm provides is more satisfactory.

Content described in this specification embodiment is only enumerating of the way of realization to inventive concept; protection scope of the present invention is not construed as being only limitted to the concrete form that embodiment is stated, protection scope of the present invention also and in those skilled in the art according to present inventive concept it is conceivable that equivalent technologies means.

Claims (4)

1. the numerical value general solution method of a thermally conductive heat source position identification indirect problem, it is characterised in that comprise the steps:

1) description of conduction of heat source item location recognition indirect problem

This conduction of heat source item location recognition indirect problem describes as follows: at thermal source q_s(x, y z), seek thermal source q temperature field θ under effect_sParameter, wherein in order to measure, point is upper given measures a temperature θ to given supplementary condition_d;

Thermal conduction under steady state is described as shown in formula (1):

$\mathrm{\λ}{\▿}^2\mathrm{\θ}+q_s=0,(x,y,z\∈\mathrm{\Ω})$

θ(x,y,z)_b=θ_b(x,y,z)(1)

$-\mathrm{\λ}{\left(\frac{\∂\mathrm{\θ}}{\∂n}\right)}_v=h(\mathrm{\θ}{|}_v-{\mathrm{\θ}}_f)$

$-\mathrm{\λ}{\left(\frac{\∂\mathrm{\θ}}{\∂n}\right)}_w=f_2---\left(2\right)$

In formula: (x, y, z) for temperature for θ;For Laplacian Differential Approach operator; Ω is the definition territory of problem; B is First Boundary Condition; V is third boundary condition; W is second kind boundary condition, and h is surface coefficient of heat transfer or convection transfer rate; θ_fFor heat exchange medium temperature; q_sFor heat source function; λ is heat conductivity, and n is border normal direction, f₂For heat flow density;

2) if step 1) described in conduction of heat source be point source, then be directly entered step 3), if step 1) described in conduction of heat source be non-point source, then adopt transfer algorithm, non-point source indirect problem be converted into point source indirect problem, enter back into step 3);

3) solve homogeneous solution and particular solution, construct numerical value general solution

Basic theories according to Solutions of Ordinary Differential Equations, formula (1) general solution is made up of homogeneous solution and particular solution, and wherein homogeneous solution is to make q in formula (1)_s=0 solves, and particular solution then makes q_s=1 solves;

Rhetoric question topic (1) has the point source of k diverse location, and thermal source is expressed as:

$q_s=q(x,y,z)=\underset{i=1}{\overset{k}{\Σ}}\left(\mathrm{\δ}\right(x_i,y_i,z_i\left){\mathrm{\η}}_i\right)---\left(3\right)$

In formula: δ (x_i,y_i,z_i) for position function, (x_i,y_i,z_i) it is location parameter; η_iFor the intensive parameter of i-th point source, steady state problem, η_iFor constant; K is the number of point source;

The position of source item, according to the character of problem or engineering significance, is provided its possible position range by a,

I-th source item q in formula (3)_siLocation variable be (x_i,y_i,z_i), this location variable sets its excursion as x_i1≤x_i≤x_i2、y_i1≤y_i≤y_i2、z_i1≤z_i≤z_i2, wherein x_i1,x_i2,y_i1,y_i2,z_i1,z_i2For given value; Introduce dimensionless location parameter variable and contribute to the simplification of follow-up formula: ${\mathrm{\α}}_i^1=(x_{i1}-x_i)/(x_{i1}-x_{i2}),{\mathrm{\α}}_i^2=(y_{i1}-y_i)/(y_{i1}-y_{i2}),{\mathrm{\α}}_i^3=(z_{i1}-z_i)/(z_{i1}-z_{i2}),$ And have $0\≤{\mathrm{\α}}_i^k\≤1,$ K=1,2,3;

B only considers a variable, such as x_i, calculate particular solution;

Calculating k source strength respectively is η_iFinite Element Numerical Solution, Finite Element Numerical Solution is meant that here: the problem definition territory Ω that formula (1) is provided, utilize computer pass through Finite Element Method to i-th point source calculate at 2 end points (x_i1,x_i2) to have intensity be η_iHeat point source effect give a definition the temperature field in the Ω of territory, be designated as numerical value particular solution

Make q in formula (1)_s=0, solve homogeneous solution θ=θ₁; Adopting dimensionless location parameter variable, being expressed as thus constructing Numerical Temperature general solution θ:

$\mathrm{\θ}={\mathrm{\θ}}_1+\underset{i=1}{\overset{k}{\Σ}}\left(\right(1-{\mathrm{\α}}_i^1){\mathrm{\θ}}_{qi}^1+{\mathrm{\α}}_i^1{\mathrm{\θ}}_{qi}^2)---\left(4\right)$

In formula:For respective coordinates x_iDimensionless location parameter, be unknown quantity to be asked;Particular solution for point source i;

By given supplementary condition, namely some measurement, point considering, numerical value general solution (4) measures temperature θ with given_dError, obtain the residuals squares cost function of problem, as follows:

$g\left({\mathrm{\α}}^1\right)=\underset{j=1}{\overset{m}{\Σ}}{({\mathrm{\θ}}_{1j}+\underset{i=1}{\overset{k}{\Σ}}(\left(1-{\mathrm{\α}}_i^1\right){\mathrm{\θ}}_{qij}^1+{\mathrm{\α}}_i^1{\mathrm{\θ}}_{qij}^2)-{\mathrm{\θ}}_{dj})}^2---\left(5\right)$

Wherein: m is measure dot number; θ_djFor measuring some the measurement temperature of j, θ_1jBe measure some j homogeneous solution,It it is the i-th point source particular solution at j point;

Source item identification indirect problem formula (1) is converted into the extreme-value problem of the function of many variables that represented by formula (5) is variable with heat source position parameter, solves extreme value, tries to achieve the solution of indirect problem;

C is to location variable y_i,z_i, respectively repeat b step, obtain similar (5) formula respectively withFor the expression formula of variable, as (6) formula provides, then go to step 4) solve;

$\begin{array}{c}g\left({\mathrm{\α}}^2\right)=\underset{j=1}{\overset{m}{\Σ}}{({\mathrm{\θ}}_{1j}+\underset{i=1}{\overset{k}{\Σ}}(\left(1-{\mathrm{\α}}_i^2\right){\mathrm{\θ}}_{qij}^1+{\mathrm{\α}}_i^2{\mathrm{\θ}}_{qij}^2)-{\mathrm{\θ}}_{dj})}^2\\ g\left({\mathrm{\α}}^3\right)=\underset{j=1}{\overset{m}{\Σ}}{({\mathrm{\θ}}_{1j}+\underset{i=1}{\overset{k}{\Σ}}(\left(1-{\mathrm{\α}}_i^3\right){\mathrm{\θ}}_{qij}^1+{\mathrm{\α}}_i^3{\mathrm{\θ}}_{qij}^2)-{\mathrm{\θ}}_{dj})}^2\end{array}---\left(6\right)$

4) solve system of linear equations, obtain heat source position parameter

Seek the extreme-value problem of formula (5), formula (6), position parametric variable is differentiated, even:

$\∂g/\∂{\mathrm{\α}}_i^p=0,i=1,2,...,k;p=1,2,3$

In formula: i correspondence i-th unknown position variables parameter, three directions of p respective coordinates, wherein 1 corresponding (5), first expression formula of 2 correspondings (6), second expression formula of 3 correspondings (6), thus obtains calculating the system of linear equations of point source location parameter:

A α=B (7)

$\begin{array}{c}B={\left(\underset{j}{\overset{m}{\Σ}}\left({\mathrm{\θ}}_{dj}-{\mathrm{\θ}}_{1j}-\underset{i}{\overset{k}{\Σ}}{\mathrm{\θ}}_{qij}^1\right)\left({\mathrm{\θ}}_{qpj}^2-{\mathrm{\θ}}_{qpj}^1\right)\right)}_{k\×1}\\ =\underset{j}{\overset{m}{\Σ}}\left[\begin{array}{c}\left({\mathrm{\θ}}_{dj}-{\mathrm{\θ}}_{1j}-\underset{i}{\overset{k}{\Σ}}{\mathrm{\θ}}_{qij}^1\right)\left({\mathrm{\θ}}_{q1j}^2-{\mathrm{\θ}}_{q1j}^1\right)\\ .\\ .\\ .\\ \left({\mathrm{\θ}}_{dj}-{\mathrm{\θ}}_{1j}-\underset{i}{\overset{k}{\Σ}}{\mathrm{\θ}}_{qij}^1\right)\left({\mathrm{\θ}}_{qkj}^2-{\mathrm{\θ}}_{qkj}^1\right)\end{array}\right]\end{array}---\left(9\right)$

Wherein: α is the dimensionless heat source position parameter vector being made up of k point source, solves each time and only correspond to some coordinate direction;

After solving formula (7), obtain the location parameter of thermal source according to the dimensionless α tried to achieve, as follows:

$x_i=x_{i1}-(x_{i1}-x_{i2}){\mathrm{\α}}_i^1$

$y_i=y_{i1}-(y_{i1}-y_{i2}){\mathrm{\α}}_i^2,(i=1,2,...,k)---\left(10\right).$

$z_i=z_{i1}-(z_{i1}-z_{i2}){\mathrm{\α}}_i^3$

2. a kind of thermally conductive heat source position according to claim 1 identifies the numerical value general solution method of indirect problem, it is characterised in that the heat flow density f in second kind boundary condition₂It is considered as border thermal source, the inverting of its parameter and thermal source q_sIdentical.

3. a kind of thermally conductive heat source position according to claim 1 identifies the numerical value general solution method of indirect problem, it is characterized in that step 2) described non-point source is converted into the transfer algorithm of point source indirect problem, specifically include following steps: non-point source indirect problem is converted it into point source indirect problem by interpolation method, then step 3 is adopted) and step 4), solve.

4. a kind of thermally conductive heat source position according to claim 3 identifies the numerical value general solution method of indirect problem, it is characterised in that the thermal source to spatial domain, and the method changing into point source indirect problem is as follows:

Space thermal source is separated into some hexahedrons or the combination of tetrahedron heat source unit, it may be assumed that

$F=\underset{i}{\Σ}{F}_i---\left(11\right)$

Using heat point source as interpolation knot, hexahedron or tetrahedron element are interpolated, with the interpolating function that point source is interpolation knot, space thermal source is described, thus converting it into the inversion problem of heat point source, obtain the location parameter of point source, thus obtaining the position of space thermal source, express by heat source unit unit interpolating function:

$F_i=\underset{j}{\overset{n}{\Σ}}{N}_j(x,y,z)P_j---\left(12\right)$

Wherein: n is the interpolation knot number of heat source unit, hexahedral element generally having 8 nodes, tetrahedron element generally has 4 nodes.