CN105677993B - A kind of numerical value general solution method of thermally conductive heat source position identification indirect problem - Google Patents
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Abstract
The present invention relates to a kind of numerical value general solution methods of thermally conductive heat source position identification indirect problem, the numerical value general solution method includes the following steps: the description of heat transfer source item position identification indirect problem, if heat transfer source is point source, then it is directly entered next step, if heat transfer source is non-point source, transfer algorithm is then used, point source indirect problem is converted by non-point source indirect problem, enters back into next step；Homogeneous solution and particular solution are solved, numerical value general solution is constructed；System of linear equations is solved, heat source position parameter is obtained.The present invention is based on Finite Element Numerical Solution, construct meet thermal conduction differential equation, using Heat-Source Parameters as the numerical value general solution of variable, by heat transfer position identification indirect problem be converted into Extreme Value Problem of Multi-Variable Functions, fast inversion goes out Heat-Source Parameters.This method not only can be with inverse point heat source position, and can be therefore wide application, adaptable with inverting arbitrary shape heat source position, has good future in engineering applications.
Description
Technical field
The present invention relates to a kind of methods of heat transfer source item position identification indirect problem, in particular to a kind of heat transfer heat source position
Set the numerical value general solution method of identification indirect problem.
Background technique
Heat conduction inverse problem application is the research hotspot in engineering in recent years, it is the emphasis of thermal conduction study basic research.Closely
Heat conduction inverse problem is just being shown good application prospect in engineering over year, is led in power engineering, metallurgy and the industry such as mold
Domain has a wide range of applications background.
Heat conduction inverse problem refers to for heat conduction problem, it is known that the structure temperature distribution after sometime, inverting structure
Thermal physical property parameter, heat source function or geometry or boundary condition etc., here it is heat conduction inverse problems.It will be apparent that heat passes
Indirect problem is led compared to heat transfer direct problem, the difficulty solved is bigger, also more time-consuming.
Existing many about the patent of solution of inverse problems algorithm at present, application surface is also relatively wider.Such as Chinese invention patent Shen
Numbers 200710051566 a kind of " research method of hydraulic and hydroelectric engineering hydraulics indirect problem " please be disclosed, propose burst spectrum
Method, Discrete Optimization Method, perturbation method, controlling metho, the component frame of waterpower hydroelectric project hydro science indirect problem.The invention category
In the method for distributed parameter system indirect problem, rather than the method for source item identification indirect problem.
Chinese invention patent application number 201210350077 discloses a kind of " numerical value of the indirect problem of solution subsonic flow
Method " carries out the numerical value of the indirect problem of solid wall surface geometry design for one kind under given solid wall surface pressure condition
Method.The invention belongs to the method for Geometric Shape Recognition indirect problem, rather than the method for source item identification indirect problem.
Chinese invention patent application number 201210366939 discloses a kind of " solved with the Eulerian equation of Lagrangian Form
The numerical method of a kind of indirect problem " proposes that a kind of Two Dimensional Euler Equations of new Lagrangian Form are several to solve solid wall surface
The indirect problem of what shape design.The method that the invention also belongs to Geometric Shape Recognition indirect problem, rather than source item identification indirect problem
Method.
Chinese invention patent application number 201410095196.7 gives " the farmland component temperature based on global optimization approach
Inversion method ", carrying out minimization to objective function using global optimization approach simulated annealing realizes farmland component temperature
Inverting.
Chinese invention patent application number 201410032593.X gives " a kind of lines of thermal conduction source strength identification indirect problem
Numerical value general solution method ", 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 that a kind of thermally conductive heat source position is known
The numerical value general solution method of other indirect problem.This method not only can be with inverse point heat source position, and can be with inverting arbitrary shape heat source position
It sets, therefore wide application, adaptable, there is good future in engineering applications.
The numerical value general solution method of a kind of thermally conductive heat source position identification indirect problem, it is characterised in that including walking as follows
It is rapid:
1) description of heat transfer source item position identification indirect problem
Heat transfer source item position identification indirect problem is described as follows: in heat source qsThere is temperature field θ (x, y, z) under effect, asks
Heat source qsParameter, wherein given supplementary condition is to give measurement temperature θ in measurement pointd；
Thermal conduction under steady state is described as shown in formula (1):
In formula: θ (x, y, z) is temperature；▽ is Laplacian Differential Approach operator；Ω is the domain of problem；B is first kind side
Boundary's condition；V is third boundary condition；W is second kind boundary condition, and h is surface coefficient of heat transfer or convection transfer rate；θfFor
Heat exchange medium temperature；qsFor heat source function；λ is thermal coefficient, and n is boundary 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 includes
Intensity and position；
2) if the heat transfer source described in step 1) is point source, it is directly entered step 3), if heat transfer described in step 1)
Source is non-point source, then uses transfer algorithm, convert point source indirect problem for non-point source indirect problem, enter back into step 3)；
3) homogeneous solution and particular solution are solved, numerical value general solution is constructed
According to the basic theories of Solutions of Ordinary Differential Equations, formula (1) general solution is made of homogeneous solution and particular solution, and wherein homogeneous solution is in formula
(1) q is enabled ins=0 solves, and particular solution then enables qs=1 solves；
Rhetoric question topic (1) has the point source of k different location, heat source expression are as follows:
In formula: δ (xi,yi,zi) it is position function, (xi,yi,zi) it is location parameter；ηiFor the intensity ginseng of i-th of point source
Number, steady state problem, ηiFor constant；K is the number of point source；
A provides its possible position range to the position of source item according to the property or engineering significance of problem,
I-th of source item q in formula (3)siLocation variable be (xi,yi,zi), which sets its variation
Range are as follows: xi1≤xi≤xi2、yi1≤yi≤yi2、zi1≤zi≤zi2, wherein xi1,xi2,yi1,yi2,
zi1,zi2For given value；Introducing dimensionless location parameter variable facilitates the simplification of subsequent formula:And have
B only considers a variable, such as xi, calculate particular solution；
Calculating separately k source strength is ηiFinite Element Numerical Solution, Finite Element Numerical Solution is meant that here: to formula
(1) the problem of providing domain Ω calculates in 2 endpoint (x i-th of point source by finite element method using computeri1,
xi2) have intensity for ηiHeat point source act on the temperature field given a definition in the Ω of domain, be denoted as numerical value particular solution
Enable q in formula (1)s=0, solve homogeneous solution θ=θ1；Using dimensionless location parameter variable, to construct temperature number of fields
Being worth general solution θ indicates are as follows:
In formula:For respective coordinates xiDimensionless location parameter, be unknown quantity to be asked；For the particular solution of point source i；
By giving supplementary condition, i.e., numerical value general solution (4) and given measurement temperature θ are considered in several measurement pointsdMistake
Difference obtains the residuals squares cost function of problem, as follows:
Wherein: m is measurement points；θdjFor the measurement temperature of measurement point j, θ1jBe measurement point j homogeneous solution,
It is particular solution of i-th of point source in j point；
Source item identification indirect problem formula (1) is converted into represented by formula (5) one using heat source position parameter as the polynary of variable
Function Extreme Value problem, solves extreme value, acquires the solution of indirect problem；
C is to location variable yi,zi, repeatedly b is walked respectively, obtain similar (5) formula respectively withFor the expression of variable
Formula, such as (6) formula provide, and then go to step 4) solution；
4) system of linear equations is solved, heat source position parameter is obtained
The extreme-value problem for asking formula (5), formula (6), differentiates to position parametric variable, even:
In formula: corresponding i-th of the unknown position variables parameter of i, three directions of p respective coordinates, wherein 1 corresponding (5), 2
First expression formula of corresponding (6), second expression formula of 3 correspondings (6) thus obtain the line for calculating point source location parameter
Property equation group:
A α=B (7)
Wherein: α is the dimensionless heat source position parameter vector being made of k point source, solve each time only correspond to it is a certain
A coordinate direction；
After solution formula (7), the location parameter of heat source is obtained according to the dimensionless α acquired, as follows:
A kind of numerical value general solution method of thermally conductive heat source position identification indirect problem, it is characterised in that the second class boundary
Heat flow density f in condition2It is considered as boundary heat source, the inverting of parameter and heat source qsIt is identical.
The numerical value general solution method of a kind of thermally conductive heat source position identification indirect problem, it is characterised in that step 2) is described
Non-point source is converted into the transfer algorithm of point source indirect problem, specifically comprises the following steps: non-point source indirect problem by interpolation method
Point source indirect problem is converted it into, then using step 3) and step 4), is solved.
The numerical value general solution method of a kind of thermally conductive heat source position identification indirect problem, it is characterised in that spatial domain
Heat source, the method for being converted to point source indirect problem are as follows:
Space heat source is separated into the combination of several hexahedrons or tetrahedron heat source unit, it may be assumed that
Using heat point source as interpolation knot, interpolation is carried out to hexahedron or tetrahedron element, is interpolation knot with point source
Interpolating function describes space heat source, to convert it into the inversion problem of heat point source, obtains the location parameter of point source, thus
To the position of space heat source, i.e., heat source unit unit interpolating function is expressed:
Wherein: n is the interpolation knot number of heat source unit, generally there is 8 nodes, tetrahedron element for hexahedral element
Generally there are 4 nodes.
Using the notably dimension of each physical quantity when this method, that is, want dimension unified.Such as use International System of Units SI, base
This dimension are as follows: rice m, kilogram kg, second s, then heat source strength is W (watt), the dimension of material thermal conductivity is W/ (m DEG C), convection current
The dimension of the coefficient of heat transfer is W/ (m2·℃)；And the dimension of temperature uses DEG C (degree Celsius).
By using above-mentioned technology, compared with prior art, beneficial effects of the present invention are as follows:
1) building method of the invention by using restriction, according to the property or engineering significance of problem, to the position of source item
Its possible position range is provided, heat source is applied, solves the homogeneous solution and particular solution of equation (1)；
2) the numerical value general solution method of a kind of thermally conductive heat source position identification indirect problem proposed by the present invention, is based on finite element number
Value solution, construct meet thermal conduction differential equation, using Heat-Source Parameters as the numerical value general solution of variable, heat transfer position is identified anti-
Problem is converted into Extreme Value Problem of Multi-Variable Functions, and fast inversion goes out Heat-Source Parameters.This method not only can with inverse point heat source position, and
Can be therefore wide application, adaptable with inverting arbitrary shape heat source position, there is good future in engineering applications.
Detailed description of the invention
Fig. 1 is the flow chart for solving point of heat transfer source position identification indirect problem；
Fig. 2 is the x of case study on implementation of the present inventioni1Act on the Finite-Element Solution of heat point source；
Fig. 3 is the x of case study on implementation of the present inventioni2Act on the Finite-Element Solution of heat point source.
Specific embodiment
With reference to the accompanying drawings of the specification and embodiment the invention will be further described, but protection scope of the present invention is not
It is only limitted to this:
As shown in Figure 1, the numerical value general solution method of identification indirect problem in thermally conductive heat source position of the present invention, including it is as follows
Step:
1) description of heat transfer source item position identification indirect problem
Heat transfer source item position identification indirect problem is described as follows: in heat source qsThere is temperature field θ (x, y, z) under effect, asks
Heat source qsParameter, wherein given supplementary condition is to give measurement temperature θ in measurement pointd；
Thermal conduction under steady state is described as shown in equation (1):
In formula: θ (x, y, z) is temperature；▽ is Laplacian Differential Approach operator；Ω is the domain of problem；B is first kind side
Boundary's 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, h
For surface coefficient of heat transfer or convection transfer rate；θfFor heat exchange medium temperature；qsFor heat source function (containing intensity and position)；λ
For thermal coefficient, n is boundary normal direction, f2For heat flow density；In the present invention, to the heat flow density f in second kind boundary condition2, can
It is considered as boundary heat source, the inverting of parameter and heat source qsIt is identical, it can be solved with this method, therefore this method is to Heat Conduction
Source item position identifies that the description of indirect problem can be expressed with equation (1), does not consider second kind boundary condition；
2) if the heat transfer source described in step 1) is point source, it is directly entered step 3), if heat transfer described in step 1)
Source is non-point source, then uses transfer algorithm, convert point source indirect problem for non-point source indirect problem, enter back into step 3)；
3) homogeneous solution and particular solution are solved, numerical value general solution is constructed
According to the basic theories of Solutions of Ordinary Differential Equations, equation (1) general solution is made of homogeneous solution and particular solution, wherein homogeneous solution be
(1) formula enables qs=0 solves, particular solution then enables qs=1 solves；
Heat source qsForm can generally be described with the heat source of point, line, surface and body, midpoint is basic, other shapes
The heat source of formula can be described by the combination of point source certain rule.Therefore following that heat point source is first discussed.Rhetoric question topic (1) has k
The point source of different location, heat source can be expressed as:
In formula: δ (xi,yi,zi) it is position function, (xi,yi,zi) it is location parameter；ηiFor the intensity ginseng of i-th of point source
Number, steady state problem, ηiFor constant；K is the number of point source.
The core of numerical value general solution method is using the homogeneous of method of value solving such as finite element method numerical solution equation (1)
Solution and particular solution, so that construction is using heat source strength parameter or location parameter as the numerical value general solution of variable.It is each in intensity inversion problem
The location parameter of point source is known, therefore is easy to apply heat source when the homogeneous solution and particular solution of solution equation (1).But in position
The intensive parameter of each point source is known in inversion problem, and position be it is undetermined, heat source can not be applied in this way, also can not just be solved
The homogeneous solution and particular solution of equation (1).
Therefore it is a feature of the present invention that providing following building method:
A provides its possible position range to the position of source item according to the property (or engineering significance) of problem.
I-th of source item q in formula (3)siLocation variable be (xi,yi,zi), which can set its variation range are as follows:
xi1≤xi≤xi2、yi1≤yi≤yi2、zi1≤zi≤zi2, wherein xi1,xi2,yi1,yi2,zi1,zi2For given value.Introduce dimensionless
Location parameter variable facilitates the simplification of subsequent formula: And have
B only considers a variable, such as xi, calculate particular solution.
Calculating separately k source strength is ηiFinite Element Numerical Solution.Here Finite Element Numerical Solution is meant that: to (1)
The problem of formula provides domain Ω calculates in 2 endpoint (x i-th of point source by finite element method using computeri1,xi2)
Having intensity is ηiHeat point source act on the temperature field given a definition in the Ω of domain, be denoted as numerical value particular solution
Enable q in formula (1)s=0, solve homogeneous solution θ=θ1；Using dimensionless location parameter variable, to construct temperature number of fields
Being worth general solution θ indicates are as follows:
In formula:For respective coordinates xiDimensionless location parameter, be unknown quantity to be asked；For the particular solution of point source i.
By giving supplementary condition, i.e., numerical value general solution (4) and given measurement temperature θ are considered in several measurement pointsdMistake
Difference obtains the residuals squares cost function of problem, as follows:
Wherein: m is measurement points；θdjFor the measurement temperature of measurement point j, θ1jBe measurement point j homogeneous solution,
It is particular solution of i-th of point source in j point.
In this way, source item identification indirect problem formula (1) translates into represented by formula (5) one using heat source position parameter as variable
The function of many variables extreme-value problem, it is easy to extreme value is solved, to acquire the solution of indirect problem.
C is to location variable yi,zi, repeatedly 2.2 step respectively, obtain similar (5) formula respectively withFor the expression of variable
Formula, such as (6) formula provide, and then go to step 4) solution.
4) system of linear equations is solved, heat source position parameter is obtained
The extreme-value problem of (5), (6) formula is sought, can be differentiated to position parametric variable, even:
In formula: corresponding i-th of the unknown position variables parameter of i, three directions of p respective coordinates, 1 corresponding (5) formula, 2 is corresponding
(6) first expression formula of formula, second expression formula of 3 corresponding (6) formulas.Thus obtain calculating the linear side of point source location parameter
Journey group:
A α=B (7)
Wherein: α be made of k point source dimensionless heat source position parameter vector (solve each time only correspond to it is a certain
A coordinate direction).
It is as follows according to the location parameter of the available heat source of dimensionless α acquired after solving equation (7):
Indirect problem is identified for the heat transfer position of non-point source, point source rhetorical question can be converted it by interpolation method
Then topic uses above-mentioned steps 2 and 3, is solved.Such as to the heat source of spatial domain, it is converted to the method for point source indirect problem
It is as follows:
Space heat source is separated into the combination of several hexahedrons or tetrahedron heat source unit, it may be assumed that
Using heat point source as interpolation knot, interpolation is carried out to hexahedron or tetrahedron element, i.e., by heat source unit unit
Interpolating function expression:
Wherein: n is the interpolation knot number of heat source unit, generally there is 8 nodes, tetrahedron element for hexahedral element
Generally there are 4 nodes.
Space heat source is described with the interpolating function that point source is interpolation knot in this way, to convert it into the anti-of heat point source
Drill problem.Through the above steps, the location parameter of point source can be obtained, to obtain the position of space heat source.
Embodiment: by taking point source as an example
Step 1: heat transfer source item position identifies indirect problem description
The example of One-dimensional heat transfer source item position identification indirect problem can be described as: one-dimensional problem, a length of 1m, midpoint effect
5W heat point source, 10 DEG C of one end assigned temperature, the other end is placed in the air that environment temperature is 25 DEG C；Material thermal conductivity 20,
Convection transfer rate 25.It is required that the position of inverting heat point source.
General finite element analysis software at present can be used in finite element analysis involved in this example, as Ansys,
Cosmos etc. is completed；
Step 2: the relevant parameter of point source inverting is determined
The inversion algorithm proposed according to the present invention, it is first determined the relevant parameter of point source inverting is as follows:
1) the possible range in setpoint source position, i.e., to xi1≤xi≤xi2, take l/4≤xi≤3l/4；
2) a measurement point is taken in the position x=1, provides supplementary condition θ after considering deviation according to accurate solution11=138 (units
It is degree Celsius DEG C, similarly hereinafter, slightly)；
Step 3: solution homogeneous solution and particular solution construct general solution
Numerical value general solution method according to the present invention, using FInite Element measurement point calculate separately homogeneous solution,
xi1、xi1The particular solution of point effect 5W heat point source, is shown in Fig. 1 and Fig. 2；The temperature value for extracting measurement point, has:
Step 4: equation is solved, heat source position parameter is obtained
It, can invocation point source position parameter according to formula (7)~(9) are as follows:
According to formula (10), it is known that the characteristic of point source location parameter is 0.5, therefore the inversion result that this algorithm provides
It is more satisfactory.
Content described in this specification embodiment is only enumerating to the way of realization of inventive concept, protection of the invention
Range should not be construed as being limited to the specific forms stated in the embodiments, and protection scope of the present invention is also and in art technology
Personnel conceive according to the present invention it is conceivable that equivalent technologies mean.
Claims (4)
1. a kind of numerical value general solution method of thermally conductive heat source position identification indirect problem, it is characterised in that include the following steps:
1) description of heat transfer source item position identification indirect problem
Heat transfer source item position identification indirect problem is described as follows: in heat source qsThere is temperature field θ (x, y, z) under effect, seeks heat source qs
Parameter, wherein given supplementary condition is to give measurement temperature θ in measurement pointd；
Thermal conduction under steady state is described as shown in formula (1):
In formula: θ (x, y, z) is temperature；For Laplacian Differential Approach operator；Ω is the domain of problem；B is first boundary item
Part；V is third boundary condition；W is second kind boundary condition, and h is surface coefficient of heat transfer or convection transfer rate；θfFor heat exchange
Medium temperature；qsFor heat source function；λ is thermal coefficient, and n is boundary normal direction, f2For heat flow density；
2) if the heat transfer source described in step 1) is point source, it is directly entered step 3), if heat transfer source described in step 1) is
Non-point source then uses transfer algorithm, converts point source indirect problem for non-point source indirect problem, enter back into step 3)；
3) homogeneous solution and particular solution are solved, numerical value general solution is constructed
According to the basic theories of Solutions of Ordinary Differential Equations, formula (1) general solution is made of homogeneous solution and particular solution, and wherein homogeneous solution is in formula (1)
In enable qs=0 solves, and particular solution then enables qs=1 solves；
Rhetoric question topic (1) has the point source of k different location, heat source expression are as follows:
In formula: δ (xi,yi,zi) it is position function, (xi,yi,zi) it is location parameter；ηiFor the intensive parameter of i-th of point source, stable state
Problem, ηiFor constant；K is the number of point source；
A provides its possible position range to the position of source item according to the property or engineering significance of problem,
I-th of source item q in formula (3)siLocation variable be (xi,yi,zi), which sets its variation range
Are as follows: xi1≤xi≤xi2、yi1≤yi≤yi2、zi1≤zi≤zi2, wherein xi1,xi2,yi1,yi2,zi1,
zi2For given value；Introducing dimensionless location parameter variable facilitates the simplification of subsequent formula:And have
B only considers variable xiCalculate particular solution；
Calculating separately k source strength is ηiFinite Element Numerical Solution, Finite Element Numerical Solution is meant that here: being provided to formula (1)
The problem of domain Ω, i-th point source is calculated in 2 endpoint (x by finite element method using computeri1,xi2) there is intensity
For ηiHeat point source act on the temperature field given a definition in the Ω of domain, be denoted as numerical value particular solution
Enable q in formula (1)s=0, solve homogeneous solution θ=θ1；Using dimensionless location parameter variable, so that it is logical to construct Numerical Temperature
Solving θ indicates are as follows:
In formula:For respective coordinates xiDimensionless location parameter, be unknown quantity to be asked；For the particular solution of point source i；
By giving supplementary condition, i.e., numerical value general solution (4) and given measurement temperature θ are considered in several measurement pointsdError, obtain
It is as follows to the residuals squares cost function of problem:
Wherein: m is measurement points；θdjFor the measurement temperature of measurement point j, θ1jBe measurement point j homogeneous solution,It is i-th
Particular solution of a point source in j point；
Source item identification indirect problem formula (1) is converted into represented by formula (5) one using heat source position parameter as the function of many variables of variable
Extreme-value problem, solve extreme value, acquire the solution of indirect problem；
C is to location variable yi,zi, y is used respectivelyiAnd ziSubstitute the x in b stepiThe calculating process in b step is repeated, similar (5) formula is obtained
Respectively withFor the expression formula of variable, such as (6) formula is provided, and then goes to step 4) solution；
4) system of linear equations is solved, heat source position parameter is obtained
The extreme-value problem for asking formula (5), formula (6), differentiates to position parametric variable, even:
In formula: corresponding i-th of the unknown position variables parameter of i, three directions of p respective coordinates, wherein 1 corresponding (5), 2 is corresponding
First expression formula of formula (6), second expression formula of 3 correspondings (6) thus obtain the linear side for calculating point source location parameter
Journey group:
A α=B (7)
Wherein: α is the dimensionless heat source position parameter vector being made of k point source, solves only correspond to some seat each time
Mark direction；
After solution formula (7), the location parameter of heat source is obtained according to the dimensionless α acquired, as follows:
2. a kind of numerical value general solution method of thermally conductive heat source position identification indirect problem according to claim 1, feature exist
Heat flow density f in second kind boundary condition2It is considered as boundary heat source, the inverting of parameter and heat source qsIt is identical.
3. a kind of numerical value general solution method of thermally conductive heat source position identification indirect problem according to claim 1, feature exist
It is converted into the transfer algorithm of point source indirect problem in the step 2) non-point source, is specifically comprised the following steps: non-point source indirect problem
Point source indirect problem is converted it by interpolation method, then using step 3) and step 4), is solved.
4. a kind of numerical value general solution method of thermally conductive heat source position identification indirect problem according to claim 3, feature exist
In the heat source to spatial domain, the method for being converted to point source indirect problem is as follows:
Space heat source is separated into the combination of several hexahedrons or tetrahedron heat source unit, it may be assumed that
Using heat point source as interpolation knot, interpolation is carried out to hexahedron or tetrahedron element, is the interpolation of interpolation knot with point source
Function describes space heat source, to convert it into the inversion problem of heat point source, obtains the location parameter of point source, to obtain sky
Between heat source position, i.e., by heat source unit with unit interpolating function express:
Wherein: n is the interpolation knot number of heat source unit, there is 8 nodes for hexahedral element, and tetrahedron element has 4 sections
Point.
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