CN105808820B - A kind of high order accurate numerical method of solution interval thermal convection diffusion problem - Google Patents
A kind of high order accurate numerical method of solution interval thermal convection diffusion problem Download PDFInfo
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- CN105808820B CN105808820B CN201610101396.8A CN201610101396A CN105808820B CN 105808820 B CN105808820 B CN 105808820B CN 201610101396 A CN201610101396 A CN 201610101396A CN 105808820 B CN105808820 B CN 105808820B
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- 238000000034 method Methods 0.000 title claims abstract description 43
- 238000009792 diffusion process Methods 0.000 title claims abstract description 32
- 238000012546 transfer Methods 0.000 claims abstract description 25
- HBMJWWWQQXIZIP-UHFFFAOYSA-N silicon carbide Chemical compound [Si+]#[C-] HBMJWWWQQXIZIP-UHFFFAOYSA-N 0.000 claims abstract description 18
- 229910010271 silicon carbide Inorganic materials 0.000 claims abstract description 18
- 230000004044 response Effects 0.000 claims abstract description 17
- 239000000463 material Substances 0.000 claims description 5
- 239000000758 substrate Substances 0.000 claims description 3
- 239000013529 heat transfer fluid Substances 0.000 claims description 2
- 238000004458 analytical method Methods 0.000 description 9
- 230000006870 function Effects 0.000 description 9
- 238000004519 manufacturing process Methods 0.000 description 5
- 238000011160 research Methods 0.000 description 4
- 230000008569 process Effects 0.000 description 3
- 238000012360 testing method Methods 0.000 description 3
- 238000000342 Monte Carlo simulation Methods 0.000 description 2
- 238000010586 diagram Methods 0.000 description 2
- 238000009826 distribution Methods 0.000 description 2
- 239000012530 fluid Substances 0.000 description 2
- 238000005259 measurement Methods 0.000 description 2
- 238000005070 sampling Methods 0.000 description 2
- 238000013076 uncertainty analysis Methods 0.000 description 2
- 241000208340 Araliaceae Species 0.000 description 1
- 235000005035 Panax pseudoginseng ssp. pseudoginseng Nutrition 0.000 description 1
- 235000003140 Panax quinquefolius Nutrition 0.000 description 1
- 238000004364 calculation method Methods 0.000 description 1
- 230000008859 change Effects 0.000 description 1
- 238000012512 characterization method Methods 0.000 description 1
- 230000019771 cognition Effects 0.000 description 1
- 238000001816 cooling Methods 0.000 description 1
- 230000007812 deficiency Effects 0.000 description 1
- 238000013461 design Methods 0.000 description 1
- 230000000694 effects Effects 0.000 description 1
- 235000008434 ginseng Nutrition 0.000 description 1
- 230000006872 improvement Effects 0.000 description 1
- 238000011089 mechanical engineering Methods 0.000 description 1
- 230000004048 modification Effects 0.000 description 1
- 238000012986 modification Methods 0.000 description 1
- 230000003068 static effect Effects 0.000 description 1
- 239000000126 substance Substances 0.000 description 1
- 238000004448 titration Methods 0.000 description 1
- 239000011800 void material Substances 0.000 description 1
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/30—Circuit design
- G06F30/36—Circuit design at the analogue level
- G06F30/367—Design verification, e.g. using simulation, simulation program with integrated circuit emphasis [SPICE], direct methods or relaxation methods
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
- G06F2119/08—Thermal analysis or thermal optimisation
Abstract
The invention discloses a kind of high order accurate numerical methods of solution interval thermal convection diffusion problem, and steps are as follows: establishing the heat transfer governing equation of thermal convection diffusion problem；It is introduced into interval variable to characterize the uncertainty for inputting parameter in heat transfer model, establishes the range restraint equation of thermal convection diffusion problem；Approximate representation is carried out to the temperature-responsive in range restraint equation using Legnedre polynomial；According to tensor product operation, determines in higher dimensional space and match point set；All temperature-responsives at point are calculated using finite element program, establish the system of linear equations about expansion coefficient in temperature-responsive approximate expression, and solved using least square method；Slickness based on polynomial function, temperature respond the extreme point of approximate expression, obtain the bound of silicon carbide response.The present invention can systematization solve the thermal convection diffusion problem containing section uncertain parameter, effectively increase the computational accuracy of section numerical method.
Description
Technical field
The invention belongs to mechanical engineering fields, and in particular to a kind of high-precision numerical value of solution interval thermal convection diffusion problem
Method.
Background technique
In nature and various production technical fields, the transmitting for carrying out heat by fluid flowing is a kind of extremely general
Time physical phenomenon.Especially in the production of the sophisticated products such as the industrial equipments such as aerospace, derived energy chemical and electronic device
In manufacturing process, how heat transfer is effectively realized, it has also become an importance of product design.It is existing about heat analysis
Many researchs carry out both for deterministic models, do not account for the uncertainty of mode input parameter.In Practical Project,
Due to the limitation of the limitation of manufacturing process, measurement error and cognition, material properties, external load and the boundary condition of structure etc.
Physical parameter is inevitably influenced by a variety of uncertain factors so that the temperature-responsive of system also show it is certain not
Certainty.
Stochastic modeling and calculation method based on probability theory and mathematical statistics uncertain system analysis, test and set
Meter aspect has been achieved for many research achievements, successfully solves many engineering problems.But with random theory Solve problems
When, often assume that uncertain factor is stochastic variable or random process, this just needs a large amount of Test Information to determine that it is general in advance
The rate regularity of distribution.However, obtaining sufficient test data under most actual conditions and often costing dearly.In this way, information
Lack prevent probabilistic model from really reflecting objective reality, also allows for random uncertainty analysis method and loses meaning.It is practical
Most cases in engineering, designer often only focus on the amplitude of variation of certain responses, and obtain ambiguous model input parameter
Value range relative probability density function for want much easier, and required uncertain information also greatly reduces, this just has
Effect has widened the application range of uncertainty analysis.In recent years, being handled using interval theory and analysis method in engineering not
Determine that factor is more and more paid attention to by scholars.It is derived currently, interval theory is combined with finite element method
The interval finite element method come has been achieved for many research achievements in terms of the static and dynamic characteristics analysis of uncertain structure, but right
The document of heat transfer problem analysis with interval parameter is but very rare.In addition, traditional Novel Interval Methods are because of interval arithmetic institute
For caused interval extension problem also than more serious, computational accuracy is also in urgent need to be improved.Therefore, high-precision interval analysis how is established
Method carries out numerical solution to uncertain heat transfer problem, is a research hotspot of current sphere of learning, for making up existing biography
The deficiency of hot numerical computation method has important engineering application value.
Summary of the invention
The technical problems to be solved by the invention are as follows: overcome the prior art thermal convection diffusion problem solve present in not
Foot, fully considers the section uncertain factor in heat transfer problem, based on approximation by polynomi-als thought and with point analysis theory, proposes
A kind of highly-accurate nephelometric titrimetry method of predicted temperature response section variation range, can systematization solve containing the uncertain ginseng in section
Several temperature field prediction problems, while guaranteeing computational accuracy, the calculating that further reduced traditional methods of sampling expends.
The technical solution that the present invention uses to solve above-mentioned technical problem are as follows: a kind of solution interval thermal convection diffusion problem
High order accurate numerical method, comprising the following steps:
Step 1: the heat transfer governing equation of thermal convection diffusion problem is established according to heat transfer model；
Step 2: it is introduced into interval variable and the uncertainty for inputting parameter in heat transfer model is characterized, according to step 1
In heat transfer governing equation establish the range restraint equation of thermal convection diffusion problem；
Step 3: the temperature-responsive in step 2 range restraint equation is approached using Legnedre polynomial, is obtained
The approximate expression of silicon carbide response；
Step 4: according to tensor product operation, by the one-dimensional space with point set determine entire higher dimensional space with point set；
The higher-dimension refers to more than one-dimensional, number of the dimension equal to interval variable；
Step 5: step 4 is calculated with the temperature-responsives at point all in point set using finite element program, establishes and closes
The system of linear equations of expansion coefficient in step 3 silicon carbide response approximate expression, and it is linear to this using least square method
Equation group is solved, and a class value of expansion coefficient is obtained；
Step 6: returned to the approximate expression of step 3 temperature-responsive the class value generation of expansion coefficient obtained in step 5
In formula, the slickness based on polynomial function determines the extreme point of this approximate expression, and then obtains the upper of silicon carbide response
Lower bound.
Wherein, approximate representation, polynomial truncation are carried out to temperature-responsive using Legnedre polynomial in the step 3
Order is not fixed and invariable, and requires to choose according to approximation accuracy, truncation order is higher, and approximation accuracy is higher.
Wherein, the selection with point scheme in the step 4 is not fixed and invariable, and expends and calculate essence according to calculating
The requirement of degree come choose match point quantity, with point quantity it is more, computational accuracy is higher, and calculate expend it is bigger.
Wherein, the higher-dimension in the step 4 refers to more than one-dimensional, number of the dimension equal to interval variable.
Above steps specifically includes following procedure:
Step 1: the heat transfer governing equation of thermal convection diffusion problem is established according to heat transfer model:
Wherein x indicates that physical coordinates, T (x) indicate temperature-responsive, ρ, c, and k respectively indicates the density, specific heat capacity and heat of material
The coefficient of conductivity, u are the flowing velocity of heat-transfer fluid, and Q (x) indicates the heat source strength of system.
Step 2: n interval variable is introducedTable is carried out to the uncertain of parameter is inputted in heat transfer model
Sign, and it is denoted as the form of vectorWherein subscript I is interval symbol,WithIt indicates
Interval variableLower bound and the upper bound,WithReferred to as interval variableMidpoint and radius,For standard interval variableThe range restraint side of thermal convection diffusion problem is established according to the governing equation in step 1
Journey:
Step 3: using limited rank Legnedre polynomial in step 2 range restraint equation temperature-responsive T (x,
αI) approached, obtain the approximate expression of silicon carbide response:
Wherein Φi(αI) it is Legendre's orthogonal polynomial substrate selected in advance, wiIt (x) is corresponding expansion coefficient, i=
(i1,i2,...,in) indicate multidimensional index, and meet | i |=i1+i2+...+in, N polynomial truncation order thus.According to more
Item formula is theoretical, and the number Available Variables number n and truncation order N that item is unfolded in above-mentioned approximate expression being expressed as
Step 4: according to tensor product operation, by the one-dimensional space with point set determine entire higher dimensional space with point set.
Firstly, for one-dimensional interval variableFor, it is set with quantity as mi, then each with the specific location put
Are as follows:
Secondly, using point setIndicate one-dimensional interval variableInterior all collection with point composition
It closes, then directly utilizing tensor product operation for the higher dimensional space of n variable composition, it can be obtained with point set Θ
And with a sum M are as follows:
Therefore Θ is rewritten asForm, for indicate higher dimensional space it is all with point
Step 5: step 4 is calculated with the temperature-responsives at point all in point set using finite element program, establishes and closes
The system of linear equations of expansion coefficient in step 3 silicon carbide response approximate expression, and it is linear to this using least square method
Equation group is solved, and a class value of expansion coefficient is obtained.Firstly, the range restraint equation established in step 2 is matching pointPlace can be rewritten as:
Above-mentioned equation is solved using finite element program, available all temperature-responsives matched at point
Secondly, responding approximate expression based on silicon carbide in step 3, can establish about all expansion coefficient wi(x)
System of linear equations:
WhereinRepresentative polynomial basis function Φi(αI) with pointThe value at place.
Then, above-mentioned equation group is solved using least square method, obtains expansion coefficient wi(x) a class value.
Step 6: the expansion coefficient w that will be calculated in step 5i(x) in class value generation, returns to step 3 temperature-responsive
Approximate expression in, be based on polynomial function TN(x,αI) slickness, enabling its first derivative is zero, easily determine its extreme point,
It is denoted as together with boundary pointWherein r is the total number of extreme point and boundary point.Compare at this r point
Temperature value size, final determination section temperature-responsive T (x, αI) lower boundThe upper bound and
The advantages of the present invention over the prior art are that:
(1) compared with traditional thermal convection diffusion problem analysis method, the numerical computation method proposed fully considers reality
The bounded-but-unknown uncertainty of material properties, external load and boundary condition in the engineering of border, calculated result have more temperature field analysis
Important directive significance.
(2) approximate representation is carried out to temperature-responsive using high-order Legnedre polynomial, approximation accuracy can be effectively improved.Together
When, using the slickness of polynomial function, it can quickly determine its extreme point, and then obtain the bound of temperature-responsive.
(3) based on point theory, the finite element program of original deterministic models can be made full use of without to it
Further modification is done, ensure that the portability of calculating.
(4) operation of the present invention is simple, easy to implement, effectively increases computational accuracy.
Detailed description of the invention
Fig. 1 is a kind of high order accurate numerical method flow chart of solution interval thermal convection diffusion problem of the invention；
Fig. 2 is three-dimensional heat exchanger model schematic of the invention；
Fig. 3 is that upper panel center line silicon carbide responds schematic diagram；
Fig. 4 is that flow tube center line silicon carbide responds schematic diagram.
Specific embodiment
The present invention will be further described with reference to the accompanying drawings and examples.
The present invention can systematization solve the thermal convection diffusion problem containing section uncertain parameter, effectively increase interval number
The computational accuracy of value method.The present invention is suitable for the temperature field prediction of the thermal convection diffusion problem containing section uncertain parameter.
Embodiment of the present invention illustrates a kind of solution interval thermal convection diffusion and asks by taking certain three-dimensional heat exchanger model as an example
The high order accurate numerical method of topic.In addition, the silicon carbide response numerical computation method of this three-dimensional heat exchanger model can be promoted
Contain in the thermal convection diffusion problem temperature field prediction of interval parameter to other.
A kind of calculating process of the high order accurate numerical method of solution interval thermal convection diffusion problem is as shown in Figure 1, according to tool
The heat transfer model of body establishes the heat transfer governing equation of thermal convection diffusion problem, is introduced into interval variable to inputting parameter in heat transfer model
Uncertainty characterized, establish the range restraint equation of thermal convection diffusion problem, using Legnedre polynomial to temperature ring
Approximate representation should be carried out, while according to tensor product operation, determining and matching point set, calculates all match at point using finite element program
Temperature-responsive solves expansion coefficient in temperature-responsive approximate expression, and the slickness based on function, quickly obtains area
Between temperature-responsive bound.The following steps progress can be divided into:
Step 1: considering the heat exchanger heat transfer model of a length of 300mm shown in Fig. 2, and section is the square of 40mm × 20mm
There are the cooling air that the square hole of 30mm × 15mm is u by speed in shape, centre, and the air themperature of inlet 8 is Ts, structure it is upper
It is q that panel 7, which bears density,s=20000 × sin (ω) W/m2Hot-fluid load, on top panel center line select number be 1~3
Three points, observation point of three points as temperature field that number is 4~6 is selected on flow tube center line.According to heat transfer model
Establish the heat transfer governing equation of thermal convection diffusion problem:
Wherein x, y, z indicate that the physical coordinates on three direction in spaces, T (x, y, z) indicate temperature-responsive, ρ, c, k difference
Indicate that density, specific heat capacity and the coefficient of heat conduction of air, u are the flowing velocity of air.
Step 2: due to the error of limitation and the measurement of material fabrication process, all mode input parameters contain one
Fixed bounded-but-unknown uncertainty introduces six interval variables and carries out characterization ρ to uncertaintyI=[1.3,1.5] kg/m3, cI=
[900,1100] J/ (kg DEG C), kI=[0.023,0.029] W/ (m DEG C), uI=[1.8,2.2] m/s, Ts I=[18,22]
DEG C, ωI=[1.5,2.5].All interval variables are collectively expressed as vector form
Wherein subscript I is interval symbol,WithIndicate interval variableLower bound and the upper bound,WithReferred to as interval variableMidpoint and radius,For standard interval variableAccording in step 1
Governing equation establish the range restraint equation of thermal convection diffusion problem:
Step 3: according to the characteristic distributions of interval variable, select Legnedre polynomial in step 2 range restraint equation
Temperature-responsive T (x, y, z, αI) approached, truncation order is set as N=3, obtains the approximate expression of silicon carbide response
Formula:
Wherein Φi(αI) it is Legendre's orthogonal polynomial substrate selected in advance, wi(x, y, z) is corresponding expansion coefficient,
I=(i1,i2,...,i6) indicate multidimensional index, and meet | i |=i1+i2+...+i6.It is unfolded in above-mentioned approximate expression at this time
Number be
Step 4: according to tensor product operation, by the one-dimensional space with point set determine entire higher dimensional space with point set.
Firstly, for one-dimensional interval variableFor, it is set with quantity as mi=5, then it is each with the specific location putAre as follows:
Secondly, using point setIndicate one-dimensional interval variableInterior all collection with point composition
It closes, then directly utilizing tensor product operation for the sextuple space of 6 variables composition, it can be obtained with point set Θ:
And it is with a sumΘ is rewritten asForm, be used to table
Show sextuple space it is all with point
Step 5: step 4 is calculated with the temperature-responsives at point all in point set using finite element program, establishes and closes
The system of linear equations of expansion coefficient in step 3 silicon carbide response approximate expression, and it is linear to this using least square method
Equation group is solved, and a class value of expansion coefficient is obtained.Firstly, the range restraint equation established in step 2 is matching pointPlace can be rewritten as:
Above-mentioned heat transfer problem is solved using the finite element program in software Nastran, it is available all with point
The temperature-responsive at place
Secondly, responding approximate expression based on silicon carbide in step 3, establish about all expansion coefficient wi(x,y,z)
System of linear equations:
WhereinRepresentative polynomial basis function Φi(αI) with pointPlace takes
Value.
Then, above-mentioned equation group is solved using least square method, obtains expansion coefficient wiOne class value of (x, y, z).
Step 6: the expansion coefficient w that will be calculated in step 5iIn the one class value generation of (x, y, z), returns to step 3 temperature
In the approximate expression of response, polynomial function T is utilizedN(x,y,z,αI) slickness, enabling its first derivative is zero, determines it
Extreme point is denoted as together with boundary pointWherein r is the total number of extreme point and boundary point.Compare this r
Temperature value size at point, final determination section temperature-responsive T (x, y, z, αI) lower boundThe upper bound and
The calculated result of temperature-responsive is as shown in table 1 at six observation points.It is 10 with sample number6Traditional Monte Carlo take out
As can be seen that the calculating error of the method for the present invention is less than 1%, computational accuracy fully meets engineering demand for quadrat method comparison.In addition,
From sample size, the sample number of the method for the present invention is 15625, calculates and expends far smaller than monte carlo method.
Silicon carbide upper and lower bounds of responses at 1 observation point of table
Other than above-mentioned six observation points, along the x-axis direction, upper panel and flow tube center line silicon carbide response such as Fig. 3
With shown in Fig. 4, abscissa indicates spatial position along the x-axis direction, the temperature value at ordinate representation space position, solid line and void
Line respectively indicates the result that the Monte Carlo methods of sampling and the method for the present invention are calculated.As can be seen that the method for the present invention calculates
Obtained temperature-responsive bound curve and the reference value degree of agreement that traditional Monte Carlo is sampled are fine, and calculated result is true
It is real credible.Can solve the thermal convection diffusion problem containing section uncertain input parameter with the method for the present invention, computational accuracy is high,
This function is that general business software institute is irrealizable.
Above-described is only presently preferred embodiments of the present invention, and the present invention is not limited solely to above-described embodiment, all
Part change, equivalent replacement, improvement etc. made by within the spirit and principles in the present invention should be included in protection of the invention
Within the scope of.
Claims (5)
1. a kind of high order accurate numerical method of solution interval thermal convection diffusion problem, it is characterised in that the following steps are included:
Step 1: the heat transfer governing equation of thermal convection diffusion problem is established according to heat transfer model:
Wherein x indicates that physical coordinates, T (x) indicate temperature-responsive, ρ, c, and k respectively indicates the density, specific heat capacity and heat transfer of material
Coefficient, u are the flowing velocity of heat-transfer fluid, and Q (x) indicates the heat source strength of system；
Step 2: n interval variable is introducedThe uncertainty for inputting parameter in heat transfer model is characterized, and is remembered
For the form of vectorWherein subscript I is interval symbol,α iWithIndicate interval variableLower bound and the upper bound,WithReferred to as interval variableMidpoint and radius,For standard
Interval variableAnd then the range restraint side of thermal convection diffusion problem is established according to the heat transfer governing equation in step 1
Journey:
Step 3: using Legnedre polynomial to temperature-responsive T (x, α involved in step 2 range restraint equationI) forced
Closely, the approximate expression of silicon carbide response is obtained:
Wherein Φi(αI) it is Legendre's orthogonal polynomial substrate selected in advance, wiIt (x) is corresponding expansion coefficient, i=(i1,
i2,...,in) indicate multidimensional index, and meet | i |=i1+i2+...+in, N polynomial truncation order thus；
Step 4: point set is usedIndicate one-dimensional interval variableInterior all collection with point composition
It closes, wherein miIt indicates with point quantity,It indicates to match point in the one-dimensional space；According to tensor product operation, by one-dimensional sky
Between match point setIt determines in the n-dimensional space being made of n variable and matches point set Θ
Wherein it is with a sum M
And then it will be rewritten as in n-dimensional space with point set ΘForm, wherein
It indicates to match point in n-dimensional space；
Step 5: point set Θ, the range restraint equation established in step 2 are matched based on the n-dimensional space established in step 4
It is all with point at itsIt rewrites at place are as follows:
Above-mentioned equation is solved using finite element program, step 4 n-dimensional space is calculated and matches at point with all in point set Θ
Temperature-responsive, establish the system of linear equations about expansion coefficient in step 3 silicon carbide response approximate expression, and use
Least square method solves this system of linear equations, obtains a class value of expansion coefficient；
Step 6: returned to the approximate expression of step 3 temperature-responsive the class value generation of expansion coefficient obtained in step 5
In, the slickness based on polynomial function determines the extreme point of this approximate expression, and then obtains the upper and lower of silicon carbide response
Boundary.
2. a kind of high order accurate numerical method of solution interval thermal convection diffusion problem according to claim 1, feature exist
In: it is therein more when being approached using Legnedre polynomial the temperature-responsive in range restraint equation in the step 3
The truncation order of item formula is not fixed and invariable, and requires to choose according to approximation accuracy, truncation order is higher, and approximation accuracy is got over
It is high.
3. a kind of high order accurate numerical method of solution interval thermal convection diffusion problem according to claim 1, feature exist
In: according to tensor product operation in the step 4, by the one-dimensional space with point set determine entire higher dimensional space with point set
In with point scheme selection be not fixed and invariable, according to calculate expend and computational accuracy requirement come choose match points
Amount, more with point quantity, computational accuracy is higher, and it is bigger to calculate consuming.
4. a kind of high order accurate numerical method of solution interval thermal convection diffusion problem according to claim 1, feature exist
In: the n dimension in the step 4 refers to more than one-dimensional, number of the dimension equal to interval variable.
5. a kind of high order accurate numerical method of solution interval thermal convection diffusion problem according to claim 1, feature exist
In: step 4 is calculated with the temperature-responsives at point all in point set using finite element program in the step 5.
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| CN103366065A (en) * | 2013-07-17 | 2013-10-23 | 北京航空航天大学 | Size optimization design method for aircraft thermal protection system based on section reliability |
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| US20100094603A1 (en) * | 2001-12-31 | 2010-04-15 | George Danko | Multiphase physical transport modeling method and modeling system |
| CN103366065A (en) * | 2013-07-17 | 2013-10-23 | 北京航空航天大学 | Size optimization design method for aircraft thermal protection system based on section reliability |
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| Chong Wang等.Uncertainty analysis for heat convection-diffusion problem with large uncertain-but-bounded parameters.《Acta Mech》.2015, * |
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