CN105787169B - A kind of section thermal convection diffusion problem method for solving theoretical with point based on sparse grid - Google Patents
A kind of section thermal convection diffusion problem method for solving theoretical with point based on sparse grid Download PDFInfo
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- 238000000034 method Methods 0.000 title claims abstract description 40
- 238000009792 diffusion process Methods 0.000 title claims abstract description 23
- 238000012546 transfer Methods 0.000 claims abstract description 18
- HBMJWWWQQXIZIP-UHFFFAOYSA-N silicon carbide Chemical compound [Si+]#[C-] HBMJWWWQQXIZIP-UHFFFAOYSA-N 0.000 claims abstract description 16
- 229910010271 silicon carbide Inorganic materials 0.000 claims abstract description 16
- 230000004044 response Effects 0.000 claims description 12
- 238000009826 distribution Methods 0.000 claims description 5
- 238000010276 construction Methods 0.000 claims description 4
- 239000000758 substrate Substances 0.000 claims description 3
- 239000013529 heat transfer fluid Substances 0.000 claims description 2
- 239000000463 material Substances 0.000 claims description 2
- 230000006870 function Effects 0.000 description 9
- 238000001816 cooling Methods 0.000 description 5
- 238000004458 analytical method Methods 0.000 description 3
- 238000000342 Monte Carlo simulation Methods 0.000 description 2
- 238000010586 diagram Methods 0.000 description 2
- 238000004519 manufacturing process Methods 0.000 description 2
- 238000005259 measurement Methods 0.000 description 2
- 230000008569 process Effects 0.000 description 2
- 238000005070 sampling Methods 0.000 description 2
- 238000012360 testing method Methods 0.000 description 2
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- 238000013461 design Methods 0.000 description 1
- 230000000694 effects Effects 0.000 description 1
- 238000005516 engineering process Methods 0.000 description 1
- 230000020169 heat generation Effects 0.000 description 1
- 230000006872 improvement Effects 0.000 description 1
- 238000011089 mechanical engineering Methods 0.000 description 1
- 238000011160 research Methods 0.000 description 1
- 239000007787 solid Substances 0.000 description 1
- 238000010183 spectrum analysis Methods 0.000 description 1
- 230000003068 static effect Effects 0.000 description 1
- 238000013076 uncertainty analysis Methods 0.000 description 1
- 239000011800 void material Substances 0.000 description 1
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Abstract
The invention discloses a kind of section thermal convection diffusion problem method for solving theoretical with point based on sparse grid, and steps are as follows: indicating the uncertainty of the input in heat transfer system with interval variable；Establish the range restraint equation of thermal convection diffusion problem；Silicon carbide is responded using orthogonal polynomial and carries out approximate representation；According to tensor product rule and Smolyak formula, constructs sparse grid and match point set；All temperature-responsives matched at point are calculated, and acquire the expansion coefficient in temperature-responsive approximate expression using least square method；The bound of slickness computation interval temperature-responsive based on polynomial function.The present invention can systematization solve the thermal convection diffusion problem containing section uncertain input parameter and further improve the computational efficiency of point collocation under the premise of guaranteeing that computational accuracy meets engineering demand, this is that general business software institute is irrealizable.
Description
Technical field
The invention belongs to mechanical engineering fields, and in particular to a kind of to be expanded based on sparse grid with the theoretical section thermal convection of point
Dissipate problem solving method.
Background technique
Heat transfer problem is generally existing in engineering, especially in aerospace field, how to improve structure in hot environment
In reliability, it has also become a main aspect of Flight Vehicle Design.And such issues that premise and key be exactly determining first
The regularity of distribution of system temperature field.In Practical Project, due to the error of limitation, the measurement of manufacturing process and the variation of environment,
Uncertain factor is ubiquitous.For complex heat transfer system, even if the uncertain factor of very little, by between each subsystem
It propagates and spreads, it is also possible to which apparent disturbance is generated to final temperature-responsive.The complexity of system leads to physical problem number
The difficulty in statement is learned, often has to make simplification, this allows for traditional heat transfer numerical value calculating side based on deterministic models
Method is not accurate enough.
Stochastic modeling and numerical computation method have played important function in uncertainty analysis field, but are solved with random theory
When problem, a large amount of Test Information is needed to determine the probability distribution rule of mode input parameter in advance.In practical projects, it obtains
Sufficient test data often costs dearly.In this way, the shortage of information prevent probabilistic model from really describing objective reality
Border, this limits the application of stochastic model to a certain extent.How system objectively responded by limited data information
Uncertain feature, becomes many scholar's urgent problems to be solved.Interval model, it is only necessary to know the bound of uncertain variables i.e.
Can, good convenience and economy are embodied in terms of uncertain modeling.Currently, by interval theory and finite element method
The interval finite element method being derived is combined to have been achieved for much grinding in terms of the static and dynamic characteristics analysis of uncertain structure
Study carefully achievement, but very rare to the document of the heat transfer problem analysis with interval parameter.In addition, traditional Novel Interval Methods because
For interval extension problem caused by interval arithmetic also than more serious, computational accuracy is also in urgent need to be improved；It is theoretical with point based on whole mesh
Spectral analysis method calculate expend can be sharply increased with the expansion of Instable Space dimension.Therefore, accurate height how is established
The Novel Interval Methods of effect carry out numerical solution to uncertain heat transfer problem, are a research hotspots of current sphere of learning, right
In the deficiency for making up existing Heat Transfer Numerical, there is 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 bounded-but-unknown uncertainty that parameter is inputted in heat transfer system, matches point based on approximation by polynomi-als theory and sparse grid
Technology proposes a kind of numerical computation method of effective solution interval thermal convection diffusion problem, can systematization solution contain section
The heat transfer model temperature field prediction problem of uncertain parameter further improves point collocation while guaranteeing computational accuracy
Computational efficiency.
A kind of technical solution that the present invention uses to solve above-mentioned technical problem are as follows: area theoretical with point based on sparse grid
Between thermal convection diffusion problem method for solving, comprising the following steps:
Step 1: it is introduced into interval variable and the uncertainty for inputting parameter in heat transfer system is characterized；
Step 2: using the interval variable introduced in step 1, the range restraint equation of thermal convection diffusion problem is established；
Step 3: temperature-responsive involved in step 2 range restraint equation is approached using orthogonal polynomial, is obtained
The approximate expression responded to silicon carbide；
Step 4: according to tensor product rule and Smolyak formula (this Mo Liyake), sparse grid is established with point set；
Step 5: step 4 is calculated with the temperature-responsives at point all in point set using existing program or software, is built
The vertical system of linear equations about expansion coefficient in step 3 temperature-responsive 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 function determines the extreme point of this approximate expression, and then obtains the bound of silicon carbide response.
Wherein, in the step 3 using orthogonal polynomial to temperature-responsive carry out approximate representation, polynomial type and
Truncation order is not fixed and invariable, and requires to be chosen according to variable distribution pattern and approximation accuracy, for being uniformly distributed
Interval variable for, it is general to choose Legendre's orthogonal polynomial, be in addition truncated that order is higher, and approximation accuracy is higher.
Wherein, sparse grid is not fixed and invariable with the foundation of point set in the step 4, is expended according to calculating
Requirement with computational accuracy matches point level to choose, and horizontal higher with point, computational accuracy is higher, and it is bigger to calculate consuming.
Above steps specifically includes following procedure:
Step 1: it is introduced into n interval variable and carries out table to the uncertain of parameter is inputted in heat transfer system
Sign, and be denoted as the form of vector wherein subscript I is interval symbol,α iAnd expression
The lower bound of interval variable and the upper bound, and midpoint and the radius of referred to as interval variable are standard interval variable
Step 2: using the interval variable introduced in step 1, the range restraint equation of thermal convection diffusion problem is established:
Wherein x indicates that physical coordinates, T indicate temperature-responsive, ρ, c, and the density, specific heat capacity and heat that k respectively indicates material pass
Coefficient is led, u is the flowing velocity of heat-transfer fluid, and Q indicates the heat source strength of system.
Step 3: according to variable characteristic distributions, select suitable orthogonal 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 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 is polynomial truncation order.Above-mentioned approximate expression
The number Available Variables number n and truncation order N that item is unfolded in formula are expressed as
Step 4: according to tensor product rule and Smolyak formula, sparse grid is established with point set.Firstly, setting is whole
With horizontal k is put, L=k+n is enabled, point set is matched according to the sparse grid of tensor product rule and Smolyak formula construction n-dimensional space
Θ:
Wherein ijJ=1,2 ..., n indicate the horizontal with point of jth dimension space (corresponding to j-th of interval variable),Table
Show j-th of one-dimensional interval variableIt is corresponding to match the horizontal i of pointjIt is all with point composition a set, with point quantity
A position is put with matchingIt is respectively as follows:
Secondly, indicating that above-mentioned sparse grid is matched in point set with M matches point quantity, therefore Θ can be denoted asForm, for indicate n-dimensional space it is all with pointWherein subscript node table
Show with point.
Step 5: step 4 is calculated with the temperature-responsives at point all in point set using existing program or software, is built
The vertical system of linear equations about expansion coefficient in step 3 temperature-responsive 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 existing program or software, 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:
Wherein| i | 1≤j of≤N≤M representative polynomial basis function Φi(αI) with pointThe value at place.
Then, above-mentioned system of linear equations 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, utilize polynomial function TN(x,αI) slickness, determine its extreme point, and then obtain TN(x,αI)
Minimum value min TN(x,αI) and maximum value max TN(x,αI), final approximation obtains silicon carbide response T (x, αI) lower boundT
(x,αI) and the upper bound
The advantages of the present invention over the prior art are that:
(1) compared with traditional Heat Transfer Numerical, the Novel Interval Methods proposed fully consider heat transfer system
The uncertainty of parameter is inputted, calculated result has prior directive significance to temperature field analysis.
(2) approximate representation is carried out to temperature-responsive using orthogonal polynomial, approximation accuracy can be effectively improved.Meanwhile it utilizing
The slickness of polynomial function can quickly determine its extreme point, and then obtain the bound of temperature-responsive.
(3) it is improved using Smolyak formula to dot grid, can effectively reduce with quantity, improve point collocation
Computational efficiency.
(4) operation of the present invention is simple, easy to implement, while guaranteeing computational accuracy, effectively reduces traditional sampling side
The calculating of method expends.
Detailed description of the invention
Fig. 1 is a kind of section thermal convection diffusion problem method for solving process theoretical with point based on sparse grid of the invention
Figure；
Fig. 2 is air cooling system model schematic of the invention；
Fig. 3 is flow tube center line along z-axis silicon carbide response schematic diagram；
Fig. 4 is cylindrical outer side along z-axis silicon carbide response schematic diagram.
Specific embodiment
The present invention will be further described with reference to the accompanying drawings and examples.
The present invention is suitable for the temperature field prediction of the thermal convection diffusion problem containing section uncertain parameter.The present invention is implemented
It is right to illustrate a kind of section heat based on sparse grid with point theory by taking certain air cooling system model as an example for mode
Flow diffusion problem method for solving.In addition, the silicon carbide response numerical computation method of this air cooling system model can be promoted
Contain in the thermal convection diffusion problem temperature field prediction of section uncertain parameter to other.
A kind of calculating process such as Fig. 1 institute based on sparse grid with the theoretical section thermal convection diffusion problem method for solving of point
Show, the input being introduced into section argument table sign heat transfer system is uncertain, and then establishes the range restraint of thermal convection diffusion problem
Equation responds silicon carbide using orthogonal polynomial and carries out approximate representation, while public according to tensor product rule and Smolyak
Formula, construction sparse grid match point set, calculate all temperature-responsives at point using existing program or software, and using minimum
Square law acquires the expansion coefficient in temperature-responsive approximate expression, finally the slickness computation interval temperature based on polynomial function
Spend the bound of response.The following steps progress can be divided into:
Step 1: consider the cylindrical empty air cooling system of a height of 100cm shown in Fig. 2, it is 10cm's that, which there is diameter in centre,
Circular hole passes through cooling air.Along the z-axis direction, three points that number is 1~3 are selected on flow tube center line, are selected in cylindrical outer side
Observation point of three points that the number of delimiting the organizational structure is 4~6 as temperature field.Due to the error and environment of limitation, the measurement of manufacturing process
Variation, the partial parameters of this heat transfer model contain certain uncertainty, introduce interval variable and characterize to it, hollow
The density p of gasI=[1.3,1.5] kg/m3, specific heat capacity cI=[900,1100] J/ (kg DEG C), coefficient of heat conduction kI=
[0.0242,0.0282] W/ (m DEG C), the flowing velocity u of airI=[4.6,5.4] m/s, the air themperature of inlet 7In addition, there is volumetric heat generation in the solid structure 8 shown in dash area, heat source strength be Q=50000 ×
sin(ω)W/m3, wherein parameter ω variation in section [1.5,2.5].Above-mentioned six interval variables are expressed as vector formWherein subscript I is interval symbol,α iWithIndicate interval variableLower bound and
The upper bound,WithReferred to as interval variableMidpoint and radius,For standard interval variable
Step 2: using the interval variable introduced in step 1, the range restraint equation of thermal convection diffusion problem is established:
Wherein x, y, z indicate that the physical coordinates on three direction in spaces, T indicate temperature-responsive, ρ, c, and k respectively indicates air
Density, specific heat capacity and the coefficient of heat conduction, u be air flowing velocity, Q indicate system heat source strength.
Step 3: the characteristics of being uniformly distributed according to interval variable selects Legendre's orthogonal polynomial to control step 2 section
Temperature-responsive T (x, y, z, α in equation processedI) approached, truncation order is set as N=3, obtains the close of silicon carbide response
Like expression 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 rule and Smolyak formula, sparse grid is established with point set.Firstly, setting is whole
With horizontal k=2 is put, L=k+n=8 is enabled, according to the sparse grid of this sextuple space of tensor product rule and Smolyak formula construction
With point set Θ:
Wherein ijJ=1,2 ..., 6 indicates the horizontal with point of j-th of interval variable,Indicate that j-th of one-dimensional section becomes
AmountIt is corresponding to match the horizontal i of pointjIt is all with point composition a set, with point quantityA position is put with matchingRespectively
Are as follows:
Above-mentioned sparse grid is M=85 with the point quantity of matching in point set, therefore Θ is denoted as
Form, for indicate sextuple space it is all with pointWherein subscript node indicates to match point.
Step 5: step 4 is calculated with the temperature-responsives at point all in point set using existing program or software, is built
The vertical system of linear equations about expansion coefficient in step 3 temperature-responsive 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 the finite element program in software Nastran, it is available it is all with point at
Temperature-responsive
Secondly, responding approximate expression based on silicon carbide in step 3, establish about all expansion coefficient wi(x,y,z)
System of linear equations:
Wherein| i | the representative polynomial basis function of≤31≤j≤85 Φi(αI) with pointThe value at place.
Then, above-mentioned system of linear equations 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, determine its extreme point, and then obtain TN
(x,y,z,αI) minimum value minTN(x,y,z,αI) and maximum value maxTN(x,y,z,αI), final approximation obtains silicon carbide and rings
Answer T (x, y, z, αI) lower boundT(x, y, z, αI) and the upper bound
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 only 85, 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 z-axis direction, flow tube center line and cylindrical outer side silicon carbide response such as Fig. 3
With shown in Fig. 4, abscissa indicates spatial position along the z-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,
It calculates and expends less, 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 (4)
1. a kind of section thermal convection diffusion problem method for solving theoretical with point based on sparse grid, it is characterised in that including following
Step:
Step 1: it is introduced into interval variable and the uncertainty for inputting parameter in heat transfer system is characterized；
Specifically: introduce n interval variableThe uncertainty for inputting parameter in heat transfer system is characterized, and is remembered
For the form of vectorWherein subscript I is interval symbol,α iWithIndicate that section becomes
AmountLower bound and the upper bound,WithReferred to as interval variableMidpoint and radius,For mark
Quasi- interval variable
Step 2: using the interval variable introduced in step 1, the range restraint equation for establishing thermal convection diffusion problem is as follows:
Wherein x indicates that physical coordinates, T indicate temperature-responsive, ρ, c, and υ respectively indicates the density, specific heat capacity and heat transfer system of material
Number, u are the flowing velocity of heat-transfer fluid, and Q indicates the heat source strength of system；
Step 3: temperature-responsive involved in step 2 range restraint equation is approached using orthogonal polynomial, obtains area
Between temperature-responsive approximate expression:
Wherein Φi(αI) it is 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 is polynomial truncation order, is opened up in above-mentioned approximate expression
The number Available Variables number n and truncation order N for opening item are expressed as
Step 4: according to tensor product rule and Smolyak formula, sparse grid is established with point set；
Specifically: firstly, setting is whole with the horizontal k of point, L=k+n is enabled, is tieed up according to tensor product rule and Smolyak formula construction n
The sparse grid in space matches point set Θ:
Wherein ij, j=1,2 ..., n indicate the horizontal with point of jth dimension space,Indicate j-th of one-dimensional interval variableIt is corresponding to match the horizontal i of pointjIt is all with point composition a set, with point quantityA position is put with matchingIt is respectively as follows:
Secondly, indicating above-mentioned sparse grid with, with point quantity, Θ is denoted as in point set with MShape
Formula, for indicate n-dimensional space it is all with pointWherein subscript node indicates to match point；
Step 5: step 4 is calculated with the temperature-responsives at point all in point set, is established about step 3 temperature-responsive
The system of linear equations of expansion coefficient in approximate expression, and this system of linear equations is solved using least square method, it obtains
One 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 function determines the extreme point of this approximate expression, and then obtains the bound of silicon carbide response.
2. a kind of section thermal convection diffusion problem solution side theoretical with point based on sparse grid according to claim 1
Method, it is characterised in that: temperature-responsive involved in range restraint equation is forced using orthogonal polynomial in the step 3
When close, polynomial type and truncation order are not fixed and invariable, and according to polynomial variable distribution pattern and approach essence
Degree requires to be chosen, and for equally distributed interval variable, chooses Legendre's orthogonal polynomial, and in addition truncation order is higher,
Approximation accuracy is higher.
3. a kind of section thermal convection diffusion problem solution side theoretical with point based on sparse grid according to claim 1
Method, it is characterised in that: sparse grid is not fixed and invariable with the foundation of point set in the step 4, is expended according to calculating
Requirement with computational accuracy matches point level to choose, and horizontal higher with point, computational accuracy is higher, and it is bigger to calculate consuming.
4. a kind of section thermal convection diffusion problem solution side theoretical with point based on sparse grid according to claim 1
Method, it is characterised in that: calculate step 4 using existing program or software in the step 5 and match at point with all in point set
Temperature-responsive.
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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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| Publication number | Priority date | Publication date | Assignee | Title |
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| US6850876B1 (en) * | 1999-05-04 | 2005-02-01 | Smithkline Beecham Corporation | Cell based binning methods and cell coverage system for molecule selection |
| 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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| 含区间参数的结构－声耦合***可靠性优化设计;王冲等;《振动工程学报》;20141031;第27卷(第5期);第728-733页 * |
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