CN105677995A - Method for numerical solution of fuzzy steady state heat conduction problem based on full grid point collocation theory - Google Patents

Method for numerical solution of fuzzy steady state heat conduction problem based on full grid point collocation theory Download PDF

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CN105677995A
CN105677995A CN201610018353.3A CN201610018353A CN105677995A CN 105677995 A CN105677995 A CN 105677995A CN 201610018353 A CN201610018353 A CN 201610018353A CN 105677995 A CN105677995 A CN 105677995A
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fuzzy
finite element
equation
temperature
steady state
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邱志平
陈贤佳
孙佳丽
吕�峥
朱静静
王鹏博
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Beihang University
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    • GPHYSICS
    • G06COMPUTING; CALCULATING OR COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00Computer-aided design [CAD]
    • G06F30/20Design optimisation, verification or simulation
    • G06F30/23Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
    • GPHYSICS
    • G06COMPUTING; CALCULATING OR COUNTING
    • G06NCOMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N7/00Computing arrangements based on specific mathematical models
    • G06N7/02Computing arrangements based on specific mathematical models using fuzzy logic
    • G06N7/06Simulation on general purpose computers

Abstract

The invention discloses a method for numerical solution of a fuzzy steady state heat conduction problem based on a full grid point collocation theory. The method comprises the first step of carrying out finite element modeling of a steady state heat conduction structure; the second step of utilizing fuzzy variables to represent uncertain input parameters and then obtaining a fuzzy finite element equation of the heat conduction problem; the third step of utilizing cut set operation to rewrite the fuzzy finite element equation to be a group of interval finite element equations; the fourth step of performing approximate representation on temperature response functions in the interval finite element equations based on a polynomial theory; the fifth step of quickly solving the interval variation range of a temperature response approximate functions according to the full grid point collocation theory; the sixth step of utilizing a fuzzy resolution theory to recombine temperature response intervals under all cut set levels, and finally obtaining a membership function of the fuzzy temperature response. According to the method, the prediction of structure heat conduction temperature field containing fuzzy uncertain parameters can be achieved systematically, and the calculation accuracy can be effectively improved on the premise of guaranteeing that the calculation efficiency meets the engineering requirements, which cannot be achieved through common commercial software.

Description

A kind of join a theoretical fuzzy thermal conduction under steady state method of value solving based on whole mesh
Technical field
The invention belongs to mechanical engineering field, be specifically related to a kind of join a theoretical fuzzy thermal conduction under steady state method of value solving based on whole mesh.
Background technology
How more efficiently heat transfer problem is ubiquity in engineering, in fields such as automobile making, electronic engineering, derived energy chemicals, realize heat transmission and make structure and electronic equipment normal operation also be the encountered major issues of engineer. Particularly in aerospace field, heat transfer is more notable with the coupling of structure. Along with the continuous progress of science and technology, aircraft heat transfer system is increasingly sophisticated, and the requirement of performance is also more and more higher, and this allows for becoming more meticulous and analyzes the important development trend becoming Flight Vehicle Design.
The little exploitation to electronic component, the big design to Aircraft structure, uncertain factor is ubiquitous. In the last few years, indetermination theory obtain and study widely and be applied in many Practical Project problems, the social and economic effects brought is huge. This reflect on the one hand indetermination theory in practical engineering application, played important function, also study indeterminacy of calculation method on the other hand for us further and optimum theory provide the foundation guarantee and power producer. Specific in heat transfer problem, structure temperature change can cause a series of derivative phenomenon such as thermal stress, thermal deformation, studies the premise of this kind of physical problem and the crucial temperature distributing rule first having to determine structure exactly. In Practical Project, the complexity of system causes the difficulty on physical problem formulation and inconvenience, often have to make simplification, uncertain problem in objective is similar to and is converted into subjective certain problem and processes, which results in the calculating of classics and method for designing increasingly can not meet complex heat transfer system and become more meticulous and analyze and the requirement of design. Obtain the result of calculation of more objective reality, it is necessary to by indetermination theory heat transfer system is modeled and solves. The ambiguity of things refers to that its concept itself is difficult to determine, is a kind of uncertainty brought owing to concept extension is fuzzy, it is common to exist with in engineering reality. Fuzzy theory is achieved with some achievements in structure static analysis, but the application in field of heat transfer is just at the early-stage. How quick and precisely to predict the fuzzy uncertainty feature of heat conduction problem structure temperature field, be a focus of current sphere of learning, for making up the deficiency of existing heat transfer numerical method, there is important engineer applied and be worth.
Summary of the invention
The technical problem to be solved is: overcome the deficiency that prior art exists in thermal conduction under steady state solves, take into full account the impact of system ambiguous input parameter, a theory is joined, it is proposed that the numerical computation method of a kind of Accurate Prediction structure temperature field fuzzy uncertain feature based on approximation by polynomi-als thought and whole mesh.
This invention address that the technical scheme that above-mentioned technical problem adopts is: a kind of join a theoretical fuzzy thermal conduction under steady state method of value solving based on whole mesh, comprise the following steps:
Step one: for thermal conduction under steady state, chooses suitable cell type and carries out discrete to structure, obtain the FEM (finite element) model of this heat transfer structure, sets up conduction of heat finite element discretization equation;
Step 2: utilize fuzzy vector to characterize all uncertain parameter in heat conduction problem, obtain the fuzzy finite element equation of heat conduction problem according to the discrete equation in step one further;
Step 3: utilize cut set computing, is converted into the fuzzy vector in step 2 a class interval vector, and then the fuzzy finite element equation in step 2 can be rewritten as a class interval finite element equation;
Step 4: based on polynomial theory, carries out approximate representation to the temperature-responsive function of interval Finite Element Method equation in step 3;
Step 5: join a theory based on whole mesh, utilizes the interval Finite Element Method equation in step 3 and the polynomial expression in step 4, the interval excursion of temperature-responsive is carried out rapid solving;
Step 6: utilize fuzzy resolution theorem to be recombinated in all temperature-responsive intervals obtained in step 5, finally gives the membership function of Fuzzy temperature response.
Wherein, utilizing polynomial theory that temperature-responsive function is carried out approximate representation in described step 4, multinomial basis function is selected in advance according to types of variables; Polynomial exponent number is then chosen flexibly according to required precision.
Wherein, it is not changeless for joining choosing of a scheme in described step 5; Quantity and the requirement of computational accuracy according to multinomial coefficient are chosen flexibly and are joined a level.
It addition, the present invention joins a theoretical fuzzy thermal conduction under steady state method of value solving based on whole mesh specifically includes following steps:
Step one: for thermal conduction under steady state, chooses suitable cell type and carries out discrete to structure, obtain the FEM (finite element) model of this heat transfer structure, sets up conduction of heat finite element discretization equation:
KT=R
Wherein T is temperature vector, and K is conduction of heat matrix, and R is the thermal force vector of equivalence.
Step 2: utilize fuzzy vector to characterize all uncertain parameter α=(α in heat conduction problemi)m=(α12,...,αn), wherein n is the number of fuzzy parameter. The fuzzy finite element equation of heat conduction problem is obtained further according to the discrete equation in step one:
K (α) T (α)=R (α)
Step 3: utilize cut set computing, is converted into a class interval vector by the fuzzy vector in step 2:
( α ) λ = α λ I = [ α ‾ λ , α ‾ λ ] = ( α i , λ I ) n = ( [ α ‾ i , λ , α ‾ i , λ ] ) n = ( α i , λ c + Δα i , λ I ) n = ( α i , λ c + Δα i , λ δ i I ) n
Wherein λ is Truncated set level selected in 0 to 1 scope,Represent interval variable,α i,λWithFor its lower bound and the upper bound, α i , λ c = ( α ‾ i , λ + α ‾ i , λ ) / 2 With Δα i , λ = ( α ‾ i , λ - α ‾ i , λ ) / 2 Represent interval midpoint and radius,For standard interval variable δ i I = [ - 1 , 1 ] .
Thus, it is possible to the fuzzy finite element equation in step 2 is rewritten as a class interval finite element equation:
K ( α λ I ) T ( α λ I ) = R ( α λ I )
WhereinFor interval variableTemperature-responsive vector under impact.
Step 4: utilize polynomial theory, it is possible to by the node temperature response of interval Finite Element Method equation in step 3With limited rank multinomialIt is approximately represented as:
T ( α λ I ) ≈ T N ( α λ I ) = Σ | i | ≤ N T i Φ i ( α λ I )
WhereinFor orthogonal polynomial substrate function, TiFor corresponding expansion coefficient, i=(i1,i2,...,in) represent multidimensional index, and meet | i |=i1+i2+...+in, N polynomial blocks exponent number for this. The number Available Variables number n of this polynomial expansion item and block exponent number N and be expressed as
Step 5: join a theory based on whole mesh, asks for the interval excursion of temperature-responsive.First, introducing index k represents joins a level, then single interval variableInterior joins a quantity miWith join a positionIt is respectively as follows:
Secondly, point set is usedRepresent single interval variableIn all set joining some composition, then for the higher dimensional space of n variable composition, directly utilizing tensor product computing, its whole mesh is joined and is gathered a Θ and be represented by:
Θ = Θ 1 m 1 × Θ 2 m 2 × ... × Θ n m n
And join a sum M and be:
M = Π i = 1 n m i = m 1 × m 2 × ... × m n
Therefore Θ is rewritten as Θ = { ξ 1 n o d e , ξ 2 n o d e , ... , ξ M n o d e } , It is used for representing all of to join a little ξ i n o d e , i = 1 , 2 , ... , M .
Make the interval Finite Element Method equation set up in step 3 set up at all places of joining, be based further on the polynomial expression in step 4 and can obtain about expansion coefficient TiAn equation group:
WhereinRepresentative polynomial basis functionJoining a littleThe value at place,Represent node temperature responseJoining a littleThe value at place.
Utilize least square method to solve above-mentioned equation group, obtain expansion coefficient TiA class value, in its generation, is returned in the multinomial in step 4, utilizes functionSlickness, and then obtain lower bound and the upper bound of temperature-responsive:
T ‾ ( α λ I ) ≈ minT N ( α λ I ) T ‾ ( α λ I ) ≈ maxT N ( α λ I )
Step 6: utilize fuzzy resolution theorem by interval for all temperature-responsives obtained in step 5Recombinate, finally give the membership function T (α) that the Fuzzy temperature under fuzzy parameter vector α impact responds:
T ( α ) = ∪ j = 1 , ... , m { λ j T I ( α λ J I ) }
Wherein m is cut set quantity selected in fuzzy cut-set computing.
Present invention advantage compared with prior art is in that:
(1) compared with traditional thermal conduction under steady state method for solving, the computation model set up fully takes into account the fuzzy uncertainty of material parameter in Practical Project, load and boundary condition, and temperature field analysis and structural design are had prior directive significance by result of calculation.
(2) based on polynomial theory, temperature-responsive function is similar to, blocks exponent number and from main separation, approximation accuracy can be effectively improved. It addition, utilize the slickness of polynomial function, can quickly determine the temperature-responsive extreme point about interval variable, and then obtain the bound of temperature-responsive.
(3) in the process that multinomial coefficient solves, whole mesh is utilized to join point methods, it is possible to make full use of the calculation procedure of original deterministic models without master mould makes further processing and amendment. By selecting a suitable scheme of joining, a small amount of joining a quantity and just can farthest ensure computational accuracy, this advantage is that tradition Monte-carlo Simulation Method can not be compared.
(4) present invention is simple to operate, and it is convenient to implement, and is effectively increased computational accuracy.
Accompanying drawing explanation
Fig. 1 is that a kind of of the present invention joins a theoretical fuzzy thermal conduction under steady state method of value solving flow chart based on whole mesh;
Fig. 2 is the flat late heat transfer structural finite element model schematic diagram of the present invention;
Fig. 3 is the membership function schematic diagram of characteristic point 2 place Fuzzy temperature response;
Fig. 4 is the membership function schematic diagram of characteristic point 4 place Fuzzy temperature response.
Detailed description of the invention
Below in conjunction with drawings and Examples, the present invention will be further described.
The present invention is applicable to the structure temperature field prediction of the thermal conduction under steady state containing fuzzy uncertain parameter. Present embodiment, for the temperature field prediction of certain flat late heat transfer structure, illustrates described fuzzy thermal conduction under steady state method of value solving. It addition, the uncertain temperature-responsive numerical computation method of this flat late heat transfer structural fuzzy can be generalized in other thermal conduction under steady state temperature field predictions containing fuzzy parameter.
Fuzzy thermal conduction under steady state numerical procedure is as shown in Figure 1, FEM (finite element) model based on structure, set up the discrete equation of heat transfer system, utilize fuzzy variable that uncertain parameter is carried out quantitative description, and based on cut set computing, uncertain parameter is further processed, and whole mesh theoretical according to approximation by polynomi-als is joined point methods and the interval excursion of temperature-responsive under all Truncated set level is carried out rapid solving, utilize fuzzy resolution theorem that it is recombinated, finally give the membership function of fuzzy uncertain temperature-responsive.Following several step can be divided into carry out:
Step one: set up flat late heat transfer structural finite element model, as shown in Figure 2: rectangular area and border circular areas are respectively divided into 100 quadrilateral units and 188 triangular elements, shadow region 8 has volumetric heat Q to generate, and applies hot-fluid load q along border 9 bottom plate simultaneouslys, left border 10 gives temperature value Ts, select 7 nodes being numbered 1~7 characteristic point as this example. Conduction of heat finite element discretization equation is set up according to FEM (finite element) model:
KT=R
Wherein T is temperature vector, and K is conduction of heat matrix, and R is the thermal force vector of equivalence.
Step 2: in this heat conduction problem, by the impact of the restriction of material processing technique and measurement error, all parameters are fuzzy number, and membership function meets angular distribution rule, wherein heat conductivity k=(174,204,234) W/ (m DEG C), heat flow density qs=(750,900,1050) W/m2, heat source strength Q=(1700,2000,2300) W/m3, boundary temperature Ts=(42,50,58) DEG C. For convenience, all fuzzy parameters involved in this computation model are expressed as the form α of vector α=(k, qs,Q,Ts), the fuzzy finite element equation of heat conduction problem is obtained further according to the discrete equation in step one:
K (α) T (α)=R (α)
Step 3: define 11 Truncated set level λq=(q-1) × 0.1q=1 ..., 11, utilize cut set computing to be rewritten as by the fuzzy vector α in step 2:
( α ) λ = α λ I = [ α ‾ λ , α ‾ λ ] = ( α i , λ I ) n = ( [ α ‾ i , λ , α ‾ i , λ ] ) n = ( α i , λ c + Δα i , λ I ) n = ( α i , λ c + Δα i , λ δ i I ) n
WhereinRepresent interval variable,α I, λWithFor its lower bound and the upper bound,WithRepresent interval midpoint and radius,For standard interval variable
Thus, it is possible to the fuzzy finite element equation in step 2 is rewritten as a class interval finite element equation:
K ( α λ I ) T ( α λ I ) = R ( α λ I )
WhereinFor interval variableTemperature-responsive vector under impact.
Step 4: the characteristic distributions according to interval variable, select Le allow type orthogonal polynomial as basis function, block exponent number and be set as N=3, then in step 3 interval Finite Element Method equation node temperature responseCan be approximately represented as:
T ( α λ I ) ≈ T N ( α λ I ) = Σ | i | ≤ N T i Φ i ( α λ I )
WhereinType orthogonal polynomial substrate function is allowed to obtain, T for stranglingiFor corresponding expansion coefficient, i=(i1,i2,...,in) represent multidimensional index, and meet | i |=i1+i2+...+in, now the number of polynomial expansion item is
Step 5: join a little based on whole mesh, asks for the interval range of temperature-responsive. Firstly, for Truncated set level λ9=0.8, set and join a little horizontal k=3, then single interval variableInterior joins a quantity miWith join a positionIt is respectively as follows:
mi=5
β j ( i ) = α i , λ c - c o s π ( j - 1 ) m i - 1 · Δα i , λ , j = 1 , 2 , ... , 5
Secondly, point set is usedRepresent single interval variableIn all set joining some composition, then for the higher dimensional space of 4 variablees compositions, directly utilizing tensor product computing, its whole mesh is joined a set Θ and is represented by:
Θ = Θ 1 5 × Θ 2 5 × ... × Θ 4 5
And join a sum M and be:
M = Π i = 1 4 5 = 625
Therefore Θ is rewritten as Θ = { ξ 1 n o d e , ξ 2 n o d e , ... , ξ 625 n o d e } , It is used for representing all of to join a little ξ i n o d e , i = 1 , 2 , ... , 625.
Make the interval Finite Element Method equation set up in step 3 set up at all places of joining, be based further on the polynomial expression in step 4 and can obtain about expansion coefficient TiAn equation group:
Utilize least square method to solve above-mentioned equation group, obtain expansion coefficient TiA class value, in its generation, is returned in the multinomial in step 4, utilizes functionSlickness, and then obtain Truncated set level λ9The lower bound of=0.8 time temperature-responsive and the upper bound:
T ‾ ( α 0.8 I ) ≈ minT N ( α 0.8 I ) T ‾ ( α 0.8 I ) ≈ maxT N ( α 0.8 I )
Same, for Truncated set level λ6=0.5, set and join a little horizontal k=4, repeat aforesaid operations, obtain Truncated set level λ6The lower bound of=0.5 time temperature-responsive and the upper bound.
λ9=0.8 and λ6Shown in the result of calculation such as table 2 and table 3 of characteristic point place temperature-responsive under=0.5 two Truncated set level.It is 10 with sample number5Monte-carlo Simulation Method result of calculation compare, the relative error that temperature-responsive interval bound is predicted by this method maintains a reduced levels, can meet the required precision of Practical Project completely. It addition, from sample size, the computational efficiency of this method is significantly larger than monte carlo method.
Table 2 Truncated set level λ9=0.8 time temperature-responsive bound
Table 3 Truncated set level λ6=0.5 time temperature-responsive bound
Step 6: utilize fuzzy resolution theorem to be recombinated in all temperature-responsive intervals obtained in step 5, finally gives the membership function T (α) that the Fuzzy temperature under fuzzy parameter vector α impact responds:
T ( α ) = ∪ j = 1 , ... , 11 { λ j T I ( α λ j I ) }
For characteristic point 2 and characteristic point 4, the membership function of Fuzzy temperature response is as shown in Figure 3 and Figure 4. It can be seen that the calculated result of this method is fine with the reference value degree of agreement that Monte Carlo sampling obtains, computational accuracy fully meets engine request. Can solving the Steady-State Thermal Field forecasting problem containing fuzzy uncertain parameter by this method, computational accuracy is high, calculates and expends less, and this function is that general business software institute is irrealizable.
The above-described presently preferred embodiments of the present invention that is only, the present invention is not limited solely to above-described embodiment, and all local made within the spirit and principles in the present invention are changed, equivalent replacement, improvement etc. should be included within protection scope of the present invention.

Claims (3)

1. join a theoretical fuzzy thermal conduction under steady state method of value solving based on whole mesh for one kind, it is characterised in that comprise the following steps:
Step one: for thermal conduction under steady state, chooses suitable cell type and carries out discrete to structure, obtain the FEM (finite element) model of this heat transfer structure, sets up conduction of heat finite element discretization equation;
Step 2: utilize fuzzy vector to characterize all uncertain parameter in heat conduction problem, obtain the fuzzy finite element equation of heat conduction problem according to the discrete equation in step one further;
Step 3: utilize cut set computing, is converted into the fuzzy vector in step 2 a class interval vector, and then the fuzzy finite element equation in step 2 can be rewritten as a class interval finite element equation;
Step 4: based on polynomial theory, carries out approximate representation to the temperature-responsive function of interval Finite Element Method equation in step 3;
Step 5: join a theory based on whole mesh, utilizes the interval Finite Element Method equation in step 3 and the polynomial expression in step 4, the interval excursion of temperature-responsive is carried out rapid solving;
Step 6: utilize fuzzy resolution theorem to be recombinated in all temperature-responsive intervals obtained in step 5, finally gives the membership function of Fuzzy temperature response.
2. according to claim 1 a kind of join a theoretical fuzzy thermal conduction under steady state method of value solving based on whole mesh, it is characterized in that: utilizing polynomial theory that temperature-responsive function is carried out approximate representation in described step 4, multinomial basis function is selected in advance according to types of variables; Polynomial exponent number is then chosen flexibly according to required precision.
3. according to claim 1 a kind of join a theoretical fuzzy thermal conduction under steady state method of value solving based on whole mesh, it is characterised in that: it is not changeless for joining choosing of a scheme in described step 5; Quantity and the requirement of computational accuracy according to multinomial coefficient are chosen flexibly and are joined a level.
CN201610018353.3A 2016-01-12 2016-01-12 Method for numerical solution of fuzzy steady state heat conduction problem based on full grid point collocation theory Pending CN105677995A (en)

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CN114018972A (en) * 2021-11-04 2022-02-08 株洲国创轨道科技有限公司 Method and system for measuring surface heat flow of solid-liquid phase change material based on dispersion fuzzy inference mechanism

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CN106897520A (en) * 2017-02-27 2017-06-27 北京航空航天大学 A kind of heat transfer system analysis method for reliability containing fuzzy parameter
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CN106909747A (en) * 2017-03-06 2017-06-30 北京航空航天大学 Fuzzy parameter membership function recognition methods in a kind of thermal convection current diffusion system
CN106909747B (en) * 2017-03-06 2020-06-30 北京航空航天大学 Fuzzy parameter membership function identification method in heat convection diffusion system
CN111125928A (en) * 2019-12-31 2020-05-08 湖南大学 Probability-interval mixed accelerometer bias uncertainty analysis method
CN114018972A (en) * 2021-11-04 2022-02-08 株洲国创轨道科技有限公司 Method and system for measuring surface heat flow of solid-liquid phase change material based on dispersion fuzzy inference mechanism

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