CN105893698B - A kind of affine arithmetic for the Multidisciplinary systems index solving Structural Engineering - Google Patents
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Abstract
A kind of affine arithmetic for the Multidisciplinary systems index solving Structural Engineering, is related to fail-safe analysis and the algorithm improvement field of Structural Engineering.Mainly include the following steps: to establish Structural functional equation；The section input variable of receptance function is converted to the Affine Incentive of single noise symbol by routinely affine arithmetic；Correlated expression item in expression formula containing same noise member is subjected to overall Affine Incentive and converts union；Calculate its value of finally returning that；Power function is substituted into, Multidisciplinary systems index is obtained.Compared to tradition interval algorithm common in current Structural Engineering fail-safe analysis and Affine arithmetic, multiple continuous items of same noise member expression can be effectively treated in algorithm of the invention, advantageously reduce error, in the reliability index operation of strong nonlinearity power function, energy acquisition is more compact, also closer to the compartmental results of true value.The Multidisciplinary systems index of multifunctional sowing and watering, fertilizing mechanic operating mechanism pull rod is calculated with the present invention, computational accuracy is increased dramatically.
Description
Technical field
The present invention relates to the analytical calculation field of Structural Engineering, relate generally to a kind of more accurately seek Multidisciplinary systems
Index calculating method.
Background technique
In Structural Engineering fail-safe analysis calculating, the Measure Indexes of interval model Multidisciplinary systems are often used,
Conventional section operation method and conventional affine arithmetic solution interval model Multidisciplinary systems index in conventional method.
Currently, to seek method be based on replacing point variable progress with interval variable for Non-probabilistic Reliability Index general
The intervl mathematics of operation, it is convenient to the uncertainty of processing data, automatically records and produce in computer floating number arithmetical operation
Raw truncation and rounding error determines the value range etc. of function, is widely used in computational science, engineering, finance, enterprise's pipe
The fields such as reason, traffic.However the correlation between the irrationality of its operation rule and complete omitted variables, cause operation knot
Fruit is easily expanded and overflows.When non linear efficacy function degree is higher or nested deeper, it can also cause error explosion, cause to count
It calculates result and loses practical value.In order to overcome these deficiencies, there is interval-truncation approach, interval finite element method and subinterval to take the photograph
A variety of solutions such as dynamic method, but the treatment process of these methods is relatively cumbersome, computationally intensive, and calculating process is not sufficiently stable,
So that the scope of application is received it greatly and limits (calculation flow chart is shown in Fig. 1).
A kind of improvement of the Affine arithmetic as interval arithmetic can consider the correlation of calculating and input data, and will be this
Correlation automatically records and is applied in independent calculate, and operation rule has Properties of Optimization in addition, also more stable, thus can obtain
Narrower more accurate compartmental results are obtained, advantage is more obvious in long calculating chain for this, is widely used to artificial intelligence system point
Analysis, system stability analysis, circuit response circle's analysis and computer graphics.However its multiplication and division operation uses close approximation；No
It can consider the primary, secondary of same noise member expression or even the repeatedly correlation between item, there are calculating error (calculation flow charts
See Fig. 2).
(1) interval arithmetic method or conventional affine arithmetic seek the mode of Non-probabilistic Reliability Index
If x=(x1,x2,…,xn)For section related with structural response input vector.Consider
Response to structure to input parameter, if the section allowable of structural response are as follows:
y∈yI∈[yl,yu] (1)
The receptance function of structure is set again are as follows:
F (x)=f (x1,x2,…,xn) (2)
The then power function of structure are as follows:
M=g (x, y)=y-f (x)=y-f (x1,x2,…,xn) (3)
As f (x1,x2,…,xn) it is xiWhen the continuous function of (i=1,2 ..., n), M, f (x) are inevitable also to be become for a section
Amount, interval arithmetic method is directly to bring interval variable into above formula, M, f (x) is calculated according to standard section algorithm, if Mu、Ml
And fu、flIt is the bound of M Yu f (x) respectively, then Non-probabilistic Reliability Index may be defined as:
In formula: yc=(yu+yl)/2, yr=(yu-yl)/2, fc=(fu+fl)/2, fr=(fu-fl)/2, Mc=(Mu+Ml)/2
=yc-fc, Mr=(Mu-Ml)/2=yr+fr, it is referred to as center and the radius in corresponding section.
And conventional affine arithmetic is then the section input variable x for first calculating receptance function1~xnMidpointAs the central value of corresponding Affine Incentive, even:
By x1~xnSection radius value as single noise symbol ε1(ε1∈ [- 1,1]) coefficient, it may be assumed that
So far, just by x1~xnIt is converted to the n Affine Incentives for possessing single noise symbol:
With interval variable x corresponding in above formula substituted (2)i(i=1,2 ..., n), so that it may by receptance function (and area
Between function) f (x)=f (x1,x2,…,xn) it is converted to affine form.Its value of finally returning that can be calculated using Affine arithmetic ruleIf finally returning that value are as follows:
ε in formula2~εt+1It is the t new noise symbols introduced in calculating process as needed.
It calculatesBound, it may be assumed that
fu=f0+|f1|+|f2|+…+|ft+1| (9)
fl=f0-|f1|-|f2|-…-|ft+1| (10)
Similarly, the bound y in the section allowable of structural response can be calculatedu,yl.Formula (4) are substituted them in, can be obtained non-
Probabilistic reliability index:
In formula
Summary of the invention
The object of the present invention is to provide a kind of affine calculations of modified of Multidisciplinary systems index for solving Structural Engineering
Method is a kind of new method for seeking Multidisciplinary systems index, can improve the computational accuracy in mechanical mechanism analytical calculation.
The present invention is achieved by the following technical solutions.
When occurring multiple expression items containing same noise member in Affine arithmetic, determined by interval algorithm or tensor operation
Primary, secondary or even repeatedly item totality value the bound of same noise member expression carries out Affine arithmetic in Affine arithmetic
It improves, is then based on bound information, introduces new noise symbol, the Affine Incentive that these continuous items carry out totality is converted and transported
It calculates.The present invention can preferably consider the primary, secondary of same noise member expression or even the repeatedly correlation between item, improve
Computational accuracy.
Step of the invention is as follows:
Step 1: establishing Structural functional equation: M=g (x, y)=y-f (x)=y-f (x1,x2,…,xn)；
Step 2: routinely affine arithmetic, by the section input variable x of receptance function1~xnIt is converted to n and possesses single make an uproar
Sound symbol εiAffine Incentive:
Thus by receptance function f (x)=f (x1,
x2,…,xn) it is converted to affine form；
Step 3: when there are multiple expression items containing same noise member in receptance function expression formula in substitution calculating process,
The bound that continuous item totality value is determined first with interval algorithm or tensor operation is then based on bound information, introduces new
Noise symbol, these continuous items are subjected to overall Affine Incentive and convert union；
Step 4: introducing new noise metasymbol εn+1~εs+1, it is whole that the continuous item in former expression formula is replaced with new Affine Incentive
Body simultaneously carries out subsequent arithmetic, using Affine arithmetic rule, calculates its value of finally returning that
Step 5: it is consistent with conventional affine arithmetic, it calculatesBound fu,flAnd its center and radius fc,fr, together
Reason calculates the bound y in the section allowable of structural responseu,ylAnd its center and radius yc,yr.Power function is substituted into again, is obtained non-
Probabilistic reliability index:
It is different from conventional affine arithmetic in step 3 of the present invention.Conventional affine arithmetic is not accounted for same noise
Correlation between multiple projects of member expression, so that traditional affine arithmetic is inevitably present error.The present invention is to borrow
Interval algorithm or tensor operation is helped to determine that bound improves Affine arithmetic, it is multiple containing same noise member when occurring in operation
Expression item when, the bound of continuous item totality value is determined first with interval algorithm or tensor operation, because of noise symbol value
In [- 1,1], therefore the determination of bound, simple possible.
It is that new noise symbol is introduced, by this based on section bound information in step 3 and step 4 of the present invention
A little continuous items carry out overall Affine Incentive and convert union.Same noise member expression can be effectively treated in obvious improved algorithm
Multiple continuous items, advantageously reduce error, the interval model applied to the explicit power function with complex nonlinear it is non-general
The calculating of rate reliability index, can obtain it is more compact, also closer to the compartmental results of true value.
By taking the tensor computation of polynary quadratic term as an example:
If
Then:
In formula, hc=(hu+hl)/2, hr=(hu-hl)/2, εkFor the noise newly introduced.
This processing mode to secondary continuous item is also suitable for the processing for extending to multiple item and nonlinear terms.
Multiple continuous item bounds of the same noise member expression of tensor form determine that method is as follows:
1. determining ternary interval polynomial bound
If ternary interval polynomial:
Enable X=(1, x ..., xn), Y=(1, y ..., ym), Z=(1, z ..., zl), AijkFor tensor coefficient.Then formula (13)
It is rewritten as tensor product form:
By section [xl,xu]、[yl,yu] and [zl,zu] it is converted to the Affine Incentive of single noise symbol:
ε in formulax,εy,εz∈ [- 1,1] is three mutually independent noises, in which:
Define noise power vector εx=(1, εx,…,εx n),εy=(1, εy,…,εy m)T,εz=(1, εz,…,εz l) and square
Battle array:
New tensor G is re-defined, and is enabledAgain because of X=εxB, Y=C εyAnd Z=εzD, then the affine form (or affine function) of f (x, y, z) can be denoted as:
Then the bound difference of f (x, y, z) is as follows:
2. determining binary interval polynomial bound
If X=(1, x, x2,…,xn), Y=(1, y, y2,…,ym).Binary interval polynomialIt can be denoted as:
If x ∈ [xl,xu], y ∈ [yl,yu], εxAnd εyIt is noise member, εx,εy∈[-1,1].Enable x0=(xu+xl)/2, x1=
(xu-xl)/2；y0=(yu+yl)/2, y1=(yu-yl)/2.X and y are converted into Affine Incentive:
Define noise member power vectorWithAnd matrix B and C are as follows:
Order matrix D=BAC, the then affine function of equal value of f (x, y) are as follows:
If i and j are even numbers,OtherwiseThenUpper bound fuWith lower bound flIt can
It is acquired by following formula:
3. determining unitary interval polynomial bound
Enable the X=(1,1,1 in formula (22)2,…,1n), then formula (22) just becomes unitary interval polynomial:
Correspondingly, the Affine Incentive of x reforms intoThat is x0=1, x1=0；It remains unchanged.
At this point, the B in formula (16), (17) and (18) is deformed are as follows:
C is remained unchanged.Enable D=[Dij]=BAC, then
The then upper bound f of f (y)uWith lower bound flIt can be acquired by following formula
The advantages of present invention incorporates interval arithmetic, tensor operation and Affine arithmetics, could not for conventional Affine arithmetic
Consider the innovatory algorithm of correlation between the first order, quadratic term or even multiple item, nonlinear terms of same noise member expression, this
A little features determine computational accuracy advantage of the invention.Calculation process is shown in Fig. 3.
When being therefore applied to the Multidisciplinary systems index calculating of the explicit power function interval model of complex nonlinear, this hair
The accuracy of bright calculated result is not only better than interval arithmetic method, and also the same Affine arithmetic better than routine is (see specific embodiment party
Formula).
Detailed description of the invention
Fig. 1 is that interval arithmetic method seeks Non-probabilistic Reliability Index calculation flow chart.
Fig. 2 is that conventional affine arithmetic seeks Non-probabilistic Reliability Index calculation flow chart.
Fig. 3 is that the present invention seeks Non-probabilistic Reliability Index calculation flow chart.
Fig. 4 is Tiebar structure of the present invention.
Specific embodiment
The present invention will be described further by following embodiment.Structure work is more accurately sought the present invention relates to a kind of
The new method of Multidisciplinary systems index in journey, can be significantly compared with traditional interval arithmetic method and conventional Affine arithmetic method
Improve computational accuracy.
The pull rod of certain multifunctional sowing and watering, fertilizing mechanic operating mechanism is the pipe cross-section component acted on by Tensile or Compressive Loading,
As shown in Figure 3.Load F ∈ [167.4,172.6] kN, the outside diameter d of pipe cross-section1∈ [34.825,35.175] mm, internal diameter d0=
0.7d1+ 0.5mm, tensile strength values y ∈ [389,411] MPa of material.Examination calculates the Multidisciplinary systems index of this pull rod.
Power function are as follows:
M=y-f (34)
The functional relation of the tensile stress of the pull rod and all of above parameter are as follows:
(1) standard interval algorithm:
By interval variable F, d1And d0Value interval be directly substituted into formula (35), obtain the tensile stress f of pull rod section expression
Formula:
Therefore fc=360.84105640551854, fr=9.34805723440935.yc=400, yr=11.Again by them
Substitution formula (4) obtains the standard section algorithm values of Multidisciplinary systems index:
(2) improvement affine arithmetic of the invention:
1. the affine method of improvement that interval arithmetic is delimited
Because load and pull rod cross-sectional outer diameter are mutually indepedent, therefore can be according to by F and d1It transforms into respectively and possesses independent noise symbol
Number Affine Incentive, i.e.,WithThe two is substituted into formula (35), is obtained:
Because of ε2∈ [- 1,1] itself is perfectly correlated, therefore the denominator part in above formulaItem meets:
That is:
Therefore new noise symbol ε can be introduced3The Affine Incentive 1885.004659904134+ of ∈ [- 1,1]
19.242255003237460ε3Instead, it obtains:
If the bound of denominator part is [a, b], then a=1865.762404900897, b=
1904.246914907372。
If
α=- 1/b2=-2.7577410174875 × 10-7,
ξ=(a+b-a2bα-ab2α)/(2ab)=0.001050393420806,
δ=(a-b+a2bα-ab2α)/(2ab)=- 1.094558727055978 × 10-7,
Affine according to derivative action approaches, then
Because of ε1,ε3,ε4∈ [- 1,1], and it is mutually indepedent, but ε1ε3With ε1、ε3Correlation, ε1ε4With ε1、ε4Correlation, ε1ε3、ε1ε4
And ε1Correlation then has:
That is:
0.351635071459507≤f≤0.370036398089323 (44)
F ∈ [f can be obtainedl,fu]=[0.351635071459507,0.370036398089323], unit GPa, fc=
0.360835734774415, fr=0.009200663314908, then formula (11) are substituted them in, obtain pull rod Multidisciplinary systems
The improvement affine arithmetic value η of this paper of index:
2. the affine method of improvement that tensor operation is delimited
The calculating process that power function is converted into Affine Incentive is delimited with interval arithmetic proposed in this paper and improves affine method phase
Together.Therefore by substituting into formula (35) after converting Affine Incentive for interval variable, obtain and the result of formula (38), it may be assumed that
The denominator of formula (46)In, accorded with containing same noise
Number ε2.Because of ε2∈ [- 1,1], to avoid operation to generate interval extension as far as possible, it is contemplated that ε2WithCorrelation, therefore willWith 19.242255003237482 ε2Item carry out new Affine Incentive replacement together, transported by tensor
It calculates and bound is determined to them, tensor operation can carry out operation with multinomial, therefore together by constant term 1884.955592153876
It is placed in a multinomial and considers, is i.e. tensor computation:
By ε2It is converted into Affine Incentive:Because formula (47) is unitary interval polynomial, tensor operation
Method obtains matrix D.
The diagonal entry A of A00=1884.955592153876, A11=19.242255003237482, A22=
0.049067750258256, remaining element is equal to 0, then the first row element of D is as follows:
D00=A00·1·1·1+A11·1·y0·1+A22·1·y0 2·1
D01=A11·1·1·y1+A22·2·y0·y1
D02=A22·1·1·y1 2
And the other elements of D are 0.
Work as ε2When [- 1,1] ∈,Then obtain D00=1884.955592153876, D01=
19.242255003237482 and D02=0.049067750258256.Delimiting method according to tensor operation can obtainBound U and L be respectively as follows:
Therefore new noise symbol ε can be introduced3Affine Incentive 1884.9804554138900+19.2667888783667 ε3Instead ofTherefore
If the bound of denominator part is [a, b], then a=1865.7136665355200, b=
1904.2472442922600。
If
α=- 1/b2=-2.757740 × 10-7,
ξ=(a+b-a2bα-ab2α)/(2ab)=0.0010503935213,
δ=(a-b+a2bα-ab2α)/(2ab)=- 1.097380 × 10-7,
Affine according to derivative action approaches, then
Then,
Because of ε1,ε3,ε4∈ [- 1,1], and it is mutually indepedent, but ε1ε3With ε1、ε3Correlation, ε1ε4With ε1、ε4Correlation, ε1ε3、ε1ε4
And ε1Correlation then has:
That is:
0.3516327280856≤f≤0.3700460646151 (54)
F ∈ [f can be obtainedl,fu]=[0.3516327280856,0.3700460646151], unit GPa, fc=
0.3608393963503, fr=0.0092066682647 substitutes them in formula (11) again, obtains pull rod Multidisciplinary systems index
This paper improvement affine arithmetic value η:
(3) conventional affine arithmetic
The process front half section of conventional affine arithmetic is identical as this paper algorithm.Therefore by converting Affine Incentive for interval variable after
Substitution formula (35) obtains and the result of formula (38).
Because of ε2∈ [- 1,1], therefore in above formula denominator partTherefore new noise symbol ε can be introduced3The Affine Incentive of ∈ [- 1,1]
0.024533875129128+0.024533875129128ε3Instead.Therefore denominator part is writeable are as follows:
19.242255003237482ε2+0.024533875129128ε3+1884.980126029005
Therefore
If setting the bound of denominator part as [a, b], then a=1865.713337150638, b=
1904.246914907372.If: α=- 1/b2=-2.757812079050777 × 10-7, ξ=(a+b-a2bα-ab2α)/
(2ab)=0.001050407239042, δ=(a-b+a2bα-ab2α)/(2ab)=- 1.097423199501504 × 10-7.Root
Affine according to derivative action approaches, then:
Therefore
Because of ε1,ε2,ε3,ε4∈ [- 1,1] contains quadratic term 5.518776212703734 × 10 in formula-5ε1ε2,-
5.518776212703734×10-5≤-5.518776212703734×10-5ε1ε2≤5.518776212703734×10-5,
Therefore new noise symbol ε can be introduced5The Affine Incentive 5.518776212703734 × 10 of ∈ [- 1,1]-5ε5Instead.Similarly quadratic term
7.036439671197323×10-8ε1ε3Introduce new noise symbol ε6The Affine Incentive 7.036439671197323 of ∈ [- 1,1] ×
10-8ε6Substitution, 1.141275710847393 × 10-6ε1ε4Introduce new noise symbol ε7The Affine Incentive of ∈ [- 1,1]
1.141275710847393×10-6ε7Substitution, obtains:
F ∈ [f can be obtainedl,fu]=[0.351522272655038,0.370046129945341], unit GPa, fc=
0.360784201300189, fr=0.009261928645151 substitutes them in formula (12) again, obtains Multidisciplinary systems index
Standard section algorithm values η:
Through calculate pull rod tensile stress f really respond section be [0.351635071459507,
0.370036398089322]MPa.That is: fc=360.835734774415, fr=9.200663314908.Substitution formula (4), meter
Calculate the true value of the Multidisciplinary systems index of pull rod are as follows: ηTRUE=3.060155397972450.
The solving result of table 1 the method for the present invention and other methods
As it can be seen from table 1 there are large errors for the resultant error of interval arithmetic, and its reliability index calculated is less than
True value ηTRUE, i.e. the reliability index value result of interval arithmetic is relatively conservative, and reason is calculatingWhen, standard interval arithmetic can not embodyWith d1Correlation.
And conventional Affine arithmetic, error is smaller for opposite interval arithmetic, and the reliability index value calculated is slightly larger than
ηTRUE, as a result partially optimistic.Method of the invention improves conventional Affine arithmetic, can effectively deal with same noise member table
Correlation between the first order, quadratic term or even the multiple item that reach, tensor operation, which is delimited, improves affine method reliability index value calculating
Error only 0.039046%, and the affine method reliability index value of improvement that interval arithmetic is delimited calculates error and levels off to 0.Upper table foot
To prove advantage place of the invention.
Claims (1)
1. a kind of multifunctional sowing and watering, fertilizing mechanic operating mechanism pull rod Multidisciplinary systems appraisal procedure, the pull rod be by
The pipe cross-section component of Tensile or Compressive Loading effect, load F ∈ [Fl,Fu] kN, the outer diameter of pipe cross-sectionInternal diameter d0
=δ1d1+δ2Mm, δ1With δ2For internal diameter and outer diameter coefficient of relationship, the tensile strength values y ∈ [y of materiall,yu]MPa；
Then load F, outside diameter d1, internal diameter d0For the related section input of the tensile stress f that is born with pull rod, according to Tiebar structure with by
Situation is carried, the functional relation of tensile stress f and all of above parameter that pull rod is born are as follows:
It is characterized in that according to the following steps:
Step 1: establishing the power function of Tiebar structure: M=y-f；
Wherein, M is the power function of Tiebar structure, and y is the limiting value that Tiebar structure can bear tensile stress, and f is what pull rod was born
Practical tensile stress interval value, unit are megapascal；
Step 2: by load F, outside diameter d1Section input be converted to two and possess single noise symbol ε1With ε2Affine Incentive, ε1∈
[- 1,1], ε2∈ [- 1,1]:
To which tensile stress f is converted to affine form:
Step 3: determining the bound of continuous item totality value using interval algorithm or tensor operation, be then based on bound letter
Breath introduces the Affine Incentive of new noise metasymbol and determining continuous item totality:
ε contained by tensile stress f expression formula denominator part is determined first with interval algorithm or tensor operation2Quadratic term and first order it
WithBound, be set as [kl,
ku], it is based on the bound information, introduces new noise metasymbol ε3,ε3∈ [- 1,1], to contain ε3Expression formula to the part
It is replaced, i.e.,
Then
Step 4: successively introducing new noise metasymbol ε4~εs+1, ε4~εs+1Equal ∈ [- 1,1] is drawn so that new Affine Incentive replacement is former
Continuous item entirety in stress f expression formula simultaneously carries out subsequent arithmetic, using Affine arithmetic rule, calculates finally returning for tensile stress f
Return value f=f0+f1ε1+f2ε2+…+fs+1εs+1；
Step 5: routinely affine method calculates the bound f of tensile stress fu,fl, fu=f0+f1|+|f2|+…+|fs+1|, fl
=f0-|f1|-|f2|-…-|fs+1| and its center and radius fc,fr, similarly calculate the allowable of the tensile strength y of rod material
The bound y in sectionu,ylAnd its center and radius yc,yr；Power function is substituted into again, obtains Multidisciplinary systems index η are as follows:
Tensile stress f is interval variable, f in formulau、flFor the bound of its value interval；yc=(yu+yl)/2, yr=(yu-yl)/2,
fc=(fu+fl)/2, fr=(fu-fl)/2, Mc=(Mu+Ml)/2=yc-fc, Mr=(Mu-Ml)/2=yr+fr, respectively pull rod knot
Structure can bear power function M three's value interval of the limit y of tensile stress, the practical tensile stress f that pull rod is born, Tiebar structure
Center and radius.
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| 结构系统非概率可靠性算法研究;江涛;《中国优秀硕博士学位论文全文数据库（博士） 工程科技Ⅱ辑》;20070515(第5期);正文第65-88页 |
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