CN106919786B - A kind of concrete fractional order creep model - Google Patents
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Abstract
A kind of concrete fractional order creep model, in the case where considering hydrated reaction of cement.A kind of expression formula of concrete fractional order creep model concrete fractional order creep model provided by the invention isIn formula: C (te, τe) it be loading age is τe, hold the lotus time be te‑τeCreep degree, f1、g1、p1、r1, β be coefficient, and 0 < β≤1.A kind of concrete fractional order creep model provided by the invention, it sufficiently reacted concrete creep and load age, held the relevant characteristic of lotus time, hydrated reaction of cement, and the new model established also overcomes the more disadvantage of common eight parameters creep model parameter, to known to the fitting effect analysis of this model, fractional model meets concrete creep rule, fitting precision with higher.
Description
Technical field
The present invention relates to prediction concrete creep field, especially a kind of concrete fractional order creep models.
Background technique
Concrete is a kind of Xu's variant material, i.e., under normal stress, with the extension of time, strain will constantly increase
Add.Concrete creep is not only related with the lotus time is held, but also related with load age etc., and load is more early, creeps bigger.Not only such as
This, temperature also has large effect to concrete creep performance, and temperature influences relationship complexity to concrete creep, not simple
Single higher bigger monotonic relationshi of creeping of temperature.
Concrete creep has a significant impact the stress and deformation performance of concrete, how to select one can accurately predict Xu
The prediction model of change has been a hot spot of research.Creep prediction model expression is to reflect Creep Coefficient or time deformation at any time
The mathematical functional expression of changing rule, the validity and accuracy of creep prediction are heavily dependent on the choosing of creep prediction model
It takes.For the mathematical function of model construction, the prediction model of domestic and international concrete creep can be roughly divided into three categories: (1) multiplying
Product module type the product for being expressed as several subitems of creeping is described, such as to concrete creep without subdivision from whole
ACI209R series model etc.；(2) and formula model it, i.e., will creep and be expressed as the sum of several subitems, such as CEB-FIP series model, B-
P series model etc.；(3) mixed model is mixed by product model and with formula model, such as GL-2000 model.
These above-mentioned models are each defective, if CEB-FIP (1978) model accuracy is lower, creep calculated value and reality of creeping
It is not high to be worth degree of agreement；ACI209 model-test data to meet situation poor, usually underestimate contraction and creep beharior；Eight ginsengs
Though exponential model can meet rule of creeping very well, it needs to be determined that parameter it is more.Provide separately below each model expression formula and
Major defect, see Table 1 for details, table 2.
1 existing concrete creep model of table
The major defect of existing creep model is listed in detail for these above-mentioned model limitations and applicability, the following table 2
And the scope of application.
The existing model scope of application of table 2 and major defect
For defect existing for above-mentioned common creep model, established below using fractional calculus theory a kind of new
Concrete creep model.
Summary of the invention
Technical problem to be solved by the invention is to provide a kind of concrete fractional order creep models, by fractional calculus
Rheological model, the creep property of Cemented and aquation topology degree combine, and establish a kind of concrete fractional order for considering degree of hydration
Creep model.Compared with traditional creep model, new model established by the present invention has sufficiently reacted concrete creep and load age
Phase holds lotus time, the relevant characteristic of hydrated reaction of cement, and the new model established also overcomes common eight parameter and creeps
The more disadvantage of model parameter, the fitting effect analysis to this model is it is found that fractional model meets concrete creep rule, tool
There is higher fitting precision.
In order to solve the above technical problems, the technical scheme adopted by the invention is that:
A kind of concrete fractional order creep model,
In the case that stress is constant, according to Riemann-Liouville type fractional calculus operator theory, base is established
It is in the creep model expression formula of fractional calculus
In formula: C (t, τ) is loading age to be τ, hold the creep degree that the lotus time is t- τ, f1、g1、p1、r1, β be coefficient, and 0
<β≤1。
Concrete creep degree is fitted using the concrete creep model of fractional calculus, unknown ginseng in formula (1)
Number has 5, respectively f1、g1、p1、r1,β；
Each undetermined parameter is denoted as X, i.e. X=[x1,x2,x3,x4,x5]T, and Prescribed Properties: 0 < x5≤ 1, xi≥0(i
=1~4), then have
It is crept with experiment and value and calculates the residual sum of squares (RSS) of value of creeping, as the objective function of parametric inversion optimization problem,
It is crept 5 parameters of expression formula with seeking the fractional calculus of concrete, i.e.,
In formula: C (t, τ) is creep degree calculated value, and C ' (t, τ) is creep degree experiment value, and F (X) is objective function, xiFor to
Determine parameter,
To acquire each undetermined parameter.
In the case where considering hydrated reaction of cement, convolution (1) obtains considering the concrete score of hydrated reaction of cement
The expression formula of rank creep model is
In formula: C (te,τe) it be loading age is τe, hold the lotus time be te-τeCreep degree, f1、g1、p1、r1, β be coefficient,
And 0 < β≤1
In the case where considering hydrated reaction of cement, convolution (1) obtains considering the concrete score of hydrated reaction of cement
Rank creep model the step of it is as follows:
Step 1: degree of hydration is described using aquation thermal method, that is, it is the heat that has discharged of certain moment and final for defining degree of hydration
The ratio of the heat discharged completely, hydrated cementitious can be accelerated as the temperature rises,
The solution reaction rate constant β proposed based on Arrhenius equationTRelational expression with temperature is
UhFor aquation activity energy；R is gas constant, is 80315J/K；T is concrete actual temperature；T0For reference temperature；
Step 2: the concept based on equivalent age proposes the equivalent age of Arrhenius functional form, and expression formula is
U in formulahFor aquation activity energy；R is gas constant, is 80315J/K；T is concrete actual temperature；T0For with reference to temperature
Degree, usually takes 293K, i.e., 20 DEG C；
Wherein, as follows about the computation model of activation energy:
Step 3: by carrying out regression analysis to Adiabatic temperature rise of concrete data under existing different temperatures, it is believed thatWith
Concrete actual temperature T meets following relationship
Using the principle of above-mentioned equivalent age, by t, τ is respectively for te,τe, convolution (1) may be accounted cement water
The expression formula of concrete fractional order creep model for changing reaction is
In formula: C (te,τe) it be loading age is τe, hold the lotus time be te-τeCreep degree, f1、g1、p1、r1, β be coefficient,
And 0 < β≤1.
Concrete creep degree is fitted using the concrete creep model of fractional calculus, unknown ginseng in formula (1)
Number has 5, respectively f1、g1、p1、r1,β；
Each undetermined parameter is denoted as X, i.e. X=[x1,x2,x3,x4,x5]T, and Prescribed Properties: 0 < x5≤ 1, xi≥0(i
=1~4), then have
It is crept with experiment and value and calculates the residual sum of squares (RSS) of value of creeping, as the objective function of parametric inversion optimization problem,
It is crept 5 parameters of expression formula with seeking the fractional calculus of concrete, i.e.,
In formula: C (t, τ) is creep degree calculated value, and C ' (t, τ) is creep degree experiment value, and F (X) is objective function, xiFor to
Determine parameter；
Similarly, with the theory of above-mentioned equivalent age, by t, τ is respectively for te,τe, convolution (3) is available, considers
The objective function of concrete creep model parameter inverting in the case where hydrated reaction of cement, i.e.,
xiFor 5 parameters of fractional model,
To acquire each undetermined parameter.
The method for acquiring each undetermined parameter are as follows:
Step 1: determining that variable number is 5, apexes of complex number k takes 6, and precision value is 1e-6；
Step 2: intial compound form is generated, in restriction rangeMiddle k random point of generation is constituted just
Beginning complex；
Step 3: convergence is carried out to formula (1) and (9), it is true in conjunction with specific example according to the convergence of infinite series
Determine the value of n；
Step 4: being iterated calculating by the MATLAB program of complex method, utilize each vertex function value size of complex
Relationship judges the descent direction of target function value, constantly loses worst point, shrinks complex constantly to optimum point, Zhi Daoman
Until sufficient convergence precision,
To obtain each undetermined parameter.
A kind of concrete fractional order creep model provided by the invention, by fractional calculus rheological model, concrete Xu
Become characteristic and aquation topology degree combines, establishes a kind of concrete fractional order creep model for considering degree of hydration.With traditional Xu
Varying model is compared, and new model established by the present invention has sufficiently reacted concrete creep and load age, held lotus time, cement water
Change and react relevant characteristic, and the new model established also overcomes that common eight parameters creep model parameter is more to be lacked
Point, the fitting effect analysis to this model is it is found that fractional model meets concrete creep rule, fitting essence with higher
Degree.
Specific embodiment
Embodiment one (the fractional order concrete creep model for not considering hydrated reaction of cement)
A kind of concrete fractional order creep model,
In the case that stress is constant, according to Riemann-Liouville type fractional calculus operator theory, base is established
It is in the creep model expression formula of fractional calculus
In formula: C (t, τ) is loading age to be τ, hold the creep degree that the lotus time is t- τ, f1、g1、p1、r1, β be coefficient, and 0
<β≤1。
Concrete creep degree is fitted using the concrete creep model of fractional calculus, unknown ginseng in formula (1)
Number has 5, respectively f1、g1、p1、r1,β；
Each undetermined parameter is denoted as X, i.e. X=[x1,x2,x3,x4,x5]T, and Prescribed Properties: 0 < x5≤ 1, xi≥0(i
=1~4), then have
It is crept with experiment and value and calculates the residual sum of squares (RSS) of value of creeping, as the objective function of parametric inversion optimization problem,
It is crept 5 parameters of expression formula with seeking the fractional calculus of concrete, i.e.,
In formula: C (t, τ) is creep degree calculated value, and C ' (t, τ) is creep degree experiment value, and F (X) is objective function, xiFor to
Determine parameter,
To acquire each undetermined parameter.
In order to verify the practicability and accuracy of the above method, divided using Gong's mouth Gravity Dam Foundation part concrete
Analysis.For Gong's mouth gravity dam concrete creep degree, its specific value is given below, see the table below 3.
3 Gong's mouth gravity dam concrete creep degree experiment value of table
The complex method principle and calculating step introduced according to above-mentioned Section 3, write calculating journey using MATLAB language
Sequence carries out parametric inversion to used fractional order expression formula of creeping in conjunction with the concrete creep experiment value of Gong's mouth gravity dam.?
When calculating, the present patent application to 180d before Gong's mouth gravity dam concrete load time, hold the lotus time before 720d be fitted.Pass through
Convergence is carried out to fractional model expression formula, when n value is greater than 40, this model expression is substantially at convergence, is
This this patent takes n=40 when writing MATLAB program, finds by sensitivity analysis, f1、r1, β compared with other two parameters sensitivity,
Initial composite type is debugged, obtaining the corresponding gentle F of residual error (X) is 4.87.It is as shown in table 4 below to join inversion result, score
Rank fitting formula such as formula (11), fractional model creep degree calculated value the results are shown in Table 5.
4 fractional model parametric inversion result of table
5 Gong's mouth gravity dam concrete creep degree fractional model calculated value of table
Inverting is carried out to eight parameter models, is obtained shown in creep degree expression formula such as following formula (12), meanwhile, eight parameter models Xu
The calculated value of variation is as shown in table 6 below.
C (t, τ)=(7+64.4 τ-0.45)[1-e-0.3(t-τ)]+(16+27.2τ-0.45)[1-e-0.005(t-τ)]
(12)
6 Gong's mouth gravity dam concrete creep degree fractional order of table, eight parameter model calculated value
By comparing concrete creep degree experiment value, fractional model calculated value, eight parameter models: fractional order mould
Type, eight parameter models can preferably fit concrete creep rule, and fractional model fitting precision is in general with eight
Parameter model fitting precision is not much different, but the parameter determined needed for fractional model is less than the ginseng determined needed for eight parameter models
Number.
Precision analysis is carried out to the fractional model that embodiment one determines below, ifWith f/f0Value as precision judge standard, by fractional model calculated value and creep degree experiment value
Bring f and f into0In expression formula, f/f can be obtained0=8.8% < 10%, value is less than the upper limit value of defined in engineering, illustrates above formula
(11) accuracy being fitted is high, can be engineering services.
Embodiment two (the fractional order concrete creep model for considering hydrated reaction of cement)
In the case where considering hydrated reaction of cement, convolution (1) obtains considering the concrete score of hydrated reaction of cement
The expression formula of rank creep model is
In formula: C (te,τe) it be loading age is τe, hold the lotus time be te-τeCreep degree, f1、g1、p1、r1, β be coefficient,
And 0 < β≤1
In the case where considering hydrated reaction of cement, convolution (1) obtains considering the concrete score of hydrated reaction of cement
Rank creep model the step of it is as follows:
Step 1: degree of hydration is described using aquation thermal method, that is, it is the heat that has discharged of certain moment and final for defining degree of hydration
The ratio of the heat discharged completely, hydrated cementitious can be accelerated as the temperature rises,
The solution reaction rate constant β proposed based on Arrhenius equationTRelational expression with temperature is
UhFor aquation activity energy；R is gas constant, is 80315J/K；T is concrete actual temperature；T0For reference temperature；
Step 2: the concept based on equivalent age proposes the equivalent age of Arrhenius functional form, and expression formula is
τe=∫ βTdt (6)
U in formulahFor aquation activity energy；R is gas constant, is 80315J/K；T is concrete actual temperature；T0For with reference to temperature
Degree, usually takes 293K, i.e., 20 DEG C；
Wherein, as follows about the computation model of activation energy:
Step 3: by carrying out regression analysis to Adiabatic temperature rise of concrete data under existing different temperatures, it is believed thatWith
Concrete actual temperature T meets following relationship
Using the principle of above-mentioned equivalent age, by t, τ is respectively for te,τe, convolution (1) may be accounted cement water
The expression formula of concrete fractional order creep model for changing reaction is
In formula: C (te,τe) it be loading age is τe, hold the lotus time be te-τeCreep degree, f1、g1、p1、r1, β be coefficient,
And 0 < β≤1.
Concrete creep degree is fitted using the concrete creep model of fractional calculus, unknown ginseng in formula (1)
Number has 5, respectively f1、g1、p1、r1,β；
Each undetermined parameter is denoted as X, i.e. X=[x1,x2,x3,x4,x5]T, and Prescribed Properties: 0 < x5≤ 1, xi≥0(i
=1~4), then have
It is crept with experiment and value and calculates the residual sum of squares (RSS) of value of creeping, as the objective function of parametric inversion optimization problem,
It is crept 5 parameters of expression formula with seeking the fractional calculus of concrete, i.e.,
In formula: C (t, τ) is creep degree calculated value, and C ' (t, τ) is creep degree experiment value, and F (X) is objective function, xiFor to
Determine parameter；
Similarly, with the theory of above-mentioned equivalent age, by t, τ is respectively for te,τe, convolution (3) may be accounted water
The objective function of concrete creep model parameter inverting in the case where mud hydration reaction, i.e.,
xiFor 5 parameters of fractional model,
To acquire each undetermined parameter.
The method for acquiring each undetermined parameter are as follows:
Step 1: determining that variable number is 5, apexes of complex number k takes 6, and precision value is 1e-6；
Step 2: intial compound form is generated, in restriction rangeMiddle k random point of generation is constituted just
Beginning complex；
Step 3: convergence is carried out to formula (1) and (9), it is true in conjunction with specific example according to the convergence of infinite series
Determine the value of n；
Step 4: being iterated calculating by the MATLAB program of complex method, utilize each vertex function value size of complex
Relationship judges the descent direction of target function value, constantly loses worst point, shrinks complex constantly to optimum point, Zhi Daoman
Until sufficient convergence precision,
To obtain each undetermined parameter.
The content of the test of the practicability and accuracy of verifying the above method is as follows:
1, experiment of creeping is compressed
This experiment compresses rule of creeping to study concrete, and comparison has carried out under more age natural curings and standard curing
Lower water conservancy project concrete in uniaxial compresses creep test, is carried out below using the fractional order creep model of consideration degree of hydration to test data
Analysis.
(1) concrete creep test parameters and instrument and equipment
It is 150mm × 150mm × 550mm concrete prism test specimen, stress ratio that concrete in uniaxial, which compresses test specimen of creeping,
For 0.3 (Compression be failing load 30%)；Loading device is produced using Shanghai Hua Jing Trade Co., Ltd.
XBJ-500 type Creep Apparatus；Load transducer model BLR-2 type；Since LVDT and outer patch foil gauge are by outside environmental elements shadow
Sound is larger, buries S-100 type difference resistive strain gauge using interior, strain gauge uses the digital bridge measurement of SO-5.
(2) concrete raw material and match ratio
The main physico-mechanical performance for the raw material that the test of this concrete creep is selected is as shown in table 7, the compounding of C30 second level
The match ratio for coagulating soil is as shown in table 8.Wherein, the ratio of mud 0.5, sand coarse aggregate ratio 35%, doping quantity of fly ash 35%.
The basic physical property of 7 raw material of table
Table 8 compresses creep test concrete mix Kg/m3
(3) concrete in uniaxial compression test scheme
2 groups of comparative tests of creeping are devised, as shown in table 9, natural curing is (i.e. outdoor to support in winter for battery of tests test specimen
Shield), another battery of tests test specimen is conserved (i.e. standard curing) in standard curing room, respectively maintenance 3d, 7d, 14d, 21d,
After 28d age, compression creep test is carried out in laboratory of creeping, and when load is 2 months a length of.3 test specimens of creeping in table 3, wherein 2
A be used as loads test specimen of creeping, and in addition 1 is used as compensation test specimen；3 standard cube test specimens measure concrete crushing strength.Its
In, the temperature range during field curing is 6~12 DEG C, and mean temperature is 9.2 DEG C, and relative humidity is 80% or so；Standard is supported
Protecting temperature is 20 ± 2 DEG C, 95% or more relative humidity；When conserving load age, it is further applied load and starts creep test, reality of creeping
The temperature of room is tested between 11 DEG C~12 DEG C, relative humidity is basically stable at 90% or more.
9 concrete in uniaxial of table compresses experimental program table of creeping
2, parametric inversion calculates
By table 4 as it can be seen that since the creep test time is long, the equipment that is put to the test and place limitation under the conditions of standard curing, add
When load age is 21d, only holding lotus 1 month terminates to test；And the concrete sample test result that load age is 28d does not conform to
Reason, therefore do not use related data.
Due to considering that the fractional order creep model of degree of hydration is an infinite series expression formula, by fractional model table
Convergence is carried out up to formula, when n value is greater than 50, this model expression is substantially at convergence, and this patent is being write thus
When MATLAB program, n=50 is taken.It is found by sensitivity analysis, f1、r1, β compared with other two parameters sensitivity.To initial composite type into
Row debugging calculates and considers that the gentle F of fractional model residual error (X) that temperature influences is 1.81 under the standard curing of acquisition, and calculating obtains
Consider that the fractional order creep model parameter of degree of hydration is shown in Table 10, considers the score of degree of hydration under standard curing under the standard curing obtained
The creep degree calculated value of rank creep model is shown in Table 12；Calculate the fractional order creep model that degree of hydration is considered under the field curing obtained
Parameter is shown in Table 11, and corresponding residual sum of squares (RSS) F (X) is 6.46, and the fractional order creep model of degree of hydration is considered under field curing
Creep degree calculated value be shown in Table 13.
Fractional model parametric fitting results under 10 standard curing operating condition of table
The fractional order that the temperature of the considerations of according to 10 parametric inversion of table as a result, under the conditions of available standard curing influences is crept
Model expression is
Fractional model parametric fitting results under natural curing operating condition outside table Room 11
According to 11 parametric inversion of table as a result, the fractional order that the consideration temperature under available outdoor natural curing maintenance influences
Creep model expression formula is
The fractional order creep degree calculated value of typical time under 12 standard curing of table each age
Outside table Room 13 under natural curing each age typical time fractional order creep degree calculated value
Similarly, parametric inversion is carried out to eight parameter models under the conditions of standard curing, obtains considering what temperature on creep influenced
Eight parameter expressions as shown in following formula 15, meanwhile, consider temperature influence eight parameter model creep degrees calculated value such as the following table 14
It is shown.
C(te,τe)=(22.65+15 τe -300)[1-e-0.0405(t-τ)]+(3.19+92.54τe -1.42)[1-e-0.6(t-τ)] (15)
Parametric inversion is carried out to eight parameter models under the conditions of outdoor natural curing, obtains considering temperature on creep influences eight
Parameter expression as shown in following formula 16, meanwhile, consider temperature influence eight parameter model creep degrees calculated value such as the following table 15 institute
Show.
C(te,τe)=(15.38+74.18 τe -1.06)[1-e-0.06(t-τ)]+(11.18+186.77τe -1.9)[1-e-1.83(t-τ)] (16)
The equivalent load age of typical time and eight parameter creep degree calculated values under 14 standard curing of table each age
The equivalent load age of typical time and eight parameter creep degree calculated values under natural curing each age outside table Room 15
Creep degree experiment value under the conditions of comparative analysis standard curing and under the conditions of outdoor natural curing, fractional order mould respectively
Known to type calculated value, eight parameter model calculated values: under the conditions of standard curing, when loading age is 3d, 7d, 14d, fractional order is crept
Model and 8 parameter creep model fitting effects are all fine, relatively coincide with experiment value；But loading age be 21d when, fractional order Xu
Varying model is better than 8 parameter creep model fitting effects.
Claims (3)
1. a kind of concrete fractional order creep model, it is characterised in that:
In the case that stress is constant, according to Riemann-Liouville type fractional calculus operator theory, establishes and be based on dividing
Number rank calculus creep model expression formula be
In formula: C (t, τ) is loading age to be τ, hold the creep degree that the lotus time is t- τ, f1、g1、p1、r1, β be coefficient, and 0 < β≤
1；
Concrete creep degree is fitted using the concrete creep model of fractional calculus, unknown parameter has 5 in formula (1)
It is a, respectively f1、g1、p1、r1,β；
Each undetermined parameter is denoted as X, i.e. X=[x1,x2,x3,x4,x5]T, and Prescribed Properties: 0 < x5≤ 1, xi>=0, i=1
~4, then have
It is crept with experiment and value and calculates the residual sum of squares (RSS) of value of creeping, as the objective function of parametric inversion optimization problem, to seek
The fractional calculus of concrete is asked to creep 5 parameters of expression formula, i.e.,
In formula: C (t, τ) is creep degree calculated value, and C ' (t, τ) is creep degree experiment value, and F (X) is objective function, xiIt is to be determined
Parameter,
To acquire each undetermined parameter；
In the case where considering hydrated reaction of cement, convolution (1) obtains considering concrete fractional order Xu of hydrated reaction of cement
The expression formula of varying model is
In formula: C (te,τe) it be loading age is τe, hold the lotus time be te-τeCreep degree, f1、g1、p1、r1, β be coefficient, and 0 <
β≤1；
In the case where considering hydrated reaction of cement, convolution (1) obtains considering concrete fractional order Xu of hydrated reaction of cement
Varying model the step of it is as follows:
Step 1: degree of hydration is described using aquation thermal method, that is, define degree of hydration be the heat that has discharged of certain moment and it is final completely
The ratio of the heat of release, hydrated cementitious can be accelerated as the temperature rises,
The solution reaction rate constant β proposed based on Arrhenius equationTRelational expression with temperature is
UhFor aquation activity energy；R is gas constant, is 80315J/K；T is concrete actual temperature；T0For reference temperature；
Step 2: the concept based on equivalent age proposes the equivalent age of Arrhenius functional form, and expression formula is
τe=∫ βTdt (6)
U in formulahFor aquation activity energy；R is gas constant, is 80315J/K；T is concrete actual temperature；T0For reference temperature,
Take 293K, i.e., 20 DEG C；
Wherein, the computation model about aquation activity energy is as follows:
Step 3: by carrying out regression analysis to Adiabatic temperature rise of concrete data under existing different temperatures, it is believed thatWith coagulation
Native actual temperature T meets following relationship
Using the principle of above-mentioned equivalent age, by t, τ is respectively for te,τe, it is anti-to may be accounted hydrated cementitious convolution (1)
The expression formula for the concrete fractional order creep model answered is
In formula: C (te,τe) it be loading age is τe, hold the lotus time be te-τeCreep degree, f1、g1、p1、r1, β be coefficient, and 0 <
β≤1。
2. a kind of concrete fractional order creep model according to claim 1, it is characterised in that:
With the theory of above-mentioned equivalent age, by t, τ is respectively for te,τe, it is anti-that convolution (3) may be accounted hydrated cementitious
The objective function of concrete creep model parameter inverting in the case where answering, i.e.,
xiFor 5 parameters of fractional model, to acquire each undetermined parameter.
3. a kind of concrete fractional order creep model according to claim 2, it is characterised in that acquire each undetermined parameter
Method are as follows:
Step 1: determining that variable number is 5, apexes of complex number k takes 6, and precision value is 1e-6；
Step 2: intial compound form is generated, in restriction rangeMiddle k random point of generation is constituted initial multiple
Close shape；
Step 3: convergence being carried out to formula (1) and (9) and determines n's in conjunction with specific example according to the convergence of infinite series
Value；
Step 4: being iterated calculating by the MATLAB program of complex method, utilize the pass of each vertex function value size of complex
System, judges the descent direction of target function value, constantly loses worst point, shrinks complex constantly to optimum point, until meeting
Until convergence precision, to obtain each undetermined parameter.
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| CN110516405B (en) * | 2019-09-11 | 2023-04-18 | 新疆农业大学 | Construction method of hydration heat presumption-free prediction model of portland cement-based cementing material system |
| CN111261234B (en) * | 2020-03-24 | 2023-04-07 | 西安理工大学 | Concrete creep prediction method considering damage influence |
| CN111625887B (en) * | 2020-05-18 | 2022-08-30 | 中国石油大学(华东) | Steam-cured concrete damp-heat elastic response prediction method, system and storage medium |
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