Research On Universal Models Of Layered Structures Existing In Natural And Artificial Environments

Layered structure substances exist widely in our natural and artificial environments.It is challenging to establish a universally unified mathematical model of layered structures,especially when the model requires uniform description on continuous parameters for any adjacent interlayer interface of any multilayer of different substances.In addition,the continuity of specific parameters in each interface is met in advance,and the characteristics of each layer’s recursive influence are reflected.The ubiquitous layered structure substances from different fields were first listed in both natural and artificial environments,such as geography,biology,environment,materials,chemistry and physics,etc.,having illustrated illustrating the extensive existence of layered structure substances;then an objective characteristic has been demonstrated that specific parameters on the interface between layers are continuous and each layer is recursively affected in turn.Next the Legendre polynomial and its derivative recursion were introduced,having successfully created two core functions L1(ξk)and L2(ξk)with excellent properties required for the universal model in this paper.The unified expression of local coordinates of each layer of any multilayer structure has been skillfully constructed,realizing the dimensionless local coordinatesξk and the global coordinates Zk.Moreover,a parametric universal model function fk(ξk)of layered structures widely existing in natural and artificial environments has been successfully constructed,which can ensure that the specific parameter is continuous between any adjacent layers in advance.Finally,the water quality problem of multi-reach rivers in environmental problems was selected as a general problem of multilayer layered structure,and the model was applied to multi-reach river problems to verify the accuracy and applicability of the universal model.The application method of the parametric universal model was exemplified,and the parameter’s continuous boundary condition was extended to a parametric equilibrium equation of the boundary,that is,mutations are allowed on the both sides of the boundary,further verifying the universality of the parametric universal model.The main conclusions were drawn as follows:(1)The dimensionless local coordinate ξk in the successfully constructed universal model realizes the correlation with the global coordinate Zk:ξk=Zk-Σi=1k-1hi/hk,and ξk∈[0,1]The uniform expressions of the local coordinates of each layer of any multilayer structure can be obtained,which can not only describe the relationship between the local and the whole,but also meet the consistency requirements for the domains of the independent variables by the created core functions L1(ξk)and L2(ξk).Its application is more universal and convenient.(2)The two core functions L1(ξk)and L2(ξk)with excellent properties required by the universal model were created in this paper:L1(ξk)1/4n+3[(4n+3)(P1(ξk)-P0(ξk))-(P2n+2(ξk)-P2n-2(ξk))]’L2(ξk)=-1/2n+1[(2n+1(P1(1-ξk)-P0(1-ξk))-(P2n+1(1-ξk)-P2n-1(1-ξk))]They meet the requirements for the value of the boundary of the independent variable domain:(?)(?) And its function curve corresponds to n waveforms for different orders n,and this fluctuation is suitable for the wave expression and fitting of the parameter function curve.(3)The creative parametric universal model function of the layered structure widely existing in natural and artificial environments was constructed successfully:fk(ξk)=L1(ξk)Bk+L2(ξk)·Bk+1|ξk+1=0It can be guaranteed in advance that the specific parameter is continuous between any adjacent layers,and Bk and Bk+1 are included in the formula,which reflects the mutual influence and recursive relationship of the parameters of each layer.Since k is arbitrary,the parameter characteristic function Bk of each layer can be different according to its characteristic law,and can be any specific parameter continuous at the interface.Besides,the total number of layers n is also not limited and it can be of any multilayers,so the model has a good universality.(4)All examples have verified that the parametric universal model in this paper has correct calculation results and it can adapt to various complex river conditions.The universal model of layered structures then has been successfully extended to general problems of multilayer layered structure(the water quality problem of multi-reach rivers in environmental problems),and an application method was given,the conditions and purposes needed to create the parametric universal model in this paper were achieved.(5)The equilibrium equation of the flow Q and the amount of pollutants W at the section can be treated as a parametric continuous condition.The universal model function created can be applied to a single continuous parameter,so for multiple continuous parameters,the only necessary operation is to apply the universal model one by one to establish its universal expression(such as f Qk(ξk)and fWk(ξk)).Finally,each parametric universal model is substituted and applied into the global function model or other parametric calculation model,for the concentration of pollutant fLk(ξk)for example,and the formula can be concluded as:f Lk(ξk))-fWk(ξk)/fQk(ξk)This formula can represent the concentration of pollutants at any location in the river.Meanwhile:The universal model for the calculation of the discharge parameters fQk(ξk)of multi-reach sections at any position Zk is fQk(ξk)=L1(ξk)Q2,k-1+L2(ξk)(Q2,k-1-Q3k+Qk)The universal model for the calculation of pollutant quantity parameters fWk(ξk)of multi-reach sections at any location Zk is fwk(ξk)=L1(ξk)Bwk+L2(ξk)Bw(k+1)|ξk+1=0 Meanwhile:(?)(?)(6)With the help of the model formula obtained in conclusion(5),the change and regularity of the influence of any parameter change in the model formula,for any multi-reach rivers with any pollutant quantity and diversion,on the pollutant concentration in the whole river can be analyzed.