Fractional Noether theorem and fractional Lagrange equation of multi-scale mechano-electrophysiological coupling model of neuron membrane

Noether theorem is applied to a variable order fractional multiscale mechano-electrophysiological model of neuron membrane dynamics.The variable orders fractional Lagrange equation of a multiscale mechano-electrophysiological model of neuron membrane dynamics is given.The variable orders fractional Noether symmetry criterion and Noether conserved quantities are given.The forms of variable orders fractional Noether conserved quantities corresponding to Noether symmetry generators solutions of the model under different conditions are discussed in detail,and it is found that the expressions of variable orders fractional Noether conserved quantities are closely dependent on the external nonconservative forces and material parameters of the neuron.

A review of multiscale expansion of low permeability reservoir cracks

The study of rock crack propagation by multi-scale method is of great significance to comprehensively and accurately understand the law of rock crack evolution.In this paper,the theoretical,experimental and numerical methods from macroscale,mesoscale and microscale used for crack propagation in recent years are summarized and analyzed.Firstly,the evolution mechanism of the crack and the related research status are analyzed from a single scale.Secondly,multi-scale theory,modeling,meshing algorithm and macro-mesoscopic parameters are reviewed in the multi-scale coupling method.Through the analysis of the results published in recent years,it is considered that the following aspects need to be further studied:the characteristic parameters of the rock are different at different scales,so the extraction of the characteristic parameters under different scales is essential to modeling and coupling;the heterogeneity of rock and the prefabrication of cracks are greatly affected by human factors,so that 3D printing will be a good breakthrough to build the model of crack owing to its accurate control on the distribution and the size of cracks.The internal stress field of the rock is complex and varied,and the generation and expansion of the microcracks in the process of crack propagation are closely related to the surrounding environment.Therefore,it is of great importance to combine theoretical,experimental and numerical research with practical engineering.