Study on the critical stress threshold of weakly cemented sandstone damage based on the renormalization group method

During the microstructural analysis of weakly cemented sandstone,the granule components and ductile structural parts of the sandstone are typically generalized.Considering the contact between granules in the microstructure of weakly cemented sandstone,three basic units can be determined:regular tetrahedra,regular hexahedra,and regular octahedra.Renormalization group models with granule-and pore-centered weakly cemented sandstone were established,and,according to the renormalization group transformation rule,the critical stress threshold of damage was calculated.The results show that the renormalization model using regular octahedra as the basic units has the highest critical stress threshold.The threshold obtained by iterative calculations of the granule-centered model is smaller than that obtained by the pore-centered model.The granule-centered calculation provides the lower limit(18.12%),and the pore-centered model provides the upper limit(36.36%).Within this range,the weakly cemented sandstone is in a phase-like critical state.That is,the state of granule aggregation transforms from continuous to discrete.In the relative stress range of 18.12%-36.36%,the weakly cemented sandstone exhibits an increased proportion of high-frequency signals(by 83.3%)and a decreased proportion of low-frequency signals(by 23.6%).The renormalization calculation results for weakly cemented sandstone explain the high-low frequency conversion of acoustic emission signals during loading.The research reported in this paper has important significance for elucidating the damage mechanism of weakly cemented sandstone.

Renormalization Group Theory and Its Application to Thermally-Induced Turbulence

Renormalization group theory applied to turbulence will be reviewed in this article.Techniques associated are used for analyzing thermally-induced turbulence.Transport properties such as effective viscosity and thermal diffusivity are derived.

Derivation of a second-order model for Reynolds stress using renormalization group analysis and the two-scale expansion technique

With the two-scale expansion technique proposed by Yoshizawa,the turbulent fluctuating field is expanded around the isotropic field.At a low-order two-scale expansion,applying the mode coupling approximation in the Yakhot-Orszag renormalization group method to analyze the fluctuating field,the Reynolds-average terms in the Reynolds stress transport equation,such as the convective term,the pressure-gradient-velocity correlation term and the dissipation term,are modeled.Two numerical examples：turbulent flow past a backward-facing step and the fully developed flow in a rotating channel,are presented for testing the efficiency of the proposed second-order model.For these two numerical examples,the proposed model performs as well as the Gibson-Launder （GL） model,giving better prediction than the standard k-ε model,especially in the abilities to calculate the secondary flow in the backward-facing step flow and to capture the asymmetric turbulent structure caused by frame rotation.

The Homotopy Perturbation Renormalization Group Method to Solve the WKB Problem with Turn Points

In this paper,we give the homotopy perturbation renormalization group method,this is a new method for turning point problem.Using this method,the independent variables are introduced by transformation without introducing new related variables and no matching is needed.The WKB approximation method problem can be solved.

Renormalization group methods for a Mathieu equation with delayed feedback

This paper presents the application of the renormalization group （RG） methods to the delayed differential equation. By analyzing the Mathieu equation with time delay feedback, we get the amplitude and phase equations, and then obtain the approximate solutions by solving the corresponding RG equations. It shows that the approximate solutions obtained from the RG method are superior to those from the conventionally perturbation methods.

RENORMALIZATION GROUP RECURSIONS BY VARIATIONAL CUMULANT EXPANSION APPROXIMATION

Renormalization group recursions are obtained by virtue of the variational cumulant expansion method. Good qualitative estimates are obtained for the d=2 square Ising system.

Memory,Time and Technique Aspects of Density Matrix Renormalization Group Method

We present the memory size,computational time,and technique aspects of density matrix renormalization group (DMRG) algorithm.We show how to estimate the memory size and computational time before starting a large scale DMRG calculation.We propose an implementation of the Hamiltonian wavefunction multiplication and a wavefunction initialization in DMRG with block matrix data structure.One-dimensional Heisenberg model is used to illustrate our study.

Specific Heat and Magnetic Susceptibility of Graphene:A Renormalization Group Study

In the present paper, we study effect of the long-range Coulomb interaction on the thermodynamic propertiesof graphene by renormalization group methods.Our calculations show that both the specific heat and the magneticsusceptibility of the material behave differently from the Landau Fermi liquid.More precisely, we find that thesequantities are logarithmically suppressed with respect to its noninteracting counterpart when temperature is low.

Real-space parallel density matrix renormalization group with adaptive boundaries

We propose an improved real-space parallel strategy for the density matrix renormalization group(DMRG)method,where boundaries of separate regions are adaptively distributed during DMRG sweeps.Our scheme greatly improves the parallel efficiency with shorter waiting time between two adjacent tasks,compared with the original real-space parallel DMRG with fixed boundaries.We implement our new strategy based on the message passing interface(MPI),and dynamically control the number of kept states according to the truncation error in each DMRG step.We study the performance of the new parallel strategy by calculating the ground state of a spin-cluster chain and a quantum chemical Hamiltonian of the water molecule.The maximum parallel efficiencies for these two models are 91%and 76%in 4 nodes,which are much higher than the real-space parallel DMRG with fixed boundaries.

Improved hybrid parallel strategy for density matrix renormalization group method

We propose a new heterogeneous parallel strategy for the density matrix renormalization group(DMRG)method in the hybrid architecture with both central processing unit(CPU)and graphics processing unit(GPU).Focusing on the two most time-consuming sections in the finite DMRG sweeps,i.e.,the diagonalization of superblock and the truncation of subblock,we optimize our previous hybrid algorithm to achieve better performance.For the former,we adopt OpenMP application programming interface on CPU and use our own subroutines with higher bandwidth on GPU.For the later,we use GPU to accelerate matrix and vector operations involving the reduced density matrix.Applying the parallel scheme to the Hubbard model with next-nearest hopping on the 4-leg ladder,we compute the ground state of the system and obtain the charge stripe pattern which is usually observed in high temperature superconductors.Based on simulations with different numbers of DMRG kept states,we show significant performance improvement and computational time reduction with the optimized parallel algorithm.Our hybrid parallel strategy with superiority in solving the ground state of quasi-two dimensional lattices is also expected to be useful for other DMRG applications with large numbers of kept states,e.g.,the time dependent DMRG algorithms.

Study on critical conditions for rock failure by means of group renormalization

A study of the characteristics of the accumulative rock failure and its evolution byapplication of the group renormalization method were presented. In addition, the interactionand long-range correlated effects between the immediate neighboring units was studied.The concept of mechanical transference for model OFC, employed in the study ofself-organized criticality, and the coefficient a were introduced into the calculation model forgroup renormalization. With the introduction, mechanisms for the drastic increase and decrease of failure intensity of rocks were investigated under similar macro-conditions.

Time-Dependent Density Matrix Renormalization Group Coupled with n-Mode Representation Potentials for the Excited State Radiationless Decay Rate:Formalism and Application to Azulene

We propose a method for calculating the nonradiative decay rates for polyatomic molecules including anharmonic effects of the potential energy surface(PES)in the Franck-Condon region.The method combines the n-mode repre-sentation method to construct the ab initio PES and the nearly exact time-dependent density matrix renormalization group method(TD-DMRG)to simulate quantum dynamics.In addition,in the framework of TD-DMRG,we further develop an algorithm to calculate the final-state-resolved rate coefficient which is very useful to analyze the contribution from each vibrational mode to the transition process.We use this method to study the internal conversion(IC)process of azulene after taking into account the anharmonicity of the ground state PES.The results show that even for this semi-rigid molecule,the intramode anharmonicity enhances the IC rate significantly,and after considering the two-mode coupling effect,the rate increases even further.The reason is that the anharmonicity enables the C-H vibrations to receive electronic energy while C-H vibrations do not contribute on the harmonic PES as the Huang-Rhys factor is close to 0.

Phase Diagrams of the Semi-Infinite Blume-Capel Model with Mixed Spins (SA = 1 and SB = 3/2) by Migdal Kadanoff Renormalization Group

We study the mixed spin-1 and spin-3/2 Blume-Capel model under crystal field in the tridimensional semi-infinite case. This has been done by using the real-space renormalization group approximation and specifically the Migdal-Kadanoff technique. As a function of the ratio R of bulk and surface interactions and the ratios R1 and R2 of bulk and surface crystals fields on the spin-1 and spin-3/2 respectively, we have determined various types of phase diagrams. Besides second- order transition lines, first-order phase transition lines terminating at tricritical points are obtained. We found that there existed nine main types of phase diagram showing a variety of phase transitions associated with the surface, including ordinary, extraordinary, surface and special phase transitions.

Progress in ab initio in-medium similarity renormalization group and coupled-channel method with coupling to the continuum

Over the last decade,nuclear theory has made dramatic progress in few-body and ab initio many-body calculations.These great advances stem from chiral efective feld theory(xEFT),which provides an efcient expansion and consistent treatment of nuclear forces as inputs of modern many-body calculations,among which the in-medium similarity renormalization group(IMSRG)and its variants play a vital role.On the other hand,signifcant eforts have been made to provide a unifed description of the structure,decay,and reactions of the nuclei as open quantum systems.While a fully comprehensive and microscopic model has yet to be realized,substantial progress over recent decades has enhanced our understanding of open quantum systems around the dripline,which are often characterized by exotic structures and decay modes.To study these interesting phenomena,Gamow coupled-channel(GCC)method,in which the open quantum nature of few-body valence nucleons coupled to a deformed core,has been developed.This review focuses on the developments of the advanced IMSRG and GCC and their applications to nuclear structure and reactions.

Evaluating First-Order Molecular Properties of Delocalized Ionic or Excited States in Molecular Aggregates by Renormalized Excitonic Method

In the past few years,the renormalized excitonic model(REM)approach was developed as an efficient low-scaling ab initio excited state method,which assumes the low-lying excited states of the whole system are a linear combination of various single monomer excitations and utilizes the effective Hamiltonian theory to derive their couplings.In this work,we further extend the REM calculations for the evaluations of first-order molecular properties(e.g.charge population and transition dipole moment)of delocalized ionic or excited states in molecular aggregates,through generalizing the effective Hamiltonian theory to effective operator representation.Results from the test calculations for four different kinds of one dimensional(1D)molecular aggregates(ammonia,formaldehyde,ethylene and pyrrole)indicate that our new scheme can efficiently describe not only the energies but also wavefunction properties of the low-lying delocalized electronic states in large systems.

Renormalization-group theory of first-order phase transition dynamics in field-driven scalar model

Through a detailed study of the mean-field approximation, the Gaussian approximation, the perturbation expansion, and the field-theoretic renormalization-group analysis of a φ^3 theory, we show that the instability fixed points of the theory, together with their associated instability exponents, are quite probably relevant to the scaling and universality behavior exhibited by the first-order phase transitions in a field-driven scalar Ca model, below its critical temperature and near the instability points. Finite- time scaling and leading corrections to the scaling are considered. We also show that the instability exponents of the first-order phase transitions are equivalent to those of the Yang-Lee edge singularity, and employ the latter to improve our estimates of the former. The outcomes agree well with existing numerical results.

The symmetry group of Feynman diagrams and consistency of the BPHZ renormalization scheme

We study the relation between the symmetry group of a Feynman diagram and its reduced diagrams.We then prove that the counterterms in the BPHZ renormalization scheme are consistent with adding counterterms to the interaction Hamiltonian in all cases,including that of Feynman diagrams with symmetry factors.

A MONTE CARLO RENORMALIZATION GROUP STUDY OF ISING MODEL ON 2-DIMENSIONAL RANDOM TRIANGLE LATTICE

Ⅰ. INTRODUCTIONMonte Carlo renormalization group method (MCRG) has become a powerful technical tool for the study of critical behaviour and continuum limit of discrete systems since 1976. The method is therefore widely applied to the study of fixed point and critical exponents in statistical models, and to the study of phase transitions, β function and scaling behaviour for some gauge fields in lattice gauge theories.

No-core Monte Carlo shell model calculations with unitary correlation operator method and similarity renormalization group

The unitary correlation operator method （UCOM） and the similarity renormalization group theory （SRG） are compared and discussed in the framework of the no-core Monte Carlo shell model （MCSM） calculations for ^3H and ^4He. The treatment of spurious center-of-mass motion by Lawson＇s prescription is performed in the MCSM calculations. These results with both transformed interactions show good suppression of spurious center-of-mass motion with proper Lawson＇s prescription parameter βc.m. values. The UCOM potentials obtain faster convergence of total energy for the ground state than that of SRG potentials in the MCSM calculations, which differs from the cases in the no-core shell model calculations （NCSM）. These differences are discussed and analyzed in terms of the truncation scheme in the MCSM and NCSM, as well as the properties of the potentials of SRG and UCOM.

Renormalization group theory for temperature-driven first-order phase transitions in scalar models

We study the scaling and universal behavior of temperature-driven first-order phase transitions in scalar models. These transitions are found to exhibit rich phenomena, though they are controlled by a single complex-conjugate pair of imaginary fixed points of φ3 theory. Scaling theories and renormalization group theories are developed to account for the phenomena, and three universality classes with their own hysteresis exponents are found： a field-like thermal class, a partly thermal class, and a purely thermal class, designated, respectively, as Thermal Classes I, II, and III. The first two classes arise from the opposite limits of the scaling forms proposed and may cross over to each other depending on the temperature sweep rate. They are both described by a massless model and a purely massive model, both of which are equivalent and are derived from φ3 theory via symmetry. Thermal Class III characterizes the cooling transitions in the absence of applied external fields and is described by purely thermal models, which include cases in which the order parameters possess different symmetries and thus exhibit different universality classes. For the purely thermal models whose free energies contain odd-symmetry terms, Thermal Class III emerges only at the mean-field level and is identical to Thermal Class II. Fluctuations change the model into the other two models. Using the extant three- and two- loop results for the static and dynamic exponents for the Yang-Lee edge singularity, respectively, which falls into the same universality class as φ3 theory, we estimate the thermal hysteresis exponents of the various classes to the same precision. Comparisons with numerical results and experiments are briefly discussed.