Catastrophe theory-based risk evaluation model for water and mud inrush and its application in karst tunnels

This paper presents a risk evaluation model of water and mud inrush for tunnel excavation in karst areas.The factors affecting the probabilities of water and mud inrush in karst tunnels are investigated to define the dangerousness of this geological disaster.The losses that are caused by water and mud inrush are taken into consideration to account for its harmfulness.Then a risk evaluation model based on the dangerousness-harmfulness evaluation indicator system is constructed,which is more convincing in comparison with the traditional methods.The catastrophe theory is used to evaluate the risk level of water and mud inrush and it has great advantage in handling problems involving discontinuous catastrophe processes.To validate the proposed approach,the Qiyueshan tunnel of Yichang-Wanzhou Railway is taken as an example in which four target segments are evaluated using the risk evaluation model.Finally,the evaluation results are compared with the excavation data,which shows that the risk levels predicted by the proposed approach are in good agreements with that observed in engineering.In conclusion,the catastrophe theory-based risk evaluation model is an efficient and effective approach for water and mud inrush in karst tunnels.

PRICING CATASTROPHE OPTIONS WITH COUNTERPARTY CREDIT RISK IN A REDUCED FORM MODEL

In this paper, we study the price of catastrophe Options with counterparty credit risk in a reduced form model. We assume that the loss process is generated by a doubly stochastic Poisson process, the share price process is modeled through a jump-diffusion process which is correlated to the loss process, the interest rate process and the default intensity process are modeled through the Vasicek model： We derive the closed form formulae for pricing catastrophe options in a reduced form model. Furthermore, we make some numerical analysis on the explicit formulae.

Research on traffic flow forecasting model based on cusp catastrophe theory

This paper intends to describe the relationship between traffic parameters by using cusp catastrophe theory and to deduce highway capacity and corresponding speed forecasting value through suitable transformation of catastrophe model. The five properties of a catastrophe system are outlined briefly, and then the data collected on freeways of Zhujiang River Delta, Guangdong province, China are examined to ascertain whether they exhibit qualitative properties and attributes of the catastrophe model. The forecasting value of speed and capacity for freeway segments are given based on the catastrophe model. Furthermore, speed-flow curve on freeway is drawn by plotting out congested and uncongested traffic flow and the capacity value for the same freeway segment is also obtained from speed-flow curve to test the feasibility of the application of cusp catastrophe theory in traffic flow analysis. The calculating results of catastrophe model coincide with those of traditional traffic flow models regressed from field observed data, which indicates that the deficiency of traditional analysis of relationship between speed, flow and occupancy in two-dimension can be compensated by analysis of the relationship among speed, flow and occupancy based on catastrophe model in three-dimension. Finally, the prospects and problems of its application in traffic flow research in China are discussed.

The Study of Chamber Rockburst by the CUSP Model of Catastrophe Theory

By means of CUSP model of catastrophe theory. this paper has studied thephysics process of rockburst occured on circular chamber. The present paper has nolonly described the instability process of rockburst more deeply. but also got the crilicaldepth of plastic softening area of chamber that is valuable in the controlling engineering of rockburst. the chamber displacement jump and energy liberation have been derived. the influence of rock parameters on the rockburst has been discussed .

A MODEL OF BUTTERFLY CATASTROPHE FOR ANODE EFFECT IN ALUMINA ELECTROLYSIS

A model of butterfly catastrophe is confirmed based on the matching and fitting the previous experimental data from alumina electrolysis when anode effect occurs.The complicated behaviour of the cryolite-alumina melt system with varying parameters could be generally described by this model.Therefore,the anode effect and its occurrence may be thoroughly understood.

Fold catastrophe model of strike-slip fault earthquake

Using a differential form of the potential energy function and taking the effect of work applied by external force in far field into account, the mechanism of strike-slip fault earthquake is analyzed. The research indicates that each characteristic displayed with a fold catastrophe model in the catastrophe theory corresPonds to a specific primary characteristic of the strike-slip fault earthquake. The fold catastrophe can describe the positions of starting and end points of a fault failure and the distance of fault dislocation. These include the description of stability of the surrounding rock-fault system before and after the earthquake. Two different illustrations about elastic energy releasing amount of the surrounding rock with the fault failure are shown with the primary characteristics mutually demonstrated. The intensity of strike-slip fault earthquake is related to the surrounding rock press and the stiffness ratio of surrounding rock and fault. The larger the surrounding rock press, the smaller the stiffness ratio. The larger the included angle between the tangential stress axis and the causative fault surface, the stronger the earthquake.

Cusp Catastrophe Polynomial Model: Power and Sample Size Estimation

Guastello’s polynomial regression method for solving cusp catastrophe model has been widely applied to analyze nonlinear behavior outcomes. However, no statistical power analysis for this modeling approach has been reported probably due to the complex nature of the cusp catastrophe model. Since statistical power analysis is essential for research design, we propose a novel method in this paper to fill in the gap. The method is simulation-based and can be used to calculate statistical power and sample size when Guastello’s polynomial regression method is used to do cusp catastrophe modeling analysis. With this novel approach, a power curve is produced first to depict the relationship between statistical power and samples size under different model specifications. This power curve is then used to determine sample size required for specified statistical power. We verify the method first through four scenarios generated through Monte Carlo simulations, and followed by an application of the method with real published data in modeling early sexual initiation among young adolescents. Findings of our study suggest that this simulation-based power analysis method can be used to estimate sample size and statistical power for Guastello’s polynomial regression method in cusp catastrophe modeling.

Stability Analysis of a SDOF Mechanical Model with Distinct Critical Points: II. Catastrophe Theory Approach

In a recent publication [1], the fully nonlinear stability analysis of a Single-Degree-of Freedom (SDOF) model with distinct critical points was dealt with on the basis of bifurcation theory, and it was demonstrated that this system is associated with the butterfly singularity. The present work is the companion one, tackling the problem via the Theory of Catastrophes. After Taylor expanding the original potential energy function and introducing Padè approximants of the trigonometric expression involved, the resulting truncated potential is a universal unfolding of the original one and an extended canonical form of the butterfly catastrophe potential energy function. Results in terms of equilibrium paths, bifurcation sets and manifold hyper-surface projections fully validate the whole analysis, being in excellent agreement with the findings obtained via bifurcation theory.

Catastrophe Theory Models for Stress-dependent Behaviour in a NiTi Shape Memory Alloy

The measurements by Huibin XU et al of the stress-dependence ot hysteresis in a NiTi shape memo ry alloy are modeled by catastrophe theory. The cusp catastrophe is used with the strain as the behaviour variable and the control parameters being functions of the stress and the temperature. A two constant model is found to be preferred to a four constant model.

Instability Analysis of Strike-Slip Fault Based on Cusp Catastrophe Model

The distribution of many active faults in western China is an important reasonfor the frequent earthquakes. With the rapid development of the western region, manymajor projects have been built there and the existence of active faults is bound to have aninfluence on the safety of the engineering structure. Therefore, it is of great significanceto study the mechanism of fault slip instability for evaluating the geological stability ofthe region and for the site selection of major projects. In this paper, cusp catastrophetheory is used to establish a cusp catastrophe model with general softened form ofstrike-slip faults on the basis of strike-slip faults. In this model, the influence of thesoftening property of fault zone on fault instability is considered. Based on this model,the conditions of slip instability of strike-slip faults are derived and further the half-slipdistance, far-field displacement and energy release equation of sliding-slip fault arerevealed. The influences of the system stiffness ratio and the softening property of thefault zone on the half-wave displacement, the far-field displacement and the energyrelease are shown. Which lays a good foundation for further research on activefault-induced earthquake mechanism.

Catastrophe model and its experimental verification ofstatic loading rock system under impact load

According to the catastrophe model for impact buckling of static loading structures, a new catastrophe model for impact loading failure of a static loading rock system was established, and one dimension (1D) catastrophe model was analyzed. The analysis results indicate that the furcation collection where catastrophe may take place is not only decided by mechanical system itself but also relates to exterior loading, which is different from the results obtained under mono-static loading where the bifurcation collection is only determined by mechanics of the system itself and has nothing to do with exterior loading. In addition, the corresponding 1D coupled static-dynamic loading experiment is designed to verify the analysis results of catastrophe model. The test is done with Instron 1342 electro-servo controlled testing system, in which medium strain rate is caused by monotony rising dynamic load. The parameters are obtained combining theoretical model with experiment. The experimental and theoretical curves of critical dynamic load vs static load are rather coincided, thus the new model is proved to be correct.

Application of Catastrophe Theory in 3D Seismic Data Interpretation of Coal Mine

In order to detect fault exactly and quickly, cusp catastrophe theory is used to interpret 3D coal seismic data in this paper. By establishing a cusp model, seismic signal is transformed into standard form of cusp catastrophe and catastrophe parameters, including time-domain catastrophe potential, time-domain catastrophe time, frequency-domain catastrophe potential and frequency- domain degree, are calculated. Catastrophe theory is used in 3D seismic structural interpretation in coal mine. The results show that the position of abnormality of the catastrophe parameter profile or curve is related to the location of fault, and the cusp catastrophe theory is effective to automatically pick up geology information and improve the interpretation precision in 3D seismic data.

Assessment of Flood Catastrophe Risk for Grain Production at the Provincial Scale in China Based on the BMM Method

Flood catastrophe risk assessment is imperative for the steady development of agriculture under the context of global climate change,and meanwhile,it is an urgent scientific issue need to be solved in agricultural risk assessment discipline.This paper developed the methodology of flood catastrophe risk assessment,which can be shown as the standard process of crop loss calculation,Monte Carlo simulation,the generalized extreme value distribution（GEV） fitting,and risk evaluation.Data on crop loss were collected based on hectares covered by natural disasters,hectares affected by natural disasters,and hectares destroyed by natural disasters using the standard equation.Monte Carlo simulation based on appropriate distribution was used to expand sample size to overcome the insufficiency of crop loss data.Block maxima model（BMM） approach based on the extreme value theory was for modeling the generalized extreme value distribution（GEV） of flood catastrophe loss,and then flood catastrophe risk at the provincial scale in China was calculated.The Type III Extreme distribution（Weibull） has a weighted advantage of modeling flood catastrophe risk for grain production.The impact of flood catastrophe to grain production in China was significantly serious,and high or very high risk of flood catastrophe mainly concentrates on the central and eastern regions of China.Given the scenario of suffering once-in-a-century flood disaster,for majority of the major-producing provinces,the probability of 10% reduction of grain output is more than 90%.Especially,the probabilities of more than 15% decline in grain production reach up to 99.99,99.86,99.69,and 91.60% respectively in Anhui,Jilin,Liaoning,and Heilongjiang.Flood catastrophe assessment can provide multifaceted information about flood catastrophe risk that can help to guide management of flood catastrophe.

DESCRIPTION AND PREDICTION OF CATASTROPHE OF VIBRATION STATE FOR FAULTY ROTORS

A sudden increase of vibration amplitude with no foreboding often results in an abrupt breakdown of a mechanical system.The catastrophe of vibration state of a faulty rotor is a typical nonlinear phenomenon,and very difficult to be described and predicted with linear vibration theory.On the basis of nonlinear vibration and catastrophe theory,fhe eatastrophe of the vibration amplitude of the faulty rotor is described;a way to predict its emergence is developed.

MECHANISM AND CATASTROPHE THEORY ANALYSIS OF CIRCULAR TUNNEL ROCKBURST

Mechanism of circular tunnel rockburst is that, when the carrying capacity of the centralized zone of plastic deformation in limiting state reduces, the comparatively intact part in rock mass unloads by way of elasticity; rockburst occurs immediately when the elastic energy released by the comparatively intact part exceeds the energy dissipated by plastic deformation. The equivalent strain was taken as a state variable to establish a catastrophe model of tunnel rockburst, and the computation expression of the earthquake energy released by tunnel rockburst was given. The analysis shows that, the conditions of rockburst occurrence are relative to rock＇s ratio of elastic modulus to descendent modulus and crack growth degree of rocks; to rock mass with specific rockburst tendency, there exists a corresponding critical depth of softened zone, and rockburst occurs when the depth of softened zone reaches.

Analysis of main shock of thrust fault earthquake by catastrophe theory

The relationship between work and energy increment of a thrust fault system with quasi-static deformation can be decomposed into two parts： volume strain energy and deviation stress energy. The relationship between work and energy increment of the deviation stress of a simplified thrust fault system is analyzed based on the catastrophe theory. The research indicates that the characteristics displayed by the fold catastrophe model can appropriately describe the condition of earthquake generation, the evolvement process of main shock of thrust fault earthquake, and some important aftershock proper- ties. The bigger the surrounding press of surrounding rock is, the bigger the maximum principal stress is, the smaller the incidences of the potential thrust fault surface are, and the smaller the ratio between the tangential stiffness of surrounding rock and the slope is, which is at the inflexion point on the softened zone of the fault shearing strength curve. Thus, when earthquake occurrs, the larger the elastic energy releasing amount of sur- rounding rock is, the bigger the earthquake magnitude is, the larger the half distance of fault dislocation is, and the larger the displacement amplitude of end face of surrounding rock is. Fracturing and expanding the fault rock body and releasing the volume strain energy of surrounding rock during the earthquake can enhance the foregoing effects to- gether.

Research on Instability Mechanism and Type of Ore Pillar Based on the Fold Catastrophe Theory

The stability of ore pillar in mine is essential for the safe and efficient mining.Based on the energy evolvement rule in ore pillar and roadway roof system,the roadway roof and ore pillar are treated as energy release body and energy dissipation body,respectively.Therefore,the double-block mechanical model is established with energy dissipation body and energy release body,and the energy mechanism of ore pillar instability is obtained,based on the fold catastrophe mathematical theory.The research result indicates that the dynamic instability of ore pillar is a physical instability problem caused by the strain softening property of ore mass,and the instability type of ore pillar is determined by the stiffness parameter of double-block mechanical system.When the systematical stiffness parameter is greater than or equal to 1,the equilibrium position of double-block mechanical system passes through shaft K-1 or the origin,from branch 1 of unstable equilibrium state to branch 2 of stable equilibrium state by smooth transition,and the ore pillar shows quasi-static fracture.When the systematical stiffness parameter is less than 1,the equilibrium position of double-block system mutates from branch 1 of unstable equilibrium state to branch 2 of stable equilibrium state by jumping transition,and the ore pillar shows dynamic instability.The research result could provide theoretical guidance for the prevention measures of ore pillar instability.

An Electrostatic Catastrophe Machine as an Attosecond Pulse Generator

The generation of an attosecond pulse in the ultraviolet range is described in the terms of the catastrophe theory. A simple criterion of tunneling is proposed. The criterion allows constructing the quasiclassical model of the generator of attosecond laser pulses based on the interaction of an electric field of extremely powerful femtosecond pulse with the valence electron in the potential well of the gas atom.

Grey forewarning and prediction for mine water inflowing catastrophe periods

Based on the theory of grey system, established GM （1, 1） grey catastrophe predict model for the first time in order to forecast the catastrophe periods of mine water inflowing （not the volume of water inflowing）. After establishing the grey predict system of the catastrophe regularity of 10 month-average volume of water inflowing, the grey forewarning for mine water inflowing catastrophe periods was established which was used to analyze water disaster in 400 meter level of Wennan Colliery. Based on residual analysis, it shows that the result of grey predict system is almost close to the actual value. And the scene actual result also shows the reliability of prediction. Both the theoretical analysis and the scene actual result indicate feasibility and reliability of the method of grey catastrophe predict system.

THE CATASTROPHE FORECAST OF THE ANNUAL SIDEMENT TRANSPORT BY DEBRIS FLOW IN JIANGJIA GULLY,YUNNAN PROVINCE, CHINA

On the basis of the observational data on the annual sediment transport by debris flow in recent 8 years, appling the catastrophe forecast method of Grey System Theory, this study has established the catastrophe model of the annual sediment transport by debris flow in Jiangjia Gully. It has forecasted the next potential catastrophic year in which the annual sediment transport will be over the catastrophic. threshold 2 million m3. Furthermore, it has introduced the 'equal dimension-new information model', which makes the forecast be done continuously.