• Volume 43,Issue 8,2015 Table of Contents
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    • >土木工程与建筑学
    • A New Method for Experimental Investigation of Microstructure Morphology of Lunar Soil Simulant

      2015, 43(8):1123-1128. DOI: 10.11908/j.issn.0253-374x.2015.08.001

      Abstract (1311) HTML (41) PDF 1.16 M (1058) Comment (0) Favorites

      Abstract:In practice, the advanced test equipment for soil microstructure morphology are very expensive and complicated to operate. Conventional studies on microstructure morphology generally focus on sands with one single fraction. A new test method, which can derive the microstructure parameters (e.g. aspect ratio, roughness) of TJ 1 lunar soil simulant consisting of various fractions, was introduced in this paper. First, high definition images of particles in different fractions were obtained by using a scanning electron microscopy (SEM). Then the microstructure parameters of each fraction were obtained by using the AutoCAD and Matlab software. Finally, the weight of morphological parameters of each fraction were gained using the distinct element method analyses, and the corresponding representative parameters of the whole specimen were calculated by using the weighted average method.

    • Damage Theory Based Analysis of Crack Opening Width for RC Structures

      2015, 43(8):1129-1134. DOI: 10.11908/j.issn.0253-374x.2015.08.002

      Abstract (1526) HTML (40) PDF 1.01 M (1429) Comment (0) Favorites

      Abstract:Starting from the damage model proposed by the design code of the RC structure, a model to evaluate the crack opening width was proposed considering the dependence between damage and crack. By implementing the proposed method into the nonlinear finite element package, the crack propagation and opening within the RC structure could be simulated in a subtle way. The agreement between the simulating results and the experimental data validates and verifies the proposed method which lays a solid foundation for the engineering analysis and design of RC structures.

    • Hysteretic Behavior of Q690D High Strength Steel Box Section Beam Columns

      2015, 43(8):1135-1143. DOI: 10.11908/j.issn.0253-374x.2015.08.003

      Abstract (2103) HTML (52) PDF 1.39 M (1068) Comment (0) Favorites

      Abstract:In order to study the hysteretic behavior of Q690D high strength steel (HSS) welded box section columns under cyclic loading, experiments and finite element simulations were conducted to study the hysteretic response of box section columns subjected to cyclic loading. From the obtained hysteretic curves and skeleton curves, it is observed that the Q690D HHS welded box section columns show a good energy dissipation capacity and seismic performance. After being verified by comparing with experimental results, the finite element(FE) model was employed to investigate the influence of axial compression ratios, slenderness ratios, width thickness ratios of plate on the hysteretic behavior of Q690D box section columns. It is notices that the second order effect cannot be neglected for Q690D HHS welded box section columns. Finally, hysteretic models for Q690D HSS welded box section columns were proposed based on the results of experimental study and FE analysis, providing the foundation for nonlinear seismic analysis of high strength steel structures. The proposed hysteretic models are demonstrated to be able to accurately predict the hysteretic behavior of Q690D high strength steel column by comparing the experimental and numerical results.

    • Response Analysis of Seismic Isolated Bridge Under Influence of Fault Crossing Groundmotions

      2015, 43(8):1144-1152. DOI: 10.11908/j.issn.0253-374x.2015.08.004

      Abstract (1487) HTML (34) PDF 1.37 M (1122) Comment (0) Favorites

      Abstract:This paper was based on the engineering background of a simple bridge across the fault. According to the proposed pulse models, a set of synthetic fault crossing strong ground motions was generated and the finite element model of a simple bridge across the fault was built, utilizing the non uniform excitation nonlinear time history analysis method to investigate the effect of the fault rupture on a seismic isolated bridge. Two different bearing systems, without and with accounting for differential ground movement across the fault rupture, were considered in the analysis to evaluate and compare the performances of the structure. The analysis indicate that the fault crossing is an important factor in the earthquake response of seismically isolated bridges which increases the earthquake response of the structure larger.

    • Composite Stiffness Matrix for a Mixed Retaining Structure of Soil Cement Wall and Concrete Piles

      2015, 43(8):1153-1158. DOI: 10.11908/j.issn.0253-374x.2015.08.005

      Abstract (1197) HTML (37) PDF 998.68 K (1090) Comment (0) Favorites

      Abstract:A composite stiffness matrix was proposed to describe the flexural performance of the soil cement wall and concrete piles mixed retaining structure. Based on Euler beam theory, both vertical and horizontal displacement functions were established by introducing a new degree of freedom—friction distance between concrete piles and cemented soil, and the flat section assumption and pile shear displacement method were used. Considering the potential energy generated by the friction of cement wall—concrete pile and pile tension, a composite stiffness matrix was derived. The proposed matrix was verified by an example and the displacement was in good agreement with the field data.

    • Lateral Equivalent Stiffness of Large Diameter Shield Tunnel Lining Structure at Various Buried Depths

      2015, 43(8):1159-1166. DOI: 10.11908/j.issn.0253-374x.2015.08.006

      Abstract (1971) HTML (53) PDF 1.32 M (1228) Comment (0) Favorites

      Abstract:The overall performance of shield tunnel lining structure is affected by the discontinuities of its longitudinal and circumferential joints. The load acting on the joints varies with the variation of buried depth of lining structure. As a result, the joint stiffness and the lateral equivalent stiffness of the overall structure also vary with the variation of buried depth, especially for large diameter shield tunnels. Taking the lining structure of Shanghai Yan Jiang Tunnel as an example, the lateral stiffness of large diameter shield tunnel lining structure at different buried depth was studied based on the lateral deformation equivalent principle and the structure test results of longitudinal joint. The influences of foundation stiffness and staggered assembling on the lateral stiffness were also studied. Numerical analyses reveal that the lateral stiffness of the structure is dependent on the buried depth. For the shallow buried tunnel, the lateral effective stiffness varies greatly. When the buried depth is greater than 1.0D (D is the tunnel diameter), the lateral effective stiffness first increases and then decreases with the increase of buried depth. When the buried depth is greater than 2.0D, the lateral effective stiffness stops decreasing because of the arching effect of the ground. In addition, the lateral effective stiffness is also influenced by the ground condition. The findings of this paper can serve as a reference for the design and analysis of large diameter shield tunnel lining structures.

    • System Reliability Method to Analyze Overturning Stability of Demolition Construction of Continuous Beam Bridges

      2015, 43(8):1167-1173. DOI: 10.11908/j.issn.0253-374x.2015.08.007

      Abstract (841) HTML (31) PDF 1.26 M (991) Comment (0) Favorites

      Abstract:The overturning stability of cantilever beam bridges in demolition process was assessed by the method of reliability, the causality chains method of system reliability was proposed, and the application of total probability formula was developed. Moreover, the system reliability formula of causality chains was derived with the developed total probability formula, and the calculated results by the formula were found to be well anastomosed with the series parallel connection. Furthermore, the application of Bayes formula was developed, by which the most important failure mode was analyzed. The sensitivity of parameters of the most important failure mode was analyzed. It is concluded that the lifting rope strength, rope number and temporary support strength have a significant influence on the overturning stability of the demolition system.

    • Standard Galerkin Finite Element Method Considering Compensation Stiffness in Fluid Mechanics

      2015, 43(8):1174-1179. DOI: 10.11908/j.issn.0253-374x.2015.08.008

      Abstract (1130) HTML (33) PDF 957.74 K (1057) Comment (0) Favorites

      Abstract:The wave problems of the finite element solution of one dimensional steady state convection diffusion equation was analyzed. It is indicated that the continuity improvement of the shape function between elements is an effective method to decrease the amplitude of wave. Based on the standard Galerkin (SG) finite element method, the compensation term considering the elastic stiffness was introduced by adding an “unbalanced force” to the nodes between elements, to form a complemental standard Galerkin (CSG) finite element equation. The expressions of compensation stiffness of linear Lagrange based interpolation (LLBI) function and exponential function based interpolation (EFBI) function were discussed. The results obtained by CSG and SG were compared with those of the analytical solution. It shows that the continuity of the shape function is improved and the amplitude of wave is decreased when the compensation term is introduced into the SG finite element. Compared with the SG finite element using LLBI, the CSG finite element using EFBI is more effective to control the wave phenomenon since it has a better continuity both inside the element and between two elements.

    • Characteristics of Wind Pressure Distribution on Billboards on High Rise Building Roof

      2015, 43(8):1180-1186. DOI: 10.11908/j.issn.0253-374x.2015.08.009

      Abstract (1513) HTML (47) PDF 1.17 M (1124) Comment (0) Favorites

      Abstract:The wind pressure on four types of billboards on the high rise building roof, the arrangements of which were unilateral, adjacent, trilateral and peripheral, was measured with the rigid model pressure tests. The distribution characteristics of mean and fluctuating wind pressure coefficients on different types of billboards and the frequency domain characteristics of the fluctuating wind pressure of typical pressure taps on the surfaces of the panels were discussed. The results show that the mean wind pressure coefficients and fluctuating wind pressure coefficients on the surfaces of the panels of various arrangements are relatively larger in skew wind case, with the coefficients near the edge larger than those on the inner areas. The distribution of extreme wind pressure coefficients on billboards of various arrangements are also different. The maximum extreme wind pressure appears on the unilateral model and the minimum pressure appears on the peripheral model. Aerodynamic interference is observed on multi panel billboards.

    • >交通运输工程
    • Effect of Subgrade Differential Settlement on Dynamic Response of Vehicle and Slab Track Vertical Coupled System

      2015, 43(8):1187-1193. DOI: 10.11908/j.issn.0253-374x.2015.08.010

      Abstract (1845) HTML (48) PDF 1.23 M (1218) Comment (0) Favorites

      Abstract:Based on the theory of vehicle track coupling dynamics, the vibration differential equations of rail, track plate and concrete footing in the slab track system were deduced considering the gravity of track structure. Combined with the vehicle dynamics model of multi rigid bodies, a vehicle and slab track vertical coupled dynamics model was established considering the gravity of track structure. The method of controlling subgrade reaction force was proposed to analyze the dynamic response of vehicle slab track system induced by a wide range of subgrade differential settlement, and the coupled model and control method were verified to be accurate and reliable. The factors of subgrade differential settlement affecting the dynamic response of the system, including driving speed, magnitude of the settlement and wavelength of the settlement, were analyzed. The result shows that the dynamics response increases with the driving speed and the magnitude of the settlement. The comfort index should be treated as the primary index which is more important than the safety index when facing differential settlement. Along with the wavelength of settlement increasing, the relative response first increases and then decreases. There exists a sensitive wavelength of the differential settlement which is related with the length of concrete base.

    • A Section Speed Guiding Method for Bus Operation Control

      2015, 43(8):1194-1199. DOI: 10.11908/j.issn.0253-374x.2015.08.011

      Abstract (1600) HTML (43) PDF 1.01 M (1106) Comment (0) Favorites

      Abstract:By analyzing the location data of automatic vehicles, this paper revealed that bus headway deviation tends to increase with vehicles traveling further along the route and there exists a turning point where non directional amplification is transformed to directional amplification influenced by the random factors. Then, this paper proposed a section speed guiding method considering the dynamic threshold of headway deviation and established the control logic and the model for optimizing the guidance speed. The case application showed that the proposed control method can restrain the amplification trend of the headway deviation along the route, decrease the travelling time of passengers to some extent, and be conducive to maintaining the operation order and service reliability of the fleet.

    • >机械、车辆与能源工程
    • Path Tracking for Evasive Collision Avoidance Based on Active Disturbance Rejection Control

      2015, 43(8):1200-1204. DOI: 10.11908/j.issn.0253-374x.2015.08.012

      Abstract (1744) HTML (42) PDF 737.44 K (1160) Comment (0) Favorites

      Abstract:The automatic evasive collision avoidance system can promote the vehicle driving safety. A robust path tracking controller based on the active disturbance rejection control algorithm was proposed to improve the path tracking performance of vehicles with different mass parameters and different wheelbase parameters. The change of vehicle model parameters can be observed and compensated by the third order extended state observer. The cubic B spline was used to solve the curvature discontinuity and high lateral acceleration problems in path planning. Carsim and Simulink were used to verify the effectiveness of the proposed path tracking controller. The simulation results show that the proposed controller can control different vehicles to follow the predesigned evasive path very well, which can guarantee the stability of vehicles.

    • A Vehicle Stability Control Criterion Based on Time Domain Disparity of Stability Variation Between Dynamic Differential Systems and Actual Vehicle

      2015, 43(8):1205-1212. DOI: 10.11908/j.issn.0253-374x.2015.08.013

      Abstract (1987) HTML (70) PDF 1.61 M (958) Comment (0) Favorites

      Abstract:A complementary criterion for vehicle electronic stability control system (ESC) was proposed for qualitative estimation of the rear axle sliding conditions which might appear in vehicle planar motion. Taking into consideration the comprehensive effect of tangential force and axle load transfer on tire cornering characteristics, the stability criteria of 2DOF lateral dynamic differential systems and 3DOF systems with longitudinal acceleration were studied. By introducing the vehicle transient stability analysis approach, the quantized disparity between the stability variation of dynamic differential systems and that of the actual vehicle was obtained in time domain. Based on the lag effect, an anticipatory stability control solution was developed, which took the system equilibriums properties of the nonlinear reference model as the criterion. Under certain conditions which considering longitudinal dynamics, the stability boundary held by the maximum of yaw rate was close to actual stability variation of the vehicle. Thus, the response time of ESC system is limited. With the anticipatory criterion, ESC could predict vehicle status earlier.

    • Side Impact Sled Test Method Based on Multipoint Impact

      2015, 43(8):1213-1218. DOI: 10.11908/j.issn.0253-374x.2015.08.014

      Abstract (1317) HTML (36) PDF 1.00 M (1109) Comment (0) Favorites

      Abstract:Based on the analysis of the movement of the door trim during the side impact and the simplification of the door trim intrusion velocity, a mass spring side impact sled test method which could reproduce multipoint movement of the door was established. Then a side impact sled model with dummy was established and its computer simulation showed that there was a high correlation of dummy injury between the injury gaining from the side impact sled test based on multiple movement reproduction and the result of the real side impact test. The results of the dummy’s injury showed that the side impact sled test method could well reproduce the interaction between the door trim, the seat and the dummy. The new method can be easily implemented in engineering and can satisfy engineering application.

    • Estimation Model of Pump Wheel Dynamic Torque Based on Load of Torque Converter

      2015, 43(8):1219-1225. DOI: 10.11908/j.issn.0253-374x.2015.08.015

      Abstract (1294) HTML (36) PDF 1.30 M (1048) Comment (0) Favorites

      Abstract:In order to solve the non association between the traditional torque converter pump wheel static model and its actual load characteristics, and to achieve the dynamic performance matching between the torque converter and the engine in initial configuration design, a pump wheel dynamic torque estimation model based on the load characteristics of torque converter was proposed. Based on the analysis of the existing torque converter model, the one dimension flow theory model, which took the load characteristics of the torque converter into account, was more comprehensive than the mass spring damper system model. Besides, the concept of load fluctuations of torque converter pump wheel torque model was proposed. The validity of pump wheel dynamic torque estimation model based on the load characteristics of torque converter was proved by comparing the analysis of load fluctuations through the method of controlling the variables with the results of the comprehensive CFD simulation experiments. This model provides guidance for the torque converter dynamic initial configuration design, which is associated with the load characteristics of the machine.

    • >电子、计算机、控制与系统
    • Stereo Image Coding Based on Histogram Pair Reversible Data Hiding

      2015, 43(8):1226-1233. DOI: 10.11908/j.issn.0253-374x.2015.08.016

      Abstract (1228) HTML (56) PDF 1.50 M (1035) Comment (0) Favorites

      Abstract:Using reversible data hiding based on histogram pair for stereo image coding, two cutting edge researches—data hiding and 3D TV were combined. One frame of the stereo image pair was compressed and embedded into the other frame without any loss, and the bandwidth and storage were halved, providing the possibility of compatibility of 3D and 2D TV. The dependency of the two frames was utilized for optimal search of four parameters, and the new compression limitation was achived. Lossless embedding of higher capacity and better image quality was completed by histogram pair reversible data hiding which adopts the multi cycle method.

    • A New Bayesian Method for Reliability Assessment of Products with Wiener Process Degradation

      2015, 43(8):1234-1238. DOI: 10.11908/j.issn.0253-374x.2015.08.017

      Abstract (1679) HTML (43) PDF 854.62 K (1210) Comment (0) Favorites

      Abstract:For the reliability assessment of products with Wiener process degradation, a new method of Bayesian estimation was proposed by analyzing two factors—prior distributions and loss functions. First, the posterior distribution of the parameters was theoretically derived with the assumption that the parameters of Wiener process follow the conjugate prior distribution. Then the Bayesian estimation results were calculated based on the quadratic loss function, absolute loss function and 0 1 loss function respectively. Finally, the reliability assessment of aeroengines was analyzed as a practical example. It is shown that this new method has a wider range of application, and its calculation process is simpler compared to existing methods. Moreover, for the reliability assessment of aeroengines, the parameter estimation based on the absolute loss function has the best fitting of these three loss functions.

    • >测绘科学与技术
    • Tunnel Point Cloud Denoising Algorithm Based on Centerline

      2015, 43(8):1239-1245. DOI: 10.11908/j.issn.0253-374x.2015.08.018

      Abstract (2207) HTML (51) PDF 1.20 M (1313) Comment (0) Favorites

      Abstract:The noise in the point cloud data of a tunnel affects the accuracy of analysis. Effectively removing the noise data has become the key factor in point cloud based tunnel deformation analysis. A tunnel point cloud denoising algorithm based on centerline was proposed in this paper. First, the gesture of the point cloud was obtained by projecting it onto the horizontal plane and vertical plane respectively, and fitting two curves by high order polynomial equations from the planar data out of which the control points of the centerline were then interpolated. The control points were are used to express the centerline were densified according to the intersection angle between the spatial lines. Meanwhile, the tangent planes at each control point were computed to segment the tunnel point cloud. In addition, the distances from each point to the centerline in each block were computed to compare with the given distance threshold to accomplish data denoising. The feasibility and accuracy of the proposed method were demonstrated by two experiments. The first test is to fit the centerline by simulating the tunnel point cloud while the fitting accuracy of the centerline was analyzed by comparing it with the given one. The second test was to analyze the practical tunnel point cloud. Data denoising was finally achieved by implementing the proposed method.

    • An Improved Maximum Consistency Geometric Primitives Fitting Algorithm for Point Cloud

      2015, 43(8):1246-1253. DOI: 10.11908/j.issn.0253-374x.2015.08.019

      Abstract (1644) HTML (41) PDF 1.29 M (1153) Comment (0) Favorites

      Abstract:Based on the idea of MCMD_Z algorithm, this paper presented a robust high precision fitting algorithm for plane, quadric surface primitives(sphere, cylinder, cone). According to the minimum sum of distance criteria, the algorithm obtained the best subset from the point cloud to fit the reliable initial value of the geometric primitive, removed the outliers cyclically using the robust Z score method, and fitted the inliers by using the weighted least square iteration method. Experimental results show that this algorithm can effectively remove outliers and precisely fit the geometric primitive in the point cloud with high content of outliers.

    • Two Step Method for 3D Coordinate Transformation in Local Area

      2015, 43(8):1254-1258. DOI: 10.11908/j.issn.0253-374x.2015.08.020

      Abstract (1751) HTML (42) PDF 937.62 K (1293) Comment (0) Favorites

      Abstract:Because the rotation parameters and the translation parameters are related, the ill posed problem will easily occur in the solution model in 3D coordinate transformation in the data of local area based on the Bursa Wolf model. The errors of the original system were not taken into account by the common regularization solution. In this paper, a two step approach was proposed for solving this problem, which first computed the rotation parameters with the least square adjustment after the translation parameters were centered by an idempotent matrix. Then, the weighted total least squares adjustment was employed to estimate the scale and the translation parameters with a new four parameter model. The suggested method not only considers the error of the original system but also avoids solving the complex ill posed total least squares problem. A simulation experiment was conducted to test the new method. The result shows that the proposed method can improve the relative accuracy of the scale and the translation parameters, particularly of the scale parameter, and significantly improve the transformation accuracy of external points.

    • >海洋与地球科学
    • Secondary Field Based Two Dimensional Magnetotelluric Numerical Simulation by Finite Element Method

      2015, 43(8):1259-1265. DOI: 10.11908/j.issn.0253-374x.2015.08.021

      Abstract (1610) HTML (51) PDF 1.08 M (963) Comment (0) Favorites

      Abstract:The Helmholtz equations for the secondary fields are almost the same as those for the total fields; the main differences are the addition of source terms involving the primary fields and the conductivity (TM mode) or magnetic permeability (TE mode) difference between the abnormal body and the host. In this paper a forward code was proposed using the finite element method, in which not only the conductivity but also the magnetic permeability differences were considered. In order to increase the calculation accuracy and efficiency, some special technologies were adopted. First, Green’s theorem was used to treat the source term of the secondary field equations as the volume integral and boundary integrals at elements. Next, the contraction grid algorithm was designed based on the binary tree structure. The advantage of the mesh was that it greatly reduced the number of nodes with almost the same precision as the uncontracting mesh. Moreover, sparse linear system of equations was solved by using the LDLT method, in order to reduce the calculation time. The sparse matrix symbolic analysis method based on the minimum fill in element were adopted before the LDLT. Finally, the two models were tested. The results showed that the calculation accuracy and efficiency were greatly improved by using the treatment method of secondary filed source term, the domain discrete method and the LDLT.

    • Response of Offshore Sea Level to River Discharge in Flood Season in Yangtze Estuary

      2015, 43(8):1266-1272. DOI: 10.11908/j.issn.0253-374x.2015.08.022

      Abstract (1238) HTML (40) PDF 1.14 M (946) Comment (0) Favorites

      Abstract:To clarify the impacts of river discharge on offshore sea level in the Yangtze estuary. Based on the river discharge frequency of 25%, 50% and 75% as well as multi year monthly discharge statistically calculated from measured runoff from 1950 to 2011 at Datong Station in the Yangtze River, the impacts of river discharge on offshore sea level were computed and analyzed using the tidal model of the Yangtze estuary and the Hangzhou Bay. The numerical results indicate that the mean tidal levels of the whole computational region rise as the river discharge increases, and the amplification decreases from upstream to the entrance. The amplification of the southern part is greater than that of the northern part in the offshore area. The mean tidal level has a very good positive correlation with river discharge in the Yangtze estuary. When the river discharge increases by 20 000 m3·s-1, the area where the mean tidal levels increase by 0.005~0.010 m covers most of the Hangzhou Bay.

    • >经济与管理科学
    • Micro Spatial Structure of Innovation Cluster

      2015, 43(8):1273-1278. DOI: 10.11908/j.issn.0253-374x.2015.08.023

      Abstract (1514) HTML (83) PDF 1.14 M (938) Comment (0) Favorites

      Abstract:This paper investigated the spatial pattern and agglomeration of architectural design industry cluster in Yangpu District by introducing the methods of exploratory spatial data analysis. By analyzing the internal spatial structure of cluster, this paper explored the changes of the innovation agglomeration with the spatial scale and step width, and studied the spatial distribution of the cluster network from the horizontal and vertical dimensions respectively based on the analysis of influencing factors. It is discovered that the spatial distribution of innovation cluster displays an obvious core periphery pattern derived from the joint effects of centripetal force consists of correlation effects, knowledge spillovers and other external economic and centrifugal forces generated by land rent and crowdedness. Moreover, the “core” comprised of the knowledge source with Tongji University as a representative created the localized force through the exchange of knowledge and information and enhancement of institutional model. Based on the “core”, the firms make their spatial decision according to the localized suitability of product, scale and complexity of technology with traffic network structure as a constraint, which constitutes the core mechanism of spatial organization pattern of innovation clusters.

    • Transportation Management System for Construction Elevators Using Agent Based Simulation

      2015, 43(8):1279-1283. DOI: 10.11908/j.issn.0253-374x.2015.08.024

      Abstract (1219) HTML (43) PDF 910.16 K (980) Comment (0) Favorites

      Abstract:Resource scheduling, or the allocation of resources over time, poses a great challenging to the transportation management of construction elevators in mega and multiple item cross construction projects. Combined with the development of construction simulation, this paper proposed a simulation based auction protocol model for resource scheduling management, and established a transportation management system for construction elevators by using agent based simulation. The application of this system in complex and dynamic environment show that it can provide a resource driven schedule, predict completion time, and calculate the total cost. Besides, it can improve the comprehensive ability of the transportation management, from multiple aspects such as transportation work packages delay and its storage, idle elevators and so on.

    • Valuation of Zero Coupon Bonds with Credit Rating Migration Risk

      2015, 43(8):1284-1288. DOI: 10.11908/j.issn.0253-374x.2015.08.025

      Abstract (1910) HTML (59) PDF 741.07 K (1133) Comment (0) Favorites

      Abstract:A model was established to price of a zero coupon bond whose issuer has credit rating migration risk. A problem of partial differential equations with terminal conditions was obtained by using the reduced form approach. The explicit solution of the bond price in each credit rating and their relations were obtained under some further constant parameter assumptions. Numerical results were shown by graphs with analyses of sensitive factors.

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