Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and repr...Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.展开更多
Regression analysis is often formulated as an optimization problem with squared loss functions. Facing the challenge of the selection of the proper function class with polynomial smooth techniques applied to support v...Regression analysis is often formulated as an optimization problem with squared loss functions. Facing the challenge of the selection of the proper function class with polynomial smooth techniques applied to support vector regression models, this study takes cubic spline interpolation to generate a new polynomial smooth function ｜×｜ε^ 2, in g-insensitive support vector regression. Theoretical analysis shows that Sε^2 -function is better than pε^2 -function in properties, and the approximation accuracy of the proposed smoothing function is two order higher than that of classical pε^2 -function. The experimental data shows the efficiency of the new approach.展开更多
The papcr gives an approach to construct shape preserving piece wise cubic, where twocubic picces are allowed by inscrting at most a new knot in each data subinterval, and these cubicpieees is C2 continuous at each ne...The papcr gives an approach to construct shape preserving piece wise cubic, where twocubic picces are allowed by inscrting at most a new knot in each data subinterval, and these cubicpieees is C2 continuous at each new knot. Two numerical examples show that the method is effec.tive and visually pleasing.展开更多
A special class of cubic polynomials possessing decay of geometry property is studied.This class of cubic bimodal maps has generalized Fibonacci combinatorics.For maps with bounded combinatorics,we show that they have...A special class of cubic polynomials possessing decay of geometry property is studied.This class of cubic bimodal maps has generalized Fibonacci combinatorics.For maps with bounded combinatorics,we show that they have an absolutely continuous invariant probability measure.展开更多
The relationship between arts and mathematics is very close, computer graphic design is based on digital methodology. The paper reveals the mathematical backgrounds behind graphic design by the example of computer-aid...The relationship between arts and mathematics is very close, computer graphic design is based on digital methodology. The paper reveals the mathematical backgrounds behind graphic design by the example of computer-aided cubic modeling and mathematical exchange methodology. Furthermore, one can get incredible artistic effects if computer graphic designers pay more attention to the probability and use probable numbers and fractal operation in their design activities.Finally, the author also discusses the bidirections between arts and mathematics.展开更多
文摘Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.
基金Supported by Guangdong Natural Science Foundation Project(No.S2011010002144)Province and Ministry Production and Research Projects(No.2012B091100497,2012B091100191,2012B091100383)+1 种基金Guangdong Province Enterprise Laboratory Project(No.2011A091000046)Guangdong Province Science and Technology Major Project(No.2012A080103010)
文摘Regression analysis is often formulated as an optimization problem with squared loss functions. Facing the challenge of the selection of the proper function class with polynomial smooth techniques applied to support vector regression models, this study takes cubic spline interpolation to generate a new polynomial smooth function ｜×｜ε^ 2, in g-insensitive support vector regression. Theoretical analysis shows that Sε^2 -function is better than pε^2 -function in properties, and the approximation accuracy of the proposed smoothing function is two order higher than that of classical pε^2 -function. The experimental data shows the efficiency of the new approach.
文摘The papcr gives an approach to construct shape preserving piece wise cubic, where twocubic picces are allowed by inscrting at most a new knot in each data subinterval, and these cubicpieees is C2 continuous at each new knot. Two numerical examples show that the method is effec.tive and visually pleasing.
文摘A special class of cubic polynomials possessing decay of geometry property is studied.This class of cubic bimodal maps has generalized Fibonacci combinatorics.For maps with bounded combinatorics,we show that they have an absolutely continuous invariant probability measure.
文摘The relationship between arts and mathematics is very close, computer graphic design is based on digital methodology. The paper reveals the mathematical backgrounds behind graphic design by the example of computer-aided cubic modeling and mathematical exchange methodology. Furthermore, one can get incredible artistic effects if computer graphic designers pay more attention to the probability and use probable numbers and fractal operation in their design activities.Finally, the author also discusses the bidirections between arts and mathematics.
