Laser Quenching is one of main contents in laser heat treatment. At present, computer simulation on cooling course of laser quenching is the main research field and the foundation of calculating inner thermal stresses...Laser Quenching is one of main contents in laser heat treatment. At present, computer simulation on cooling course of laser quenching is the main research field and the foundation of calculating inner thermal stresses in object. It also provides theoretical basis for optimizing and controlling the course of laser quenching technology. In this paper, the difference between finite element method and finite differential method, which are two methods to calculate the laser quenching temperature field and calculation precision are studied. The unstable temperature field is solved and the configure and time are discretizcd simultaneously. About time discrete, two kinds of differential pattern are discussed. Compared the calculation results with measurement values, it shows that the differential method adopting in the paper is feasible and the calculation precision and calculation velocity can be increased to use variable step-size about time. Also, the result testifies that different calculation methods can be employed in case of variable application situation and calculation precision.展开更多
Fractional-order time-delay differential equations can describe many complex physical phenomena with memory or delay effects, which are widely used in the fields of cell biology, control systems, signal processing, et...Fractional-order time-delay differential equations can describe many complex physical phenomena with memory or delay effects, which are widely used in the fields of cell biology, control systems, signal processing, etc. Therefore, it is of great significance to study fractional-order time-delay differential equations. In this paper, we discuss a finite volume element method for a class of fractional-order neutral time-delay differential equations. By introducing an intermediate variable, the fourth-order problem is transformed into a system of equations consisting of two second-order partial differential equations. The L1 formula is used to approximate the time fractional order derivative terms, and the finite volume element method is used in space. A fully discrete format of the equations is established, and we prove the existence, uniqueness, convergence and stability of the solution. Finally, the validity of the format is verified by numerical examples.展开更多
文摘Laser Quenching is one of main contents in laser heat treatment. At present, computer simulation on cooling course of laser quenching is the main research field and the foundation of calculating inner thermal stresses in object. It also provides theoretical basis for optimizing and controlling the course of laser quenching technology. In this paper, the difference between finite element method and finite differential method, which are two methods to calculate the laser quenching temperature field and calculation precision are studied. The unstable temperature field is solved and the configure and time are discretizcd simultaneously. About time discrete, two kinds of differential pattern are discussed. Compared the calculation results with measurement values, it shows that the differential method adopting in the paper is feasible and the calculation precision and calculation velocity can be increased to use variable step-size about time. Also, the result testifies that different calculation methods can be employed in case of variable application situation and calculation precision.
文摘Fractional-order time-delay differential equations can describe many complex physical phenomena with memory or delay effects, which are widely used in the fields of cell biology, control systems, signal processing, etc. Therefore, it is of great significance to study fractional-order time-delay differential equations. In this paper, we discuss a finite volume element method for a class of fractional-order neutral time-delay differential equations. By introducing an intermediate variable, the fourth-order problem is transformed into a system of equations consisting of two second-order partial differential equations. The L1 formula is used to approximate the time fractional order derivative terms, and the finite volume element method is used in space. A fully discrete format of the equations is established, and we prove the existence, uniqueness, convergence and stability of the solution. Finally, the validity of the format is verified by numerical examples.
