Let C be an n-dimensional sphere with diameter 1 and center at the origin in E<sup>n</sup>. The view-obstruction problem for n-dimensional spheres is to determine a constant v（n） to be the lower bound of...Let C be an n-dimensional sphere with diameter 1 and center at the origin in E<sup>n</sup>. The view-obstruction problem for n-dimensional spheres is to determine a constant v（n） to be the lower bound of those α for which any half-line L, given by x<sub>i</sub>=a<sub>i</sub>t（i=1, 2,...,n） where parameter t≥0 and a<sub>i</sub>（i=1, 2,...,n） are positive real numbers, intersects Δ（C, α）={αC+（m<sub>1</sub>+（1/2）, m<sub>2</sub>+（1/2）,…,m<sub>n</sub>+（1/2））:m<sub>1</sub>, m<sub>2</sub>,…m<sub>n</sub> nonnegative integers}. In this paper, for n=3, the following result is proved. For α】1/5<sup>1/2</sup> we have that any half-line L, given by x<sub>i</sub>=a<sub>i</sub>t（i=1,2,3）, intersects Δ（C, α）, where parameter t≥0 and a<sub>i</sub>（i=1,2,3） are positive real numbers such that |a|+|b|+|c|≠3 whenever aa<sub>1</sub>+ba<sub>2</sub>+ca<sub>3</sub>=0 for three integers a, b, c.展开更多
In order to solve the problem of truncating the open boundaries of cylindrical waveguides used in the simulation of high power microwave （HPM） sources, this paper studies the convolutional PML （CPML） in the cylind...In order to solve the problem of truncating the open boundaries of cylindrical waveguides used in the simulation of high power microwave （HPM） sources, this paper studies the convolutional PML （CPML） in the cylindrical coordinate system. The electromagnetic field＇s FDTD equations and the expressions of axis boundary conditions are presented. Numerical experiments are conducted to validate the equations and axis boundary conditions. The performance of CPML is simulated when it is used to truncate the cylindrical waveguides excited by the sources with different frequencies and modes in the 2.5-dimensional problems. Numerical results show that the maximum relative errors are all less than -90 dB. The CPML method is introduced in the 2.5-dimensional electromagnetic PIC software, and the relativistic backward wave oscillator is simulated by using this method. The results show that the property of CPML is much better than that of the Mur-type absorbing boundary condition when they are used to truncate the open boundaries of waveguides. The CPML is especially suitable for truncating the open boundaries of the dispersive waveguide devices in the simulation of HPM sources.展开更多
Both numerical simulation and theoretical analysis of seawater intrusion in coastal regions are of great theoretical importance in environmental sciences. The mathematical model can be described as a problem of the in...Both numerical simulation and theoretical analysis of seawater intrusion in coastal regions are of great theoretical importance in environmental sciences. The mathematical model can be described as a problem of the initial boundary values for a system of 3-dimensional nonlinear parabolic partial differential equations, one being the pressure flow equation and the other is the concentration convection-dispersion equation of the salt contained. For a generic case of a 3-dimensional bounded region, a backward-difference time-stepping scheme is defined. It approximates the pressure by the standard Galerkin procedure and the concentration by a Galerkin method of charederistics, where calculus of variations, theory of prior estimates and techniques are made use of Optimal order estimates in H1 norm are derived for the errors in the approximate solution.展开更多
文摘Let C be an n-dimensional sphere with diameter 1 and center at the origin in E<sup>n</sup>. The view-obstruction problem for n-dimensional spheres is to determine a constant v（n） to be the lower bound of those α for which any half-line L, given by x<sub>i</sub>=a<sub>i</sub>t（i=1, 2,...,n） where parameter t≥0 and a<sub>i</sub>（i=1, 2,...,n） are positive real numbers, intersects Δ（C, α）={αC+（m<sub>1</sub>+（1/2）, m<sub>2</sub>+（1/2）,…,m<sub>n</sub>+（1/2））:m<sub>1</sub>, m<sub>2</sub>,…m<sub>n</sub> nonnegative integers}. In this paper, for n=3, the following result is proved. For α】1/5<sup>1/2</sup> we have that any half-line L, given by x<sub>i</sub>=a<sub>i</sub>t（i=1,2,3）, intersects Δ（C, α）, where parameter t≥0 and a<sub>i</sub>（i=1,2,3） are positive real numbers such that |a|+|b|+|c|≠3 whenever aa<sub>1</sub>+ba<sub>2</sub>+ca<sub>3</sub>=0 for three integers a, b, c.
文摘In order to solve the problem of truncating the open boundaries of cylindrical waveguides used in the simulation of high power microwave （HPM） sources, this paper studies the convolutional PML （CPML） in the cylindrical coordinate system. The electromagnetic field＇s FDTD equations and the expressions of axis boundary conditions are presented. Numerical experiments are conducted to validate the equations and axis boundary conditions. The performance of CPML is simulated when it is used to truncate the cylindrical waveguides excited by the sources with different frequencies and modes in the 2.5-dimensional problems. Numerical results show that the maximum relative errors are all less than -90 dB. The CPML method is introduced in the 2.5-dimensional electromagnetic PIC software, and the relativistic backward wave oscillator is simulated by using this method. The results show that the property of CPML is much better than that of the Mur-type absorbing boundary condition when they are used to truncate the open boundaries of waveguides. The CPML is especially suitable for truncating the open boundaries of the dispersive waveguide devices in the simulation of HPM sources.
文摘Both numerical simulation and theoretical analysis of seawater intrusion in coastal regions are of great theoretical importance in environmental sciences. The mathematical model can be described as a problem of the initial boundary values for a system of 3-dimensional nonlinear parabolic partial differential equations, one being the pressure flow equation and the other is the concentration convection-dispersion equation of the salt contained. For a generic case of a 3-dimensional bounded region, a backward-difference time-stepping scheme is defined. It approximates the pressure by the standard Galerkin procedure and the concentration by a Galerkin method of charederistics, where calculus of variations, theory of prior estimates and techniques are made use of Optimal order estimates in H1 norm are derived for the errors in the approximate solution.
