Fast High Order Algorithm for Three-Dimensional Helmholtz Equation Involving Impedance Boundary Condition with Large Wave Numbers

Acoustic fields with impedance boundary conditions have high engineering applications, such as noise control and evaluation of sound insulation materials, and can be approximated by three-dimensional Helmholtz boundary value problems. Finite difference method is widely applied to solving these problems due to its ease of use. However, when the wave number is large, the pollution effects are still a major difficulty in obtaining accurate numerical solutions. We develop a fast algorithm for solving three-dimensional Helmholtz boundary problems with large wave numbers. The boundary of computational domain is discrete based on high-order compact difference scheme. Using the properties of the tensor product and the discrete Fourier sine transform method, the original problem is solved by splitting it into independent small tridiagonal subsystems. Numerical examples with impedance boundary conditions are used to verify the feasibility and accuracy of the proposed algorithm. Results demonstrate that the algorithm has a fourth- order convergence in and -norms, and costs less CPU calculation time and random access memory.

Study on Hydrodynamic Characteristics of Wave Turbulent Bottom Boundary Layer Using A Large-Sized Wave Flume

Experimental studies were conducted in a super-large wave flume,aiming at uncovering the hydrodynamic characteristics involved in the turbulent wave boundary layer of full scale environment.An explicit formula of boundary layer thickness on rough turbulent flow was presented based on the measured velocity data of the present study and collected experimental data on wave boundary layer.It was found that the bottom wave-associated nominal stresses under the conditions of prototype scale tests suppress the vertical turbulence scattering upward over the boundary layer,which accounts for thickening of the boundary layer under wave condition.Such effect has yet not been reported in the literatures using oscillatory U-tube or small-sized wave flume.The phase inconsistency in the turbulent boundary layer to the free stream velocity(velocity immediately outside the boundary layer)is within15°,which is remarkably smaller than the results from oscillatory U-tubes,as well as the larger wave flume experiment presented by Xie et al.(2021),showing that the coarser bed would further reduce the phase lead.The intensity of the vertical turbulent component is approximately 1/2 of the horizontal component,which has larger ratio compared with the value of 1/5 reported by previous studies.Especially,it was also found that the vertical turbulent energy was approximately 3/4 of the turbulent energy in spanwise directions(y-direction).This means that the turbulent fluctuation has similar order in all three-directions(x,y,z)in a full scale environment and highlights that the turbulent components in all the three directions should not be neglected when calculating the total turbulent energy.

NUMERICAL ANALYSIS OF NONLINEAR WAVE FORCE ON LARGE BODY WITH ARBITRARY SHAPE

In this paper a nonlinear diffraction theory due to Stoke's 2nd-order wave for computing the wave force on the large body is presented. The radiation condition as r-∞ for 2nd-order scattered potential has been studied in connection with asymptotic solutions. A numerical procedure has been developed for the purpose of calculating the nonlinear wave force on the large body with arbitrary shape.

TWO-DIMENSIONAL FINITE ELEMENT METHOD FOR DETERMINING NONLINEAR WAVE FORCES ON LARGE SUBMERGED BODIES

In this paper, we first develop the far field asymptotic solutions of the second-order scattering waves for the vertical plane problem taking the second-order Stokes waves as the incident waves. The asymptotic solutions satisfy the Laplace equation, the sea bed and free surface boundary conditions and are the out-going waves. Then the radiation conditions of the second-order mattering waves are derived by using the asymptotic solutions. By using the two-dimensinal finite clement method with the radiation conditions imposed on the ar- tificial boundaries, the computer program, known as 'NWF2', for determining nonlinear wave forces on large submerged bodies has been written. As a numerical example, nonlinear wave forces on a semi-circu- lar cylinder lying on the sea bed arc presented.

A Numerical Analysis of the Weak Galerk in Method for the Helmholtz Equation with High Wave Number

We study the error analysis of the weak Galerkin finite element method in[24,38](WG-FEM)for the Helmholtz problem with large wave number in two and three dimensions.Using a modified duality argument proposed by Zhu and Wu,we obtain the pre-asymptotic error estimates of the WG-FEM.In particular,the error estimates with explicit dependence on the wave number k are derived.This shows that the pollution error in the broken H1-norm is bounded by O(k(kh)^(2p))under mesh condition k^(7/2)h^(2)≤C0 or(kh)^(2)+k(kh)^(p+1)≤C_(0),which coincides with the phase error of the finite element method obtained by existent dispersion analyses.Here h is the mesh size,p is the order of the approximation space and C_(0) is a constant independent of k and h.Furthermore,numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the WG-FEM in reducing the pollution effect.

Nonlinear stability of traveling waves for a multi-type SIS epidemic model

The nonlinear stability of traveling waves for a multi-type SIS epidemic model is inves- tigated in this paper. By using the comparison principle together with the weighted energy function, we obtain the exponential stability of traveling wavefront with large wave speed. The initial perturbation around the traveling wavefront decays exponen- tially as x → -∞, but it can be arbitrarily large in other locations.

The Large Scale Circulation of the Snow Disaster in Southern China in the Beginning of 2008

This study analyzed the heaviest snowfalls or icy-rainfalls occurring in southern China from January to the beginning of February 2008.The results are summarized as follows：the disaster was induced by the persistent front of warm/cold air masses in southern China,which displayed an interaction between the weather systems in higher and lower latitudes.There was an adjustment for circulation at hemisphere scale during mid January by a variation of sign of the Arctic Oscillation（AO）index from negative to positive.The long lasting precipitation well coincided with a blocking situation centered near 80°E from mid January to the beginning of February.A diagnostic analysis indicates that stationary waves with an energy dispersion accompanying the blocking high propagated from upstream region in high latitudes to the south of the Yangtze River,which formed a maintaining energy source for the cyclonic circulation in situ.This resulted in that the large mass of cold air in high latitudes could not easily invade into the south but slowly shifted southward.On the other hand,the sea surface temperature（SST）over the warm pool of the western Pacific increased with a new history record due to the effect of the strong La Nina episode,which also blocked the cold air mass from the north.The blocking high collapsed around 30 January and the energy source for the local cyclonic circulation was cut off.Thus,the precipitation in southern China ceased after 1 February.

Iterative Pure Source Transfer Domain Decomposition Methods for Helmholtz Equations in Heterogeneous Media

We extend the pure source transfer domain decomposition method(PSTDDM)to solve the perfectly matched layer approximation of Helmholtz scattering problems in heterogeneous media.We first propose some new source transfer operators,and then introduce the layer-wise and block-wise PSTDDMs based on these operators.In particular,it is proved that the solution obtained by the layer-wise PSTDDM in R2 coincides with the exact solution to the heterogeneous Helmholtz problem in the computational domain.Second,we propose the iterative layer-wise and blockwise PSTDDMs,which are designed by simply iterating the PSTDDM alternatively over two staggered decompositions of the computational domain.Finally,extensive numerical tests in two and three dimensions show that,as the preconditioner for the GMRES method,the iterative PSTDDMs are more robust and efficient than PSTDDMs for solving heterogeneous Helmholtz problems.

A Numerical Study on the Weak Galerkin Method for the Helmholtz Equation

A weak Galerkin(WG)method is introduced and numerically tested for the Helmholtz equation.This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property.At the same time,the WG finite element formulation is symmetric and parameter free.Several test scenarios are designed for a numerical investigation on the accuracy,convergence,and robustness of the WG method in both inhomogeneous and homogeneous media over convex and non-convex domains.Challenging problems with high wave numbers are also examined.Our numerical experiments indicate that the weak Galerkin is a finite element technique that is easy to implement,and provides very accurate and robust numerical solutions for the Helmholtz problem with high wave numbers.