The extended Durbin method and its application for piezoelectric wave propagation problems

Based on the well-known Durbin method, an efficient numerical method was developed for the inversion of the two-sided Laplace transform. The accuracy of the method was verified using examples. As an application of the method, transient elastic waves propagating in a two-layered piezoelectric medium subjected to anti-plane concentrated loading and in-plane electric displacement loading were investigated. One-sided and two-sided Laplace transforms were applied to determine the shear stresses and electric displacements in the double Laplace transform domain. Subsequently, the Durbin method for one-sided Laplace transform inversion and the extended Durbin method for two-sided Laplace transform inversion were used to implement the numerical inversions. Additionally, the numerical results of the transient stresses and electric displacements were evaluated and discussed. It showed that the arrival time of transient waves satisfies physical phenomena, and the transient solution oscillates near the static solution and rapidly approximates the static solution.     

Multilayer open flow model: A simple pressure correction method for wave problems

This paper describes a pressure correction method for single‐ and multilayer open flow models. The method does not require any complex procedures to solve the discretization of the Poisson equation and is distinguished by a high computational efficiency. The algorithm can easily be adapted to irregular meshes and parallelized. Parabolic interpolation of the pressure profile is used for the free surface. The discretization of the Poisson equation is written in a matrix form, allowing its usage also in the case of basic function expansion of the depth pressure profile. The paper presents the results of algorithm verification where experimental data sensitive to the numerical dissipation of the calculation model was used. Iteration convergence is high including problems with dry‐bed flooding. The complete described technique of pressure correction is implemented in OpenCL on the GPU. Computation time for a test problem solved using CPU and GPU is compared.     

Finite difference method of transinet nonlinear free surface wave problems

A finite difference method is developed for computing the two-dimensional transient potential flow generated by an impulse on the free surface. Both the dynamic and kinematic free surface conditions are considered in nonlinear version. the primary features of the present paper include the use of special coordinates transformations so that the geometry of the flow field is transformed into a time-invariant region, presents an iteration process, by which the velocity potential is computed as the solution of a Poisson equation, the application of fast Fourier transform (FFT) technique results in a tri-diagonal system of equations which can be readily solved by the Thomas algorithm, the computing time is significantly reduced. Thus an efficient technique for handling the transient potential problems is well justified. The feasibility of the present method has been verified by two examples including different initial disturbances respectively.     

A wideband fast multipole boundary element method for half-space/plane-symmetric acoustic wave problems

This paper presents a novel wideband fast multipole boundary element approach to 3D half-space/planesymmetric acoustic wave problems.The half-space fundamental solution is employed in the boundary integral equations so that the tree structure required in the fast multipole algorithm is constructed for the boundary elements in the real domain only.Moreover,a set of symmetric relations between the multipole expansion coefficients of the real and image domains are derived,and the half-space fundamental solution is modified for the purpose of applying such relations to avoid calculating,translating and saving the multipole/local expansion coefficients of the image domain.The wideband adaptive multilevel fast multipole algorithm associated with the iterative solver GMRES is employed so that the present method is accurate and efficient for both lowand high-frequency acoustic wave problems.As for exterior acoustic problems,the Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem involved in the conventional boundary integral equation method.Details on the implementation of the present method are described,and numerical examples are given to demonstrate its accuracy and efficiency.     

A finite element method for two‐dimensional water‐wave problems

In this paper, the authors treat the free‐surface waves generated by a moving disturbance with a constant speed in water of finite and constant depth. Specifically, the case when the disturbance is moving with the critical speed is investigated. The water is assumed inviscid and its motion irrotational. The surface tension is neglected. It is well‐known that the linear theory breaks down when a disturbance is moving with the critical speed. As a remedy to overcome the invalid linear theory, approximate non‐linear theories have been applied with success in the past, i.e. Boussinesq and Korteweg de Vries equations, for example. In the present paper, the authors describe a finite element method applied to the non‐linear water‐wave problems in two dimensions. The present numerical method solves the exact non‐linear formulation in the scope of potential theory without any additional assumptions on the magnitude of the disturbances. The present numerical results are compared with those obtained by other approximate non‐linear theories. Also presented are the discussions on the validity of the existing approximate theories applied to two types of the disturbances, i.e. the bottom bump and the pressure patch on the free‐surface at the critical speed. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite difference method for 2-D and 3-D nonlinear free surface wave problems

A finite difference method is developed for the numerical modelling of the 2-D and 3-D unsteady potential flow generated by transient disturbances on the free surface, on which the nonlinear boundary conditions are fully satisfied. The unknown function is computed with an iteration scheme processing in a transformed time-invariant space. After the velocity is calculated, the location of the free surface is renewed and so is the value of velocity on it. The boundary-value problem of the governing equation is then solved at the next time step. The present method incorporates the FFT. Consequently, a tri-diagonal equation system is obtained which could be readily solved. The feasibility of this method has been demonstrated by 2-D and 3-D examples corresponding to different initial disturbances. This work is supported by the science foundation of Academia Sinica. The paper had been accepted by the XVIth International Congress of IUTAM, Lyngby, Denmark, August, 1984.     

Hydrodynamic coefficients for vertical plane wave motion problems

In this paper a numerical method for calculating the hydro-dynamic coefficients for vertical plane wave motion problems in deep water is described.This procedure is developed by using the wave source method based on Green‘s theorem.The applications of the method to the cases of semi-circular and rectangular section bodies subjected to linear waves are pre-sented here,and the computed results are compared with the earlier experimental data of Vugts.     

A general numerical method for solution of gravity wave problems. Part 2: Steady nonlinear gravity waves

A type of numerical scheme for 2D and 3D steady non-linear water wave problems is described. It is based on the finite process method and is insensitive to initial solutions. The relationship between the finite process method and iterative techniques is discussed. As a numerical example the flow past a submerged vortex is solved and the results are compared with those of other authors.     

Tikhonovs regularization method for ill-posed problems

Frequently the determination of material characteristic functions, such as the molecular mass distribution of a polymeric sample or the relaxation spectrum of a viscoelastic fluid, leads to an ill-posed problem. When Tikhonov regularization is applied to such a problem the problem of an appropriate choice of the regularization parameter arises. Well-known methods to determine this parameter, such as the discrepancy principle, and a method based on the minimization of the predictive mean-square signal error are compared with a self-consistence method. Monte Carlo simulations have been carried out for the determination of the relaxation spectrum from small amplitude oscillatory shear flow data. The self-consistence method has proven to be much more robust and reliable.     

Stabilization meshless method for convection-dominated problems

It is weN-known that the standard Galerkin is not ideally suited to deal with the spatial discretization of convection-dominated problems. In this paper, several techniques are proposed to overcome the instabilitY issues in convection-dominated problems in the simulation with a meshless method. These stable techniques included nodal refinement, enlargement of the nodal influence domain, full upwind meshless technique and adaptive upwind meshless technique. Numerical results for sample problems show that these techniques are effective in solving convection-dominated problems, and the adaptive upwind meshless technique is the most effective method of all.     

Strip method for steady heat-conduction problems

Summary An approximate solution of heat-conduction problems can be obtained by the strip method. The method consists of an application of the finite-difference approximation in one physical coordinate and an analytic solution in other coordinates. A simple illustrative example is given and the result is compared with that obtained by the exact solution. By application of this method, an approximate solution is given for the steady heat conduction through a rectangular parallel composite wall with different rates of heat generation.     

A boundary-type finite element model for water surface wave problems

A new combinative method of boundary-type finite elements and boundary solutions is presented to study wave diffraction-refraction and harbour oscillation problems. The numerical model is based on the mild-slope equation. The key feature of this method is that the discretized matrix equation can be formulated only by the calculation of a line integral, since the interpolation equation which satisfies the governing equation in each element is used. The numerical solutions are compared with existing analytical, experimental, observed and other numerical results. The present method is shown to be an effective and accurate method for water surface wave problems.     

Solving the velocity parameter of the 2-D wave inverse problems with the integration-characteristic method

For the 2 -D wave inverse problems introduced from geophysical exploration, in this paper, the author presents integration-characteristic method to solve the velocity parameter, and then applies it to common shotpoint model data, in noise-free case. The accuracy is quite good.     

Boundary-type finite element method for wave propagation analysis

The boundary-type finite element method has been investigated and applied to the Helmholz and mild-slope equations. Four types of interpolation function are examined based on trigonometric function series. Three-node triangular, four-node quadrilateral, six-node triangular and eight-node quadrilateral elements are tested; these are all non-conforming elements. Three types of numerical example show that the three-node triangular and four-node quadrilateral elements are useful for practical analysis.     

The semi-discrete method for solving high-dimension wave equation

The article gives a semi-discrete method for solving high-dimension wave equation. By the method, high-dimension wave equation is converted by means of discretization into I-D wave equation system which is well-posed. The convergence of the semidiscrete method is given. The numerical calculating results show that the speed of convergence is high.