A Fourier approximation for finite amplitude short‐crested waves

Abstract

A numerical scheme using Fourier series approximation is presented to calculate short‐crested waves of finite amplitude in water of arbitrary uniform depth. The numerical model preserves the water surface elevation in an implicit form which retains the nonlinear nature of the dynamic and kinematic free surface boundary conditions. Accurate solutions can be derived for the variations in frequency, wave profiles and properties of pressures. The present model is directly reducible to the two‐dimensional limiting cases of progressive and standing waves.     

Initial conditions contribution in a BEM formulation based on the convolution quadrature method

This work presents a two‐dimensional boundary element method (BEM) formulation for the analysis of scalar wave propagation problems. The formulation is based on the so‐called convolution quadrature method (CQM) by means of which the convolution integral, presented in time‐domain BEM formulations, is numerically substituted by a quadrature formula, whose weights are computed using the Laplace transform of the fundamental solution and a linear multistep method. This BEM formulation was initially developed for scalar wave propagation problems with null initial conditions. In order to overcome this limitation, this work presents a general procedure that enables one to take into account non‐homogeneous initial conditions, after replacing the initial conditions by equivalent pseudo‐forces. The numerical results included in this work show the accuracy of the proposed BEM formulation and its applicability to such kind of analysis. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite element solution of free‐surface ship‐wave problems

An unstructured finite element solver to evaluate the ship‐wave problem is presented. The scheme uses a non‐structured finite element algorithm for the Euler or Navier–Stokes flow as for the free‐surface boundary problem. The incompressible flow equations are solved via a fractional step method whereas the non‐linear free‐surface equation is solved via a reference surface which allows fixed and moving meshes. A new non‐structured stabilized approximation is used to eliminate spurious numerical oscillations of the free surface. Copyright © 1999 John Wiley & Sons, Ltd.     

The spectral element method for elastic wave equations—application to 2‐D and 3‐D seismic problems

A spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two‐ and three‐dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, defined with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor‐product of a Gauss–Lobatto–Legendre 1‐D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy‐momentum conserving scheme that can be put into a classical explicit‐implicit predictor/multicorrector format. Long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two‐dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three‐dimensional model. Very accurate modelling of Rayleigh wave propagation and surface diffraction is obtained at a low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three‐dimensional surface topographies. Copyright © 1999 John Wiley & Sons, Ltd.     

A new particle method for simulation of incompressible free surface flow problems

A new Lagrangian particle method called the consistent particle method (CPM), which solves the Navier–Stokes equations in a semi‐implicit time stepping scheme, is proposed in this paper. Instead of using kernel function as in some particle methods, partial differential operators are approximated in a way consistent with Taylor series expansion. A boundary particle recognition method is applied to help define the changing liquid domain. The incompressibility condition of free surface particles is enforced by an adjustment scheme. With these improvements, the CPM is shown to be robust and accurate in long time simulation of free surface flow particularly for smooth pressure solution. Two types of free surface flow problems are presented to verify the CPM, that is, two‐dimensional dam break and liquid sloshing in a rectangular tank. In the dam break example, the CPM solutions of pressure and wave elevation are in good agreement with published experimental results. In addition, an experimental study of water sloshing in tank on a shake table was conducted to verify the CPM solutions. Copyright © 2011 John Wiley & Sons, Ltd.     

An implicit boundary element solution with consistent linearization for free surface flows and non‐linear fluid–structure interaction of floating bodies

In this work, a new comprehensive method has been developed which enables the solution of large, non‐linear motions of rigid bodies in a fluid with a free surface. The application of the modern Eulerian–Lagrangian approach has been translated into an implicit time‐integration formulation, a development which enables the use of larger time steps (where accuracy requirements allow it). Novel features of this project include: (1) an implicit formulation of the rigid‐body motion in a fluid with a free surface valid for both two or three dimensions and several moving bodies; (2) a complete formulation and solution of the initial conditions; (3) a fully consistent (exact) linearization for free surface flows valid for any boundary elements such that optimal convergence properties are obtained when using a Newton–Raphson solver. The proposed framework has been completed with details on implementation issues referring mainly to the computation of the complete initial conditions and the consistent linearization of the formulation for free surface flows. The second part of the paper demonstrates the mathematical and numerical formulation through numerical results simulating large free surface flows and non‐linear fluid structure interaction. The implicit formulation using a fully consistent linearization based on the boundary element method and the generalized trapezoidal rule has been applied to the solution of free surface flows for the evolution of a triangular wave, the generation of tsunamis and the propagation of a wave up to overturning. Fluid–structure interaction examples include the free and forced motion of a circular cylinder and the sway, heave and roll motion of a U‐shaped body in a tank with a flap wave generator. The presented examples demonstrate the applicability and performance of the implicit scheme with consistent linearization. Copyright © 2001 John Wiley & Sons. Ltd.     

Comparative topographic study of surface micro‐relief of primary martensite plates in shape memory alloys with different crystalline structures

The surface micro‐reliefs of primary martensite plates, representative for two shape memory alloys (SMAs) with different crystalline structures were compared from qualitative and quantitative point of view by scanning electron microscopy and atomic force microscopy, respectively. Qualitative evaluations revealed larger widths and heights of the primary plates of ε hexagonal close packed (hcp) martensite, in an Fe‐Mn‐Si‐Cr‐Ni SMA than those of β₂′ orthorhombic (9R) martensite, in a Cu‐Zn‐Al SMA. Quantitative evaluations were based on systematic dimensional measurements of the width and height of primary plate profiles. The measurements were performed on one hundred and twenty five profiles, five on each martensite plate belonging to five typical groups of primary plates, with length above 50 micrometers, of both ε hcp and β₂′ 9R martensites. In order to compare the topographies of the two types of plates a statistical evaluation of the dimensional intervals of width and height of measured plates was performed.     

Mesh‐free radial point interpolation method for the buckling analysis of Mindlin plates subjected to in‐plane point loads

In this paper, a mesh‐free approach is employed for buckling analysis of Mindlin plates that are subjected to in‐plane point loads. The radial point interpolation method (RPIM) is used to approximate displacements based on nodes. Variational forms of the system equations are established. Two‐step solution procedures are implemented. The non‐uniform pre‐stress distribution of plate is first obtained using the RPIM based on a two‐dimensional (2D) elastic plane stress problem. This predetermined non‐uniform pre‐stress distribution is then used to compute buckling loads of plate using the RPIM based on Mindlin's plate assumption. The RPIM can easily handle any number and location of nodes in the plate domain for a desired computational accuracy without major difficulties in solving the initial stresses and buckling loads. Numerical examples considered here include circular and rectangular Mindlin plates that are subjected to in‐plane uniform and point loads with different aspect ratios and boundary conditions. The present results are validated against the available analytical and numerical solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

A decoupled augmented IIM strategy for incompressible two‐phase flows with interfaces on irregular domains

A decoupled augmented immersed interface method for solving incompressible two‐phase flows involving both irregular domains and interfaces is presented. In order to impose the prescribed velocity at the boundary of the irregular domain, singular force as one set of augmented variables is introduced. The velocity components at the two‐fluid interface as another set of augmented variables are introduced to satisfy the continuity condition of the velocity across the interface so that the jump conditions for the velocity and pressure are decoupled across the interface. The augmented variables and/or the forces along the interface/boundary are related to the jumps in both pressure and velocity and the jumps in their derivatives across the interface/boundary and applied to the fluid through jump conditions. The resulting augmented equation is a couple system of these two sets of augmented variables, and the direct application of the GMRES is impractical due to larger iterations. In this work, the novel decoupling of two sets of the augmented variables is proposed, and the decoupled augmented equation is then solved by the LU or the GMRES method. The Stokes equations are discretized via the finite difference method with the incorporation of jump contributions on a staggered Cartesian grid and solved by the conjugate gradient Uzawa‐type method. The numerical results show that second‐order accuracy for the velocity is confirmed. The present method has also been applied to solve for incompressible two‐phase Navier–Stokes flow with interfaces on irregular domains. Copyright © 2011 John Wiley & Sons, Ltd.     

Vibration isolation by a row of piles using a 3‐D frequency domain BEM

The problem of vibration isolation by a row of piles is numerically solved in a three‐dimensional context by an advanced frequency domain boundary element method (BEM). Both the piles and the soil are modelled by boundary elements and coupled together through equilibrium and compatibility at their interfaces. Linear elastic or viscoelastic material behaviour is assumed for both the piles and the soil. The piles can be tubular or solid and have circular or square cross‐section. The vibration source is a vertical harmonically varying with time force and the row of piles acts as a wave barrier in a passive way. The boundary element method is first validated for accuracy by solving two three‐dimensional wave diffraction problems dealing with spheres and trenches as scatterers for which there are analytical and highly accurate numerical solutions available in the literature. Numerical examples dealing with passive vibration isolation by a row of piles are then solved and the screening effectiveness of these wave barriers is assessed and compared against that of trenches. Copyright © 1999 John Wiley & Sons, Ltd.     

Two‐dimensional unsteady heat conduction analysis with heat generation by triple‐reciprocity BEM

If the initial temperature is assumed to be constant, a domain integral is not needed to solve unsteady heat conduction problems without heat generation using the boundary element method (BEM).However, with heat generation or a non‐uniform initial temperature distribution, the domain integral is necessary. This paper demonstrates that two‐dimensional problems of unsteady heat conduction with heat generation and a non‐uniform initial temperature distribution can be solved approximately without the domain integral by the triple‐reciprocity boundary element method. In this method, heat generation and the initial temperature distribution are interpolated using the boundary integral equation. Copyright © 2001 John Wiley & Sons, Ltd.     

The boundary layer phenomena in two‐dimensional transversely isotropic piezoelectric media by exact symplectic expansion

A separable variable method is introduced to find the exact homogeneous solutions of a two‐dimensional transversely isotropic piezoelectric media to handle general boundary conditions. The usual method of separable variables for partial differentiation equations cannot be readily applicable due to the tangling of the unknowns and their derivatives. Introducing dual variables of stresses, we obtain a set of first‐order Hamiltonian equations whose eigensolutions are symplectic spanning over the solution space to cover all possible boundary conditions. The solutions consist of two parts. The first part is the derogative zero‐eigenvalue solutions of the Saint Venant type together with all their Jordan chains. The second part is the decaying non‐zero‐eigenvalue solutions describing the boundary layer effects. The classical solutions are actually the zero‐eigenvalue solutions representing the simple extension, bending, equipotential field, and the uniform electric displacement. On the other hand, the non‐zero‐eigenvalue solutions represent the localized solutions, which are sensitive to the boundary conditions and are decaying rapidly with respect to the distance from the boundaries. Some rate‐of‐decay curves of the newly found non‐zero‐eigenvalue solutions are shown by numerical examples. Finally, the complete boundary layer effects are quantified for the first time. Copyright © 2006 John Wiley & Sons, Ltd.     

Mesh independent analysis of shell‐like structures

Mesh independent analysis is motivated by the desire to use accurate geometric models represented as equations rather than approximated by a mesh. The trial and test functions are approximated or interpolated on a background mesh that is independent of the geometry. This background mesh is easy to generate because it does not have to conform to the geometry. Essential boundary conditions can be applied using the implicit boundary method where the trial and test functions are constructed utilizing approximate step functions such that the boundary conditions are guaranteed to be satisfied. This approach has been demonstrated for two‐dimensional (2D) and three‐dimensional (3D) structural analysis and is extended in this paper to model shell‐like structures. The background mesh consists of 3D elements that use uniform B‐spline approximations, and the shell geometry is assumed to be defined as parametric surfaces to allow arbitrarily complex shell‐like structures to be modeled. Several benchmark problems are used to study the validity of these 3D B‐spline shell elements. Copyright © 2012 John Wiley & Sons, Ltd.     

Crack–surface displacements for cracks emanating from a circular hole under various loading conditions

The purpose of this paper is to calculate and develop equations for crack–surface displacements for two‐symmetric cracks emanating from a circular hole in an infinite plate for use in strip‐yield crack‐closure models. In particular, the displacements were determined under two loading conditions: (1) remote applied stress and (2) uniform stress applied to a segment of the crack surface (partially loaded crack). The displacements were calculated by an integral‐equation method based on accurate stress–intensity factor equations for concentrated forces applied to the crack surfaces and those for remote applied stress or for a partially loaded crack surface. A boundary‐element code was also used to calculate crack–surface displacements for some selected cases. Comparisons made with crack–surface displacement equations previously developed for the same crack configuration and loading showed significant differences near the location where the crack intersected the hole surface. However, the previous equations were fairly accurate near the crack‐tip location. Herein an improved crack–surface displacement equation was developed for the case of remote applied stress. For the partially loaded crack case, only numerical comparisons were made between the previous equations and numerical integration. A rapid algorithm, based on the integral‐equation method, was developed to calculate these displacements. Because cracks emanating from a hole are quite common in the aerospace industry, accurate displacement solutions are crucial for improving life‐prediction methods based on the strip‐yield crack‐closure models.     

Non‐reflecting artificial boundaries for transient scalar wave propagation in a two‐dimensional infinite homogeneous layer

This paper presents an exact non‐reflecting boundary condition for dealing with transient scalar wave propagation problems in a two‐dimensional infinite homogeneous layer. In order to model the complicated geometry and material properties in the near field, two vertical artificial boundaries are considered in the infinite layer so as to truncate the infinite domain into a finite domain. This treatment requires the appropriate boundary conditions, which are often referred to as the artificial boundary conditions, to be applied on the truncated boundaries. Since the infinite extension direction is different for these two truncated vertical boundaries, namely one extends toward x →∞ and another extends toward x→‐ ∞, the non‐reflecting boundary condition needs to be derived on these two boundaries. Applying the variable separation method to the wave equation results in a reduction in spatial variables by one. The reduced wave equation, which is a time‐dependent partial differential equation with only one spatial variable, can be further changed into a linear first‐order ordinary differential equation by using both the operator splitting method and the modal radiation function concept simultaneously. As a result, the non‐reflecting artificial boundary condition can be obtained by solving the ordinary differential equation whose stability is ensured. Some numerical examples have demonstrated that the non‐reflecting boundary condition is of high accuracy in dealing with scalar wave propagation problems in infinite and semi‐infinite media. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical simulation of fluid flow and temperature field in keyhole double‐sided arc welding process on stainless steel

Double‐sided arc welding process powered by a single supply is a type of novel high‐production process. In comparison with the conventional single‐sided arc welding, this process has remarkable advantages in enhancing penetration, minimizing distortion and improving welding production. In this paper, a three‐dimensional steady numerical model is developed for the heat transfer and fluid flow in plasma arc (PA)–gas tungsten arc (GTA) double‐sided keyhole welding process. The model considers the surface tension gradient, electromagnetic force and buoyancy force. A CCD camera is used to observe the size and shape of the keyhole and weld pool. The acquired images are analysed through image processing to obtain the surface diameters of the keyhole on the two sides. A double‐V‐shaped keyhole geometry is then proposed and its characteristic parameters are derived from the images and cross‐section of weld bead. In the numerical model, the keyhole cavum within the weld pool is treated as a whole quality, whose temperature is fixed at the boiling point of the workpiece material. The heat exchange between the keyhole and weld pool is treated as an interior boundary of the workpiece. Based on the numerical model, the distributions of the fluid flow and temperature field are calculated. A comparison of cross‐section of the weld bead with the experimental result shows that the numerical model's accuracy is reasonable. Copyright © 2005 John Wiley & Sons, Ltd.     

Large‐eddy simulation of separated leading‐edge flow in general co‐ordinates

An incompressible separated transitional boundary‐layer flow on a flat plate with a semi‐circular leading edge has been simulated and a very good agreement with the experimental data has been obtained, demonstrating how this technique may be applied even when finite difference formulae are used in the periodic direction. The entire transition process has been elucidated and vortical structures have been identified at different stages during the transition process. Efficient numerical methods for the large‐eddy simulation (LES) of turbulent flows in complex geometry are developed. The methods used are described in detail: body‐fitted co‐ordinates with the contravariant velocity components of the general Navier–Stokes equations discretized on a staggered mesh with a dynamic subgrid‐scale model in general co‐ordinates. The main source of computational expense in simulations for incompressible flows is due to the solution of a Poisson equation for pressure. This is especially true for flows in complex geometry. Fourier techniques can be employed to speed up the pressure solution significantly for a flow which is periodic in one dimension. With simple conditions fulfilled, it is possible to Fourier transform a discrete elliptic equation such as the Poisson equation for the pressure field, decomposing the problem into a set of two‐dimensional problems of similar type (Poisson‐like). Even when a complex geometry and body‐fitted curvilinear co‐ordinates are used in the other two dimensions, as in the present case, the resulting Fourier‐transformed 2D problems are much more efficiently solved than the 3D problem by iterative means. Copyright © 2000 John Wiley & Sons, Ltd.     

The meshless Galerkin boundary node method for Stokes problems in three dimensions

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines an equivalent variational formulation of boundary integral equations for governing equations and the moving least‐squares (MLS) approximations for generating the trial and test functions. In this approach, boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Besides, the resulting formulation inherits the symmetry and positive definiteness of the variational problems. The GBNM is developed in this paper for solving three‐dimensional stationary incompressible Stokes flows in primitive variables. The numerical scheme is based on variational formulations for the first‐kind integral equations, which are valid for both interior and exterior problems simultaneously. A rigorous error analysis and convergence study of the method for both the velocity and the pressure is presented in Sobolev spaces. The capability of the method is also illustrated and assessed through some selected numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.     

Wave propagation in presence of oscillators on the free surface

This article is devoted to wave propagation in presence of a periodic distribution of oscillators (3D sprung-damped mass) on the surface of an elastic medium. The case of an elastic half space loaded by quasi-periodic surface forces is studied at first. The equivalent boundary conditions at the macro-scale are derived at the first and second order, by assuming the existence of a boundary layer and using a two-dimensional homogenisation method. When the force distribution results from attached oscillators, it is shown that boundary conditions can be expressed in terms of equivalent surface impedance at the first order, with local and non-local correctors at the second order. These results are used to study wave refraction by an oscillator layer. The main phenomena—atypical redistribution of mode and mode conversion, frequency range of efficiency, characteristic time of response—are identified. The first correctors are determined analytically for plane waves of oblique incidence. Finally the validity range of the modelling is discussed.     

Hypersingular integral formulation of elastic wave scattering

A hypersingular boundary integral formulation for calculating two dimensional elastic wave scattering from thin bodies and cracks is described. The boundary integral equation for surface displacement is combined with the hypersingular equation for surface traction. The difficult part in employing the traction equation, the derivation of analytical formulas for the hypersingular integral by means of a limit to the boundary, is easily handled by means of symbolic computation. In addition, the terms containing an integrable logarithmic singularity are treated by a straightforward numerical method, bypassing the use of Taylor series expansions. Example wave scattering calculations for cracks and thin ellipses are presented.