Numerical modeling of wave breaking induced by fixed or moving boundaries

In this paper, several numerical aspects of an existing model for fully nonlinear waves are improved and validated to study wave breaking due to shoaling over a gentle plane slope and wave breaking induced by a moving lateral boundary.The model is based on fully nonlinear potential flow theory and combines a higher-order Boundary Element Method (BEM) for solving Laplace's equation at a given time and Lagrangian Taylor expansions for the time updating of the free surface position and potential. An improved numerical treatment of the boundary conditions at the intersection between moving lateral boundaries and the free surface (corner) is implemented and tested in the model, and the free surface interpolation method is also improved to better model highly curved regions of the free surface that occur in breaking waves. Finally, a node regridding technique is introduced to improve the resolution of the solution close to moving boundaries and in breaker jets.Examples are presented for solitary wave propagation, shoaling, and breaking over a 1:35 slope and for wave breaking induced by a moving vertical boundary. Using the new methods, both resolution and extent of computations are significantly improved compared to the earlier model, for similar computational efforts. In all cases computations can be carried out up to impact of the breaker jets on the free surface.     

Implicit formulation with the boundary element method for nonlinear radiation of water waves

An accurate and efficient numerical method is presented for the two-dimensional nonlinear radiation problem of water waves. The wave motion that occurs on water due to an oscillating body is described under the assumption of ideal fluid flow. The governing Laplace equation is effectively solved by utilizing the GMRES (Generalized Minimal RESidual) algorithm for the boundary element method (BEM) with quadratic approximation. The intersection or corner singularity in the mixed Dirichlet–Neumann problem is resolved by introducing discontinuous elements. The fully implicit trapezoidal rule is used to update solutions at new time-steps, by considering stability and accuracy. Traveling waves generated by the oscillating body are absorbed downstream by the damping zone technique. To avoid the numerical instability caused by the local gathering of grid points, the re-gridding technique is employed, so that all the grids on the free surface may be re-distributed with an equal distance between them. The nonlinear radiation force is evaluated by means of the acceleration potential. For a mixed Dirichlet–Neumann problem in a computational domain with a wavy top boundary, the present BEM yields numerical solutions for the quadratic rate of convergence with respect to the number of boundary elements. It is also demonstrated that the present time-marching and radiation condition work successfully for nonlinear radiation problems of water waves. The results obtained from this study concur reasonably well with other numerical computations.     

Numerical simulation of the dynamic response of floating structures

This paper summarizes a computational procedure for the simulation of wave-structure interactions. The procedure combines the Boundary Element Method (BEM) for potential flows and a predictor-corrector scheme for the time-integration of the nonlinear free-surface condition. In order to have a good representation of a domain of infinite extent, a significant part of the procedure deals with the formulation of a nonreflecting boundary condition for multidirectional waves. This formulation is based on generalizations of modal-superposition in a least-squares sense. In spite of the assumption that the linearized free-surface condition is satisfied in the far field, the proposed nonreflecting boundary condition gives reasonably good performance in absorbing the nonlinear free-surface waves from the interior. The verification and validation of the time–domain computations show very good agreement between the numerical and experimental data from a physical wave basin. Financial support for this research, provided by the Offshore Technology Research Center and the Texas Advanced Research Program, is gratefully acknowledged. Computing resources have been made available by the Offshore Technology Research Center and the Department of Civil Engineering at The University of Texas at Austin.     

A boundary-integral method for water wave motion over irregular beds

This paper describes a numerical method for solving the fully nonlinear equations of motion for gravity waves propagating over an irregular rigid bed, in two dimensions. The scheme is based on a boundary-integral method developed by Dold and Peregrine¹, and extended by Tanaka et al.². The method quickly and accurately computes the position of the free surface of an inviscid, incompressible fluid in irrotational motion. The schemes of Dold and Peregrine¹, and Tanaka, Dold, Lewy and Peregrine², describe waves in a flat-bottomed domain. In this paper the method is extended to describe wave propagation over an irregular bed contour, and follows², in that the domain is unbounded in the horizontal, with static flow at infinity. A conformal transformation is used to map the irregular bed at each time-step to a flat-bedded domain. In the transformed plane symmetry is used so that discretization points need only be used on the free surface. The conformal map is a relatively small contribution to the run-time of the computer program. The method is more efficient than schemes which use discretization points to describe the bed contour. The method's capability is demonstrated with computations of a wave breaking on a submerged elliptical shoal.     

Development of a 3D numerical wave tank for modeling tsunami generation by underwater landslides

A three-dimensional (3D) numerical wave tank (NWT) solving fully nonlinear potential flow theory, with a higher-order boundary element method (BEM), is modified to simulate tsunami generation by underwater landslides. New features are added to the NWT to model underwater landslide geometry and motion and specify corresponding boundary conditions in the BEM model. In particular, a new snake absorbing piston boundary condition is implemented to remove reflection from the onshore and offshore boundaries of the NWT. Model results are favorably compared to recent laboratory experiments. Sensitivity analyses of numerical results to the width and length of the discretization are conducted, to determine optimal numerical parameters. The effect of landslide width on tsunami generated is estimated. Results show that the two-dimensional approximation is applicable when the ratio of landslide width over landslide length is greater than 2. Numerical accuracy is examined and found to be excellent in all cases.     

Applicability of the method of fundamental solutions to 3-D wave–body interaction with fully nonlinear free surface

A numerical model for three-dimensional fully nonlinear free-surface waves is developed by applying a boundary-type meshless approach with a leap-frog time-marching scheme. Adopting Gaussian Radial Basis Functions to fit the free surface, a non-iterative approach to discretize the nonlinear free-surface boundary is formulated. Using the fundamental solutions of the Laplace equation as the solution form of the velocity potential, free-surface wave problems can be solved by collocations at only a few boundary points since the governing equation is automatically satisfied. The accuracy of the present method is verified by comparing the simulated propagation of a solitary wave with an exact solution. The applicability of the present model is illustrated by applying it to the problem of a solitary wave running up on a vertical surface-piercing cylinder and the problem of wave generation in infinite water depth by a submerged moving object.     

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.     

Singular enrichment functions for Helmholtz scattering at corner locations using the boundary element method

In this paper, we use an enriched approximation space for the efficient and accurate solution of the Helmholtz equation to solve problems of wave scattering by polygonal obstacles. This is implemented in both boundary element method (BEM) and partition of unity BEM (PUBEM) settings. The enrichment draws upon the asymptotic singular behaviour of scattered fields at sharp corners, leading to a choice of fractional-order Bessel functions that complement the existing Lagrangian (BEM) or plane wave (PUBEM) approximation spaces. Numerical examples consider configurations of scattering objects, subject to the Neumann “sound hard” boundary conditions, demonstrating that the approach is a suitable choice for both convex scatterers and also for multiple scattering objects that give rise to multiple reflections. Substantial improvements are observed, significantly reducing the number of degrees of freedom required to achieve a prescribed accuracy in the vicinity of a sharp corner.     

A least squares procedure for the evaluation of multiple generalized stress intensity factors at 2D multimaterial corners by BEM

Detailed characterization of linear elastic stress states at corners and crack tips requires knowledge of the stress singularity orders, the characteristic angular functions and the generalized stress intensity factors (GSIF). Typically a high accuracy is found in the literature for the evaluation of the stress singularity orders and characteristic angular functions (numerically computed from analytical expressions in most cases). Nevertheless, GSIF values, evaluated by means of a numerical model using FEM or BEM and usually by postprocessing the results, are often reported with a lower level of confidence. A robust procedure is presented in this work for the evaluation of the GSIF at multimaterial corners. The procedure is based on a simple least squares technique involving stresses and/or displacements, computed by BEM, at the neighborhood of the corner tip. A careful verification of the robustness and accuracy of the procedure using a few benchmark problems in the literature has been carried out. Applications of the procedure developed to the evaluation of GSIFs appearing at corners in metal-composite adhesive lap joints are presented.     

A second-order reduced asymptotic homogenization approach for nonlinear periodic heterogeneous materials

An efficient second-order reduced asymptotic homogenization approach is developed for nonlinear heterogeneous media with large periodic microstructure. The two salient features of the proposed approach are (i) an asymptotic higher-order nonlinear homogenization that does not require higher-order continuity of the coarse-scale solution and (ii) an efficient model reduction scheme for solving higher-order nonlinear unit cell problems at a fraction of computational cost in comparison to the direct computational homogenization. The former is a consequence of a sequential solution of increasing order solutions, which permits evaluation of higher-order coarse-scale derivatives by postprocessing from the zeroth-order solution. The efficiency and accuracy of the formulation in comparison to the classical zeroth-order homogenization and direct numerical simulations are assessed on hyperelastic and elastoplastic periodic structures.     

Boundary element analysis of inclusions with corners

Boundary element method (BEM) has proven to have very good resolution of large stress gradients such as those that may arise at material interface and reentrant corners. There is, however, a paucity of literature in usage of BEM when the inclusion has a corner. The stress singularity at the corner creates numerical difficulties that need to be addressed. This paper describes: application of BEM to inclusion with and without corners; the numerical modeling difficulties; a methodology for calculation of eigenvalues and stress intensity factors without elaborate analytical expressions; and the future research that is needed for the growth of the boundary element methodology for application to inclusion problems. Numerical results for a rectangular inclusion with sharp and fillet corners that in the limit becomes a circular inclusion demonstrate the potential of the proposed methodology in the analysis of inclusion problems.     

3D non-linear time domain FEM–BEM approach to soil–structure interaction problems

Dynamic soil–structure interaction is concerned with the study of structures supported on flexible soils and subjected to dynamic actions. Methods combining the finite element method (FEM) and the boundary element method (BEM) are well suited to address dynamic soil–structure interaction problems. Hence, FEM–BEM models have been widely used. However, non-linear contact conditions and non-linear behavior of the structures have not usually been considered in the analyses. This paper presents a 3D non-linear time domain FEM–BEM numerical model designed to address soil–structure interaction problems. The BEM formulation, based on element subdivision and the constant velocity approach, was improved by using interpolation matrices. The FEM approach was based on implicit Green's functions and non-linear contact was considered at the FEM–BEM interface. Two engineering problems were studied with the proposed methodology: the propagation of waves in an elastic foundation and the dynamic response of a structure to an incident wave field.     

Nonlinear waves in a prestressed elastic tube filled with a layered fluid

In this work, we studied the propagation of weakly nonlinear waves in a prestressed thin elastic tube filled with an incompressible layered fluid, where the outer layer is assumed to be inviscid whereas the cylindrical core is considered to be viscous. Using the reductive perturbation technique, the propagation of weakly nonlinear waves in the longwave approximation is studied. The governing equation is shown to be the Korteweg-de Vries-Burgers' (KdV-B) equation. A travelling wave type of solution to this evolution equation is sought and it is observed that the formation of shock wave becomes evident with increasing core radius parameter.     

Comparative study of different nonconserving time integrators for wave propagation in hyperelastic waveguides

This paper addresses the formulation and numerical efficiency of various numerical models of different nonconserving time integrators for studying wave propagation in nonlinear hyperelastic waveguides. The study includes different nonlinear finite element formulations based on standard Galerkin finite element model, time domain spectral finite element model, Taylor–Galerkin finite element model, generalized Galerkin finite element model and frequency domain spectral finite element model. A comparative study on the computational efficiency of these different models is made using a hyperelastic rod model, and the optimal computational scheme is identified. The identified scheme is then used to study the propagation of transverse and longitudinal waves in a Timoshenko beam with Murnaghan material nonlinearity.     

A 3D time domain numerical model based on half-space Green’s function for soil–structure interaction analysis

This paper presents a numerical method based on a three dimensional boundary element–finite element (BEM–FEM) coupled formulation in the time domain. The proposed model allows studying soil–structure interaction problems. The soil is modelled with the BEM, where the radiation condition is implicitly satisfied in the fundamental solution. Half-space Green’s function including internal soil damping is considered as the fundamental solution. An effective treatment based on the integration into a complex Jordan path is proposed to avoid the singularities at the arrival time of the Rayleigh waves. The efficiency of the BEM is improved taking into account the spatial symmetry and the invariance of the fundamental solution when it is expressed in a dimensionless form. The FEM is used to represent the structure. The proposed method is validated by comparison with analytical solutions and numerical results presented in the literature. Finally, a soil–structure interaction problem concerning with a building subjected to different incident wave fields is studied.     

Thermoelastic Lamb waves in a transversely isotropic plate bordered with layers of inviscid liquid

Analysis for the propagation of thermoelastic waves in a homogeneous, transversely isotropic, thermally conducting plate bordered with layers of inviscid liquid or half space of inviscid liquid on both sides, is investigated in the context of coupled theory of thermoelasticity. Secular equations for homogeneous transversely isotropic plate in closed form and isolated mathematical conditions for symmetric and anti-symmetric wave modes in completely separate terms are derived. The results for isotropic materials and uncoupled theories of thermoelasticity have been obtained as particular cases. It is shown that the purely transverse motion (SH mode), which is not affected by thermal variations, gets decoupled from rest of the motion of wave propagation and occurs along an in-plane axis of symmetry. The special cases, such as short wavelength waves and thin plate waves of the secular equations are also discussed. The secular equations for leaky Lamb waves are also obtained and deduced. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for transversely isotropic plate of zinc material bordered with water. The dispersion curves for symmetric and anti-symmetric wave modes, attenuation coefficient and amplitudes of displacement and temperature change in case of fundamental symmetric (S₀) and skew symmetric (A₀) modes are presented in order to illustrate and compare the theoretical results. The theory and numerical computations are found to be in close agreement.     

一种由横波激发的特殊非线性声现象

目前对非线性超声的研究多集中在纵波激发的谐波性质以及对材料微观结构变化的实验检测上,横波激发的非线性声波性质少有研究。对横波激发的一维非线性声波方程入手,利用摄动法求解该方程,并改写为一阶偏微分方程,然后利用交错网格的有限差分形式进行数值求解。结果表明:采用横波激发,能产生线性横波和非线性纵波,且纵波的高次谐波内有两个信号,分别以纵波和横波两种速度传播。若采用较长的激发信号,纵波谐波能形成"拍"现象,成为一种奇特的声传播现象。     

Depth measurement for cracks in corners using ultrasonic Rayleigh waves

The scattering of ultrasonic Rayleigh waves incident normally on corners containing cracks is considered by using elastodynamic ray theory. Detailed calculations are presented for vertical and horizontal cracks in right-angle corners in aluminium. It is shown that crack depth can be measured simply from the spacing of interference fringes in the high-frequency spectra of either the back- or forward-scattered Rayleigh waves, given only a knowledge of the Rayleigh wave speed. Use of the back-scattered wave is preferable because its fringes show stronger modulation, and because an experiment requires a single transducer and access to only one face of the specimen. The technique is applicable without modification to the more general case of a crack at any angle in a corner of any angle.     

Adaptive fast multipole boundary element method for three-dimensional half-space acoustic wave problems

A new adaptive fast multipole boundary element method (BEM) for solving 3-D half-space acoustic wave problems is presented in this paper. The half-space Green's function is employed explicitly in the boundary integral equation (BIE) formulation so that a tree structure of the boundary elements only for the boundaries of the real domain need to be applied, instead of using a tree structure that contains both the real domain and its mirror image. This procedure simplifies the implementation of the adaptive fast multipole BEM and reduces the CPU time and memory storage by about a half for large-scale half-space problems. An improved adaptive fast multipole BEM is presented for the half-space acoustic wave problems, based on the one developed recently for the full-space problems. This new fast multipole BEM is validated using several simple half-space models first, and then applied to model 3-D sound barriers and a large-scale windmill model with five turbines. The largest BEM model with 557470 elements was solved in about an hour on a desktop PC. The accuracy and efficiency of the BEM results clearly show the potential of the adaptive fast multipole BEM for solving large-scale half-space acoustic wave problems that are of practical significance.