The simulation of 3D elastic scattering produced by thin rigid inclusions using the traction boundary element method

A mixed formulation that uses both the traction boundary element method (TBEM) and the boundary element method (BEM) is proposed to compute the three-dimensional (3D) propagation of elastic waves scattered by two-dimensional (2D) thin rigid inclusions. Although the conventional direct BEM has limitations when dealing with thin-body problems, this model overcomes that difficulty. It is formulated in the frequency domain and, taking into account the 2-1/2D configuration of the problem, can be expressed in terms of waves with varying wavenumbers in the zdirection, k_z. The elastic medium is homogeneous and unbounded and it should be noted that no restrictions are imposed on the geometry and orientation of the internal crack.     

Rayleigh wave dispersion due to spatial (FEM) discretization of a thin elastic solid having non-curved boundary

The grid dispersion for a harmonic Rayleigh wave propagating along a straight boundary of a thin elastic solid, modelled by finite elements, is investigated. It is shown that with an increase of dimensionless wavenumber γ · q, a phase velocity of Rayleigh waves increases from the long wave limit C_R. For sufficiently short waves it follows the dispersion curve referring to quasitransverse waves in an unbounded discretized medium.     

Lagrangian finite element modelling of dam–fluid interaction: Accurate absorbing boundary conditions

The dynamic dam–fluid interaction is considered via a Lagrangian approach, based on a fluid finite element (FE) model under the assumption of small displacement and inviscid fluid. The fluid domain is discretized by enhanced displacement-based finite elements, which can be considered an evolution of those derived from the pioneering works of Bathe and Hahn [Bathe KJ, Hahn WF. On transient analysis of fluid–structure system. Comp Struct 1979;10:383–93] and of Wilson and Khalvati [Wilson EL, Khalvati M. Finite element for the dynamic analysis of fluid–solid system. Int J Numer Methods Eng 1983;19:1657–68]. The irrotational condition for inviscid fluids is imposed by the penalty method and consequentially leads to a type of micropolar media. The model is implemented using a FE code, and the numerical results of a rectangular bidimensional basin (subjected to horizontal sinusoidal acceleration) are compared with the analytical solution. It is demonstrated that the Lagrangian model is able to perform pressure and gravity wave propagation analysis, even if the gravity (or surface) waves are dispersive. The dispersion nature of surface waves indicates that the wave propagation velocity is dependent on the wave frequency.For the practical analysis of the coupled dam–fluid problem the analysed region of the basin must be reduced and the use of suitable asymptotic boundary conditions must be investigated. The classical Sommerfeld condition is implemented by means of a boundary layer of dampers and the analysis results are shown for the cases of sinusoidal forcing.The classical Sommerfeld condition is highly efficient for pressure-based FE modelling, but may not be considered fully adequate for the displacement-based FE approach. In the present paper a high-order boundary condition proposed by Higdom [Higdom RL. Radiation boundary condition for dispersive waves. SIAM J Numer Anal 1994;31:64–100] is considered. Its implementation requires the resolution of a multifreedom constraint problem, defined in terms of incremental displacements, in the ambit of dynamic time integration problems. The first- and second-order Higdon conditions are developed and implemented. The results are compared with the Sommerfeld condition results, and with the analytical unbounded problem results.Finally, a number of finite element results are presented and their related features are discussed and critically compared.     

Hydrodynamic loadings due to the motion of large offshore structures

A very effective non-reflecting boundary condition is proposed for the three-dimensional finite element analysis of hydrodynamic loads on offshore structures under motion due to earthquake, waves or machinery forces. The effect of surface waves is considered and the analysis is conducted in the frequency domain. The structure is assumed to be sufficiently large such that the non-linear effect of drag force is negligible. The unbounded extent of water surrounding the structure is assumed to be incompressible. The boundary condition, which is derived for the general analysis of structure-water interaction, is found to depend on the frequency of excitation, the location of the truncation boundary and the depth of water in the far field. Through some simplified numerical examples, it is shown that the proposed technique is very efficient for a wide range of the frequency of excitation. Incorporation of the boundary condition into a finite element program requires practically no extra effort.     

Flow field computation for the high voltage gas blast circuit breaker with the moving boundary

We studied the gas dynamics for the ideal gas in the simplified high voltage (HV) gas blast circuit breaker with the moving boundary. The piston and the electric contact are moving. Since the boundary is moving, it is difficult for the ordinary finite difference (FD) method or the finite element (FE) method to compute the solution. For the purpose of numerical simplicity and efficiency, we introduced an upwind meshfree scheme which is an excellent scheme for the time varying domain. Despite the low coding and computational cost, the numerical simulation is successfully conducted. Our method is even more efficient when considering a three-dimensional computation with a moving boundary.     

Absorbing boundary conditions for time harmonic wave propagation in discretized domains

While many successful absorbing boundary conditions (ABCs) are developed to simulate wave propagation into unbounded domains, most of them ignore the effect of interior discretization and result in spurious reflections at the artificial boundary. We tackle this problem by developing ABCs directly for the discretized wave equation. Specifically, we show that the discrete system (mesh) can be stretched in a non-trivial way to preserve the discrete impedance at the interface. Similar to the perfectly matched layers (PML) for continuous wave equation, the stretch is designed to introduce dissipation in the exterior, resulting in a PML-type ABC for discrete media. The paper includes detailed formulation of the new discrete ABC, along with the illustration of its effectiveness over continuous ABCs with the help of error analysis and numerical experiments. For time-harmonic problems, the improvement over continuous ABCs is achieved without any computational overhead, leading to the conclusion that the discrete ABCs should be used in lieu of continuous ABCs.     

A Pseudospectral Penalty Scheme for 2D Isotropic Elastic Wave Computations

In this paper, we present a pseudospectral scheme for solving 2D elastic wave equations. We start by analyzing boundary operators leading to the well-posedness of the problem. In addition, equivalent characteristic boundary conditions of common physical boundary conditions are discussed. These theoretical results are further employed to construct a Legendre pseudospectral penalty scheme based on a tensor product formulation for approximating waves on a general curvilinear quadrilateral domain. A stability analysis of the scheme is conducted for the case where a straight-sided quadrilateral element is used. The analysis shows that, by properly setting the penalty parameters, the scheme is stable at the semi-discrete level. Numerical experiments for testing the performance of the scheme are conducted, and the expected p- and h-convergence patterns are observed. Moreover, the numerical computations also show that the scheme is time stable, which makes the scheme suitable for long time simulations. This work is supported by National Science Council grant No. NSC 95-2120-M-001-003.     

Hybrid Spectral Difference Methods for Elliptic Equations on Exterior Domains with the Discrete Radial Absorbing Boundary Condition

The hybrid spectral difference methods (HSD) for the Laplace and Helmholtz equations in exterior domains are proposed. We consider the fictitious domain method with the absorbing boundary conditions (ABCs). The HSD method is a finite difference version of the hybridized Galerkin method, and it consists of two types of finite difference approximations; the cell finite difference and the interface finite difference. The fictitious domain is composed of two subregions; the Cartesian grid region and the boundary layer region in which the radial grid is imposed. The boundary layer region with the radial grid makes it easy to implement the discrete radial ABC. The discrete radial ABC is a discrete version of the Bayliss–Gunzburger–Turkel ABC without pertaining any radial derivatives. Numerical experiments confirming efficiency of our numerical scheme are provided.     

Boundary second-order sliding-mode control of an uncertain heat process with unbounded matched perturbation

The primary concern of the present paper is the regulation of an uncertain heat process with collocated boundary sensing and actuation. The underlying heat process is governed by an uncertain parabolic partial differential equation (PDE) with Neumann boundary conditions. It exhibits an unknown constant diffusivity parameter and it is affected by a smooth boundary disturbance, which is not available for measurements and which is possibly unbounded in magnitude. The proposed robust synthesis is formed by the linear feedback design and by the ”Twisting” second-order sliding-mode control algorithm, suitably combined and re-worked in the infinite-dimensional setting. A non-standard Lyapunov functional is invoked to establish the global asymptotic stability in a Sobolev space, involving spatial state derivatives of the same order as that of the plant equation. The stability proof is accompanied by a set of simple tuning rules for the controller parameters. The effectiveness of the developed control scheme is supported by simulation results.     

Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation

One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non-tangential angles-of-incidence and of all non-zero frequencies. This paper develops the PML concept for time-harmonic elastodynamics in Cartesian coordinates, utilising insights obtained with electromagnetics PMLs, and presents a novel displacement-based, symmetric finite-element implementation of the PML for time-harmonic plane-strain or three-dimensional motion. The PML concept is illustrated through the example of a one-dimensional rod on elastic foundation and through the anti-plane motion of a two-dimensional continuum. The concept is explored in detail through analytical and numerical results from a PML model of the semi-infinite rod on elastic foundation, and through numerical results for the anti-plane motion of a semi-infinite layer on a rigid base. Numerical results are presented for the classical soil–structure interaction problems of a rigid strip-footing on a (i) half-plane, (ii) layer on a half-plane, and (iii) layer on a rigid base. The analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by PML models even with small bounded domains.     

Real-time finite element modeling for surgery simulation: an application to virtual suturing

Real-time finite element (FE) analysis can be used to represent complex deformable geometries in virtual environments. The need for accurate surgical simulation has spurred the development of many of the new real-time FE methodologies that enable haptic support and real-time deformation. These techniques are computationally intensive and it has proved to be a challenge to achieve the high modeling resolutions required to accurately represent complex anatomies. The authors present a new real-time methodology based on linear FE analysis that is appropriate for a wide range of surgical simulation applications. A methodology is proposed that is characterized by high model resolution, low preprocessing time, unrestricted multipoint surface contact, and adjustable boundary conditions. These features make the method ideal for modeling suturing, which is an element common to almost every surgical procedure. This paper describes constraints in the context of a Suturing Simulator currently being developed by the authors.     

A new procedure for evaluating the time domain boundary influence matrix of unbounded media

The finite element method can only deal with finite domains with well defined boundaries. For dynamic problems involving unbounded media, the boundaries of the finite model distort the real physical behaviour of the problem if they remain untreated. For many problems it is possible to formulate so-called silent boundary conditions which perfectly simulate the effect of the truncated unbounded medium. Unfortunately, most of these conditions are properly formulated in the frequency domain.

The present paper introduces a new procedure which employs these frequency-dependent boundary conditions to calculate the time domain influence matrix of the truncated unbounded medium. This matrix, which was introduced in a previous publication, is used to calculate the reflection-free response of the truncation boundary one time step ahead of the present time station. The known boundary response is then used as a prescribed condition for the finite model. A one-dimensional example with a frequency-dependent boundary condition is presented to examine the effectiveness of the new procedure. Two other silent boundary conditions formulated directly in the time domain are also examined.     

Mathematical modeling of waves in layered media near a caustic

Problems associated with the modeling of wave fields in an acoustic medium near a caustic in a nonstationary statement are considered. A mathematical model is proposed making it possible explicitly to mark a caustic as a boundary of a solution domain for an arbitrary change in the sound velocity. An effectively realizable boundary condition is established such as boundedness of the solution (pressure) on a caustic and Green’s function of a boundary-value problem is constructed. An auxiliary Goursat problem is studied and a system of its partial solutions is constructed based on hypergeometric functions. The integral Volterra equation in Green’s function is obtained and an algorithm for its expansion in the smoothness is presented. A difference scheme is proposed approximating the solution of a differential problem with an unbounded coefficient. The results of the numerical simulation are presented.     

A FV-TD electromagnetic solver using adaptive Cartesian grids

A second-order finite-volume (FV) method has been developed to solve the time-domain (TD) Maxwell equations, which govern the dynamics of electromagnetic waves. The computational electromagnetic (CEM) solver is capable of handling arbitrary grids, including structured, unstructured, and adaptive Cartesian grids, which are topologically arbitrary. It is argued in this paper that the adaptive Cartesian grid is better than a tetrahedral grid for complex geometries considering both efficiency and accuracy. A cell-wise linear reconstruction scheme is employed to achieve second-order spatial accuracy. Second-order time accuracy is obtained through a two-step Runge-Kutta scheme. Issues on automatic adaptive Cartesian grid generation such as cell-cutting and cell-merging are discussed. A multi-dimensional characteristic absorbing boundary condition (MDC-ABC) is developed at the truncated far-field boundary to reduce reflected waves from this artificial boundary. The CEM solver is demonstrated with several test cases with analytical solutions.     

全波场地震波数值模拟中的边界处理实践研究

在用有限差分法或有限元法模拟无界区域中的波动时,需要对计算区域的边界做特殊处理,以消除由于把地震波的传播设定在有限区域而产生的边界反射。为了这一目的,人们研究出了多种人工边界处理方法,完全匹配层（PML）吸收边界条件就是理想的方法之一,现已被广泛应用。本文将PML吸收边界条件应用于全波场地震波的数值模拟,数值计算实验表明,对qP波,匹配层的厚度为5个网格间距即可达到要求,而对qSV波与qSH波,为达到理想的吸收效果,匹配层的厚度应当增大,当厚度为13个网格间距时达到了理想的吸收效果。     

High Order Weighted Essentially Non-oscillation Schemes for Two-Dimensional Detonation Wave Simulations

In this paper, we demonstrate the detailed numerical studies of three classical two dimensional detonation waves by solving the two dimensional reactive Euler equations with species with the fifth order WENO-Z finite difference scheme (Borges et al. in J. Comput. Phys. 227:3101?C3211, 2008) with various grid resolutions. To reduce the computational cost and to avoid wave reflection from the artificial computational boundary of a truncated physical domain, we derive an efficient and easily implemented one dimensional Perfectly Matched Layer (PML) absorbing boundary condition (ABC) for the two dimensional unsteady reactive Euler equation when one of the directions of domain is periodical and inflow/outflow in the other direction. The numerical comparison among characteristic, free stream, extrapolation and PML boundary conditions are conducted for the detonation wave simulations. The accuracy and efficiency of four mentioned boundary conditions are verified against the reference solutions which are obtained from using a large computational domain. Numerical schemes for solving the system of hyperbolic conversation laws with a single-mode sinusoidal perturbed ZND analytical solution as initial conditions are presented. Regular rectangular combustion cell, pockets of unburned gas and bubbles and spikes are generated and resolved in the simulations. It is shown that large amplitude of perturbation wave generates more fine scale structures within the detonation waves and the number of cell structures depends on the wave number of sinusoidal perturbation.     

Discrete artificial boundary conditions for transient scalar wave propagation in a 2D unbounded layered media

This paper presents an efficient numerical method for direct time-domain solution of the transient scalar wave propagation in a two-dimensional unbounded multi-layer soil. The unbounded domain is truncated by an artificial boundary which demands the corresponding boundary conditions. In the new approach, only the artificial boundary is discretized into one-dimensional finite elements, yielding a new time-dependent partial differential equation (PDE) for displacements with respect to only one spatial coordinate. Factorization of the PDE and introduction of the residual radiation functions, there results a linear first-order ordinary differential equation (ODE). Its stability is ensured. The time-dependent discrete artificial boundary conditions are determined by the solution of the ODE. In general, it is local in time, but it is non-local in space. Several numerical examples are given to verify the superiority of the proposed method.     

Cutoff wave numbers for energy-orthogonal six-node triangular plane-strain elements

This paper studies the propagation of plane harmonic waves in unbounded media discretized by the standard six-node triangular finite element. The element stiffness matrix is split into basic and higher order components which are obtained from mean and deviatoric strain fields, respectively. This decomposition is applied to the elastic energy. Based on the properties of the higher order energy, two values of the wave number are selected. Depending on the desired precision one of those values can be used as optimum cutoff wave number to properly capture a wave field.     

Imaging of layered media in inverse scattering problems for an acoustic wave equation

Two-dimensional (2D) inverse scattering problems for the acoustic wave equation consisting of obtaining the density and acoustic impedance of the medium are considered. A necessary and sufficient condition for the unique solvability of these problems in the form of the law of energy conservation has been established. It is proved that this condition is that for each pulse oscillation source located on the boundary of a half-plane, the energy flow of the scattered waves is less than the energy flux of waves propagating from the boundary of this half-plane. This shows that for inverse dynamic scattering problems in acoustics and geophysics when the law of energy conservation holds it is possible to determine the elastic density parameters of the medium. The obtained results significantly increase the class of mathematical models currently used in solving multidimensional inverse scattering problems. Some specific aspects of interpreting inverse problems solutions are considered.     

Consistent transmitting boundary with continued-fraction absorbing boundary conditions for analysis of soil–structure interaction in a layered half-space

A continued-fraction absorbing boundary condition (CFABC) is combined with a consistent transmitting boundary and applied to an analysis of soil–structure interaction in a layered half-space. In order to be a perfect absorber for in-plane waves with an arbitrary horizontal wavenumber in a layered half-space, it is proposed that the CFABC be used in the form of a 2-layer. A method for determination of parameters of the 2-layers was proposed for evanescent waves as well as propagating waves. Combined with consistent transmitting boundaries, the 2-layers of CFABC are applied to problems of dynamics of rigid strip and circular foundations on a layered half-space. The results demonstrate that the proposed 2-layers of the CFABC can represent a layered half-space accurately in the entire frequency range and absorb both evanescent and propagating P and S waves effectively.