Finite element analysis of time‐dependent semi‐infinite wave‐guides with high‐order boundary treatment

A new finite element (FE) scheme is proposed for the solution of time‐dependent semi‐infinite wave‐guide problems, in dispersive or non‐dispersive media. The semi‐infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC), based on the Higdon non‐reflecting operators, is developed and applied on ??. The new NRBC does not involve any high derivatives beyond second order, but its order of accuracy is as high as one desires. It involves some parameters which are chosen automatically as a pre‐process. A C⁰ semi‐discrete FE formulation incorporating this NRBC is constructed for the problem in the finite domain bounded by ??. Augmented and split versions of this FE formulation are proposed. The semi‐discrete system of equations is solved by the Newmark time‐integration scheme. Numerical examples concerning dispersive waves in a semi‐infinite wave guide are used to demonstrate the performance of the new method. Copyright © 2003 John Wiley & Sons, Ltd.     

Explicit finite element artificial boundary scheme for transient scalar waves in two‐dimensional unbounded waveguide

To simulate the transient scalar wave propagation in a two‐dimensional unbounded waveguide, an explicit finite element artificial boundary scheme is proposed, which couples the standard dynamic finite element method for complex near field and a high‐order accurate artificial boundary condition (ABC) for simple far field. An exact dynamic‐stiffness ABC that is global in space and time is constructed. A temporal localization method is developed, which consists of the rational function approximation in the frequency domain and the auxiliary variable realization into time domain. This method is applied to the dynamic‐stiffness ABC to result in a high‐order accurate ABC that is local in time but global in space. By discretizing the high‐order accurate ABC along artificial boundary and coupling the result with the standard lumped‐mass finite element equation of near field, a coupled dynamic equation is obtained, which is a symmetric system of purely second‐order ordinary differential equations in time with the diagonal mass and non‐diagonal damping matrices. A new explicit time integration algorithm in structural dynamics is used to solve this equation. Numerical examples are given to demonstrate the effectiveness of the proposed scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

High‐order Higdon‐like boundary conditions for exterior transient wave problems

Recently developed non‐reflecting boundary conditions are applied for exterior time‐dependent wave problems in unbounded domains. The linear time‐dependent wave equation, with or without a dispersive term, is considered in an infinite domain. The infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC) is imposed on ??. Then the problem is solved numerically in the finite domain bounded by ??. The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. We consider here two reformulations: one that involves high‐order derivatives with a special discretization scheme, and another that does not involve any high derivatives beyond second order. The latter formulation is made possible by introducing special auxiliary variables on ??. In both formulations the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. In contrast to previous papers using similar formulations, here the method is applied to a fully exterior two‐dimensional problem, with a rectangular boundary. Numerical examples in infinite domains are used to demonstrate the performance and advantages of the new method. In the auxiliary‐variable formulation long‐time corner instability is observed, that requires special treatment of the corners (not addressed in this paper). No such difficulties arise in the high‐derivative formulation. Published in 2005 by John Wiley & Sons, Ltd.     

A boundary condition in Padé series for frequency‐domain solution of wave propagation in unbounded domains

A boundary condition satisfying the radiation condition at infinity is frequently required in the numerical simulation of wave propagation in an unbounded domain. In a frequency domain analysis using finite elements, this boundary condition can be represented by the dynamic stiffness matrix of the unbounded domain defined on its boundary. A method for determining a Padé series of the dynamic stiffness matrix is proposed in this paper. This method starts from the scaled boundary finite‐element equation, which is a system of ordinary differential equations obtained by discretizing the boundary only. The coefficients of the Padé series are obtained directly from the ordinary differential equations, which are not actually solved for the dynamic stiffness matrix. The high rate of convergence of the Padé series with increasing order is demonstrated numerically. This technique is applicable to scalar waves and elastic vector waves propagating in anisotropic unbounded domains of irregular geometry. It can be combined seamlessly with standard finite elements. Copyright © 2006 John Wiley & Sons, Ltd.     

An optimal high‐order non‐reflecting finite element scheme for wave scattering problems

A new finite element scheme is proposed for the numerical solution of time‐harmonic wave scattering problems in unbounded domains. The infinite domain in truncated via an artificial boundary ?? which encloses a finite computational domain Ω. On ?? a local high‐order non‐reflecting boundary condition (NRBC) is applied which is constructed to be optimal in a certain sense. This NRBC is implemented in a special way, by using auxiliary variables along the boundary ??, so that it involves no high‐order derivatives regardless of its order. The order of the scheme is simply an input parameter, and it may be arbitrarily high. This leads to a symmetric finite element formulation where standard C⁰ finite elements are used in Ω. The performance of the method is demonstrated via numerical examples, and it is compared to other NRBC‐based schemes. The method is shown to be highly accurate and stable, and to lead to a well‐conditioned matrix problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical simulation of Lamb wave scattering in semi‐infinite plates

A finite element formulation is applied to study Lamb wave scattering in homogeneous and sandwich isotropic plates. Dispersion curves are calculated in a simple and automatic way by solving a quadratic eigenproblem. A meshing criterion to obtain accurate results with linear and quadratic elements is provided. An absorbing boundary condition for semi‐infinite plates is derived from this formulation by means of a truncated normal mode expansion technique, where the finite element eigenvectors are used instead of the analytical expressions for the normal modes. This non‐reflecting boundary condition is directly applicable to study Lamb wave reflection by simple obstacles such as a flat edge. In order to tackle Lamb wave diffraction problems by defects with more complex geometries, a hybrid boundary element‐finite element formulation is developed. The validity and accuracy of both formulations are checked thoroughly with a series of test problems studied by other researchers. Copyright © 2001 John Wiley & Sons, Ltd.     

Non‐reflective planar boundary condition based on Gauss filter

A non‐reflecting boundary condition based on the Gauss filter is employed for the determination of scattered potential governed by the equation. A filtering layer is used for closing infinite domain calculations. An expression for the reflection coefficient is derived and an optimal filtering layer is designed. Numerical results validate the performance of this method for unbounded wave guide problems. Copyright © 2000 John Wiley & Sons, Ltd.     

Transient non‐linear heat conduction–radiation problems—a boundary element formulation

A novel boundary‐only formulation for transient temperature fields in bodies of non‐linear material properties and arbitrary non‐linear boundary conditions has been developed. The option for self‐irradiating boundaries has been included in the formulation. Heat conduction equation has been partially linearized by Kirchhoff's transformation. The result has been discretized by the dual reciprocity boundary element method. The integral equation of heat radiation has been discretized by the standard boundary element method. The coupling of the resulting two sets of equations has been accomplished by static condensation of the radiative heat fluxes arising in both sets. The final set of ordinary differential equations has been solved using the Runge–Kutta solver with automatic time step adjustment. The algorithm proved to be robust and stable. Numerical examples are included. Copyright © 1999 John Wiley & Sons, Ltd.     

Non‐reflecting boundary conditions for acoustic propagation in ducts with acoustic treatment and mean flow

We consider a time‐harmonic acoustic scattering problem in a 2D infinite waveguide with walls covered with an absorbing material, in the presence of a mean flow assumed uniform far from the source. To make this problem suitable for a finite element analysis, the infinite domain is truncated. This paper concerns the derivation of a non‐reflecting boundary condition on the artificial boundary by means of a Dirichlet‐to‐Neumann (DtN) map based on a modal decomposition. Compared with the hard‐walled guide case, several difficulties are raised by the presence of both the liner and the mean flow. In particular, acoustic modes are no longer orthogonal and behave asymptotically like the modes of a soft‐walled guide. However, an accurate approximation of the DtN map can be derived using some bi‐orthogonality relations, valid asymptotically for high‐order modes. Numerical validations show the efficiency of the method. The influence of the liner with or without mean flow is illustrated. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite element formulation of exact non‐reflecting boundary conditions for the time‐dependent wave equation

A modified version of an exact Non‐reflecting Boundary Condition (NRBC) first derived by Grote and Keller is implemented in a finite element formulation for the scalar wave equation. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the second‐order local boundary condition derived by Bayliss and Turkel. Two alternative finite element formulations are given. In the first, the boundary operator is implemented directly as a ‘natural’ boundary condition in the weak form of the initial–boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit time‐integration schemes are presented for solution of the finite element semidiscrete equations concurrently with the first‐order differential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the finite element method. The results demonstrate that the finite element formulation of the (modified) NRBC is remarkably robust, and highly accurate. Copyright © 1999 John Wiley & Sons, Ltd.     

A non‐reflecting layer method for non‐linear wave‐type equations on unbounded domains with applications to shape memory alloy rods

In this paper a new technique is introduced and applied in solving one‐dimensional linear and non‐linear wave‐type equations on an unbounded spatial domain. This new technique referred to as the non‐reflecting layer method (NRLM) extends the computational domain with an artificial layer on which a one‐way wave equation is solved. The method will be applied to compute stress waves in long rods consisting of NiTi shape memory alloy material subjected to impact loading and undergoing detwinning and pseudo‐elastic material responses. The NRLM has been tested on model problems and it has been found that the computed solutions agree well with the exact solutions, i.e. normalized error levels are in ranges acceptable for engineering computations. Copyright © 2005 John Wiley & Sons, Ltd.     

一种高精度圆柱形人工边界条件:水-柱体相互作用问题

针对水-柱体动力相互作用问题,提出一种用于模拟无限域水体的圆柱形高精度时域人工边界条件。首先,基于三维可压缩水体的波动方程和边界条件,采用分离变量法建立了时空全局的精确人工边界条件;然后,将其动力刚度表示为外域模型和波导模型人工边界条件动力刚度的嵌套形式;之后,应用时间局部化方法得到时间局部的高精度人工边界条件;最后,离散高精度人工边界条件,并将其与近场有限元方程耦合,形成一种能够采用显式时间积分方法求解的时间二阶常微分方程组。数值算例表明:提出的三维圆柱形高精度人工边界条件精确、高效、稳定。     

离散高阶Higdon-like透射边界

基于比例边界有限元法(SBFEM)半离散思想和Higdon透射微分算子提出了一种用于模拟二维层状介质标量波传播的高效离散高阶Higdon-like透射边界。对无限介质边界进行迦辽金有限元离散后,描述标量波的偏微分方程转换为局部坐标系下半离散矩阵方程组;然后使用高阶Higdon透射算子和辅助变量,在时域内得到了一个阶数不超过2阶的离散高阶透射边界。透射边界是由一组常微分方程构成,可以采用通常的时步积分方法求解,它在截断边界上非局部,在时间域局部。算例表明:该文提出的透射边界的计算精度可以随着辅助变量的增加而提高,但计算量却呈线性化增加,因而计算效率较全局方法有了显著提高。另外,由于该文的边界条件是直接建立在离散节点上的,所以它很方便与近场有限单元法耦合。     

Rayleigh wave correction for the BEM analysis of two‐dimensional elastodynamic problems in a half‐space

A simple, elegant approach is proposed to correct the error introduced by the truncation of the infinite boundary in the BEM modelling of two‐dimensional wave propagation problems in elastic half‐spaces. The proposed method exploits the knowledge of the far‐field asymptotic behaviour of the solution to adequately correct the BEM displacement system matrix for the truncated problem to account for the contribution of the omitted part of the boundary. The reciprocal theorem of elastodynamics is used for a convenient computation of this contribution involving the same boundary integrals that form the original BEM system. The method is formulated for a two‐dimensional homogeneous, isotropic, linearly elastic half‐space and its implementation in a frequency domain boundary element scheme is discussed in some detail. The formulation is then validated for a free Rayleigh pulse travelling on a half‐space and successfully tested for a benchmark problem with a known approximation to the analytical solution. Copyright © 2004 John Wiley & Sons, Ltd.     

Transmitting boundary conditions for 1D peridynamics

The peridynamic theory reformulates the equations of continuum mechanics in terms of integro‐differential equations instead of partial differential equations. It is not straightforward to apply the available artificial boundary conditions for continua to peridynamic modeling. We therefore develop peridynamic transmitting boundary conditions (PTBCs) for 1D wave propagation. Differently from the previous method where the matching boundary condition is constructed for only one boundary material point, the PTBCs are established by considering the interaction and exchange of information between a group of boundary material points and another group of inner material points. The motion of the boundary material points is recursively constructed in terms of their locations and is determined through matching the peridynamic dispersion relation. The effectiveness of the PTBCs is examined by reflection analyses, numerical tests, and numerical convergent conditions. Furthermore, two‐way interfacial conditions are proposed. The PTBCs are then applied to simulations of wave propagation in a bar with a defect, a composite bar with interfaces, and a domain with a seismic source. All the analyses and applications demonstrate that the PTBCs can effectively remove undesired numerical reflections at artificial boundaries. The methodology may be applied to modeling of wave propagation by other nonlocal theories. Copyright © 2016 John Wiley & Sons, Ltd.     

Accurate radiation boundary conditions for the time‐dependent wave equation on unbounded domains

Asymptotic and exact local radiation boundary conditions (RBC) for the scalar time‐dependent wave equation, first derived by Hagstrom and Hariharan, are reformulated as an auxiliary Cauchy problem for each radial harmonic on a spherical boundary. The reformulation is based on the hierarchy of local boundary operators used by Bayliss and Turkel which satisfy truncations of an asymptotic expansion for each radial harmonic. The residuals of the local operators are determined from the solution of parallel systems of linear first‐order temporal equations. A decomposition into orthogonal transverse modes on the spherical boundary is used so that the residual functions may be computed efficiently and concurrently without altering the local character of the finite element equations. Since the auxiliary functions are based on residuals of an asymptotic expansion, the proposed method has the ability to vary separately the radial and transverse modal orders of the RBC. With the number of equations in the auxiliary Cauchy problem equal to the transverse mode number, this reformulation is exact. In this form, the equivalence with the closely related non‐reflecting boundary condition of Grote and Keller is shown. If fewer equations are used, then the boundary conditions form high‐order accurate asymptotic approximations to the exact condition, with corresponding reduction in work and memory. Numerical studies are performed to assess the accuracy and convergence properties of the exact and asymptotic versions of the RBC. The results demonstrate that the asymptotic formulation has dramatically improved accuracy for time domain simulations compared to standard boundary treatments and improved efficiency over the exact condition. Copyright © 2000 John Wiley & Sons, Ltd.     

Three‐dimensional finite element calculations in acoustic scattering using arbitrarily shaped convex artificial boundaries

We report on a generalization of the Bayliss–Gunzburger–Turkel non‐reflecting boundary conditions to arbitrarily shaped convex artificial boundaries. For elongated scatterers such as submarines, we show that this generalization can improve significantly the computational efficiency of finite element methods applied to the solution of three‐dimensional acoustic scattering problems. Copyright © 2001 John Wiley & Sons, Ltd.     

竖向成层介质中标量波传播问题的高精度人工边界条件

为了实现含竖向成层介质以及表面不规则地形场地中标量波传播问题的高效且高精度求解,该文基于连分式展开和扩展的一致边界,建立了一种频域下折线形高精度人工边界条件。通过在每个竖向地层内引入独立的斜角坐标变换,新的人工边界条件可以用于多起伏地表地形条件。新的折线形人工边界在频域下推导,仅含有连分式阶数一个待定实参数,用于调整计算精度,该参数不随外行波的频率和传播角度改变。人工边界条件可以与内域有限元方程无缝耦合,应用简单方便。由于新边界条件的高精度,内域尺寸可以取较小甚至可以直接将人工边界加在结构周围或者地表,从而极大提高计算效率。通过典型数值算例,将人工边界计算模型与有限元大模型的解进行了对比分析,验证了该文提出的折线形人工边界条件的有效性和高精度。     

High‐order doubly asymptotic open boundaries for scalar wave equation

High‐order doubly asymptotic open boundaries are developed for transient analyses of scalar waves propagating in a semi‐infinite layer with a constant depth and a circular cavity in a full‐plane. The open boundaries are derived in the frequency domain as doubly asymptotic continued fraction solutions of the dynamic stiffness of the unbounded domains. Each term of the continued fraction is a linear function of the excitation frequency. The constants of the continued fraction solutions are determined recursively. The continued fraction solution is expressed in the time domain as ordinary differential equations, which can be solved by standard time‐stepping schemes. No parameters other than the orders of the low‐ and high‐frequency expansions need to be selected by users. Numerical experiments demonstrate that evanescent waves and long‐time (low‐frequency) responses are simulated accurately. In comparison with singly asymptotic open boundaries, significant gain in accuracy is achieved at no additional computational cost. Copyright © 2009 John Wiley & Sons, Ltd.     

模拟P波向无穷远域传播的一致边界

将人工边界设置在半无穷层单元和内部有限元区域的交界面上,建立了半无穷层单元的刚度矩阵后,得到了边界节点的动力平衡方程。任意给定激励圆频率,将边界节点系统的动力平衡方程转化为特征值方程。求解特征值方程得出边界节点系统的特征值和特征模态,利用模态叠加原理得到体现左半无穷层单元和右半无穷层单元对内部有限元区域作用的边界矩阵,这就是该文的一致边界。将其与内部有限元区域的刚度矩阵进行组装来模拟无穷远域介质对波的传播作用。最后用数值算例来说明一致边界的精确性和可行性。