Direct implementation of high order bgt arti cial boundary. Emm engquistmajdamur absorbing boundary conditions. On the engquist majda absorbing boundary conditions for. Variational boundary conditions for molecular dynamics. The standard boundary conditions used at the sides ofs seismic section in. Pdf on the engquist majda absorbing boundary conditions. In a local scheme, the solution at any time step depends only on the current node.
Wellposedness of oneway wave equations and absorbing. Mur applied the scalar wave results of engquist and majda in the special context of staggeredgrid cartesian simulation of maxwells equations an integration method known as the yee scheme. Absorbing boundary conditions for the numerical simulation. The basic idea behind these boundary conditions is that they cancel several leading terms in an expansion of the solution.
Majda boundary conditions and baylissturkel boundary conditions 5. Selfadapting absorbing boundary conditions for the wave. Computer methods in applied mechanics and engineering, vol. In the case of the wave equation this boundary condition is valid at all frequencies.
Radiation boundary conditions for acoustic and elastic. Open boundary conditions for wave propagation problems on. Absorbing boundary conditions for acoustic and elastic waves. Assuming that the propagation is in x direction perpendicular to the boundary, traveling waves can be expressed in the form. Nonreflecting boundary conditions for elastodynamic. In such conditions, an intuitive and simple abc can be derived. Absorbing boundary conditions for the numerical simulation of waves. Radiation boundary conditions for acoustic and elastic wave calculations b engquist, a majda communications on pure and applied mathematics 32 3, 3357, 1979. On surface radiation conditions for highfrequency wave. Poinsot center for turbulence research, stanford university, stanford, california 94305.
Absorbing boundary todditions secondorder hyperbolic. Lindman in 1975 23 and by engquist and majda in 1977 10, 11 for application as approximate absorbing boundary conditions in numerical computations, where artificial boundaries must be introduced to limit a computational domain. Homogeneous dirichlet or neumann boundary conditions for the wave equation lead to a total re. Radiation boundary conditions for the numerical simulation of. Absorbing boundary conditions abcs or nonreflecting boundary conditions nrbcs have been developed since the seminal paper by engquist and majda 24 for planar boundaries, afterwards for. There follows a systematic derivation of a hierarchy of local radiating boundary conditions for the elastic wave equation. These reflections from the edges of the computational grid appear as artifacts in. A common feature to most analytical conditions is the assumption that the waves to be absorbed are traveling waves propagating with the speed of light c. In a global scheme, each boundary node is fully coupled to all other boundary nodes in both space and time. Numerical multiscale methods and effective boundary. Accurate radiation boundary conditions for the twodimensional wave equation on unbounded domains. Taflove 7 and others successfully exploited these boundary conditions for a variety of finitedifference time domain fdtd simulations. A technique to incorporate the second order engquistmajda absorbing boundary condition into the weak formulation is presented.
Absorbing boundary conditions have been introduced in the seminal papers by engquist and majda 15 and bayliss and turkel 2 to truncate in. For linear hyperbolic systems, engquist and majda 1977. In this pap er w e deriv e a general order absorbing b oundary conditions of the t yp e. In this paper we derive a gen eral order absorbing boundary conditions of the type suggested by engquist and majda. The nonlocality in time is clear from the convolution term. Nonreflecting boundary conditions for the timedependent wave.
Emm is defined as engquistmajdamur absorbing boundary conditions somewhat frequently. To derive practical absorbing boundary conditions, we will adopt the following three criteria of engquist and majda cf. An approximation of the potential is obtained without solving any system of equations. Absorbing boundary conditions for the wave equation and. The derivation utilizes a dierent methodology which is more general and simpler. A sequence of absorbing boundary conditions for maxwells. Absorbing boundary todditions secondorder hyperbolic equations. Fourth, the truncated dtn condition in elliptic and spheroidal coordinates is modified to remove difficulties. Use of the perfect electric conductor boundary conditions to. Engquist and majda 3 proposed a pseudodifferential operator as asymptotically valid absorbing boundary condition for hyperbolic equations. Fifth, a sequence of new and more accurate local boundary conditions is derived for polar coordinates in two dimensions.
In 11, engquist and majda proposed the following condition the integer n is a parameter meant to be large. They may also be called nonre ecting boundary conditions or radiating boundary. By using continued fraction expansions, a systematic sequence of stable highly absorbing boundary conditions with successively better absorbing properties as the order of the boundary conditions increases is obtained. Emm stands for engquistmajdamur absorbing boundary conditions. In this paper we present a set of absorbing boundary conditions that are based on paraxial approximations pa of the scalar and elastic wave equations. Absorbing boundary conditions and perfectly matched layers. Absorbing boundary conditions for seismic analysis in abaqus. Mahrer 1986, in his empirical study ofvarious types of absorbing boundary conditions, found the boundary conditions of clayton andengquist1977 and reynolds 1978, as well as certain type of cerjan et al. This boundary condition is derived from a taylor series expansion about the ideal onedimensional boundary. Nonreflecting boundary conditions for the timedependent. Approximation of pseudodifferential operators in absorbing.
Absorbing boundary conditions and perfectly matched layers in. In practical calculations, it is often essential to introduce artificial boundaries to limit the area of computation. The goal is to derive e ective boundary conditions, or wall laws, through high resolution simulations localized to the boundary coupled to a coarser simulation in the domain interior. Absorbing boundary conditions for the numerical simulation of waves by bjorn engquist and andrew majda abstract. By robert clayton and bjorn engquist abstract boundary conditions are derived for numerical wave simulation that minimize artificial reflections from the edges of the domain of computation. Third, approximate local boundary conditions are derived for these coordinates. Engquist and majda 1977, reynolds 1978, mur 1981, keys 1985. Absorbing boundary conditions for numerical simulation of. We develop numerical multiscale methods for viscous boundary layer ow. Absorbing boundary conditions are boundary procedures that are applied at the arti cial numerical boundaries of a computational domain to miminize or eliminate the spurious relections at these boundaries which occur in the simulations of wave propagation phenomena. The superiority of the proposed boundary condition over the first and second order engquistmajda ones is demonstrated for different angles of incidence and wave types. Similar techniques applied to the elastic wave equation con rm that an ansatz of engquist and majda 2 about the form of a low order.
Several methods to derive radiation boundary conditions for the twodimen sional wave equation are known. Radiation boundary conditions for acoustic and elastic wave. Boundary conditions for problems in aerodynamics springerlink. In particular, i will present a procedure which allows for very easy implementation of highorder boundary conditions for acoustic waves.
Open boundary conditions in stratified ocean models. In their classical paper 2, the authors presented a methodology for the derivation of far. Numerical multiscale methods and effective boundary conditions. Absorbing boundary conditions for the numerical simulation of waves authors. Finite difference time domain fdtd methods for solution of. Absorbing boundary conditions for 2d wave equation. Mahrer 1986, in his empirical study ofvarious types of absorbing boundary conditions, found the boundary conditions of clayton and engquist 1977 and reynolds 1978, as well as certain type of cerjan et al. Radiation boundary conditions for acoustic and elastic wave calculations. Our system con sists of an electromagnetic wave in vacuum, propagating into a circular array of 20 antennas. Use of the perfect electric conductor boundary conditions to discretize a diffractor in fdtdpml environment c. Unlimited viewing of the articlechapter pdf and any associated supplements and figures.
Pdf on the engquist majda absorbing boundary conditions for. Majda in 1977 see 7 to design absorbing conditions that are easy to implement and yield a small. How is engquistmajdamur absorbing boundary conditions abbreviated. On the engquist majda absorbing boundary conditions for hyperbolic systems adi ditkowski and david gottlieb this paper is dedicated to stan osher on the occasion of his 60th birthday abstract. Highlights we propose a selfadapting absorbing boundary condition for the linear wave equation. Journal of computational physics 101, 104 129 1992 boundary conditions for direct simulations of compressible viscous flows t. Absorbing boundary conditions for numerical simulation of waves bjorn engquist, andrew majda proceedings of the national academy of sciences may 1977, 74 5 17651766. Boundary conditions for direct simulations of compressible. For subsequent developments of this and related ideas, see 3, 9, 18 and recent. Absorbing boundary conditions for the numerical simulation of.
Mathematisches institut on nonreflecting boundary conditions g. Absorbing boundary conditions for waveequation migration robert w. In this section we derive the most general absorbing boundary conditions of the engquist majda type. Here, leastsquares approximation of the symbol of the pseudodifferential operator is proposed to obtain differential operators as boundary conditions. Exact boundary conditions variational boundary condition examples application to fracture simulations. Here we develop a systematic method for obtaining a hierarchy of local boundary conditions at these artificial boundaries. Engquist ma jda absorbing boundary conditions for hyp erb olic systems adi ditk o wski da vid gottlieb july 3, 2002 abstract in their classical pap er 2, the authors presen ted a metho dology for the deriv ation of far eld b oundary conditions for the absorption of w a v es that are almost p erp endicular to the b oundary. Use of the perfect electric conductor boundary conditions. Gustafsson observed that the engquist majda boundary conditions admit a gen eralized eigenvalue in the classical wave equation case. The conventional wavefield predictionbased abc adopts one way wave equation owwe to predict the wavefield at the model boundary e.
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