A. Bayliss, M. Gunzburger, and E. Turkel, "boundary conditions for the numerical solution of elliptic equations in exterior regions," SIAM J. Appl. Math., vol. 42, pp. 430-451, Apr. 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... conditions based on Sommerfeld s famous asymptotic boundary condition, 3] r u Gamma in 0 k 0 u = o(r ) Then there are the nonlocal Dirichlet to Neumann boundary conditions for separable coordinate systems [4] and [5] the family of corresponding local and asymptotic approximations [6], infinite element methods [7] and the various kinds of boundary element methods [8] However, if the exterior domain becomes inhomogeneous, e.g. by an embedded waveguide (Fig. 1) these methods do not longer work. For such cases Goldstein [9] offers a solution by constructing transparent ....

A. Bayliss, M. Gunzburger, and E. Turkel. Boundary conditions for the numerical solution of elliptic equations in exterior domains. SIAM J. Appl. Math., 42:430--451, 1982.

....radiation boundary condition (Grote, 1995) The resulting modified DtN condition is unique for any choice of N, and approximates the harmonics n N with greater accuracy than the original DtN condition. Local boundary conditions are easily constructed by extending the procedures employed in (Bayliss, 1982) for a circle or sphere, where radial terms in a multipole expansion for outgoing waves are annihilated. The generalization to spheroidal coordinates is given by the asymptotic expansion given by (Holford; Burnett, 1994) f exp(icx) cx j=0 g j (q;j;c) cx) j (25) where c = k f , is the ....

.... expansion (25) is constructed as a product of normalized radial derivatives: B j = L j (L j 1 ( L 2 (L 1 ) 26) L j = 1 f x ic 2 j 1 x (27) such that B j f = O( cx] 2 j 1 ) Setting the remainder equal to zero defines the analogue of the boundary conditions derived in (Bayliss, 1982) for a sphere. The first two boundary conditions are: B 1 f = 1 f x a 1 f = 0; on G (28) B 2 f = 1 f 2 2 x 2 a 2 x a 3 f = 0; on G (29) where a 1 = 1 icx 0 ) x 0 a 2 = 4 2icx 0 ) x 0 a 3 = 2 4icx 0 (cx 0 ) 2 =x 2 0 Applying the B 1 operator to the ....

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A. Bayliss, M. Gunzberger, and E. Turkel, `Boundary conditions for the numerical solution of elliptic equations in exterior domains', SIAM J. Appl. Math., 42, 430-451, 1982.

.... such that spherical coordinates are used, if Gamma 1 is a sphere, while cartesian coordinates are employed in the case with a rectangular boundary (see Figure 2) On a spherical boundary we use the first order and second order boundary conditions developed by Bayliss, Gunzburger, and Turkel in [5]. The first order condition is given by r 1 R1 Gamma i u = 0 (2.2) and the second order boundary condition by u r Gamma i Gamma 1 R1 u Gamma 1 2R1 Gamma i2 R 2 1 Du = 0; 2.3) where the operator D is the Laplace Beltrami operator on the unit sphere. ....

A. Bayliss, M. Gunzburger, and E. Turkel. Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J. Appl. Math., 42(2):430--451, 1982.

....the Fourier Laplace transformed domain, then yielded a sequence of approximate local differential operators. As an alternative, Bayliss, Gunzburger and Turkel constructed a sequence of local differential operators, which annihilate the leading terms of the large distance expansion of the solution [3]. Using the analogous Wilcox expansion [77] Peterson extended these boundary conditions to the vector Helmholtz equation for use in electromagnetic scattering [66] 67] which were then used and analyzed by Mittra, Ramahi et CHAPTER 1. INTRODUCTION 4 al. 59] Kang derived radiation conditions ....

....difficult or impossible to solve. Our second goal is to derive the DtN condition for elliptic and spheroidal artificial boundaries. Third, we shall derive approximate local boundary conditions for elliptic and spheroidal coordinates, which are the analogues of the Bayliss Gunzburger Turkel (BGT) [3] boundary conditions in polar and spherical coordinates. Fourth, we shall modify the truncated versions of the DtN conditions for elliptic and spheroidal boundaries. Fifth, we shall present a sequence of local boundary conditions in two dimensional polar coordinates, which are much more accurate ....

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A. Bayliss, M. Gunzburger, and E. Turkel. Boundary conditions for the numerical solution of elliptic equations in exterior domains. SIAM J. Appl. Math., 42:430-- 451, 1982.

....Selection of the coefficients ff and fi is often based on an expansion of the solution in the exterior of a circle. A sequence of approximate RBCs, fBn g, can be constructed from the boundary operators that annihilate the first m terms in the expansion. Two examples are the Bayliss Turkel RBCs [1], based on the far field expansion u = e Gammaikr p r 1 X n=0 an ( r n (8) and the Li Cendes RBCs [4] which are based on the Hankel expansion u = P 1 n=0 b n ( Hn (kr) where Hn is the n th order Hankel function of the second kind. A third set of coefficients, the ....

A. Bayliss, M. Gunzburger, E. Turkel, "Boundary conditions for the numerical solution of elliptic equations in exterior regions", SIAM J. Appl. Math., 42(1982), pp. 430--451.

....to truncate the open region as close and as conformal as possible to the scattering object. An accurate radiation boundary condition (RBC) must be applied on the truncation boundary. In recent years, a number of local RBCs for use on general boundaries have been proposed ( EM77] EM79] BT80] BGT82] Hig86] KTU87] MBTK88] KM89] KRM89] MRK 89] Jan92] LC93] MSPW92] MSPW95] Jin93] An excellent summary of the RBCs for circular and planar truncation boundaries can be found in [Giv91] The basic mathematical problem is to find a solution to the scalar Helmholtz equation ....

....OEOE (3) where (ae; OE) are polar coordinates, subscripts with ae and OE denote partial derivatives (e.g. u ae = u ae ) and the subscript i denotes the order of the RBC (either 1 or 2) The coefficients for three common families of RBCs are given in Table 1. The Bayliss Turkel (BT) BT80] BGT82] and Khebir RamahiMittra (KRM) RBC [KRM89] are both based on the two dimensional farfield expansion by Karp [Kar61] The KRM RBC has one more term in the numerators of ff 2 and fi 2 ; these coefficients can also be considered as the leading terms of a Taylor series expansion in ae Gamma1 of ....

A. Bayliss, M. Gunzburger, and E. Turkel. Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J. Appl. Math., 42(2):430--451, April 1982.

....in the boundary condition by a rational approximation. While Engquist and Majda s approach applies to wave problems of all kinds, CHAPTER 1. INTRODUCTION 31 there is another approach for obtaining local NRBC s specifically for the Helmholtz equation which is due to Bayliss, Gunzburger and Turkel [8]. This approach uses the asymptotic expansion which solutions of exterior Helmholtz problems satisfy outside of some disk (cf. 7, 110, 111, 66] By requiring that the approximate solution agree with the exact solution up to the term of the asymptotic expansion with index m on the artificial ....

A. Bayliss, M. Gunzburger, and E. Turkel. Boundary conditions for the numerical solution of elliptic equations in exterior domains. SIAM J. Appl. Math., 42:430--451, 1982.

....difficult or impossible to solve. Our second goal is to derive the DtN condition for elliptic and spheroidal artificial boundaries. Third, we shall derive approximate local boundary conditions for elliptic and spheroidal coordinates, which are the analogues of the Bayliss GunzburgerTurkel (BGT) [2] boundary conditions in polar and spherical coordinates. Fourth, we shall modify the truncated versions of the DtN conditions for elliptic and spheroidal boundaries. Fifth, we shall present a sequence of local boundary conditions in two dimensional polar coordinates, which are much more accurate ....

....condition (3.2) is unique for all N 0 . The same conclusion holds in elliptic or spheroidal coordinates, with S defined by = a and r replaced by in (3.2) As an application of theorem 3.2, let us choose B = r B 2 , where B 2 is the second BGT operator defined in (4.11) with m = 2. In [2] it was shown that (2.1) and (2.2) with B 2 U = 0 on r = a, is well posed for Gamma a sphere. Theorem 3.2 shows that the problem remains well posed when the modified DtN condition (3.2) is used with B = B 2 r . This theorem holds when Gamma is a circle, sphere, ellipse, or spheroid, which ....

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A. Bayliss, M. Gunzburger, and E. Turkel, Boundary Conditions for the Numerical Solution of Elliptic Equations in Exterior Regions. SIAM J. Appl. Math., Vol. 42, No. 2, pp. 430--451, 1982.

....requiring both differential equation and boundary integral methods. While this approach has the advantage that the radiation boundary condition is exact, the resulting system will contain fully populated submatrices. This problem can be overcome through the use of local, but approximate, RBCs [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Since the behavior of the outgoing wave is described by the fields in the local neighborhood, the resulting matrices are sparse. To be effective, the approximations involved with the use local RBCs must not introduce significant errors due to reflections from the boundary. In this paper we focus ....

....from the boundary. In this paper we focus on local RBCs. Local RBCs for planar and circular artificial boundaries are well known. Engquist and Majda [5, 6] developed a boundary condition for planar boundaries using Pad e approximations to pseudo differential operators. Bayliss and Turkel (BT) [7, 8] approximate the normal derivative of the fields over circular or spherical boundaries. The BT boundary condition is widely used and has been shown to be accurate in the far field of the scatterer. Because of the nature of the field expansion used in the derivation of these boundary conditions, ....

A. Bayliss, M. Gunzburger, and E. Turkel, "Boundary conditions for the numerical solution of elliptic equations in exterior regions", SIAM J. Appl. Math., vol. 42, pp. 430--451, April 1982.

....[32] is proportional to the lift, which is calculated by integrating the pressure along the surface. There is also another way of using the asymptotics for setting the far field ABC s. It has been proposed by Bayliss and Turkel in work [33, 34, 35] and by Bayliss, Gunzburger, and Turkel in work [36] and does not require the explicit knowledge of the coefficients. Instead, the authors of [33, 34, 35, 36] develop a set of special local differential relations that identically cancel out the prescribed number of leading terms in the corresponding series; these relations can obviously serve as ....

....surface. There is also another way of using the asymptotics for setting the far field ABC s. It has been proposed by Bayliss and Turkel in work [33, 34, 35] and by Bayliss, Gunzburger, and Turkel in work [36] and does not require the explicit knowledge of the coefficients. Instead, the authors of [33, 34, 35, 36] develop a set of special local differential relations that identically cancel out the prescribed number of leading terms in the corresponding series; these relations can obviously serve as the ABC s. However, they are typically of a high order even if the order of the original equation (system) ....

A. Bayliss, M. Gunzburger, and E. Turkel, Boundary Conditions for Numerical Solution of Elliptic Equations in Exterior Domains, SIAM J. Appl. Math., 42 (1982) pp. 430--451.

....flows can be found in Section 4 of the paper. Another group of local ABC s techniques that use the asymptotic form of the solution has been introduced by Bayliss and Turkel in work [114 116] for time dependent problems (wave type and Euler s equations) and Bayliss, Gunzburger, and Turkel in work [117] for steady state problems (Helmholtz s and Laplace s equations) The proposed methodology is independent and does not relate directly to local approximations of the previously constructed exact ABC s. Moreover, unlike some of the aforementioned asymptotic methods, the approach of [114 117] does ....

....in work [117] for steady state problems (Helmholtz s and Laplace s equations) The proposed methodology is independent and does not relate directly to local approximations of the previously constructed exact ABC s. Moreover, unlike some of the aforementioned asymptotic methods, the approach of [114 117] does not require the explicit knowledge of the coefficients in the asymptotic expansion. In fact, the authors of [114 117] do not use the asymptotic form of the solution directly to set the ABC s, they rather construct a set of special local differential relations that identically cancel out a ....

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A. Bayliss, M. Gunzburger, and E. Turkel, Boundary Conditions for Numerical Solution of Elliptic Equations in Exterior Domains, SIAM J. Appl. Math., 42 (1982) pp. 430--451.

....of separability of the problem. Another way to address the computational difficulties for the discretized Helmholtz equation is to design iterative methods. Bayliss et al. BaGoTu83] used a preconditioned conjugate gradient method applied to the normal equations for a finite element discretization [BaGuTu82]. Due to the ill conditioning of the normal equations, the unpreconditioned algorithm suffered from extremely slow convergence. The convergence rate was substantially improved through preconditioners based on symmetric successive overrelaxation [BaGoTu83] or a multigrid V cycle [BaGoTu85b] ....

....transformation this becomes u 1 (1; 2 ) 0; 0 2 1: 4) For the radiation conditions at the near and far zone boundaries, Dirichlet to Neumann (DtN) maps [KeGi89] are employed. The main reason for choosing nonlocal DtN maps, instead of the local radiation conditions described in [BaGuTu82], is that discretized DtN maps are more apt to preconditioning by fast transforms. Our design of radiation conditions follows the principles outlined in [FixMa78] where a variational formulation of DtN conditions was derived for an axially symmetric duct parametrized by cylindrical coordinates. ....

A. Bayliss, M. Gunzburger, and E. Turkel, Boundary conditions for the numerical solution of elliptic equations in exterior regions, SIAM J. Appl. Math., 42 (1982), pp. 430--451.

....Efficient numerical solution procedures for scattering problems necessitate the imposition of an RBC as close as possible to the scatterer. In order to maintain the desirable sparse characteristics of a differential equation formulation, a local RBC is often employed. Bayliss and Turkel (BT) 1] [2] have proposed RBCs which approximately absorb outgoing waves over a circular or spherical boundary. A wide variety of other local RBCs have been proposed [3, 4] and are usually similar in form to the BT expressions. In this article we consider the extension of the second order BT RBC to ....

A. Bayliss, M. Gunzburger and E. Turkel, "Boundary Conditions for the Numerical Solution of Elliptic Equations in Exterior Regions," SIAM J. Appl. Math., vol. 42, no. 2, April, 1982, pp. 430--451.

....for a normally incident wave impinging on a planar artificial boundary. These ideas have been extended and generalized in numerous ways, in particular by Higdon [7] Radiation boundary conditions designed for use on a circular artificial boundary have been introduced by Bayliss and Turkel ( 1] [2]) These boundary conditions are based on the annihilation of leading terms in an asymptotic expansion of the solution in the far field. Similar ideas making use of a modal expansion have been recently proposed by Li and Cendes [11] The Bayliss Turkel and Li Cendes RBCs have the same general ....

....ff and fi is often based on an expansion of the solution in the exterior of a circle. A sequence of approximate RBCs, fBm g, can be constructed from the boundary operators that annihilate an increasing number of the leading terms in the expansion. Two examples are the Bayliss Turkel RBCs [2], based on the far field expansion u = e Gammaikr p r 1 X n=0 a n ( r n (8) and the Li Cendes RBCs [11] based on the Hankel expansion u = 1 X n=0 b n ( H n (kr) where H n : H (2) n is the n th order Hankel function of the second kind. A third set of coefficients, the ....

A. Bayliss, M. Gunzburger, E. Turkel, Boundary conditions for the numerical solution of elliptic equations in exterior regions, SIAM J. Appl. Math., 42(1982), pp. 430--451.

....this term corresponds to the point vortex model. Except for approximating the exact ABC s, there are, of course, independent techniques for constructing the approximate local ABC s. In particular, the approach introduced by Bayliss and Turkel [53, 54, 55] and Bayliss, Gunzburger, and Turkel [56] is also based on using the far field asymptotic expansion of the solution. However, the authors do not directly use this expansion to set the ABC s but construct special local differential relations that annihilate a certain number of leading terms in the aforementioned expansion. Being applied ....

....to set the ABC s but construct special local differential relations that annihilate a certain number of leading terms in the aforementioned expansion. Being applied at the artificial boundary, these relations provide some local ABC s. It is interesting to mention that sometimes local ABC s [53, 54, 55, 56] may coincide with those obtained by means of rational approximation to the PsiDO symbols. We also note that an apparatus of asymptotic expansions for constructing the approximate ABC s was extensively used by Hagstrom [9, 10] Hagstrom and Keller [11] Hagstrom and Hariharan [57] and Hagstrom ....

A. Bayliss, M. Gunzburger, and E. Turkel, Boundary Conditions for Numerical Solution of Elliptic Equations in Exterior Domains, SIAM J. Appl. Math., 42 (1982) pp. 430--451.

....banded form of open region problems through the use of approximate boundary conditions. These boundary conditions model the behavior of the field extending to infinity on the outer boundary of the finite region. These approximate boundary conditions are mainly absorbing boundary conditions (ABC) [4, 5, 6, 7, 8] and on surface radiation conditions (OSRC) 9] used to approximate the radiation boundary condition. When these methods are applied correctly, the solution inside the truncated region approximates the problem over the entire space. However, these methods have drawbacks. Some of them impose ....

A. Bayliss and E. Turkel, "Boundary conditions for the numerical solution of elliptic equations in exterior regions," SIAM J. Appl. Math., vol. 42, no. 2, pp. 430--451, April 1982.

....in nity. Based on their concept, Bayliss and Turkel [5] obtained a sequence of radiation boundary conditions for the wave equation with axial and spherical symmetries. Their boundary conditions are based on an asymptotic expansion of the solution at large distances. Bayliss, Gunzburger, and Turkel [4] used the same boundary conditions with a nite element method. 1.6.1 In Three Dimensions The associated Helmholtz equation for time harmonic waves is #u k 2 u = 0, 1.6.1) 1.6. Radiation Boundary Conditions 19 where the wave speed c has been scaled to unity and the harmonic wave is ....

....and in fact it can easily be seen that it is not exact even for the rst term in the expansion (1.6.2) We develop a sequence of linear dioeerential operators Bm which provide more accurate extension of the condition (1.6.3) by annihilating the rst m terms in the far eld expansion (1.6. 2) [5, 4]. Thus the condition that at r = R the solution lies in the null space of the operator Bm can be considered as a procedure to match the solution to the rst m terms in the expansion (1.6.2) Let us de ne L = ik # #r . Multiplying (1.6.2) by r m and then applying the operator L m we see ....

A. Bayliss, M. Gunzburger, and E. Turkel, Boundary conditions for the numerical solution of elliptic equations in exterior regions, SIAM J. Appl. Math. 42 (1982), 430451.

.... systems, although the fast solvers in such settings are not nearly as fast, relying on generation and solution of general block tridiagonal systems [25] We also expect them to perform well for more accurate local approximations to the radiation boundary conditions (3) of the type considered in [4]. Acknowledgements. We thank Alan Sussman for a great deal of advice on using the IBM SP2, Ilya Zavorine for help with programming, and Olof Widlund for some helpful discussions. ....

A. Bayliss, M. Gunzburger, and E. Turkel, Boundary conditions for the numerical solution of elliptic equations in exterior domains, SIAM J. Appl. Math., 42 (1982), pp. 430--451.

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A. Bayliss, M. Gunzburger, and E. Turkel, "boundary conditions for the numerical solution of elliptic equations in exterior regions," SIAM J. Appl. Math., vol. 42, pp. 430-451, Apr. 1982.

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A. Bayliss, M. Gunzburger, and E. Turkel, "Boundary conditions for the numerical solution of elliptic equations in exterior regions," SIAM J. Appl. Math., vol. 42, pp. 430--451, Apr. 1982.

No context found.

A. Bayliss, M. Gunzburger, and E. Turkel. Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J. Appl. Math., 42(2):430--451, April 1982.

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A. Bayliss, M. Gunzburger, and E. Turkel. Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J. Appl. Math., 42(2):430--451, April 1982. Key: bayliss82

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A. Bayliss, M. Gunzburger, and E. Turkel, "boundary conditions for the numerical solution of elliptic equations in exterior regions," SIAM J. Appl. Math., vol. 42, pp. 430-451, Apr. 1982.

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