Extension of the PML Absorbing Boundary Condition to 3D Spherical Coordinates: Scalar Case F. L. Teixeira and W. C. Chew
Center for Computational Electromagnetics, Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, Urbana IL 61801-2991 USA
Abstract|The Perfectly Matched Layer (PML) absorbing boundary condition has shown to be an extremely ecient way to truncate the computational domain in nite-di erence time-domain (FDTD) simulation of scattering and radiation problems in open regions. In this work we extend the PML concept to spherical coordinates by using an analytic continuation of the Helmholtz equation to complex coordinates (complex coordinate stretching). A 3D PMLFDTD code is written on a spherical grid to validate the formulation against results from pseudo-analytical solutions in free-space. Index Terms | Di erential equations, FDTD methods, transient propagation. I. Introduction
The Perfectly Matched Layer (PML) absorbing boundary condition (ABC), rst devised by Berenger [1] for Cartesian coordinates, has been demonstrated to be more ecient than other local ABCs in preventing spurious re ections from the grid terminations in nite-di erence time-domain (FDTD) simulations [1]-[6]. In [2], it was shown that a PML medium can be derived ab initio by using the concept of complex coordinate stretching, whereby a modi ed set of Maxwell's equations in the frequency domain is introduced with additional degrees of freedom providing the desired absorption in speci ed Cartesian directions without any re ection in the continuum limit. Further development of this approach [7] showed that the modi ed Maxwell's equations in the PML media reduce to the ordinary ones but on a complex coordinate space. From this fact, closed form solutions obtained in the real space map directly to corresponding closed form solutions on the PML complex space through a simple analytic continuation of the spatial variables. More importantly, if the resultant solutions are still causal, this can be achieved in non-Cartesian coordinated systems, Manuscript received November 3, 1997. F. L. Teixeira, fax 1-217-333-5962,
[email protected]; W. C. Chew, fax 1-217-333-5962,
[email protected], http://www.ccem.uiuc.edu/. This work is supported by Air Force Oce of Scienti c Research under MURI grant F49620-96-1-0025, Oce of Naval Research under grant N00014 -95-1-0872, National Science Foundation under grant ECS93-02145 and by a CAPES Graduate Fellowship.
providing a systematic procedure to achieve PML ABC on these systems. In this work we derive the formulation for a PML in spherical coordinates and for scalar wave propagation through an analytic continuation on the radial variable. An FDTD code is written on a 3D spherical grid to validate the approach. Results are compared with analytical solutions in free-space. II. Helmholtz Equation in Complex Space
The modi ed Helmholtz equation inside the Cartesian PML is written as (e?i!t convention) [2], [7]: (rs rs) + k2 = 0
(1)
In the above @ + y^ 1 @ + z^ 1 @ ; rs = x^ s1 @x s @y s @z x
y
z
(2)
where sx (x), sy (y), and sz (z) are the frequency-dependent complex stretching variables [2]. With the following change of variables, Z ~ = s ( 0 ) d 0 ; (3) 0
so that @ = 1 @; @ ~ s @
(4)
where stands for x; y; z, the modi ed Helmholtz equation on a PML medium can be recast in the same form as the original Helmholtz equation but on a complex variable spatial domain, (~x; y~; z~). Therefore, closed form solutions obtained in the ordinary media can be mapped to the PML media through a simple analytic continuation of the spatial variables to a complex space. This analytic continuation can be easily generalized to other coordinate systems to obtain partial di erential equations incorporating absorbing boundary conditions. In the next section we illustrate this for spherical coordinates. By similar approach, the PML can also be derived for cylindrical coordinates.
III. Complex 3D Spherical Coordinates
In spherical coordinates, the scalar wave equation reads
2 1 @ sin @ + r2 + k2 = @@r2 + r2 @@r + r2 sin @ @ 1 @ 2 + k2 = ?(~r ? ~r0): (5) r2 sin2 @ 2 The solution of the above in free-space is ikj~r?~r j (~r) = 4e j ~r ? ~r0 j : 0
(6)
0.8 0.6 0.4 Y Axis in Meters
as
1
0.2 0 −0.2 −0.4 −0.6
When a perfectly conducting wall is placed at r = a, −0.8 the solution inside the sphere is given by ikj~r?~r j −1 −1 −0.5 0 0.5 1 (~r) = 4e j ~r ? ~r0 j ? X Axis in Meters n 1 X (1) X ( 0 ) jn (kr0 )hn (ka) ;(7) Fig. 1. Solution of a point source inside a 3D sphere of radius 1.0 ik jn (kr)Ynm ( )Ynm j (ka) 0
n
n=0 m=?n
where
s
m. The frequency is 2.2 GHz, = 0:2, and rm = 0:85 m. With this small complex stretching, one can still see re ections from the walls of the sphere.
? m)! 2n + 1 P m (cos )eim ;(8) that with this mapping, the sphere radius is inside the Ynm ( = ; ) = (n complex space, a ! ~a. (n + m)! 4 n Figures 1 and 2 show the calculation of the above series summation formula inside the sphere for two di erent and jn(), hn () are spherical Bessel and Hankel functions, values of > 0. For larger , one can see diminished rerespectively.
ections form the walls. This is because the re ected eld A complex map can be de ned such that the radial [summation term in (7)] becomes exponentially small unvariable, r, is mapped through der the mapping (9). Therefore, the resultant solution will Zr behave just as a free-space solution, as Figure 2 clearly ilr ! r~ = sr (r0 ) dr0 = r + ir (r)=!; (9) lustrates. 0 It can also be shown that this complex coordinate from which stretching at the outer boundary (concave PML) preserve the analyticity of the solutions on the upper-half complex @ = 1 @; (10) ! plane, and, therefore, the resultant solutions do not @~r sr @r violate causality [8]. However, this is not true for a comsr (r) = 1Z+ ir (r)=!; (11) plex stretching on the inner spherical boundary (convex r PML), the resultant solutions present singularities 0 0 (12) (poles) where r (r) = r (r )dr : on the upper-half complex ! plane (non-causal so0 lutions [8]). In the FDTD method, it implies a dynamical The variable r in (11) is the added degree of free- unstable time-stepping scheme for the convex case. dom that achieves the absorption for outward traveling waves. In the physical region, r must be set equal to IV. Time-Domain Formulation zero in order to reduce the modi ed equation to the original (physical) one. To avoid numerical re ections due to From the discussion of the previous section, the modithe discretization process, r should increase gradually in ed Helmholtz equation in complex spherical coordinates the PML region. Analogous to the Cartesian case [2], a reads as quadratic taper is chosen so that r is given by: (r ? rm )2 2 r (r) = (a (13) 1 @ sin @ + ? rm )2 r~ 2 + k2 = @@~r2 + r2~ @@~r + r~2 sin @ @ for rm < r < a, and r (r) = 0 for 0 < r < rm . The 1 @ 2 + k2 = 0: (14) variable rm is the inner radius of the PML region. Note r~2 sin2 @ 2
1 0.6
0.8 0.4
0.6 0.2
psi field (norm.)
Y Axis in Meters
0.4 0.2 0
0
−0.2
−0.4
−0.2 −0.6
−0.4 −0.8
−0.6 −1 0
−0.8 −1 −1
−0.5
0 X Axis in Meters
0.5
1
Fig. 2. Solution of a point source inside a 3D sphere of radius 1.0 m. The frequency is 2.2 GHz, = 2:0, and rm = 0:85 m. With this larger complex stretching, one can hardly see any re ections from the walls of the sphere. De ning (e?i!t convention)
ik~ = r~ ; and letting
(15)
0.2
0.4
0.6
0.8 1 t (sec.)
1.2
1.4
1.6
1.8 −8
x 10
Fig. 3. Pseudo-analytical solution for an in nitesimal electric dipole on free-space (dashed line) vs. 3D spherical-grid FDTD solution without PML (solid line). A spurious re ected pulse due to the grid truncation is present.
order to preserve the second-order accuracy of the overall staggered-grid scheme, the r component in (17) should be taken as an average over the half-grid points. V. Time-Domain Numerical Results
The formulation is validated by comparing the results for the PML-FDTD simulation of a point ik (16) sourceobtained against a pseudo-analytical solution. The pseudoanalytical solution is obtained by rst solving the free(17) space problem in the frequency domain for many excitaik tion frequencies and multiplying by the spectrum of the ik (18) FDTD source pulse. The resultant eld in the frequencydomain is then inverse-Fourier transformed to yield the (19) time-domain solution. ik Fig. 3 shows the normalized eld computed with both the pseudo-analytical formulation and the 3-D FDTD althen gorithm for a point source. The excitation pulse is the rst = r1 + r2 + + : (20) derivative of a slightly di erent version of the BlackmanHarris pulse [11] so that the pulse vanishes completely A time-stepping scheme for (14) is easily derived from after a time period T = 1:55=fc. The central frequency is (15)-(20) by identifying: at fc = 300 MHz (c = 1 m). The point source is located at (r; ; ) = (2:5c; 900; 00). The resultant eld is sam@ 1 (21) pled at (r; ; ) = (3c ; 900; 00). The spherical grid has a iksr ! ? c @t + r ; termination at r = 4c . From this gure, one clearly sees the direct pulse and the spurious re ected pulse due to 1 @ ik~r ! ? c r @t + r : (22) the grid termination. Fig. 4 superposes the pseudo-analytical solution with The time-domain versions of (6)-(10) are implemented on the FDTD algorithm including an 8-layer spherical PML a spherical staggered-grid with central di erencing. The region before the grid ends (outer boundary). The PML spatial discretization and the treatment of the spherical thickness is 0:8c, for a cell size r = c =10 in the ragrid singularities are similar to [9]. The elds are de- dial direction. The spurious re ection present in Fig. 1 ned over integer grid points and the vector ~ is a fore- is suppressed. Any di erences between the FDTD and vector associated with the integer grid points [10]. In the pseudo-analytical results can be attributed to mod@ r 1 @ r r1 = @~r = s @r ; r 2 r r2 = r~ ; 1 @ = r~ sin @ (sin ) ; 1 @ = r~ sin @ ;
The accuracy of the formulation is tested by computing the eld radiated from a point source using the proposed PML-FDTD algorithm and comparing it against a freespace pseudo-analytical solution. Using this approach, the PML-FDTD can also be extended to cylindrical coordinates. This can be done through a complex stretching on the radial, , and the z coordinates [7].
0.6
0.4
psi field (norm.)
0.2
0
−0.2
References
−0.4
−0.6
−0.8
−1 0
0.2
0.4
0.6
0.8 1 t (sec.)
1.2
1.4
1.6
1.8 −8
x 10
Fig. 4. Pseudo-analytical solution for an in nitesimal electric dipole on free-space (dashed line) vs. 3D spherical-grid FDTD solution with 8-layer PML (solid line). The spurious re ected pulse is suppressed.
eling errors such as the nite size of the point source, the discretization of the Maxwell's equations as well as grid dispersion e ects due to the high grid curvature in the simulation region. In particular, it is noted that the small oscillations after the passage of the incident pulse (typically a grid dispersion e ect) are also present on the simulation without the PML. VI. Conclusions
A perfectly matched layer absorbing boundary condition is derived for the scalar wave equation in spherical coordinates. The formulation is based on the complex coordinate stretching approach. A complex mapping (analytic continuation) of the radial coordinate is performed on the spherical Helmholtz equation to provide a re ectionless absorption of outgoing waves at the outer boundary. A 3D FDTD code on a spherical grid is developed based on the time domain version of the equations.
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