Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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APPROXIMATED SEPARATION FORMULA FOR THE HELMHOLTZ EQUATION

  • Received : 2018.11.11
  • Accepted : 2018.12.14
  • Published : 2019.06.25
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Abstract

The Helmholtz equation represents acoustic or electromagnetic scattering phenomena. The Method of Lines are known to have many advantages in simulation of forward and inverse scattering problems due to the usage of angle rays and Bessel functions. However, the method does not account for the jump phenomena on obstacle boundary and the approximation includes many high order Bessel functions. The high order Bessel functions have extreme blow-up or die-out features in resonance region obstacle boundary. Therefore, in particular, when we consider shape reconstruction problems, the method is suffered from severe instabilities due to the logical confliction and the severe singularities of high order Bessel functions. In this paper, two approximation formulas for the Helmholtz equation are introduced. The formulas are new and powerful. The derivation is based on Method of Lines, Huygen's principle, boundary jump relations, Addition Formula, and the orthogonality of the trigonometric functions. The formulas reduce the approximation dimension significantly so that only lower order Bessel functions are required. They overcome the severe instability near the obstacle boundary and reduce the computational time significantly. The convergence is exponential. The formulas adopt the scattering jump phenomena on the boundary, and separate the boundary information from the measured scattered fields. Thus, the sensitivities of the scattered fields caused by the boundary changes can be analyzed easily. Several numerical experiments are performed. The results show the superiority of the proposed formulas in accuracy, efficiency, and stability.

Keywords

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Fig. 1(a). Kite

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Fig. 1(b). Peanut

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Fig. 1(c). Non-symmetric

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Fig. 2(a). |us|

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Fig. 2(b). Angle(us)

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Fig. 3(a). |us|

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Fig. 3(b). Angle(us)

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Fig. 4(a). |us|

HNSHCY_2019_v41n2_403_f0009.png 이미지

Fig. 4(b). Angle(us)

TABLE 1. L2-Errors (θ = 0, ρ = 20, N = 128)

HNSHCY_2019_v41n2_403_t0001.png 이미지

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