Journal of the Computational Structural Engineering Institute of Korea (한국전산구조공학회논문집)

Computational Structural Engineering Institute of Korea (한국전산구조공학회)
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Explicit Transient Simulation of SH-waves Using a Spectral Element Method

스펙트럴 요소법을 이용한 SH파 전파의 외연적 시간이력해석

  • 윤승욱 (홍익대학교 토목공학과) ;
  • 강준원 (홍익대학교 토목공학과)
  • Received : 2018.01.18
  • Accepted : 2018.03.20
  • Published : 2018.04.30
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Abstract

This paper introduces a new explicit spectral element method for the simulation of SH-waves in semi-infinite domains. To simulate the wave motion in unbounded domains, it is necessary to reduce the infinite extent to a finite computational domain of interest. To prevent the wave reflection from the trunctated boundaries, perfectly matched layer(PML) wave-absorbing boundary is introduced. The forward problem for simulating SH-waves in PML-truncated domains can be formulated as second-order PDEs. The second-order semi-discrete form of the governing PDEs is constructed by using a mixed spectral elements with Legendre-gauss-Lobatto quadrature method, which results in a diagonalized mass matrix. Then the second-order semi-discrete form is transformed to a first-order, whose solutions are calculated by the fourth-order Runge-Kutta method. Numerical examples showed that solutions of SH-wave in the two-dimensional analysis domain resulted in stable and accurate, and reflections from truncated boundaries could be reduced by using PML boundaries. Elastic wave propagation analysis using explicit time integration method may be apt for solving larger domain problems such as three-dimensional elastic wave problem more efficiently.

이 논문에서는 스펙트럴 요소법과 외연적 시간적분법을 이용해 SH파의 전파 거동을 계산하는 수치해석 기법을 제시한다. 2차원 영역에서의 탄성파 해석을 위해 해석영역을 유한 영역으로 한정하고 파동이 반사되지 않도록 수치적 파동흡수 경계조건인 perfectly matched layer(PML)를 도입하였다. PML이 포함된 시간영역 파동방정식의 유한요소해법을 위해 스펙트럴 요소법을 적용하였고 Legendre- Gauss-Lobatto 수치적분법을 사용하여 질량행렬을 대각화하였다. 2차 미분방정식 시스템의 파동방정식을 1차 미분방정식 시스템으로 변환하였고 병렬화를 통한 탄성파 해석 성능의 최적화를 위해 외연적 시간적분법인 4차 Runge-Kutta 방법을 이용해 해석영역에서의 변위응답을 계산하였다. 2차원 해석영역에서 SH파의 전파 거동을 계산하는 수치예제를 통해 제시한 외연적 스펙트럴 요소법의 정확성을 검증하였고 PML로 인한 반사파의 감쇠효과를 확인하였다. 외연적 시간적분법을 통한 탄성파 해석 기법은 3차원 영역과 같은 대규모 문제에서의 탄성파 수치해석을 효율적으로 수행할 수 있을 것으로 기대된다.

Keywords

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