음향파동방정식의 유한요소법

목차

초 록
List of Figures
List of Tables
제 1 장 서 론
제 2 장 음향파동방정식의 유한요소법
2.1 음향파동방정식의 유한요소식 유도
2.2 음향파동방정식의 유한요소화
2.2.1 요소 강성행렬 및 질량행렬 계산
2.2.2 사각형 요소행렬 계산의 특성
2.2.3 음향파동방정식의 유한요소화
2.3 가중평균 유한요소법 개발
2.3.1 강성행렬의 확장
2.3.2 질량행렬의 확장
2.3.3 가중평균 유한요소법 적용
제 3 장 결과 및 고찰
3.1 가중평균 유한요소법의 정확도 분석
3.1.1 수치오차
3.1.2 가중평균계수의 결정
3.1.3 분산곡선 분석
3.1.4 해석해와의 비교
3.2 가중평균 유한요소법의 효율성
3.3 인공 음향파의 합성 고찰
3.3.1 전산화
3.3.1.1 모델링 변수의 결정
3.3.1.2 경계조건
3.3.1.3 로이드 미러 효과
3.3.2. 로이드미로 효과 고찰
3.3.3. 종래의 유한요소법과 결과 비교
제 4 장. 결 론

본문내용

7)
이 된다.
무한매질에서 2 차원 그린함수는
a~ -> ~ INF
경우에 얻어진다. (A.36)식과 (A.37)식 모두에서
a~ -> ~ INF
일때
~`-`` J_0 (k xi ){Y_0`(k a )} over {J_0`(k a ) }J_0`(kr)~
을 빼고
{ i} over {4 pi } J_0`(k xi)J_0`(kr)
을 더한다.
따라서,
if~~~ 0 ``<`` xi `` < ``r``,
G( r``;`xi) ~=~{1} over {4} LEFT [J_0 (k xi )Y_0 (kr)``+``i`` J_0 (k xi ) J_0 (kr) RIGHT ]
(A.38)
+
if~~~ 0 ``<``r`` < ``xi``< ``a`,
G( r``;`xi) ~=~{1} over {4} LEFT [Y_0 (k xi )J_0 (kr)``+``i`` J_0 (k xi ) J_0 (kr) RIGHT ]
(A.39)
or
G( r``;`xi) ~=~ {i} over{4}J_0 (k xi )``( J_0 (kr) ``-``i`` Y_0 (kr))
(A.40)
G( r``;`xi) ~=~ {i} over{4}J_0 (k r )``( J_0 (k xi) ``-``i`` Y_0 (k xi))##
(A.41)
이 된다.
만일
xi~ -> ~0
이라면, 2 차원 스칼라 파동방정식의 그린함수는
G( r``;0) ~=~ {i} over {4}``H_0`^(2)`(kr)##
(A.42)
이 된다.
ABSTRACT
The space-frequency domain modelling is very efficient in the simulation of roll along seismic aquisition because of it's flexibility in defining the damping matrix for seismic attenuation and dispersion effects.
The conventional operator in the space-frequency domain requires more grid points than operators in the space-time domain to achieve the same accuracy. Thus, this has been one of the problems to overcome. if one use space-frequency domain methods.
In this study, a new weighted-averaging finite-element method which can reduce the number of grid points per minimum wavelength is designed. The stiffness matrix and mass matrix, the consistent and lumped masses, obtained for four different set of finite-elements were averaged with weighting coefficients. The weighting coefficients were determined to minimize numerical dispersion and numerical anisotropy by employing the extended 25-point finite-difference operator and the Gauss-Newton optimization technique.
The weighted-averaging finite-element method, proposed in this study, can reduce the number of grid points about 3 per minimum wavelength within 1% errors of normalized group velocity. This reduction of grid points reduces computer core memory to about 1/60, compared to the conventional formula when the band type solver is used to solve the symmetric complex impedance matrix. A comparison of numerical and analytic solutions showed that the weighted-averaging finite-element method can yield more accurate results with less grid points than the conventional frequency domain finite-element method.
A 2D numerical modelling of acoustic wave equation was tested to simulate the homogeneous model. The interference effect between the direct wave from source to receiver and the wave reflected from the free surface boundary was included in the modelling for the first time in the finite element form. Hereby, the efficiency of a new weighted-averaging finite-element method was confirmed in frequency domain for the more accurate results, and the more cost-effective method.
Keywords : space-frequency domain modelling, acoustic wave
equation , weighted-averaging finite-element method,
numerical dispersion, numerical anisotropy,
numerical velocity
Student Number : 90411-811

키워드

음향파동,   강성행렬,   가중평균,   모델링 변수의 결정,   경계조건,   로이드 미러 효과