Free-Surface Boundary Condition in Finite-Difference Elastic Wave Modeling

2004 ◽  
Vol 94 (1) ◽  
pp. 237-250 ◽  
Author(s):  
D.-J. Min
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Finite Difference ◽  
Elastic Wave ◽  
Surface Boundary ◽  
Wave Modeling ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition

scholarly journals An improved vacuum formulation for 2D finite-difference modeling of Rayleigh waves including surface topography and internal discontinuities

Geophysics ◽  
2012 ◽  
Vol 77 (1) ◽  
pp. T1-T9 ◽  
Author(s):  
Chong Zeng ◽  
Jianghai Xia ◽  
Richard D. Miller ◽  
Georgios P. Tsoflias
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Finite Difference ◽  
Surface Topography ◽  
Rayleigh Waves ◽  
Surface Boundary ◽  
Near Surface ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition ◽  
Grid Nodes

Rayleigh waves are generated along the free surface and their propagation can be strongly influenced by surface topography. Modeling of Rayleigh waves in the near surface in the presence of topography is fundamental to the study of surface waves in environmental and engineering geophysics. For simulation of Rayleigh waves, the traction-free boundary condition needs to be satisfied on the free surface. A vacuum formulation naturally incorporates surface topography in finite-difference (FD) modeling by treating the surface grid nodes as the internal grid nodes. However, the conventional vacuum formulation does not completely fulfill the free-surface boundary condition and becomes unstable for modeling using high-order FD operators. We developed a stable vacuum formulation that fully satisfies the free-surface boundary condition by choosing an appropriate combination of the staggered-grid form and a parameter-averaging scheme. The elastic parameters on the topographic free surface are updated with exactly the same treatment as internal grid nodes. The improved vacuum formulation can accurately and stably simulate Rayleigh waves along the topographic surface for homogeneous and heterogeneous elastic models with high Poisson’s ratios ([Formula: see text]). This method requires fewer grid points per wavelength than the stress-image-based methods. Internal discontinuities in a model can be handled without modification of the algorithm. Only minor changes are required to implement the improved vacuum formulation in existing 2D FD modeling codes.

scholarly journals A stable free‐surface boundary condition for two‐dimensional elastic finite‐difference wave simulation

Geophysics ◽  
1986 ◽  
Vol 51 (12) ◽  
pp. 2247-2249 ◽  
Author(s):  
John E. Vidale ◽  
Robert W. Clayton
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Finite Difference ◽  
Difference Approximation ◽  
Surface Boundary ◽  
Stability Range ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition ◽  
The One ◽  
Lateral Variations

Two of the persistent problems in finite‐difference solutions of the elastic wave equation are the limited stability range of the free‐surface boundary condition and the boundary condition’s treatment of lateral variations in velocity and density. The centered‐difference approximation presented by Alterman and Karal (1968), for example, remains stable only for β/α greater than 0.30, where β and α are the shear [Formula: see text] and compressional [Formula: see text] wave velocities. The one‐sided approximation (Alterman and Rotenberg, 1969) and composed approximation (Ilan et al., 1975) have similar restrictions. The revised‐composed approximation of Ilan and Loewenthal (1976) overcomes this restriction, but cannot handle laterally varying media properly.

Coupled Dynamic Analysis of a Moored Spar Platform in Time Domain

2010 ◽  
Author(s):  
M. D. Yang ◽  
B. Teng
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Dynamic Analysis ◽  
Time Domain ◽  
Surface Boundary ◽  
Spar Platform ◽  
Mooring Lines ◽  
Coupled Dynamic Analysis ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition

A time-domain simulation method is developed for the coupled dynamic analysis of a spar platform with mooring lines. For the hydrodynamic loads, a time domain second order method is developed. In this approach, Taylor series expansions are applied to the body surface boundary condition and the free surface boundary condition, and Stokes perturbation procedure is then used to establish corresponding boundary value problems with time-independent boundaries. A higher order boundary element method is developed to calculate the velocity potential of the resulting flow field at each time step. The free-surface boundary condition is satisfied to the second order by 4th order Adams-Bashforth-Moultn method. An artificial damping layer is adopted on the free surface to avoid the wave reflection. For the mooring-line dynamics, a geometrically nonlinear finite element method using isoparametric cable element based on the total Lagrangian formulation is developed. In the coupled dynamic analysis, the motion equation for the hull and dynamic equations for mooring lines are solved simultaneously using Newmark method. Numerical results including motions and tensions in the mooring lines are presented.

scholarly journals Free-Surface Effects on Interaction of Multiple Ships Moving at Different Speeds

2019 ◽  
Vol 63 (4) ◽  
pp. 251-267 ◽  
Author(s):  
Zhi-Ming Yuan ◽  
Liang Li ◽  
Ronald W. Yeung
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Present Article ◽  
Hydrodynamic Interaction ◽  
Surface Effects ◽  
Superposition Method ◽  
Surface Boundary ◽  
Gravitational Acceleration ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition

Ships often have to pass each other in proximity in harbor areas and waterways in dense shipping-traffic environment. Hydrodynamic interaction occurs when a ship is overtaking (or being overtaken) or encountering other ships. Such an interactive effect could be magnified in confined waterways, e.g., shallow and narrow rivers. Since Yeung published his initial work on ship interaction in shallow water, progress on unsteady interaction among multiple ships has been slow, though steady, over the following decades. With some exceptions, nearly all the published studies on ship-to-ship problem neglected free-surface effects, and a rigid-wall condition has often been applied on the water surface as the boundary condition. When the speed of the ships is low, this assumption is reasonably accurate as the hydrodynamic interaction is mainly induced by near-field disturbances. However, in many maneuvering operations, the encountering or overtaking speeds are actually moderately high (Froude number Fn > 0.2, where <inline-graphic xlink:href="josr10180089inf1.tif"/>, U is ship speed, g is the gravitational acceleration, and L is the ship length), especially when the lateral separation between ships is the order of ship length. Here, the far-field effects arising from ship waves can be important. The hydrodynamic interaction model must take into account the surface-wave effects. Classical potential-flow formulation is only able to deal with the boundary value problem when there is only one speed involved in the free-surface boundary condition. For multiple ships traveling with different speeds, it is not possible to express the free-surface boundary condition by a single velocity potential. Instead, a superposition method can be applied to account for the velocity field induced by each vessel with its own and unique speed. The main objective of the present article is to propose a rational superposition method to handle the unsteady free-surface boundary condition containing two or more speed terms, and validate its feasibility in predicting the hydrodynamic behavior in ship encountering. The methodology used in the present article is a three-dimensional boundary-element method based on a Rankine-type (infinite-space) source function, initially introduced by Bai and Yeung. The numerical simulations are conducted by using an in-house‐developed multibody hydrodynamic interaction program “MHydro.” Waves generated and forces (or moments) are calculated when ships are encountering or passing each other. Published model-test results are used to validate our calculations, and very good agreement has been observed. The numerical results show that free-surface effects need to be taken into account for Fn > 0.2.

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