A Coupled Finite Difference and Weighted Least Squares Simulation of Violent Breaking Wave Impact

2012 ◽  
Author(s):  
Ole Lindberg ◽  
Harry B. Bingham ◽  
Allan Peter Engsig Karup
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Least Squares ◽  
Flow Model ◽  
Weighted Least Squares ◽  
Surface Boundary ◽  
Wave Impact ◽  
Finite Point ◽  
Point Set ◽  
Vertical Breakwater

Two model for simulation of free surface flow are presented. The first model is a finite difference based potential flow model with non-linear kinematic and dynamic free surface boundary conditions. The second model is a weighted least squares based incompressible and inviscid flow model. A special feature of this model is a generalized finite point set method which is applied to the solution of the Poisson equation on an unstructured point distribution. The presented finite point set method is generalized to arbitrary order of approximation. The two models are applied to simulation of steep and overturning wave impacts on a vertical breakwater. Wave groups with five different wave heights are propagated from offshore to the vicinity of the breakwater, where the waves are steep, but still smooth and non-overturning. These waves are used as initial condition for the weighted least squares based incompressible and inviscid model and the wave impacts on the vertical breakwater are simulated in this model. The resulting maximum pressures and forces on the breakwater are relatively high when compared with other studies and this is due to the incompressible nature of the present model.

Fourth‐order finite‐difference P-SV seismograms

Geophysics ◽  
1988 ◽  
Vol 53 (11) ◽  
pp. 1425-1436 ◽  
Author(s):  
Alan R. Levander
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Fourth Order ◽  
Equations Of Motion ◽  
Dispersion Analysis ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Order System ◽  
Surface Boundary ◽  
Lamb’S Problem

I describe the properties of a fourth‐order accurate space, second‐order accurate time, two‐dimensional P-SV finite‐difference scheme based on the Madariaga‐Virieux staggered‐grid formulation. The numerical scheme is developed from the first‐order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga‐Virieux staggered‐grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic‐elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free‐surface or within a layer and to satisfy free‐surface boundary conditions. Benchmark comparisons of finite‐difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite‐difference and reflectivity solutions for elastic‐elastic and acoustic‐elastic layered models.

Free Surface NC Machining Tool Path Optimization Algorithm Based on the Iso-Scallop Method

2015 ◽  
Vol 799-800 ◽  
pp. 1193-1196 ◽  
Author(s):  
Shu Kun Cao ◽  
Yong Hong Deng ◽  
Kun Zhang ◽  
Shi Ping Liu ◽  
Wen Jing Meng
Keyword(s):  
Free Surface ◽  
Optimization Algorithm ◽  
Tool Path ◽  
Nc Machining ◽  
Path Optimization ◽  
Surface Boundary ◽  
Step Size ◽  
Processing Efficiency ◽  
Tool Path Optimization ◽  
Point Set

In order to solve the problem of free surface processing of tool redundancy,the tool lack problem, and the demerit of low machining efficiency, etc., based on the iso-scallop method, based on the iso-scallop method, we put forward a kind of free surface NC machining tool path optimization algorithm,make the surface boundary discrete point set, which is generated by point set ring machining path, diagonal connection and then use the path of the adjacent curve, forming cutting tool machining line.finally, the calculation of step size and line spacing in machining path based on the iso-scallop method and the process of feeding direction is optimized. Proved by the simulation process, the algorithm is feasible and can effectively avoid tool redundancy and tool lack problems,concesquently, processing efficiency improved significantly.

Properties of Finite-Difference Operators for the Steady-Wave Problem

1993 ◽  
Vol 37 (01) ◽  
pp. 1-7
Author(s):  
John S. Letcher
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Radiation Condition ◽  
Free Surface Flows ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Difference Operators ◽  
Surface Boundary ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition

A feature of most implementations of Dawson's boundary-integral method for steady free-surface flows is the use of upstream finite-difference operators for the streamwise derivative occurring in the linearized free-surface boundary condition. An algebraic analysis of a family of candidate operators reveals their essential damping and dispersion error characteristics, which correlate well with their observed performance in two-dimensional example flows. Some new operators are found which perform somewhat better than Dawson's, but the general outlook for accurate results using difference operators is nevertheless bleak. It is shown that the calculation necessarily diverges as panel size is reduced, and a breakdown at higher speeds is also inevitable. More promise appears to lie in satisfying the radiation condition by several alternative ways, which are briefly discussed.

Optimizing the finite-difference implementation of three-dimensional free-surface boundary in frequency-domain modeling of elastic waves

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. T363-T379
Author(s):  
Jian Cao ◽  
Jing-Bo Chen
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Frequency Domain ◽  
Computational Cost ◽  
Good Choice ◽  
Weighted Averaging ◽  
Seismic Exploration ◽  
Surface Boundary ◽  
Domain Modeling ◽  
Near Surface

The problem of modeling seismic wave propagation for multiple sources, such as in the solution of gradient-based elastic full-waveform inversion, is an important topic in seismic exploration. The frequency-domain finite-difference (FD) method is a good choice for this purpose, mainly because of its simple discretization and high computational efficiency. However, when it comes to modeling the complete elastic wavefields, this approach has limited surface-wave accuracy because, when modeling with the strong form of the wave equation, it is not always easy to implement an accurate stress-free boundary condition. Although a denser spatial sampling is helpful for overcoming this problem, the additional discrete points will significantly increase the computational cost in the resolution of its resulting discrete system, especially in 3D problems. Furthermore, sometimes, when modeling with optimized schemes, an inconsistency in the computation precision between the regions at the free surface and inside the model volume would happen and introduce numerical artifacts. To overcome these issues, we have considered optimizing the FD implementation of the free-surface boundary. In our method, the problem was formulated in terms of a novel system of partial differential equations satisfied at the free surface, and the weighted-averaging strategy was introduced to optimize its discretization. With this approach, we can impose FD schemes for the free surface and internal region consistently and improve their discretization precision simultaneously. Benchmark tests for Lamb’s problem indicate that the proposed free-surface implementation contributes to improving the simulation accuracy on surface waves, without increasing the number of grid points per wavelength. This reveals the potential of developing optimized schemes in the free-surface implementation. In particular, through the successful introduction of weighting coefficients, this free-surface FD implementation enables adaptation to the variation of Poisson’s ratio, which is very useful for modeling in heterogeneous near-surface weathered zones.

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.

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