Free‐boundary conditions of arbitrary polygonal topography in a two‐dimensional explicit elastic finite‐difference scheme

Geophysics ◽  
1988 ◽  
Vol 53 (8) ◽  
pp. 1045-1055 ◽  
Author(s):  
Rong‐Song Jih ◽  
Keith L. McLaughlin ◽  
Zoltan A. Der
Keyword(s):  
Free Surface ◽  
Coordinate System ◽  
Practical Interest ◽  
Special Treatment ◽  
Simple Method ◽  
Present Scheme ◽  
Line Segments ◽  
First Order ◽  
Approximate Boundary Condition ◽  
Transition Points

We present a simple method for simulating 2-D elastic waves in a model with free‐surface topography of polygonal shape, i.e., a continuous but irregular surface composed of line segments. Our method requires special treatment for each of the six specific cases involving line segments of various slopes as well as transition points between the sloping segments. For brevity, only nonnegatively sloping segments are specifically included. On an inclined free surface, vanishing stress conditions are implemented using a rotated coordinate system parallel to the inclined boundary. At transition points on the topography between line segments, we use a first‐order approximate boundary condition in a locally rotated coordinate system aligned with the bisector of the corner. As in the classical one‐sided explicit approximation scheme widely used for the flat free‐surface case, these extrapolation formulas are accurate to first order in spatial increment. Numerical tests indicate that the present scheme is stable over a range of Poisson’s ratios of practical interest (v > 0.3) for fairly complicated geometric shapes consisting of ridges and valleys with both steep and gentle slopes. Stability for complicated shapes enables us to study realistic problems for which the topography plays a significant role in shaping the wave field and for which analytical solutions are not generally available.

scholarly journals A rapid simple method for the assay of renin in rabbit plasma

1968 ◽  
Vol 108 (4) ◽  
pp. 679-685 ◽  
Author(s):  
J W Ryan ◽  
J. K. McKenzie ◽  
M. R. Lee
Keyword(s):  
Rabbit Plasma ◽  
Selective Inhibitors ◽  
Angiotensin I ◽  
Simple Method ◽  
Renin Substrate ◽  
First Order ◽  
Complete Inactivation ◽  
Plasma Enzymes ◽  
First Order Kinetics ◽  
Percentage Release

1. EDTA (10mm), 2,3-dimercaptopropan-1-ol (10mm) and chlorhexidine gluconate (0·005%, w/v) cause complete inactivation of plasma enzymes that degrade angiotensin I, but have no effect on the reaction of renin with its substrate. The reagents were termed the selective inhibitors. 2. Thus it is possible to measure renin in plasma by its ability to catalyse the release of angiotensin I. 3. Sterile plasma, treated with the selective inhibitors, is incubated with renin substrate (500–1000ng. of angiotensin content/ml.) at pH6 at 42° for 6hr. 4. Under these conditions the reaction obeys first-order kinetics. Renin activity is calculated in terms of the percentage release of the angiotensin content/hr. 5. As described, the assay is sufficiently sensitive to measure renin in the plasma of all normal rabbits. By extending the length of the incubation, much lower activities can be measured.

A Perturbation Analysis of the Wavemaking of a Ship, with an Interpretation of Guilloton’s Method

1975 ◽  
Vol 19 (03) ◽  
pp. 140-148
Author(s):  
F. Noblesse
Keyword(s):  
Approximate Solution ◽  
Free Surface ◽  
Perturbation Analysis ◽  
Order Approximation ◽  
Second Order ◽  
Regular Perturbation ◽  
First Order ◽  
Perturbation Problem ◽  
Sinkage And Trim ◽  
Basic Ideas

A thin-ship perturbation analysis, suggested by Guilloton's basic ideas, is presented. The analysis may be regarded as an application of Lighthill's method of strained coordinates to a regular perturbation problem. An inconsistent second-order approximation in which the Laplace equation is satisfied to first order, and the boundary conditions both at the free surface and on the ship hull are satisfied to second order, is derived. When sinkage and trim, incorporated into the present analysis, are ignored, this approximate solution is shown to be essentially equivalent to the method of Guilloton.

Oscillatory Third-Order Perturbation Solutions for Sums of Interacting Long-Crested Stokes Waves on Deep Water

1993 ◽  
Vol 37 (04) ◽  
pp. 354-383
Author(s):  
Willard J. Pierson
Keyword(s):  
Free Surface ◽  
Deep Water ◽  
Perturbation Expansion ◽  
The United States ◽  
Order Perturbation ◽  
Naval Academy ◽  
Third Order ◽  
First Order ◽  
Stokes Waves ◽  
Perturbation Solutions

Oscillatory third-order perturbation solutions for sums of interacting long-crested Stokes waves on deep water are obtained. A third-order perturbation expansion of the nonlinear free boundary value problem, defined by the coupled Bernoulli equation and kinematic boundary condition evaluated at the free surface, is solved by replacing the exponential term in the potential function by its series expansion and substituting the equation for the free surface into it. There are second-order changes in the frequencies of the first-order terms at third order. The waves have a Stokes-like form when they are high. The phase speeds are a function of the amplitudes and wave numbers of all of the first-order terms. The solutions are illustrated. A preliminary experiment at the United States Naval Academy is described. Some applications to sea keeping are bow submergence and slamming, capsizing in following seas and bending moments.

scholarly journals Unsplit complex frequency-shifted PML implementation using auxiliary differential equations for seismic wave modeling

Geophysics ◽  
2010 ◽  
Vol 75 (4) ◽  
pp. T141-T154 ◽  
Author(s):  
Wei Zhang ◽  
Yang Shen
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Differential Equations ◽  
Seismic Wave ◽  
Numerical Scheme ◽  
Complex Frequency ◽  
Order Accuracy ◽  
First Order ◽  
Time Marching ◽  
Interior Domain

The complex-frequency-shifted perfectly matched layer (CFS-PML) technique can efficiently absorb near-grazing incident waves. In seismic wave modeling, CFS-PML has been implemented by the first-order-accuracy convolutional PML technique or second-order-accuracy recursive convolution PML technique. Both use different algorithms than the numerical scheme for the interior domain to update auxiliary memory variables in the PML and thus cannot be used directly with higher-order time-marching schemes. We work with an unsplit-field CFS-PML implementation using auxiliary differential equations (ADEs) to update the auxiliary memory variables. This ADE CFS-PML results in complete first-order differential equations. Thus, the numerical scheme for the interior domain can be used to solve ADE CFS-PML equations. We have implemented ADE CFS-PML in the finite-difference time-domain method and in anonstaggered-grid finite-difference method with the fourth-order Runge-Kutta scheme, demonstrating its straightforward implementation in different numerical time-marching schemes. We have also theoretically analyzed the role of the scalingfactor of CFS-PML; it transforms the PML to a transversely isotropic material, reducing the effective wave speed normal to the PML layer and bending the wavefront toward the normal direction of the PML layer. Our numerical tests indicate that the optimal value reduces the points per dominant wavelength at the outermost boundary to three, about half the value required by the numerical scheme. We also have found that the PML equations should be derived taking the free-surface boundary condition into account in finite-difference methods. Otherwise, the free surface in the PML layer causes instability or ineffective absorption of surface waves. Tests show that we can use a narrow-slice mesh with ADE CFS-PML to simulate full wave propagation efficiently in models with complex structure.

Irregular Frequency Removal and Convergence in Higher-Order BEM for Wave Diffraction/Radiation Analysis

2019 ◽  
Author(s):  
Tomoaki Utsunomiya
Keyword(s):  
Free Surface ◽  
Wave Diffraction ◽  
Higher Order ◽  
The Body ◽  
Diffraction Radiation ◽  
First Order ◽  
Irregular Frequency ◽  
Intersection Line ◽  
Order Quantities ◽  
Drift Forces

Abstract Higher-order boundary element method (HOBEM) for wave diffraction/radiation analysis is a powerful tool for its applicability to a general (curved) geometry. Inspired by the paper which examined the convergence of BIE code with constant panels (Martic, et al., 2018; OMAE2018-77999), the convergence characteristics of HOBEM with quadrilateral panels have been examined. Here, the effect of removal of irregular frequencies is particularly focused as discussed by Martic, et al. (2018). The irregular frequency removal has been made by the rigid-lid method which is applicable to HOBEM, where the intersection line between the body-surface and the free-surface should be carefully handled. The results show that for first order quantities the convergence is quite good for both cases with/without irregular frequency removal (except where the irregular frequencies affect for the case without irregular frequency removal). For mean drift forces, the convergence becomes poor particularly for the case without irregular frequency removal. The convergence characteristics are examined and some discussions are made.

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