A Finite Volume Scheme for Three-Dimensional Diffusion Equations

2015 ◽  
Vol 18 (3) ◽  
pp. 650-672 ◽  
Author(s):  
Xiang Lai ◽  
Zhiqiang Sheng ◽  
Guangwei Yuan
Keyword(s):  
Three Dimensional ◽  
Three Dimensions ◽  
Finite Volume Scheme ◽  
Order Convergence ◽  
Order Accuracy ◽  
First Order ◽  
Dimensional Diffusion ◽  
Material Discontinuities ◽  
The Stability ◽  
New Iterative Method

AbstractThe extension of diamond scheme for diffusion equation to three dimensions is presented. The discrete normal flux is constructed by a linear combination of the directional flux along the line connecting cell-centers and the tangent flux along the cell-faces. In addition, it treats material discontinuities by a new iterative method. The stability and first-order convergence of the method is proved on distorted meshes. The numerical results illustrate that the method appears to be approximate second-order accuracy for solution.

Monotone Finite Volume Scheme for Three Dimensional Diffusion Equation on Tetrahedral Meshes

2016 ◽  
Vol 21 (1) ◽  
pp. 162-181 ◽  
Author(s):  
Xiang Lai ◽  
Zhiqiang Sheng ◽  
Guangwei Yuan
Keyword(s):  
Diffusion Equation ◽  
Finite Volume ◽  
Three Dimensional ◽  
Finite Volume Scheme ◽  
Order Accuracy ◽  
Volume Scheme ◽  
Tetrahedral Meshes ◽  
Dimensional Diffusion ◽  
Coefficient Problems ◽  
Monotone Finite Volume Scheme

AbstractWe construct a nonlinear monotone finite volume scheme for three-dimensional diffusion equation on tetrahedral meshes. Since it is crucial important to eliminate the vertex unknowns in the construction of the scheme, we present a new efficient eliminating method. The scheme has only cell-centered unknowns and can deal with discontinuous or tensor diffusion coefficient problems on distorted meshes rigorously. The numerical results illustrate that the resulting scheme can preserve positivity on distorted tetrahedral meshes, and also show that our scheme appears to be approximate second-order accuracy for solution.

Three-Dimensional Slightly Nonplanar Cracks

1992 ◽  
Vol 59 (2) ◽  
pp. 335-343 ◽  
Author(s):  
Huajian Gao
Keyword(s):  
Stress Intensity ◽  
Crack Front ◽  
Three Dimensional ◽  
Crack Deflection ◽  
Order Accuracy ◽  
First Order ◽  
Shear Component ◽  
T Stress ◽  
Planar Crack ◽  
The Stability

Three-dimensional slightly nonplanar cracks are studied via a perturbation method valid to the first-order accuracy in the deviation of the crack shape from a perfectly planar reference crack. The Bueckner-Rice crack-face weight functions are used in the perturbation analysis to establish a relationship, within first-order accuracy, between the apparent and local stress intensity factors for the nonplanar crack. Perturbation solutions for a cosine wavy crack with arbitrary wavelengths are used to examine the effects of three T-stress components, Txx, TXZ, TZZ, on the stability of a mode 1 planar crack in the x-z plane with front lying along the z-axis. A condition for the mode 1 crack to be stable against three-dimensional wavy perturbations of wavelengths λx and λz is determined as Txx + Tzzg < 0 where g is negative, with a very small magnitude, for 0<λx/λz<1/3 and positive for 1/3<λx/λz<∞; this suggests that when Txx = 0, a compressive stress Tzz may cause crack deflection with large wavelengths parallel to the crack front and a tensile stress Tzz may cause deflection with small wavelengths parallel to the front. For comparable T-stress values, it is shown that a negative Txx always enhances the stability of a mode 1 planar crack and a negative Tzz ensures the stability of a mode 1 crack against perturbations parallel to the crack front. The shear component Txz, while not affecting the mode 1 path stability, induces a mode 3 stress intensity factor once crack deflection occurs, and thus promotes the formation of en echelon-type cracking patterns.

scholarly journals Evolution of a narrow-band spectrum of random surface gravity waves

2003 ◽  
Vol 478 ◽  
pp. 1-10 ◽  
Author(s):  
KRISTIAN B. DYSTHE ◽  
KARSTEN TRULSEN ◽  
HARALD E. KROGSTAD ◽  
HERVÉ SOCQUET-JUGLARD
Keyword(s):  
Three Dimensional ◽  
Low Frequency ◽  
Three Dimensions ◽  
Band Spectrum ◽  
Nls Equation ◽  
Random Surface ◽  
Narrow Bandwidth ◽  
Quasi Stationary State ◽  
The Stability ◽  
Three Dimensional Simulations

Numerical simulations of the evolution of gravity wave spectra of fairly narrow bandwidth have been performed both for two and three dimensions. Simulations using the nonlinear Schrödinger (NLS) equation approximately verify the stability criteria of Alber (1978) in the two-dimensional but not in the three-dimensional case. Using a modified NLS equation (Trulsen et al. 2000) the spectra ‘relax’ towards a quasi-stationary state on a timescale (ε2ω0)−1. In this state the low-frequency face is steepened and the spectral peak is downshifted. The three-dimensional simulations show a power-law behaviour ω−4 on the high-frequency side of the (angularly integrated) spectrum.

scholarly journals Dynamics of gravity–capillary solitary waves in deep water

2012 ◽  
Vol 708 ◽  
pp. 480-501 ◽  
Author(s):  
Zhan Wang ◽  
Paul A. Milewski
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Periodic Structures ◽  
Time Integration ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Full Potential ◽  
Numerical Time Integration ◽  
The Stability ◽  
Nonlinear Focusing

AbstractThe dynamics of solitary gravity–capillary water waves propagating on the surface of a three-dimensional fluid domain is studied numerically. In order to accurately compute complex time-dependent solutions, we simplify the full potential flow problem by using surface variables and taking a particular cubic truncation possessing a Hamiltonian with desirable properties. This approximation agrees remarkably well with the full equations for the bifurcation curves, wave profiles and the dynamics of solitary waves for a two-dimensional fluid domain, and with higher-order truncations in three dimensions. Fully localized solitary waves are then computed in the three-dimensional problem and the stability and interaction of both line and localized solitary waves are investigated via numerical time integration of the equations. There are many solitary wave branches, indexed by their finite energy as their amplitude tends to zero. The dynamics of the solitary waves is complex, involving nonlinear focusing of wavepackets, quasi-elastic collisions, and the generation of propagating, spatially localized, time-periodic structures akin to breathers.

Stability and Robustness of a 3D Slip Model for Walking Using Lateral Leg Placement Control

2012 ◽  
Author(s):  
Dominik Budday ◽  
Fabian Bauer ◽  
Justin Seipel
Keyword(s):  
Humanoid Robot ◽  
Performance Test ◽  
Control Mechanism ◽  
Sagittal Plane ◽  
Control Strategies ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Slip Model ◽  
The Stability

The SLIP model has shown a way to easily represent the center of mass dynamics of human walking and running. For 2D motions in the sagittal plane, the model shows self-stabilizing effects that can be very useful when designing a humanoid robot. However, this self-stability could not be found in three-dimensional running, but simple control strategies achieved stabilization of running in three dimensions. Yet, 3D walking with SLIP has not been analyzed to the same extent. In this paper we show that three-dimensional humanoid SLIP walking is also unstable, but can be stabilized using the same strategy that has been successful for running. It is shown that this approach leads to the desired periodic solutions. Furthermore, the influence of different parameters on stability and robustness is examined. Using a performance test to simulate the transition from an upright position to periodic walking we show that the stability is robust. With a comparison of common models for humanoid walking and running it is shown that the simple control mechanism is able to achieve stable solutions for all models, providing a very general approach to this problem. The derived results point out preferable parameters to increase robustness promising the possibility of successfully realizing a humanoid walking robot based on 3D SLIP.

A Numerical Solution of Three-Dimensional Problems in Dynamic Elasticity

1970 ◽  
Vol 37 (1) ◽  
pp. 116-122 ◽  
Author(s):  
W. W. Recker
Keyword(s):  
Dynamic Deformation ◽  
Three Dimensional ◽  
Elastic Solid ◽  
System Of Difference Equations ◽  
Order Accuracy ◽  
First Order ◽  
Dynamic Elasticity ◽  
Approximate Integral ◽  
Explicit Determination

The equations governing the dynamic deformation of an elastic solid are considered as a symmetric hyperbolic system of linear first-order partial-differential equations. The characteristic properties of the system are determined and a numerical method for obtaining the solution of mixed initial and boundary-value problems in elastodynamics is presented. The method, based on approximate integral relations along bicharacteristics, is an extension of the method proposed by Clifton for plane problems in dynamic elasticity and provides a system of difference equations, with second-order accuracy, for the explicit determination of the solution. Application of the method to a problem which has a known solution provides numerical evidence of the convergence and stability of the method.

scholarly journals A Systematic Study of the Fracturing of Ronne - Filchner Ice Shelf, Antarctica, Using Multisource Satellite Data from 2001 to 2016

2017 ◽  
Author(s):  
Rongxing Li ◽  
Haifeng Xiao ◽  
Shijie Liu ◽  
Xiaohua Tong
Keyword(s):  
Satellite Data ◽  
Three Dimensional ◽  
Remote Sensing Data ◽  
Three Dimensions ◽  
Landsat 8 ◽  
Ice Shelf ◽  
Sar Imagery ◽  
The Stability ◽  
Fracture Mapping ◽  
New Framework

Abstract. We propose a new framework of systematic fracture mapping and major calving event prediction for the large ice shelves in Antarctica using multisource satellite data, including optical imagery, SAR imagery, altimetric data, and stereo mapping imagery. The new framework is implemented and applied for a comprehensive study of the fracturing of Ronne-Filchner Ice Shelf (RFIS), the second largest ice shelf in Antarctica, using a long time dataset dating back to 1957. New remote sensing data that have been made available in the past decade, including Landsat 8, WV-2, ZY-3 and others, greatly enhance our abilities to detect new fractures and monitor large rifts in three dimensions. Two large rifts, Rifts 1 and 2, were newly detected and are comparable to the Grand Chasm that caused a major calving event in the region in 1986. Three-dimensional rift models generated from quasi real-time stereo ZY-3 images revealed important topographic information about the large rifts that can be used to improve the reliability of ice shelf modeling and support enhanced analyses of ice shelf stability. Based on the results of the 2D and 3D fracture mapping, the spatial and temporal analyses of the overall fracture changes and large rift evolutions, i.e., the level of fracturing in RFIS, were slightly increased, particularly at the front of the ice sheet. The overall fracture observations do not seem to suggest immediate significant impacts on the stability of the shelf. However, the most active regional fracturing activities occurred at the front of Filchner Ice Shelf (FIS). A potential upcoming major calving event of FIS is estimated to occur in 2051. The stability of the ice shelf, particularly with regard to the developments of Rifts 1 and 2, should be closely monitored.

Structural Failure Mechanisms of Common Flexor Tendon Repairs

Hand Surgery ◽  
2015 ◽  
Vol 20 (03) ◽  
pp. 369-379 ◽  
Author(s):  
Tim Sebastian Peltz ◽  
Roger Haddad ◽  
Peter James Scougall ◽  
Sean Nicklin ◽  
Mark Peter Gianoutsos ◽  
...  
Keyword(s):  
Failure Mechanisms ◽  
Biomechanical Testing ◽  
Tendon Repair ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Biomechanical Stability ◽  
Repair Method ◽  
Before And After ◽  
The Stability ◽  
Repair Techniques

Background: This study investigated the exact failure mechanisms of the most commonly used conventional tendon repair techniques. A new method, radiographing repair constructs in antero-posterior and lateral projections before and after tensioning was used. This allowed to precisely analyse failure mechanisms in regards to geometrical changes in all three dimensions. Additionally the biomechanical stability focusing on gapping was tested. Methods: Sheep fore limb deep flexor tendons were harvested and divided in eight groups of ten tendons. Three common variants of the Kessler repair method and four common 4-strand repair techniques were tested. Additionally a new modification of the Adelaide repair was tested. Results: Biomechanical testing showed no significant differences in gapping for the three tested 2-strand Kessler repair groups. Once a double Kessler or 4-strand Kessler repair was performed the stability of the repair improved significantly. Further significant improvements in biomechanical stability could be achieved by using cross locks in the repair like in the Adelaide repair method. Qualitative analysis using radiographs showed that all Kessler repair variants unfolded via rotations around the transverse suturing component, no matter which variant was used. Conclusions: Additional to the commonly described constriction of the repair construct, the rotating deformation is the main reason for repair site gapping in Kessler tendon repair methods. The term “locking” in a Kessler repair is misleading. The cruciate repairs tended to loose grip and drag (cheese-wire) through the tendon and therefore lead to gapping. The most stable repair constructs in all three dimensions were the Adelaide repair and its interlocking modification. This is due to the superior anchoring qualities of its cross locks and three dimensional stability.

The Hopf bifurcation theorem in three dimensions

1977 ◽  
Vol 82 (3) ◽  
pp. 469-483 ◽  
Author(s):  
Peter Swinnerton-Dyer
Keyword(s):  
Hopf Bifurcation ◽  
Singular Point ◽  
Differential Equations ◽  
Limit Cycle ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Bifurcation Theorem ◽  
First Order ◽  
Range Of Values ◽  
First Order Differential Equations

AbstractThe Hopf bifurcation theorem describes the creation of a limit cycle from an isolated singular point of a system of first-order differential equations depending on a parameter. This paper describes a method for determining explicitly a range of values of the parameter throughout which the Hopf configuration continues to exist; only the three-dimensional case is described in this paper, but the method can be generalized.

Spatial Distribution of Gravity-Dependent Gain Changes in the Vestibuloocular Reflex

2005 ◽  
Vol 93 (6) ◽  
pp. 3693-3698 ◽  
Author(s):  
Sergei B. Yakushin ◽  
Yongqing Xiang ◽  
Theodore Raphan ◽  
Bernard Cohen
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Head Tilt ◽  
Fundamental Property ◽  
Head Position ◽  
Three Dimensions ◽  
Vestibuloocular Reflex ◽  
Gain Adaptation ◽  
The Stability ◽  
Three Dimensional Space

This study determined whether dependence of angular vestibuloocular reflex (aVOR) gain adaptation on gravity is a fundamental property in three dimensions. Horizontal aVOR gains were adaptively increased or decreased in two cynomolgus monkeys in upright, side down, prone, and supine positions, and aVOR gains were tested in darkness by yaw rotation with the head in a wide variety of orientations. Horizontal aVOR gain changes peaked at the head position in which the adaptation took place and gradually decreased as the head moved away from this position in any direction. The gain changes were plotted as a function of head tilt and fit with a sinusoid plus a bias to obtain the gravity-dependent (amplitude) and gravity-independent (bias) components. Peak-to-peak gravity-dependent gain changes in planes containing the position of adaptation and the magnitude of the gravity-independent components were both ∼25%. We assumed that gain changes over three-dimensional space could be described by a sinusoid the amplitude of which also varied sinusoidally. Using gain changes obtained from the head position in which the gains were adapted, a three-dimensional surface was generated that was qualitatively similar to a surface obtained from the experimental data. This extends previous findings on vertical aVOR gain adaptation in one plane and introduces a conceptual framework for understanding plasticity in three dimensions: aVOR gain changes are composed of two components, one of which depends on head position relative to gravity. It is likely that this gravitational dependence optimizes the stability of retinal images during movement in three-dimensional space.

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