Simple but Efficient Method for Integrating 1/r Singularities

2013 ◽  
Vol 444-445 ◽  
pp. 615-620
Author(s):  
Feng Liu ◽  
Hong Zheng ◽  
Chun Guang Li
Keyword(s):  
Efficient Method ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Integration Schemes ◽  
Integration Techniques ◽  
Elastic Fracture ◽  
Duffy Transformation

New integration schemes are presented for integrands with singularity of 1/r. We partition the element with a singular center into several triangles sharing the center. Then, a transformation between a standard square and each of the triangles is conducted. We prove such a transformation itself brings about the Jacobian with the factor r, leading to no need to introduce any other transformation. Both two-dimensional and three-dimensional cases are considered. Compared to the Duffy transformation, the proposed methods enjoy more excellent numerical properties. Numerical examples in elastic fracture are also presented to illustrate the performance of the new integration techniques.

Curvature Theory of Envelope Curve in Two-Dimensional Motion and Envelope Surface in Three-Dimensional Motion

2015 ◽  
Vol 7 (3) ◽  
Author(s):  
Wei Wang ◽  
Delun Wang
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Spatial Motion ◽  
Envelope Curve ◽  
Contact Conditions ◽  
Numerical Examples ◽  
Envelope Surface ◽  
Straight Line ◽  
Curvature Theory ◽  
Fixed Line

The curvature theories for envelope curve of a straight line in planar motion and envelope ruled surface of a plane in spatial motion are systematically presented in differential geometry language. Based on adjoint curve and adjoint surface methods as well as quasi-fixed line and quasi-fixed plane conditions, the centrode and axode are taken as two logical starting-points to study kinematic and geometric properties of the envelope curve of a line in two-dimensional motion and the envelope surface of a plane in three-dimensional motion. The analogical Euler–Savary equation of the line and the analogous infinitesimal Burmester theories of the plane are thoroughly revealed. The contact conditions of the plane-envelope and some common surfaces, such as circular and noncircular cylindrical surface, circular conical surface, and involute helicoid are also examined, and then the positions and dimensions of different osculating ruled surfaces are given. Two numerical examples are presented to demonstrate the curvature theories.

scholarly journals Ice shelf rift propagation: stability, three dimensional effects, and the role of marginal weakening

2019 ◽  
Author(s):  
Bradley Paul Lipovsky
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Ice Shelf ◽  
Linear Elastic ◽  
Partial Contact ◽  
Rift Propagation ◽  
Elastic Fracture ◽  
Shelf Margins ◽  
Determining Factor

Abstract. Understanding the processes that govern ice shelf extent are of fundamental importance to improved estimates of future sea level rise. In present-day Antarctica, ice shelf extent is most commonly determined by the propagation of through-cutting fractures called ice shelf rifts. Here, I present the first three-dimensional analysis of ice shelf rift propagation. I present a linear elastic fracture mechanical (LEFM) description of rift propagation. The model predicts that rifts may be stabilized when buoyant flexure results in contact at the tops of the near-tip rift walls. This stabilizing tendency may be overcome, however, by processes that act in the ice shelf margins. In particular, both marginal weakening and the advection of rifts into an ice tongue are shown to be processes that may trigger rift propagation. Marginal shear stress is shown to be the determining factor that governs these types of rift instability. I furthermore show that rift stability is closely related to the transition from uniaxial to biaxial extension known as the compressive arch. Although the partial contact of rift walls is fundamentally a three-dimensional process, I demonstrate that it may be parameterized within more numerically efficient two-dimensional calculations. This study provides a step towards a description of calving physics that is based in fracture mechanics.

The Third Two-Dimensional Problem of Three-Dimensional Blade Systems of Hydraulic Machines—Part 2: Analytical Results

1981 ◽  
Vol 103 (1) ◽  
pp. 42-51
Author(s):  
P. K. Agarwal ◽  
G. V. Viktorov
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Blade Profile ◽  
Flow Conditions ◽  
Numerical Examples ◽  
The Third ◽  
Hydraulic Machines ◽  
Method Of Solution ◽  
Axisymmetric Stream

This is the second part of a study of the “third” two-dimensional problem of three-dimensional blade systems of hydraulic machines. Part I described the formulation of the problem and the proposed method of solution to determine the velocity field on surfaces orthogonal to mean axisymmetric stream surfaces. Part II presents the numerical method of solving the integral equations; a few numerical examples for actual impellers/runners are also given. The results are presented in a series of figures and tables showing the distribution of the velocity component c2 along the blade profile on the surface q1 = const. The purpose of these numerical examples is to demonstrate the method and to help create a general understanding and awareness of the flow conditions existing in the runner passage.

On the Correspondence between Two- and Three-Dimensional Elastic Solutions of Crack Problems

2016 ◽  
Vol 713 ◽  
pp. 18-21 ◽  
Author(s):  
Andrei G. Kotousov ◽  
Zhuang He ◽  
Aditya Khanna
Keyword(s):  
Scale Effects ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Two Dimensional ◽  
Governing Equations ◽  
Singular Stress ◽  
Linear Elastic ◽  
Crack Problems ◽  
Elastic Fracture ◽  
Stress States

The classical two-dimensional solutions of the theory of elasticity provide a framework of Linear Elastic Fracture Mechanics. However, these solutions, in fact, are approximations despite that the corresponding governing equations of the plane theories of elasticity are solved exactly. This paper aims to elucidate the main differences between the approximate (two-dimensional) and exact (three-dimensional) elastic solutions of crack problems. The latter demonstrates many interesting features, which cannot be analysed within the plane theories of elasticity. These features include the presence of scale effects of deterministic nature, the existence of new singular stress states and fracture modes. Furthermore, the deformation and stress fields near the tip of the crack is essentially three-dimensional and do not follow plane stress or plane strain simplifications. Moreover, in certain situations the two-dimensional solutions can provide misleading results; and several characteristic examples are outlined in this paper.

scholarly journals Extrapolation Algorithm for Computing Multiple Cauchy Principal Value Integrals

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Jin Li ◽  
Yongling Cheng
Keyword(s):  
Asymptotic Expansion ◽  
Density Function ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Cauchy Principal ◽  
Numerical Examples ◽  
Extrapolation Algorithm ◽  
Expansion Formulae ◽  
Error Functional ◽  
Principal Value

In this paper, the computation of multiple (including two dimensional and three dimensional) Cauchy principal integral with generalized composite rectangle rule was discussed, with the density function approximated by the middle rectangle rule, while the singular kernel was analytically calculated. Based on the expansion of the density function, the asymptotic expansion formulae of error functional are obtained. A series is constructed to approach the singular point, then the extrapolation algorithm is presented, and the convergence rate is proved. At last, some numerical examples are presented to validate the theoretical analysis.

A Three-Dimensional Deterministic Model for Rough Surface Line-Contact EHL Problems

2007 ◽  
Author(s):  
Ning Ren ◽  
W. Wayne Chen ◽  
Dong Zhu ◽  
Yuchuan Liu ◽  
Q. Jane Wang
Keyword(s):  
Rough Surface ◽  
Present Model ◽  
Deterministic Model ◽  
Three Dimensional ◽  
Line Contact ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Asperity Contacts ◽  
Line Contacts ◽  
2D Model

This paper reports the development of a novel three-dimensional (3D) deterministic model for rough surface line-contact mixed-EHL problems. This model is of great importance because line contacts are found in many mechanical components. The macro aspects of a line-contact problem can be simplified into a two-dimensional (2D) model, but the topography of contacting rough surfaces, micro asperity contacts, and lubricant flows around asperities are often 3D. The present model is based on Hu and Zhu’s unified mixed EHL model [1] and the mixed FFT-based approach formulated by Chen et al [2]. It is numerically verified through comparisons with results from conventional 2D line-contact EHL theories. Numerical examples involving sinusoidal roughness and digitized 3D machined surfaces are analyzed.

Capacitance Calculation for Offset Via Structures Using an Integral Approximation Approach Based on Finite Element Method

2010 ◽  
Vol 2010 (1) ◽  
pp. 000408-000412
Author(s):  
Hanfeng Wang ◽  
Yaojiang Zhang ◽  
James L. Drewniak ◽  
Jun Fan ◽  
Bruce Archambeault
Keyword(s):  
Finite Element Method ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Combined Method ◽  
Integral Approximation ◽  
Approximation Approach ◽  
Numerical Examples ◽  
Cpu Time ◽  
Fem Method ◽  
Similar Accuracy

A simple yet efficient approach is presented to extract the via-plane capacitances for an offset via structure. According to the integral approximation approach, the geometry of offset via is first divided into several segments with equally distributed angles from the origin. The two-dimensional FEM method for the concentric case is used for each segment based on its pad-stack parameters. Then, the final offset via-plane capacitance is approximated as the average of these ‘segmental’ capacitance values. Numerical examples demonstrated that the combined method has similar accuracy with a three-dimensional solver but it has much higher efficiency in both CPU time and memory cost.

Paper 8: Fixed-Direction Probes for Aerodynamic Measurements

1965 ◽  
Vol 180 (10) ◽  
pp. 141-152 ◽  
Author(s):  
W. E. Lewis
Keyword(s):  
Three Dimensional ◽  
Flow Regimes ◽  
Two Dimensional ◽  
Three Dimensional Flow ◽  
Numerical Examples ◽  
Dimensional Flow ◽  
Sources Of Error ◽  
Fixed Direction ◽  
Flow Systems ◽  
Two Dimensional Flow

This paper reviews the problems of using null-yaw and fixed-direction, multi-hole probes for aerodynamic measurements. The preparation of calibration charts for the use of fixed-direction three-hole probes in two-dimensional flow regimes and for the use of their five-hole counterparts in three-dimensional flow systems is described. Some of the sources of error in the use of these types of probes are also discussed. In an Appendix, numerical examples demonstrating the use of the calibration charts are discussed. A novel type of probe, fitted with a rotating sting is described. This probe is being developed to combine the advantages of null-yaw and fixed-direction probes.

High Resolution Microscopy of Multi-Layered Biological Crystals

1982 ◽  
Vol 40 ◽  
pp. 80-83
Author(s):  
H.A. Cohen ◽  
T.W. Jeng ◽  
W. Chiu
Keyword(s):  
High Resolution ◽  
Low Dose ◽  
Three Dimensional ◽  
Data Interpretation ◽  
Two Dimensional ◽  
Biological Objects ◽  
Density Maps ◽  
Density Map ◽  
Complex Crystal ◽  
High Resolution Microscopy

This tutorial will discuss the methodology of low dose electron diffraction and imaging of crystalline biological objects, the problems of data interpretation for two-dimensional projected density maps of glucose embedded protein crystals, the factors to be considered in combining tilt data from three-dimensional crystals, and finally, the prospects of achieving a high resolution three-dimensional density map of a biological crystal. This methodology will be illustrated using two proteins under investigation in our laboratory, the T4 DNA helix destabilizing protein gp32*I and the crotoxin complex crystal.

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