The numerical solution of some singular contact problems using a Hilbert transform

1988 ◽  
Vol 23 (3) ◽  
pp. 129-135 ◽  
Author(s):  
R W Hall
Keyword(s):  
Numerical Solution ◽  
Half Plane ◽  
Hilbert Transform ◽  
Solution Method ◽  
Contact Problems ◽  
Finite Depth ◽  
Analytic Solutions ◽  
Finite Hilbert Transform ◽  
Contact Width ◽  
Wedge Indentation

This paper presents a numerical solution method for frictionless, two dimensional contact problems involving punches with slope discontinuities or sharp corners. The solution method presented employs a finite Hilbert transform to remove computational difficulties associated with the logarithmic pressures encountered within such contacts. A complementary (smooth) problem may then be solved using a rapidly convergent Chebyshev expansion which requires relatively few terms to achieve accurate results. The algebraic (square root) pressure singularities encountered at the ends of certain contacts are also easily resolved, being implicit in the series solution used. The solutions to several problems on the half plane are compared with those obtained directly, and a comparison with analytic solutions is made. In the knowledge that the method gives efficient and accurate half plane solutions, it is then extended to problems involving layers of finite depth. In particular, symmetric wedge indentation on both unbonded and bonded layers is investigated, and results for contact width in relation to applied load are given.

On the numerical solution of line contact problems involving bonded and unbonded strips

1988 ◽  
Vol 23 (2) ◽  
pp. 67-77 ◽  
Author(s):  
M J Jaffar ◽  
M D Savage
Keyword(s):  
Contact Problem ◽  
Arbitrary Shape ◽  
Practical Interest ◽  
Contact Problems ◽  
Analytic Solutions ◽  
Line Contact ◽  
Elastic Strip ◽  
Effective Technique ◽  
Contact Width ◽  
Cylindrical Punch

This paper investigates the contact problem in which an elastic strip is indented by a rigid body (punch) of arbitrary shape. Both bonded and unbonded strips are considered. A numerical method due to Gladwell (1)† is shown to be a direct and effective technique for analysing the effect of any punch whose profile is a polynomial of degree n, over a range of a/t (semi-contact width to a depth ratio) which is of practical interest 0 ≤ a/t ≤ 10 for Poisson's ratio 0 ≤ v ≤ 0.5. For the cylindrical punch results are presented and compared with Meijers' asymptotic analytic solutions (2). For small a/t agreement is very good as expected. For a/t large, however, there are some large discrepancies which can be traced to an error in Meijers— expression for pressure distribution when v ≠ 0.5. Results are also presented for both the flat and the linear punch.

scholarly journals Linear Stability Analysis and Validation of a Unified Solution Method for Fluid-Structure-Interaction on Structural Dynamic Problems

2005 ◽  
Author(s):  
C. G. Giannopapa ◽  
G. Papadakis
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Solution Method ◽  
Fluid Structure Interaction ◽  
Analytic Solutions ◽  
Computational Domain ◽  
Structural Dynamic ◽  
Fluid Structure ◽  
Structure Interaction

In the conventional approach for fluid-structure interaction problems, the fluid and solid components are treated separately and information is exchanged across their interface. According to the conventional terminology, the current numerical methods can be grouped in two major categories: Partitioned methods and monolithic methods. Both methods use two separate sets of equations for fluid and solid. A unified solution method has been presented [1], which is different from these methods. The new method treats both fluid and solid as a single continuum, thus the whole computational domain is treated as one entity discretised on a single grid. Its behavior is described by a single set of equations, which are solved fully implicitly. In this paper, 2 time marching and one spatial discretisation scheme, widely used for fluids’ equations, are applied for the solution of the equations for solids. Using linear stability analysis, the accuracy and dissipation characteristics of the resulting difference equations are examined. The aforementioned schemes are applied to a transient structural problem (beam bending) and the results compare favorably with available analytic solutions and are consistent with the conclusions of the stability analysis. A parametric investigation using different meshes, time steps and beam sizes is also presented. For all cases examined the numerical solution was stable and robust and proved to be suitable for the next stage of application to full fluid-structure interaction problems.

Optimal Performance of Ships Under Combined Power and Sail

1982 ◽  
Vol 26 (03) ◽  
pp. 209-218
Author(s):  
John S. Letcher
Keyword(s):  
Numerical Solution ◽  
Economic Model ◽  
Analytic Solutions ◽  
Simultaneous Equations ◽  
Simultaneous Optimization ◽  
Fixed Cost ◽  
University Of Michigan ◽  
Cost Rate ◽  
Optimal Setting ◽  
Set Up

A simplified hydrodynamic and economic model is developed to describe the operation of a ship equipped with both sails and engine. In the range of light-to-moderate winds in which use of the engine is likely to be economical, the vessel is described by a characteristic speed, a characteristic fixed-cost rate, and five dimensionless parameters (four hydrodynamic, one economic). The model includes simultaneous optimization of three control variables: sail lift, throttle setting, and course angle; optimal setting of variable draft devices can be included optionally. Although no analytic solutions are attained, the simultaneous equations expressing minimization of cost per mile made good are set up, and a general algorithm is given for numerical solution of these problems. As an illustrative example, numerical values are worked out for the 30,000-dwt square-rigged bulk cargo ship from the 1975 University of Michigan study.

Thermal Distortion of an Anisotropic Elastic Half-Plane and its Application in Contact Problems Including Frictional Heating

2005 ◽  
Vol 128 (1) ◽  
pp. 32-39 ◽  
Author(s):  
Yuan Lin ◽  
Timothy C. Ovaert
Keyword(s):  
Heat Flux ◽  
Half Plane ◽  
Frictional Heating ◽  
Contact Problems ◽  
Local Heat ◽  
Extended Version ◽  
Parabolic Profile ◽  
Local Heat Flux ◽  
Anisotropic Elastic ◽  
Thermoelastic Contact Problem

The thermal surface distortion of an anisotropic elastic half-plane is studied using the extended version of Stroh’s formalism. In general, the curvature of the surface depends both on the local heat flux into the half-plane and the local temperature variation along the surface. However, if the material is orthotropic, the curvature of the surface depends only on the local heat flux into the half-plane. As a direct application, the two-dimensional thermoelastic contact problem of an indenter sliding against an orthotropic half-plane is considered. Two cases, where the indenter has either a flat or a parabolic profile, are studied in detail. Comparisons with other available results in the literature show that the present method is correct and accurate.

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