Lobatto Point Quadrature for Thermal Lubrication Problems Involving Compressible Lubricants

2005 ◽  
Author(s):  
L. Moraru ◽  
T. G. Keith
Keyword(s):  
Calculation Method ◽  
Efficient Algorithm ◽  
Legendre Polynomials ◽  
Continuity Equation ◽  
Transverse Velocity ◽  
Fine Mesh ◽  
Point Calculation ◽  
Temperature And Pressure ◽  
Grid Points

Refined solutions of thermal lubrication problems generally require fine mesh and many iteration steps. To resolve these difficulties, Elrod and Brewe (1) proposed an efficient algorithm based on the use of Lobatto point quadrature. Within this approach, the unknown temperature across the film is written in a series of Legendre polynomials. This paper presents a Lobatto point quadrature algorithm which is applicable for thermal lubrication problems with compressible lubricants. In this case both density and viscosity of the lubricant are taken to be temperature and pressure dependent. The transverse velocity is obtained from the continuity equation. Use of the Labatto point calculation method has resulted in greater accuracy without the use of a large number of grid points.

Lobatto Point Quadrature for Thermal Lubrication Problems Involving Compressible Lubricants. EHL Applications

2006 ◽  
Vol 129 (1) ◽  
pp. 194-198 ◽  
Author(s):  
L. Moraru ◽  
T. G. Keith
Keyword(s):  
Legendre Polynomials ◽  
Hydrodynamic Lubrication ◽  
Transverse Velocity ◽  
Elastohydrodynamic Lubrication ◽  
Fine Mesh ◽  
Quadrature Methods ◽  
Order Polynomial ◽  
Point Calculation ◽  
Temperature And Pressure ◽  
Grid Points

Refined solutions of thermal lubrication problems generally require fine mesh and many iteration steps. To resolve these difficulties, Elrod and Brewe proposed an efficient algorithm based on the use of a Lobatto point version of the Gauss quadrature, which is typically twice as fast as the other quadrature methods. The original Lobatto algorithm was only applied for incompressible hydrodynamic lubrication. This paper presents a Lobatto point quadrature algorithm which is applicable for thermal elastohydrodynamic lubrication (EHL) problems where both density and viscosity of the lubricant are taken to be temperature and pressure dependent and the transverse velocity term in the energy equation is obtained from the continuity equation. Within this approach, the unknown temperature across the film is written in a series of Legendre polynomials. Regardless of the order of the series expansions, the thermal Reynolds equation can explicitly contain only the information from the first three Legendre polynomials, i.e., data from up to a second-order polynomial. Use of the Lobatto point calculation method has resulted in accuracy without the use of a larger number of grid points.

Application of the Lobatto Quadrature Within Probabilistic Thermal Mixed EHL

2009 ◽  
Author(s):  
Laurentiu Moraru ◽  
Theo G. Keith
Keyword(s):  
Efficient Algorithm ◽  
Legendre Polynomials ◽  
Main Interest ◽  
Bulk Properties ◽  
Fine Mesh ◽  
Ehl Contact

Refined solutions of thermal lubrication problems generally require fine mesh and many iteration steps. To solve these difficulties, Elrod and Brewe [1–2] proposed an efficient algorithm based on the use of Lobatto point quadrature. Within this approach, the unknown temperature across the film is written in a series of Legendre polynomials. Over the years, the deterministic EHL approach has been widely applied, however, the probabilistic EHL methodology is still used when the main interest of the engineer is directed toward computations of bulk properties. This paper presents an application of the Lobatto point quadrature within a probabilistic EHL algorithm for the computation of the load supported by the fluid, the elastically deformed asperities and the plastically deformed asperities in a mixed EHL contact with either isotropic of non-isotropic roughness.

scholarly journals An Adaptive Approach to Solutions of Fredholm Integral Equations of the Second Kind

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Nebiye Korkmaz ◽  
Zekeriya Güney
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Iteration Method ◽  
Legendre Polynomials ◽  
Optimization Problem ◽  
Approximate Solutions ◽  
Basis Functions ◽  
Fredholm Integral Equations ◽  
Coarse Mesh ◽  
Fine Mesh

As an approach to approximate solutions of Fredholm integral equations of the second kind, adaptive hp-refinement is used firstly together with Galerkin method and with Sloan iteration method which is applied to Galerkin method solution. The linear hat functions and modified integrated Legendre polynomials are used as basis functions for the approximations. The most appropriate refinement is determined by an optimization problem given by Demkowicz, 2007. During the calculationsL2-projections of approximate solutions on four different meshes which could occur between coarse mesh and fine mesh are calculated. Depending on the error values, these procedures could be repeated consecutively or different meshes could be used in order to decrease the error values.

An efficient algorithm for real time collision detection involving a continuum manipulator with multiple uniform-curvature sections

Robotica ◽  
2014 ◽  
Vol 34 (7) ◽  
pp. 1566-1586 ◽  
Author(s):  
Jinglin Li ◽  
Jing Xiao
Keyword(s):  
Real Time ◽  
Collision Detection ◽  
Efficient Algorithm ◽  
Convex Polygons ◽  
Polygonal Mesh ◽  
Distance Information ◽  
Fine Mesh ◽  
Wide Range ◽  
Continuum Manipulators ◽  
Continuum Manipulator

SUMMARYA continuum manipulator, such as a multisection trunk/tentacle robot, performs manipulation tasks by continuously deforming into different concave shapes. While such a robot is promising for manipulating a wide range of objects in less-structured and cluttered environments, it poses a greater challenge to collision detection than conventional, articulated manipulators. Existing collision detection algorithms are built upon intersection checking between convex primitives, such as between two convex polygons or polyhedra, with the assumption that both the manipulator and the objects in the environment are modeled in terms of those primitives, for example, as polygonal meshes. However, to approximate a continuum manipulator with a polygonal mesh requires a fine mesh because of its concavity, and each time the manipulator changes its configuration by deforming its shape, the mesh has to be updated for the new configuration. This makes mesh-based collision detection involving such a robot much more computationally expensive than that involving an articulated manipulator with rigid links.Hence, we introduce an efficient algorithm for Collision Detection between a Continuum Manipulator (CD-CoM) and its environment based on analytical intersection checking with nonconvex primitives. Our algorithm applies to the exact model of any continuum manipulator consisting of multiple uniform-curvature sections of toroidal and (sometimes) cylindrical shapes as well as more general continuum manipulators whose sections can be approximated by toroidal and cylindrical primitives. Our test results show that using this algorithm is both more accurate and efficient in time and space to detect collisions than approximating a continuum manipulator as a polygonal mesh. Moreover, the CD-CoM algorithm also provides the minimum distance information between the continuum manipulator and objects when there is no collision. Such an efficient algorithm is essential for path/trajectory planning of continuum manipulators in real-time.

scholarly journals A SIMPLIFIED TWO-NODE COARSE-MESH FINITE DIFFERENCE METHOD FOR PIN-WISE CALCULATION WITH SP3

2021 ◽  
Vol 247 ◽  
pp. 02023
Author(s):  
Wenbo Zhao ◽  
Yingrui Yu ◽  
Xiaoming Chai ◽  
Zhonghao Ning ◽  
Bin Zhang ◽  
...  
Keyword(s):  
Finite Difference ◽  
Legendre Polynomials ◽  
Coarse Mesh ◽  
Coupling Coefficients ◽  
Fine Mesh ◽  
Davidson Method ◽  
Comparable Accuracy ◽  
Current Coupling ◽  
Node Problem ◽  
3D Problem

For accurate and efficient pin-by-pin core calculation of SP3 equations, a simplified two-node Coarse Mesh Finite Difference (CMFD) method with the nonlinear iterative strategy is proposed. In this study, the two-node method is only used for discretization of Laplace operator of the 0th moment in the first equation, while the fine mesh finite difference (FMFD) is used for the 2nd moment flux and the second equation. In the two-node problem, transverse flux is expanded to second-order Legendre polynomials. In addition, the associated transverse leakage is approximated with flat distribution. Then the current coupling coefficients are updated in nonlinear iterations. The generalized eigenvalue problem from CMFD is solved using Jacobi-Davidson method. A protype code CORCA-PIN is developed. FMFD scheme is implemented in CORCA-PIN as well. The 2D KAIST 3A benchmark problem and extended 3D problem, which are cell homogenized problems with strong absorber, are tested. Numerical results show that the solution of the simplified two-node method with 1×1 mesh per cell has comparable accuracy of FMFD with 4×4 meshes per cell, but cost less time. The method is suitable for whole core pin-wise calculation.

REGULAR ENUMERATION OF GRID POINTS IN A CONVEX POLYGON

1993 ◽  
Vol 03 (03) ◽  
pp. 305-322
Author(s):  
HSUAN-SHIH LEE ◽  
RUEI CHUAN CHANG
Keyword(s):  
Line Segment ◽  
Efficient Algorithm ◽  
Convex Polygon ◽  
Grid Points

In this paper we present an efficient algorithm for enumerating all the grid points in a convex polygon, where the enumeration is done by scanning the grid points with a set of parallel grid lines. Given a convex n-gon, enumeration can be done in [Formula: see text] time, where K is the number of the grid points reported, l is the diameter of the polygon and w is the length of the shorter sides of the rectangle enclosing the polygon with longer sides parallel to the line segment connecting the diametral pair of the polygon. We also show that the ratio of the number of the scanned grid lines to the minimal is less than some constant.

Numerical Simulation of Cavity Flow within the Flow Field

2013 ◽  
Vol 712-715 ◽  
pp. 1259-1262
Author(s):  
Jian Yu Cai ◽  
He Juan Chen
Keyword(s):  
Numerical Simulation ◽  
Temperature Field ◽  
Flow Field ◽  
Centrifugal Pump ◽  
Calculation Method ◽  
Simulation Method ◽  
Cavity Flow ◽  
Mechanical Seal ◽  
Seal Ring ◽  
Temperature And Pressure

Analysis of numerical simulation was used on flow field and temperature field in mechanical seal cavity. It borrowed a new calculation method, which was more visual and simpler than before. And this method analysed the change in temperature field of mechanical seal ring and the distribution of temperature and pressure in the flow field when mechanical seal was working. The result showed that the numerical simulation method was effect to the flow field in centrifugal pump.

scholarly journals Establishing calculation method for chemical composition of primitive magma in the Cenozoic in South Central coast region and the adjacent continental shelf of Vietnam

2019 ◽  
Vol 19 (3B) ◽  
pp. 55-70
Author(s):  
Le Duc Anh ◽  
Nguyen Hoang ◽  
Phung Van Phach ◽  
A. I. Malinovskii ◽  
Renat Shakirov ◽  
...  
Keyword(s):  
Chemical Composition ◽  
Continental Shelf ◽  
Calculation Method ◽  
Volcanic Eruptions ◽  
Magma Source ◽  
Central Coast ◽  
South Central ◽  
Primitive Magma ◽  
Temperature And Pressure ◽  
Coast Region

The calculations which determine the chemical composition of the primitive magma are simple but they show changes in the temperature and pressure states of the magma source. The method is based on the addition of the chemical composition of the Olivine to the major element composition of the eruptive rocks which follows the formula: Ci = Ci-1+ 0.1 * Ci-1Ol. In accordance with the characteristics of the study area, we have made new additions to the calculation method. The calculation results are highly accurate when tested and compared with the chemical composition of the eruptive rocks. The chemical composition of the primitive magma solution is used to calculate the temperature and pressure states in the magma source. The results show that there is a difference in temperature and pressure in the source at different tectonic positions in the study area. Accordingly, the South Central coast region and the adjacent continental shelf are divided into two main types of eruptions. The first type of volcanic eruptions occurs at locations where major faults intersect and they are located north of the study area. The second type of volcanic eruptions in the form of a single volcano is located to the south of the study area and the southeastern continental shelf, and occurs in intracontinental extension structure.

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