scholarly journals A note on the uniqueness of 2D elastostatic problems formulated by different types of potential functions

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 201-210
Author(s):  
José Luis Morales Guerrero ◽  
Manuel Cánovas Vidal ◽  
José Andrés Moreno Nicolás ◽  
Francisco Alhama López
Keyword(s):  
Boundary Conditions ◽  
Potential Function ◽  
Vector Potential ◽  
Numerical Scheme ◽  
Scalar Potential ◽  
Three Dimensional ◽  
Potential Functions ◽  
Network Method ◽  
Different Types ◽  
Physical Boundary

Abstract New additional conditions required for the uniqueness of the 2D elastostatic problems formulated in terms of potential functions for the derived Papkovich-Neuber representations, are studied. Two cases are considered, each of them formulated by the scalar potential function plus one of the rectangular non-zero components of the vector potential function. For these formulations, in addition to the original (physical) boundary conditions, two new additional conditions are required. In addition, for the complete Papkovich-Neuber formulation, expressed by the scalar potential plus two components of the vector potential, the additional conditions established previously for the three-dimensional case in z-convex domain can be applied. To show the usefulness of these new conditions in a numerical scheme two applications are numerically solved by the network method for the three cases of potential formulations.

A Thermal Network Approach for Predicting Thermal Characteristics of Three-Dimensional Electronic Packages

2006 ◽  
Author(s):  
P. L. Chen ◽  
S. F. Chang ◽  
T. Y. Wu ◽  
Y. H. Hung
Keyword(s):  
Boundary Conditions ◽  
Thermal Analyses ◽  
Three Dimensional ◽  
Numerical Approach ◽  
Electronic Packages ◽  
Network Approach ◽  
Electronic Package ◽  
Thermal Network ◽  
Network Method ◽  
Thermal Network Method

In the present study, a numerical approach for characterizing three-dimensional (3-D) electronic packages is presented, based on the steady-state solution of the thermal network method for generalized rectangular geometries, where boundary conditions are uniformly specified over specific regions of the package. As we know, the thermal-network method is very powerful on thermal analysis of electronic packaging because of its feasibility and flexibility. Accordingly, the numerical approach with thermal-network method to simulate heat transfer characteristics for 3-D package geometries becomes important in the modem microelectronic applications. The thermal analyses are presented with a general overview of the thermal network method, boundary conditions and solution procedures. Furthermore, the application of boundary conditions at the fluid-solid, package-board and layer-layer interfaces provides a means for obtaining a unique numerical result for 3-D complex electronic packages. The complex geometries found in most 3-D electronic package configurations are modeled using numerical method through the careful use of simplifying assumptions. Comparisons of the present numerical results with the existing experimental data for 3-D electronic package of pin grid arrays and multi-chip modules are made with a satisfactory agreement. Thus, the present study demonstrates that the numerical thermal-network approach can offer an accurate and efficient solution procedure for evaluating the thermal characterization of 3-D electronic packages.

Mass transport in three-dimensional water waves

1991 ◽  
Vol 231 ◽  
pp. 417-437 ◽  
Author(s):  
Mohamed Iskandarani ◽  
Philip L.-F. Liu
Keyword(s):  
Mass Transport ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Steady Flow ◽  
Water Waves ◽  
Numerical Scheme ◽  
Three Dimensional ◽  
Langmuir Circulation ◽  
Circulation Patterns ◽  
Wave Trains

A spectral scheme is developed to study the mass transport in three-dimensional water waves where the steady flow is assumed to be periodic in two horizontal directions. The velocity–vorticity formulation is adopted for the numerical solution, and boundary conditions for the vorticity are derived to enforce the no-slip conditions. The numerical scheme is used to calculate the mass transport under two intersecting wave trains; the resulting flow is reminiscent of the Langmuir circulation patterns. The scheme is then applied to study the steady flow in a three-dimensional standing wave.

Thermoelastodynamics in Transversely Isotropic Media With Scalar Potential Functions1

2013 ◽  
Vol 81 (2) ◽  
Author(s):  
Morteza Eskandari-Ghadi ◽  
Mohammad Rahimian ◽  
Stein Sture ◽  
Maysam Forati
Keyword(s):  
Heat Equation ◽  
Potential Function ◽  
Convex Domain ◽  
Scalar Potential ◽  
Transversely Isotropic ◽  
Potential Functions ◽  
Newtonian Potential ◽  
Existence Of A Solution ◽  
Axis Of Symmetry ◽  
Special Case

A complete set of potential functions consisting of three scalar functions is presented for coupled displacement-temperature equations of motion and heat equation for an arbitrary x3-convex domain containing a linear thermoelastic transversely isotropic material, where the x3-axis is parallel to the axis of symmetry of the material. The proof of the completeness theorem is based on a retarded logarithmic potential function, retarded Newtonian potential function, repeated wave equation, the extended Boggio's theorem for the transversely isotropic axially convex domain, and the existence of a solution for the heat equation. It is shown that the solution degenerates to a set of complete potential functions for elastodynamics and elastostatics under certain conditions. In a special case, the number of potential functions is reduced to one, and the required conditions are discussed. Another special case involves the rotationally symmetric configuration with respect to the axis of symmetry of the material.

Article

1998 ◽  
Vol 76 (8) ◽  
pp. 609-620
Author(s):  
M RM Witwit ◽  
N A Gordon
Keyword(s):  
Large Class ◽  
Potential Function ◽  
Three Dimensional ◽  
Potential Functions ◽  
Numerical Calculations ◽  
Dimensional System ◽  
Perturbation Parameters ◽  
The Hill ◽  
Three Dimensional System

A determination of the eigenvalues for a three-dimensional system is made by expanding the potential function V(x,y,z;Z2, λ,β)= –Z2[x2+y2+z2]+λ {x4+y4+z4+2β[x2y2+x2z2+y2z2]}, around its minimum. In this paper the results of extensive numerical calculations using this expansion and the Hill-determinant approach are reported for a large class of potential functions and for various values of the perturbation parameters Z2, λ, and β. PACS No.:03.65

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