On the nature of periodic traction boundary conditions in micromechanical FE analyses of unit cells

2011 ◽  
Vol 77 (4) ◽  
pp. 441-450 ◽  
Author(s):  
S. Li
Keyword(s):  
Boundary Conditions ◽  
Unit Cells ◽  
Traction Boundary Conditions

scholarly journals Surface Wave Speed of Functionally Graded Magneto-Electro-Elastic Materials with Initial Stresses

2014 ◽  
Vol 44 (3) ◽  
pp. 49-64 ◽  
Author(s):  
Li Li ◽  
P. J. Wei
Keyword(s):  
Surface Wave ◽  
Boundary Conditions ◽  
Elastic Material ◽  
Short Circuit ◽  
Equations Of Motion ◽  
Open Circuit ◽  
Functionally Graded ◽  
Initial Stresses ◽  
Shear Surface ◽  
Traction Boundary Conditions

Abstract The shear surface wave at the free traction surface of half- infinite functionally graded magneto-electro-elastic material with initial stress is investigated. The material parameters are assumed to vary ex- ponentially along the thickness direction, only. The velocity equations of shear surface wave are derived on the electrically or magnetically open circuit and short circuit boundary conditions, based on the equations of motion of the graded magneto-electro-elastic material with the initial stresses and the free traction boundary conditions. The dispersive curves are obtained numerically and the influences of the initial stresses and the material gradient index on the dispersive curves are discussed. The investigation provides a basis for the development of new functionally graded magneto-electro-elastic surface wave devices.

Local Solutions in Potential Theory and Linear Elasticity Using Monte Carlo Methods

2003 ◽  
Vol 70 (3) ◽  
pp. 408-417 ◽  
Author(s):  
S. S. Kulkarni ◽  
S. Mukherjee ◽  
M. D. Grigoriu
Keyword(s):  
Monte Carlo ◽  
Boundary Conditions ◽  
Potential Theory ◽  
Linear Elasticity ◽  
Neumann Boundary ◽  
Convex Domains ◽  
Local Method ◽  
Point Of Interest ◽  
Local Solutions ◽  
Traction Boundary Conditions

A numerical method called the boundary walk method is described in this paper. The boundary walk method is a local method in the sense that it directly gives the solution at the point of interest. It is based on a global integral representation of the unknown solution in the form of potentials, followed by evaluating the integrals in the resulting series solutions using Monte Carlo simulation. The boundary walk method has been applied to solve interior problems in potential theory with either Dirichlet or Neumann boundary conditions. It has also been applied to solve interior problems in linear elasticity with either displacement or traction boundary conditions. Weakly singular integral formulations in linear elasticity, to which the boundary walk method has been applied, are also derived. Finally, numerical results, which are computed by applying the boundary walk method to solve some two-dimensional problems over convex domains in potential theory and linear elasticity, are presented. These solutions are compared with the known analytical solutions (when available) or with solutions from the standard boundary element method.

Modeling Thermal Microspreading Resistance in Via Arrays

2016 ◽  
Vol 138 (1) ◽  
Author(s):  
Michael Fish ◽  
Patrick McCluskey ◽  
Avram Bar-Cohen
Keyword(s):  
Boundary Conditions ◽  
Effective Properties ◽  
Three Dimensional ◽  
Thermal Characteristics ◽  
Simulation Software ◽  
Electronic Packages ◽  
Isothermal Boundary ◽  
Unit Cells ◽  
Flow Through ◽  
Management Techniques

As thermal management techniques for three-dimensional (3D) chip stacks and other high-power density electronic packages continue to evolve, interest in the thermal pathways across substrates containing a multitude of conductive vias has increased. To reduce the computational costs and time in the thermal analysis of through-layer via (TXV) structures, much research to date has focused on defining effective anisotropic thermal properties for a pseudohomogeneous medium using isothermal boundary conditions. While such an approach eliminates the need to model heat flow through individual vias, the resulting properties are found to depend on the specific boundary conditions applied to a unit TXV cell. More specifically, effective properties based on isothermal boundary conditions fail to capture the local “microspreading” resistance associated with more realistic heat flux distributions and local hot spots on the surface of these substrates. This work assesses how the thermal microspreading resistance present in arrays of vias in interposers, substrates, and other package components can be properly incorporated into the modeling of these arrays. We present the conditions under which spreading resistance plays a major role in determining the thermal characteristics of a via array and propose methods by which designers can both account for the effects of microspreading resistance and mitigate its contribution to the overall thermal behavior of such substrate–via systems. Finite element modeling (FEM) of TXV unit cells is performed using commercial simulation software (ansys).

A numerical homogenization technique for unidirectional composites using polygonal generalized finite elements

2021 ◽  
pp. 146442072110463
Author(s):  
Murilo Sartorato
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Boundary Conditions ◽  
Computational Efficiency ◽  
Periodic Boundary ◽  
Deformation Gradient ◽  
Reaction Force ◽  
Periodic Boundary Conditions ◽  
Unit Cell ◽  
Unit Cells

The present study proposes a computational methodology to obtain the homogenized effective elastic properties of unidirectional fibrous composite materials by using the generalized finite-element method and penalization techniques to impose periodic boundary conditions on non-uniform polygonal unit cells. Each unit cell is described by a single polygonal finite element using Wachspress functions as base shape functions and different families of enrichment functions to account for the internal fiber influence on stresses and strains fields. The periodic boundary conditions are imposed using reflection laws between two parallel opposing faces using a Lagrange multiplier approach; this reflection law creates a distributed reaction force over the edges of the [Formula: see text]-gon from the direct application of a given deformation gradient, which simulates different macroscopic load cases on the macroscopic body the unit cell is part of. The methodology is validated through a comparison with results for similar unit cells found in the literature and its computational efficiency is compared to simple cases solved using a classic finite-element approach. This methodology showed computational advantages over the classic finite elements in both computational efficiency and total number of degrees of freedom for convergence and flexibility on the shape of the unit cell used. Finally, the methodology provides an efficient way to introduce non-circular fiber shapes and voids.

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