A boundary integral investigation for unsteady modified Helmholtz problems of some other classes of anisotropic functionally graded materials

Numerical solutions for a class of unsteady modified Helmholtz problems of anisotropic functionally graded materials are sought. The governing equation which is a variable coefficients equation is transformed to a constant coefficients equation. The time variable is transformed using the Laplace transform. The resulted partial differential equation of constant coefficients and time free variable is then converted to a boundary integral equation, from which boundary element solutions can be obtained. Some examples are considered to verify the accuracy, convergence and consistency of the numerical solutions. The results show that the numerical solutions are accurate, convergent and consistent.

Download Full-text

Numerical Simulation for Unsteady Anisotropic-Diffusion Convection Equation of Spatially Variable Coefficients and Incompressible Flow

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.54.2023.4069 ◽

2021 ◽

Vol 54 ◽

Author(s):

Moh.Ivan Azis

Keyword(s):

Laplace Transform ◽

Anisotropic Diffusion ◽

Numerical Solutions ◽

Boundary Integral ◽

Functionally Graded ◽

Time Variable ◽

Convection Equation ◽

Constant Coefficients ◽

Diffusion Convection ◽

Element Method

The anisotropic-diffusion convection equation of spatiallyvariable coefficients which is relevant for functionally graded mediais discussed in this paper to find numerical solutions by using acombined Laplace transform and boundary element method. The variablecoefficients equation is transformed to a constant coefficients equation.The constant coefficients equation is then Laplace-transformed sothat the time variable vanishes. The Laplace-transformed equationis consequently written in a pure boundary integral equation whichinvolves a time-free fundamental solution. The boundary integral equationis therefore employed to find numerical solutions using a standardboundary element method. Finally the results obtained are inverselytransformed numerically using the Stehfest formula to get solutionsin the time variable. The combined Laplace transform and boundaryelement method is easy to be implemented, efficient and accurate forsolving unsteady problems of anisotropic functionally graded mediagoverned by the diffusion convection equation.

Download Full-text

Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method

Computational Materials Science ◽

10.1016/j.commatsci.2003.08.006 ◽

2003 ◽

Vol 28 (3-4) ◽

pp. 494-504 ◽

Cited By ~ 118

Author(s):

J. Sladek ◽

V. Sladek ◽

Ch. Zhang

Keyword(s):

Integral Equation ◽

Heat Conduction ◽

Functionally Graded Materials ◽

Integral Equation Method ◽

Transient Heat Conduction ◽

Boundary Integral ◽

Functionally Graded ◽

Graded Materials ◽

Local Boundary ◽

Local Boundary Integral Equation

Download Full-text

Numerical solutions to Helmholtz equation of anisotropic functionally graded materials

Journal of Physics Conference Series ◽

10.1088/1742-6596/1341/8/082012 ◽

2019 ◽

Vol 1341 ◽

pp. 082012

Author(s):

Paharuddin ◽

Sakka ◽

P Taba ◽

S Toaha ◽

M I Azis

Keyword(s):

Helmholtz Equation ◽

Functionally Graded Materials ◽

Numerical Solutions ◽

Functionally Graded ◽

Graded Materials

Download Full-text

The Crack-Tip Fields of Reissner’s Plates of Radial Functionally Graded Materials

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.217-218.1319 ◽

2011 ◽

Vol 217-218 ◽

pp. 1319-1323

Author(s):

Yao Dai ◽

Jun Feng Liu ◽

Peng Zhang

Keyword(s):

Functionally Graded Materials ◽

Plate Theory ◽

Crack Tip ◽

Variable Coefficients ◽

Functionally Graded ◽

Homogeneous Material ◽

Governing Equations ◽

Crack Tip Field ◽

Graded Materials ◽

Crack Tip Fields

For homogeneous material plates and non-homogeneous material plates, the crack-tip field plays an important role in the research of fracture mechanics. However, the governing equations become the system of the sixth order partial differential ones with the variable coefficients when the material gradient is perpendicular to the thickness direction of plates. In this paper, they are derived first. Then, the crack-tip fields of the plates of radial functionally graded materials (FGMs) are studied and the higher order crack-tip fields are obtained based on the Reissner’s plate theory. The results show the effect of the non-homogeneity on the crack-tip fields explicitly and become the same as solutions of the homogeneous material plates as the non-homogeneous parameter approaches zero.

Download Full-text

A numerical boundary integral equation analysis for standard linear viscoelastic media made of functionally graded materials

International Journal of Mechanical and Materials Engineering ◽

10.1186/s40712-014-0009-4 ◽

2014 ◽

Vol 9 (1) ◽

Cited By ~ 1

Author(s):

Hossein Ashrafi ◽

Mohammad Shariyat

Keyword(s):

Integral Equation ◽

Functionally Graded Materials ◽

Boundary Integral ◽

Functionally Graded ◽

Graded Materials ◽

Viscoelastic Media ◽

Equation Analysis ◽

Linear Viscoelastic ◽

Standard Linear ◽

Linear Viscoelastic Media

Download Full-text

Transient Heat Conduction in Functionally Graded Materials by LT-MFS

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.189-193.1664 ◽

2011 ◽

Vol 189-193 ◽

pp. 1664-1669 ◽

Cited By ~ 1

Author(s):

Ning Zhao ◽

Lei Lei Cao ◽

Hui Guo

Keyword(s):

Heat Conduction ◽

Functionally Graded Materials ◽

Heat Conduction Problem ◽

Transient Heat Conduction ◽

Fundamental Solutions ◽

Functionally Graded ◽

Time Variable ◽

Graded Materials ◽

Time Space ◽

Transient Heat Conduction Problem

: The LT-MFS approach is proposed to solve two-dimensional transient heat conduction problem in functionally graded materials (FGMs). First, a Laplace transform approach is used to move the time variable. Then, a fundamental solution in Laplace space for FGMs is constructed. Next, the solution in Laplace space is approximated by the linear combination of fundamental solutions. Further, Stefest’s algorithm is employed to convert the results in Laplace space back into the time–space domain. Finally, the method is tested on several benchmark examples. The results demonstrate well the efficiency and accuracy of the proposed method.

Download Full-text

Boundary Integral Equation Analysis of an Inhomogeneous Medium Made of Functionally Graded Materials

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.685.285 ◽

2013 ◽

Vol 685 ◽

pp. 285-289

Author(s):

H. Ashrafi ◽

M.R. Bahadori ◽

H. Keshmiri ◽

M. Shariyat

Keyword(s):

Integral Equation ◽

Boundary Conditions ◽

Boundary Integral Equation ◽

Functionally Graded Materials ◽

Inhomogeneous Media ◽

Boundary Integral ◽

Functionally Graded ◽

Element Formulation ◽

Graded Materials ◽

Integral Equation Formulation

The present work develops direct graded boundary integral equation formulation for behavior investigation of the inhomogeneous media made of functionally graded materials. The isoparametric boundary elements, the elastostatic governing equations and a weighted residual technique are implemented with the material characteristics that vary continuously along a given dimension. The resulting algorithm is capable of solving the quasistatic problems for elastic functionally graded media with a variety of the boundary conditions and loadings. The inhomogeneous media is made of a ceramic–metal mixture, in which the material properties vary continuously according to a power law graded distribution in a given direction. Avoiding the use of internal elements in the graded boundary element formulation is one of the main objectives of this paper, which results only in numerical discretization of the boundaries of the considered media. Some examples with continuously inhomogeneous isotropic materials were provided under different boundary conditions to evaluate the proposed numerical formulation for the FGMs.

Download Full-text