Radial point interpolation method and high-order continuation for solving nonlinear transient heat conduction problems
Saïd Mesmoudi, Omar Askour, Bouazza Braikat
Abstract
In this work, we propose the investigation of unsteady nonlinear heat conduction by using the radial point interpolation method (RPIM) in high-order continuation coupled with homotopy transformation for the first time. In this resolution strategy, the Euler implicit time scheme is used to transform the unsteady nonlinear continuous problem into a sequence of stationary continuous problems. Moreover, by using the RPIM, we transform the sequence of stationary nonlinear continuous problems into discrete problems. Then, homotopy transformation is applied by introducing an arbitrary invertible pre-conditioner <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>*</mml:mo> </mml:msup> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> and a dimensionless parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>a</mml:mi> </mml:math> . These nonlinear problems are transformed into a sequence of linear problems thanks to Taylor series expansions used in a continuation technique to compute the whole solution branch by branch. Numerical examples have been investigated to show the accuracy and efficiency of the proposed approach in this type of problem. The results obtained by the proposed high-order homotopic continuation with the RPIM are compared with those computed by the Newton–Raphson method coupled with the RPIM, high-order homotopic continuation with moving least squares, and high-order homotopic continuation with the finite element method.