Effective boundary conditions of the heat equation on a body coated by functionally graded material

Huicong  Li

doi:10.3934/dcds.2016.36.1415

Article Contents

2016, Volume 36, Issue 3: 1415-1430. Doi: 10.3934/dcds.2016.36.1415

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Effective boundary conditions of the heat equation on a body coated by functionally graded material

Huicong Li^1,

1.
Center for Partial Dierential Equations, East China Normal University, Minhang, Shanghai, 200241

Received: November 30, 2014

Revised: April 30, 2015

Published: May 2017

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Abstract

We consider the linear heat equation on a bounded domain, which has two components with a thin coating surrounding a body (of metallic nature), subject to the Dirichlet boundary condition. The coating is composed of two layers, the pure ceramic part and the mixed part. The mixed part is considered to be functionally graded material (FGM) that is meant to make a smooth transition from being metallic to being ceramic. The diffusion tensor is isotropic on the body, and allowed to be anisotropic on the coating; and the size of diffusion tensor may differ significantly in these components. We find effective boundary conditions (EBCs) that are approximately satisfied by the solution of the heat equation on the boundary of the body. A concrete example is considered to study the effect of FGM coating. We also provide numerical simulations to verify our theoretical results.

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Mathematics Subject Classification: Primary: 35K05; Secondary: 35K20, 80A20, 74E30.

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