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# Two- and Three-Dimensional Transient Thermoelastic Analysis by the MLPG Method

Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia
Department of Mechanics, Slovak Technical University, Bratislava, Slovakia
Department of Mechanical & Aerospace Engineering, Carleton University, Ottawa, Canada
Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany

Computer Modeling in Engineering & Sciences 2009, 47(1), 61-96. https://doi.org/10.3970/cmes.2009.047.061

### Abstract

The meshless local Petrov-Galerkin (MLPG) method for transient linear thermoelastic analysis is presented. Orthotropic material properties are considered here. In uncoupled thermoelasticity, the temperature field is not influenced by displacements. Therefore, in the first step, the heat conduction equation is solved for the temperature distribution in the domain. The equations of motion are then solved with the inertial term considered. A Heaviside step function as the test functions is applied in the weak-form to derive local integral equations for solving two- and three-dimensional problems. Local integral equations are written on small sub-domains with circular or spherical shapes. They surround nodal points which are distributed over the analyzed domain. The spatial variation of the displacements and temperature are approximated by the moving least-squares (MLS) scheme. After performing the spatial integrations, a system of ordinary differential equations for certain nodal unknowns is obtained. The backward finite difference method is applied for the approximation of the diffusive term in the heat conduction equation. Then, the system of the ordinary differential equations of the second order resulting from the equations of motion is solved by the Houbolt finite-difference scheme as a time stepping method.

### Keywords

Transient thermoelasticity, orthotropic materials, moving least-squares interpolation, Houbolt finite-difference scheme, crack problems, functionally graded materials

Sladek, J., Sladek, V., Solek, P., Tan, C., Zhang, C. (2009). Two- and Three-Dimensional Transient Thermoelastic Analysis by the MLPG Method. CMES-Computer Modeling in Engineering & Sciences, 47(1), 61–96. This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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