High precision simulation of thermal-mechanical problems in functionally graded materials by spectral element differential method

Abstract

Our purpose is to establish a numerical method meeting the requirements of accuracy and easy-using for thermal-mechanical analysis of functionally graded structures. Faced with these demands, a new point collocation pseudo-spectral method named spectral element differential method (SEDM), is proposed in this paper. In this paper the Chebyshev polynomial is used to obtain the derivatives of the variables with respect to intrinsic coordinates. By using the analytical expressions of the differentiation for the shape functions which are used for geometry mapping, the first- and second- derivatives of the variables with respect to global coordinates can be obtained directly. Besides, the local equilibrium equation technique is proposed to connect the elements to form the final system of equations in this paper. Based on the spectral derivative matrix and the element mapping technique, an accurate and efficient strong-form numerical method without any variational principles or energy principles can be obtained lastly for irregular domains. Some examples with complex material properties or geometries are calculated by the proposed method to illustrate the accuracy and convergence. The thermoelastic performances of functionally graded structures with different material distribution and temperature-dependent properties are investigated.