A three-dimensional enriched finite element method for nonlinear transient heat transfer in functionally graded materials - Malek, Mustapha; Izem, Nouh; Mohamed M, Shadi; Seaid, Mohammed (International journal of heat and mass transfer, 2020)
•Development of an efficient partition of unity finite element method for the 3D nonlinear transient heat transfer in functionally graded materials.•Implementation of a class of 3D time-independent enrichment functions using multiple Gaussian approximations in unstructured meshes.•Derivation of a numerical method with the potential to accurately resolve sharp gradients in the solution without requiring very fine meshes.•Numerical assessment of the partition of unity finite element method for 3D nonlinear transient heat transfer subject to steep gradients.•Simulation of a problem of heat transfer in a 3D pump part using the partition of unity finite element method. Nonlinear transient heat transfer in functionally graded materials is being studied more popular in present. In preliminary design, this problem can be simplified as a composite, and a three-dimensional transient heat transfer analysis is used to adjust dimensions of the considered materials. This paper is concerned with the numerical modeling of transient heat transfer in composite materials where the thermal conductivity is also dependent on the temperature; hence the problem is nonlinear. We are interested in solutions with steep boundary layers where highly refined meshes are commonly needed. Such problems can be challenging to solve with the conventional finite element method. To deal with this challenge we propose an enriched finite element formulation where the basis functions are augmented with a summation of exponential functions. First, the initial-value problem is integrated in time using a semi-implicit scheme and the semi-discrete problem is then integrated in space using the enriched finite elements. We demonstrate through several numerical examples that the proposed approach can recover the heat transfer on coarse meshes and with much fewer degrees of freedom compared to the standard finite element method. Thus, a significant reduction in the computational requirements is achieved without compromising on the solution accuracy. The results also show the stability of the scheme when using tetrahedral unstructured grids.