Parametric Formulation of the Finite-Volume Theory for Functionally Graded Materials—Part I: Analysis

The recently reconstructed higher-order theory for functionally graded materials is further enhanced by incorporating arbitrary quadrilateral subcell analysis capability through a parametric formulation. This capability significantly improves the efficiency of modeling continuous inclusions with arbitrarily-shaped cross sections of a graded material’s microstructure previously approximated using discretization based on rectangular subcells, as well as modeling of structural components with curved boundaries. Part I of this paper describes the development of the local conductivity and stiffness matrices for a quadrilateral subcell which are then assembled into global matrices in an efficient manner following the finite-element assembly procedure. Part II verifies the parametric formulation through comparison with analytical solutions for homogeneous curved structural components and graded components where grading is modeled using piecewise uniform thermoelastic moduli assigned to each discretized region. Results for a heterogeneous microstructure in the form of a single inclusion embedded in a matrix phase are also generated and compared with the exact analytical solution, as well as with the results obtained using the original reconstructed theory based on rectangular discretization and finite-element analysis.