A universal array approach for finite elements with continuously inhomogeneous material properties and curved boundaries

The advent of technologies such as photo-lithography and holography has lead to accurate fabrication of devices such as the Luneburg lens. Furthermore, many real life applications involve spatial changes in material property tensors (MPTs). Clearly, integration of continuously inhomogeneous MPTs (CIMPT) inside individual finite elements (FEs) adds to the flexibility and efficiency of finite elements methods (FEMs). Curved FEs have been extensively used to mitigate geometrical non-conformities associated to rectilinear approximation of curvilinear features while CIMPT are conventionally handled by element-wise constant approximation. FE matrices are traditionally evaluated through 1) numerical cubature or 2) universal matrices (UMs). In essence, both methods rely on polynomial integration. Furthermore, complications associated with evaluation of FE matrices on elements with curved boundaries or CIMPTs are identical in nature, i.e., integrals with non-constant Jacobian or MPT terms. In this work, a generalized UM approach is proposed, which reduces the cost associated with evaluation of FE matrices with curvilinear features and CIMPTs. Motivated by a non-graded Luneburg lens, the conventional element-wise constant MPTs are replaced with polynomial representations yielding: a) more accurate physical models for problems with CIMPTs and curvature; and, b) better utilization of computer resources when higher-order curved elements replace smaller lower-order elements.