Conformal Dual Basis Functions on Curvilinear Quadrilaterals for Calderon Preconditioning of Surface Integral Equations

Abstract

We present an implementation of the Calderon Multiplicative Preconditioner in the context of conformal quadrilateral discretizations of surface scattering problems. Conformal rooftop basis functions that are traditionally defined on pairs of quadratic quadrilateral elements [1] are augmented n this work by defining a set of conformal “dual” basis functions. The conformal dual basis functions are defined for each edge in the surface mesh and have their support on all conformal elements that share the edge and its two end nodes. As such, the dual basis functions flow across multiple elements in the general direction of the common edge. Similar to the Buffa-Christiansen basis functions [2], due to their expanded support, the proposed dual basis functions of curvilinear quadrilaterals span multiple elements around each edge, care must be taken to properly define each dual basis function across the mesh. A similar approach was recently presented in [3] using barycentric representation on bilinear quadrilaterals. Here, we implement the proposed dual basis functions on bi-quadratic quadrilateral elements and demonstrate the efficacy of the approach for accelerating the electric field integral equation (EFIE) representations of surface scattering problems. Thanks to the quasi-orthogonality of dual bases and the conventional rooftop bases, the Gram projection matrix is well-conditioned and the proposed Calderon Multiplicative Preconditioning of curvilinear geometry formulations leads to significant savings in the iterative solutions of the EFIE for geometries that have been known to cause convergence issues. As an example, the computed monostatic radar cross section (RCS) of a 10-inch-long ogive geometry is shown in Fig. 1 below for 9GHz. The mesh used in this example had 1,150 bi-quadratic quadrilaterals, leading to 2,300 unknowns. The conventional EFIE formulation took 650 iterations to converge in 11.8 seconds, whereas the proposed Calderon Preconditioner converged in 121 iterations, taking only 8.1 seconds for a residual of 10−6 on an Intel i7 processor using the conjugate gradient squared (CGS) iterative solver. As seen, albeit the increased computational cost per iteration, the overall savings provided by Calderon preconditioner is significant. When the same geometry is solved at a much lower frequency of 1.18GHz, the conventional moment method matrix became ill conditioned and did not converge after 5,000 iterations and 340 seconds, however, the proposed Calderon preconditioned approach converged in 266 iterations that took 4.82 seconds.