A new consistent technique for the construction of symmetric constitutive matrices for generalized time domain algorithms

The cell method (CM) (Tonti, E., PIER monograph series, vol.32, p.1-44, 2001) is a numerical technique suitable for solving electromagnetic static and dynamic problems both on structured and unstructured grids. Topological equations (field equations) can be enforced on the cell complexes in an exact discrete form by using appropriate incidence matrices. Constitutive relations can be enforced in a discrete form by using suitable constitutive matrices. The desired features of the constitutive matrices are: (1) symmetry; (2) positive definiteness, in order to ensure stability; (3) sparsity, to save memory and ensure fast computations; (4) consistency of the algebraic constitutive relations with the local constitutive relations. We propose a new technique to build these matrices such that they satisfy the desired features in most cases. This is true for both 2D grids and 3D grids. This new technique enables us to obtain positive definite constitutive matrices in all of the practical cases we have tested.