A technique for solving multiscale problems by hybridizing frequency and time domain algorithms

Summary form only given. Despite the availability of modern supercomputers, direct solution of multiscale problems by means of conventional CEM methods-be it FEM, FDTD or MoM-is highly challenging. This is because modeling of structures with fine features, which might share the computational domain with other large objects, often requires dealing with a large number of degrees of freedom. Dealing with such multiscale problems often forces us to compromise the accuracy of the numerical discretization process when faced with the problem of capturing the small-scale features, in order to cope with the limited available resources in terms of CPU memory and time.Many fine-featured structures that are integral parts of sensors and other complex systems have dimensions that are often only a small fraction of the wavelength in the medium and require a very fine mesh to capture the nuances of their geometry. In the past, it has been demonstrated that the electromagnetic properties of small objects can be accurately and conveniently characterized by a Dipole Moment (DM) representation, which is valid both in the nearand far-field regions of the scatterer. Based on this, a hybrid FDTD method has been proposed, which models the fine features by using a DM located at the center of the FDTD cell. Since the DM is always located in close vicinity of the grid lines of the cell containing the DM, the 1/r3 term dominates within the cell, rendering the fields “time independent” and identical to those from a static charge. These fields are then passed on to the FDTD, which automatically performs the analytic continuation of the quasi-static solution into the region external to the cell containing the small object, as a consequence of the dynamic nature of Maxwell's equations. However, when modeling straight wires passing through multiple FDTD cells, the above hybrid method fails since the quasi-static approximation of the DM is no longer valid over the entire wire. To address this problem, we propose to model the wire by using basis functions which can be conveniently expressed in closed forms, not only in the frequency domain, but in the time domain as well. The scheme is simpler and also computationally more efficient than the conventional Time Domain Integral Equation (TDIE) algorithms, especially when a large number of unknowns is involved. However, we have found that the proposed algorithm becomes unstable when we generalize the field expressions in order to model bent wire geometries, for which a time domain basis function which is convenient, causal and stable cannot be found. In this work we propose a novel DM-FDTD hybrid technique when dealing with an arbitrarily shaped wire, which may span several cells. The key is to use a modified DM approach, in which all the three terms of the fields generated by the DM, corresponding to both the near and far-field regions, are expressed in the time domain in convenient closed forms. Thus, the field contributions can be calculated at any arbitrary distancefrom the DM source and can be coupled with the FDTD. The proposed method is universal and can be used for scatterers traversing through multiple cells. Also, since the fields produced from an arbitrary scatterer can always be represented as a superposition of electric and/or magnetic DMs, the method can be employed to model an arbitrarily shaped thin scatterer without altering the FDTD updating scheme or the size of the FDTD discretization, which can be the nominal λ/20, even when dealing with objects with fine features that are small fractions of the wavelength.