A method is presented that allows the use of multiresolution principles in a time domain electromagnetic modeling technique that is applicable to general structures. Specifically, methods are presented that are compatible with the multiresolution time-domain (MRTD) technique using Haar basis functions that allow the modeling of general structures without limiting the cell size to the features of the modeled structure. Existing Haar techniques require that cells be homogenous in regard to PECs and other localized effects (with the exception that permeability and permittivity can vary throughout the cell). The techniques that are presented here allow the modeling of these structures using a subcell technique that permits the modeling of these effects at individual equivalent grid points. This is accomplished by transforming the application of the effects at individual points in the grid into the wavelet domain. There are several other contributions that are provided in this work. First, the MRTD technique is derived for a general wavelet basis using a relatively compact vector notation that both makes the technique easier to understand and allows the differences and similarities between different MRTD schemes more apparent. Second, techniques such as the uniaxial perfectly matched layer (UPML) for arbitrary wavelet resolution and non-uniform gridding are presented for the first time. Using these techniques, any structure that can be simulated in Yee-FDTD can be modeled with Haar-MRTD. For the first time, results for the use of a time-and-space-adaptive grid in an MRTD simulation are presented.