Adaptive Modeling of Complex Packaging Geometries Using Haar-based MRTD Algorithms

Abstract

-The Haar-based MRTD algorithm is applied to the design and optimization of complex 3D RF Packaging geometries. Significant numerical aspects concerning the implementation of Boundary Conditions and the thresholding of the wavelet coefficients are discussed in detail and guidelines for the efficient modeling of a practical Flip-Chip package are derived. Multiresolution Time-Domain TechniquesExpansion BasisThe MultiResolution Time-Domain (MRTD) algorithms have demonstrated unparallel properties when applied to the analysis of structures with medium or large computational domains [1][2][3][4][5][6]. Through a two-fold expansion of the fields in scaling and wavelet functions with respect to time/space, memory and execution time requirements are minimized while a high resolution in areas of strong field variations or field singularities is achieved through the use of sufficiently large number of wavelet resolutions. The major advantage of the MRTD algorithms is their capability to develop real-time time and space adaptive grids through the efficient thresholding of the wavelet coefficients.Various expansion basis have been utilized for the implementation of the MRTD algorithms. The Battle-Lemarie basis offers a reduction in memory by 2-3 orders of magnitude for 3D structures. Nevertheless, the entire-domain character of these functions adds a significant computational overhead in the approximation of the field derivatives in Curl Maxwell equations. In addition, Hard Boundaries (e.g. PEC's) cannot be applied directly by zeroing out the appropriate field components; image theory has to be implemented to account for the neighboring cells' contribution. Due to their compact support, Haar expansion basis functions ( Fig.1) provide schemes that are similar to the FDTD algorithm that can be derived using pulse basis. They do not provide the drastical economies of the entire-basis schemes, but can be implemented in a much simpler way and maintain the adaptive feature. MRTD Scheme with Arbitrary Wavelet ResolutionsFor simplicity, the 1D MRTD scheme for TEM propagation will be presented. It can be extended to 2D and 3D in a straightforward way. The Electric (E x ) and the Magnetic (H y ) fields are displaced by half step in both time-and space-domains (Yee cell formulation) and are expanded in a summation of scaling (φ) and wavelet (ψ) functions in space and scaling components in time. For example, E x is given by, where φ i (z)=φ(z ∆z-i) and ψ i r,ir (z)=2 r/2 ψ 0 (2 r (z ∆z-ir)-i) represent the Haar scaling and r-resolution wavelet functions located inside the i-cell. The conventional notation m E x,i is used for the Electric field component at time t=m ∆t and z=i ∆z, where ∆t and ∆z are the time-step and the spatial cell size respectively. The notation for H y is similar.