Two recent advances in Magnetic Resonance Imaging (MRI) technology have resulted in a need for sophisticated computational electromagnetics (CEM) tools. The first is the availability of higher magnetic fields, which can be used to improve signal-to-noise ratio. The second is the availability of transmit-coil arrays, which can be used to minimize human-body heating by electric fields. The higher magnetic fields imply higher-frequency RF pulses, which complicate electromagnetic analysis, as the dimensions of the human body and wavelength become comparable. They also imply increased human-body heating, which limits the RF power used for imaging purposes. In the Computational Prototyping Group, we are leveraging known CEM techniques to address these new needs of the MRI community, working in close collaboration with the RLE MRI group. Also, we are using the MRI application to drive the development of new CEM methods.

A number of MRI-based applications are being targeted. First, we are developing fast methods for tuning and matching the transmitters to the human-body loaded MRI coils. By using scattering-matrix formalism, a frequency-domain finite-elements method and commercial RF optimization software, we have shrunk this process from days to hours and are working to reduce it further by use of integral-equation methods. Second, we are developing a hybrid method-of-moments Green’s function approach that is tailored to the problem of optimizing the geometrical configuration of the transmit coils. In this method, we are pre-computing Green’s functions for a realistic model of the human body. Using these Green’s functions we hope to be able to rapidly assess different coil configurations for a typical body. To aid this assessment, we plan to leverage our work on parameterized model-order reduction, with which models of the parametric dependence of various quantities of interest can be generated automatically. Lastly, we are working on the problem of fast computation of approximate solutions to the electromagnetic fields inside the human body, assuming a simplified 3-tissue model that can be obtained for each patient by a quick MRI scan.

Figure 1: A sample layout containing 85 nets (represented by different colours) and 14 dielectric layers (removed from image for clarity). Image courtesy of Intel Corporation.

Standard full chip capacitance extraction algorithms rely for computational efficiency on 2D scanning and table lookup algorithms. These algorithms trade off accuracy for computational efficiency and result in significant error in the extracted capacitance of complex layouts. It is therefore desirable to use accurate field solvers for full chip extraction, which in general can be divided into two types: discretization-based and discretization-free. Discretization-based methods include the finite difference methods, finite element method, and boundary element method ^[1] , while the best-known discretization-free algorithms are the floating random walk method and its variants. It is widely accepted that discretization based methods are very efficient for small and medium-size structures and that discretization free methods are more efficient for very large structures. Recently, our group has proposed a Markov Chain-based hierarchical algorithm ^[2] to combine the advantages of both discretization-based and discretization-free methods.

In this project we further refine our algorithm and implement it in C++ for very large-scale capacitance extraction. A layout is first partitioned into smaller blocks, and the Markov Transition Matrix (MTM) for each block is computed. Then, the capacitance of the full layout is extracted by simulating the Markov Chain using random walk methods. Our algorithm is efficient because the computation of an MTM is derived from the capacitance matrix associated with each block. Such a matrix is computed using the Finite Difference Method, which can handle highly inhomogeneous structures including different dielectrics and metal fills. In addition, our algorithm does not require the assembly or the solution of any linear system of equations at the level of the full layout. Consequently, the algorithm requires very modest computational time and memory to compute the capacitance matrix of the full chip to field solver accuracy. In addition, our algorithm lends itself to efficient parallelization because both the computation of MTMs and the computation of random walks are embarrassingly parallelizable. The accuracy, efficiency, and almost linear scalability of the algorithm are being tested on FastMarkov – our C++ implemented, MPI-parallelized solver, which can handle large-scale, geometrically complex, industry provided examples (such as the one shown in Figure 1) in 45 min using 4-core parallelization. We have so far also verified that around 95% of computation time is spent on MTM computation, which can be reduced significantly by larger number parallelization.

1. K. Nabors and J. White, “Fastcap: a multipole accelerated 3-d capacitance extraction program,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 10, no. 11, pp. 1447-1459, Nov. 1991.
2. T. El-Moselhy, I. Elfadel, and L. Daniel, “A hierarchical floating random walk algorithm for fabric-aware 3d capacitance extraction,” in Proc. IEEE/ACM Int. Conf. Computer-Aided Design, 2009, page 752-758, San Jose CA.