K. Nabors and J. White. Fast capacitance extraction of general three-dimensional structures. IEEE Trans. on Microwave Theory and Techniques, June 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....between far away pairs. The clustering also helps with the matrix inversion and as a result the computational complexity is reduced to O(Nm)wheremis the number of di#erent conductors and N is a measure of the size of the layout. A faster and general version of this algorithm is presented in [21]. A program named Fas t Cap [22] was developed which implements this algorithm. In the next section, the basic concepts of FastCap are discussed. A commercial tool Raphael [23] exists that can use either FEM or BEM to extract interconnect capacitance. 2.3 FastCap FastCap is a multipole ....

K. Nabros, S. Kim and J. White, "Fast Capacitance Extraction of General Three-Dimensional Structures," IEEE Transactions on Microwave Theory and Techniques, Vol. 40, July 1992, pp. 1496-1506.

....by appropriately setting the voltages at the contacts and solving (1) for the detailed current distribution. The current flowing into each contact is then obtained by summing the currents from all panels in the contact. The algorithm is similar to the standard capacitance extraction problem [14]: extraction of the full model requires m linear solves for a system with m contacts. The straightforward way to accomplish this task requires the computation of Z p using (2) In [3] it was shown that the Green s function G(x; y; x 0 ; y 0 ) for the bounded substrate with grounded backplane ....

....the PcDCT algorithm is to realize that the effect of an injected current in a panel p i on the potential of another far away panel p j , can be considered the same for small variations in the distance between panels. Similar ideas have been used extensively in multipole accelerated algorithms [17, 14, 11]. The approach developed here is reminiscent of other precorrected algorithms previously published [18] and relies on a simple coarser grid projection method, which can be used in combination with the DCT to accelerate the computation of the matrix vector product while taking into account all of ....

K. Nabors and J. White. Fast capacitance extraction of general three-dimensional structures. IEEE Transactions on Microwave Theory and Techniques, June 1992.

....is kept small, the total cost of obtaining the substrate admittance model is O(S KG n 2 ) where KG is the average number of GMRES iterations per solution. In [8] this cost is further decreased to O(S K P n) by acceleration of the matrix vector product using the hierarchical multipole algorithm [13]. However this is only possible due to the simplifications assumed in computing Z, specifically the translation invariance of the implied Green s function in this case, and is thus not a general procedure. Therefore general methods have to be devised to accelerate the submitted to DATE 98 ....

K. Nabors and J. White. Fast capacitance extraction of general three-dimensional structures. IEEE Transactions on Microwave Theory and Techniques, June 1992.

.... method known as Generalized Minimal Residual (GMRES) Saad Schultz 1986) Finally, since the system of equations is dense, the matrix vector product required at each iteration of GMRES is expensive and to reduce its cost, the fast multipole method is used (Greengard Rokhlin 1987, Nabors White 1992). The combination of these techniques has been implemented in a packaging analysis program, named FASTHENRY, whose computational complexity and memory requirements grow linearly with the number of volume elements required to discretize the conductors. In FASTHENRY each conductor in a t conductor ....

Nabors, K. & White, J. (1992). Fast capacitance extraction of general three-dimensional structures, IEEE Trans. on Microwave Theory and Techniques.

....than an order of magnitude. 1. Introduction. The recently developed multipole accelerated iterative methods for solving potential integral equations have renewed interest in using discretized integral formulations for the numerical solution of geometrically complicated three dimensional problems [1, 2]. As multipole based approaches use implicit matrix representations which can not be easily directly factored, the success of such approaches hinges on reliable convergence of the underlying iterative method. To aid in insuring rapid iteration convergence, multilevel and local inversion ....

K. Nabors and J. White, "Fast capacitance extraction of general three-dimensional structures," IEEE Trans. on Microwave Theory and Techniques, To Appear 1992.

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K. Nabors and J. White. Fast capacitance extraction of general three-dimensional structures. IEEE Trans. on Microwave Theory and Techniques, June 1992.

....to achieve convergence. If the number of GMRES iterations approaches n, then the minimization in each GMRES iteration will require order n operations, and the whole algorithm becomes order n 3 operations. This problem can be avoided easily through the use of the preconditioner described in [2, 10], which reduces the number of GMRES iterations required to achieve convergence with 1 error (tol = 0.01 in Algorithm 1) to well below n for large problems. evaluation points Figure 1: Approximately computing the potentials at d evaluation points due to a cluster of d charged panels in order d ....

K. Nabors, S. Kim, and J. White, "Fast capacitance extraction of general three-dimensional structures," IEEE Transactions on Microwave Theory and Techniques, vol. 40, pp. 1496- 1506, November 1992.

.... using the Krylov subspace method known as generalized conjugate residual (GCR) 19] can be found in a general form in [20] and specifically for a constant matrix in [17] The operations of the iterative algorithm can be reduced further by using a multipole accelerated iterative algorithm [23] whose cost and memory has been shown to Fig. 4. Two dimensional illustration of the search direction space for two different calls to the iterative algorithm. Here, the search directions Pw and P w are close so the spaces they span are similar. grow only as . Similarly, the computation ....

....expensive to compute. To follow the approach of [14] consider preconditioning with a block diagonal version of . Thus, the preconditioner will be an LU factored version of (37) where and are block diagonal. One can improve this preconditioner by noting that for fast capacitance extraction in [23], 26] it was found that block diagonal preconditioning for is not adequate to capture the strong coupling involved in charge interaction. For that reason, in [23] 26] a local inversion preconditioner was developed. Since we know this preconditioner works well for ,we wish to use it in (37) ....

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K. Nabors and J. White, "Fast capacitance extraction of general threedimensional structures," IEEE Trans. Microwave Theory Tech., June 1992.

....required to achieve convergence approaches n, then the minimization in each GMRES iteration requires order n operations. This problem is avoided through the use of a preconditioner which reduces the number of GMRES iterations required to achieve convergence to well below n for large problems [4]. The product Aq I: is, using (9) Aq= Pqk ] 12) Dqa d evaluation points Oi, Oi) Figure 2: The direct evaluation of the potential due to d panel charges at d points. d evaluation points l panels oi) Figure 3: The evaluation of a the potential due to d panel charges at d points ....

....Maintaining this efficiency for general distributions of panels while controlling error leads to the hierarchical multipole algorithm used in FASTCAP2. A detailed description of the complete multipole algorithm is given in [1] and its use in the context of capacitance extraction is described in [5, 4]. 3.2 Electric Field Evaluations The evaluation of Dq amounts to calculating the left hand side of rid equations of the form (5) or, equivalent ]y, 4) The left hand side of (4) can be approx intoted by replacing the derivatives by divided differences constructed near panel i as illustrated in ....

K. Nabors, S. Kim, J. White, and S. Senturia. Fast capacitance extraction of general three-dimensional structures. In Proceedings of the 1991.

....GMRES [3] is used to solve (2) the major cost of the algorithm is the order n operations required to form the dense matrix P and the order n operations to compute the dense matrix vector products for each GMRES iteration. Sparsification techniques, such as fast multipole algorithms [4] [5] or precorrected FFT methods [1] avoid forming P and can be used to compute densematrix vector products in order n or nlogn operations. Empirical studies on typical 3 D capacitance extraction problems indicate that precorrected FFT methods generally use the least memory and CPU time. In the ....

K. Nabors, S. Kim, and J. White, "Fast capacitance extraction of general three-dimensional structure," IEEE Trans. on Microwave Theory and Tech., vol. 40, pp. 1496-1507, 1992.

....Therefore, by using the multipole algorithm three times, L[ can be computed in order b operations. 4. 2 The Hierarchical Multipole Algorithm A complete description of the fast multipole algorithm is quite lengthy, and can be found in [9] or in the context of 3 D capacitance extracation, in [11, 12]. To see roughly what the algorithm exploits to achieve its efficiency consider the two configurations given in Figs. 4 and 5, depicted in 2 D for simplicity. In either figure, the obvious approach to determining the electrostatic potential at the n evaluation points from the n2 point charges ....

K. Nabors, S. Kim, and J. White, "Fast capacitance extraction of general three-dimensionM structures," IEEE Trans- actions on Microwave Theory and Techniques, vol. 40, pp. 1496 1506, November 1992.

....expansion. Figure 3: The evaluation point potentials are approximated with a local expansion. 3. 1 The Hierarchical Multipole Algo rithm A complete description of the fast multipole algorithm is quite lengthy, and can be found in [6] or in the context of 3 D capacitance extracation, in [7, 8]. To see roughly what the algorithm exploits to achieve its efficiency consider the two configurations given in Figs. 2 and 3, depicted in 2 D for simplicity. In either figure, the obvious approach to determining the elec trostatic potential at the n evaluation points from the n2 point charges ....

K. Nabors, S. Kim, and J. White, "Fast capacitance extraction of general three-dimensional structures," IEEE Trans. on Microwave Theory and Techniques, vol. 40, pp. 1496-1506, July 1992.

....substrate may need to be analyzed at once. Recent work on numerical modeling of substrate coupling effects has focused on obtaining an n by n impedance or admittance coupling matrix, where n is the number of contacts. Any of the algorithms developed for rapid analysis of interconnect parasitics[6, 7, 8, 9] may be adapted to solve the problem of extracting the substrate coupling information associated with a single contact(e.g. 10, 11, 12] The difficulty with these approaches is that, in the substrate coupling context, knowing how to do each of these single contact solves quickly is ....

K. Nabors, S. Kim, and J. White, "Fast capacitance extraction of general three-dimensional structure," IEEE Trans. on Microwave Theory and Techniques, vol. 40, no. 7, pp. 1496--1507, July 1992.

....treecode [15] are perhaps the most famous of all sparsification techniques. However, the technique of [12] only applies to matrices which come from the 1 r kernel, or polynomials in 1 r. This includes, for example, the panel potential from panel charge matrices used in capacitance calculation [16]. A kernel has been derived for the substrate problem, but it is not of 1 r form. In addition, it takes panel currents to panel potentials. Our conductance matrix G instead takes contact potentials to contact currents. Precorrected FFT methods [17] can be applied to a broader range of problems ....

K. Nabors, S. Kim, and J. White, "Fast capacitance extraction of general three- dimensional structure," IEEE Trans. on Microwave Theory and Techniques, vol. 40, no. 7, pp. 1496 1507, July 1992. 108

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K. Nabors, S. Kim. and J. White, "Fast capacitance extraction of general three-dimensional structures," [EEE Trans. MicrowatJe Theory Tech., vol. 40, pp. 1496-1506, July 1992.

....using method of moments [1] based discretizations of integral equation formulations are commonly used to compute these capacitances, but such approaches generate dense matrix problems which are computationally expensive to solve, and this limits the complexity of problems which can be analyzed. In [2], a rapid method, based on the hierarchical multipole algorithm[3] for computing the capaci tance of three dimensional structures was presented. In this paper, we describe a precorrected FFT approach which can replace the fast multipole algorithm for accelerating the dense matrix vector product ....

K. Nabors, S. Kim, and J. White, "Fast capacitance extraction of general three-dimensional structure," IEEE Trans. on Microwave Theory and Techniques, vol. 40, pp. 1496-1507, July 1992.

.... sizes, where the sizes are chosen so that each finest level cube has roughly the same number of panels [11] For the kinds of charge and evaluation point distributions common to boundary element problems, such an approach can sometimes require more computation than a nonadaptive algorithm [12]. Instead, a more effective approach in this setting is to avoid transformation from panel charges to multipole expansions, and to avoid forming local expansions, whenever such representations are inefficient. For example, consider computing M,j from Q2M(1, j)q,j. If the number of entries in ....

.... 5. Experimental Results. In this section, results from computational experiments using our program FASTCAP are presented to demonstrate that the preconditioned, adaptive, multipole accelerated (PAMA) 3 D capacitance extraction algorithm described above really deserves a four letter mnemonic [12]. The structures described below were created with the solid modeling program PATRAN, or by computer program, and all capacitance calculations were performed using FASTCAP. The multipole accelerated algorithms in FASTCAP use, by default, second order multipole expansions and a GMRES convergence ....

I(. Nabors and J. White, "Fast capacitance extraction of general three-dimensional structures," IEEE Trans. on Microwave Theory and Techniques, June 1992.

....previously reported methods, both in extraction speed and model size, making it possible to generate guaranteed passive low order models for efficient inclusion in a circuit simulator such as SPICE or SPECTRE. Combining our formulation with acceleration techniques such as the Fast Multipole Method [10, 11] or the Precorrected FFT [12] allows the accurate analysis of larger, more complex three dimensional geometries than previously possible. In Section 2 we discuss the integral formulation and discretization from which we derive the circuit equations that describe the interconnect effects. In ....

K. Nabors and J. White. Fast capacitance extraction of general threedimensional structures. IEEE Trans. on Microwave Theory and Techniques, June 1992.

....previously reported methods, both in extraction speed and model size, making it possible to generate guaranteed passive low order models for efficient inclusion in a circuit simulator such as SPICE or SPECTRE. Combining our formulation with acceleration techniques such as the Fast Multipole Method [10, 11] or the Precorrected FFT [12] allows the accurate analysis of larger, more complex three dimensional geometries than previously possible. In Section 2 we discuss the integral formulation and discretization from which we derive the circuit equations that describe the interconnect effects. In ....

K. Nabors and J. White. Fast capacitance extraction of general threedimensional structures. IEEE Trans. on Microwave Theory and Techniques, June 1992.

....previously reported methods, both in extraction speed and model size, making it possible to generate guaranteed passive low order models for efficient inclusion in a circuit simulator such as SPICE or SPECTRE. Combining our formulation with acceleration techniques such as the Fast Multipole Method [10, 11] or the Precorrected FFT [12] allows the accurate analysis of larger, more complex three dimensional geometries than previously possible. In Section 2 we discuss the integral formulation and discretization from which we derive the circuit equations that describe the interconnect effects. In ....

K. Nabors and J. White. Fast capacitance extraction of general threedimensional structures. IEEE Trans. on Microwave Theory and Techniques, June 1992.

....Therefore, by using the multipole algorithm three times, LI b can be computed in order b operations. 4. 2 The Hierarchical Multipole Algorithm A complete description of the fast multipole algorithm is quite lengthy, and can be found in [9] or in the context of 3 D capacitance extracation, in [11, 12]. To see roughly what the algorithm exploits to achieve its efficiency consider the two configurations given in Figs. 4 and 5, depicted in 2 D for simplicity. In either figure, the obvious approach to determining the electrostatic potential at the n 1 evaluation points from the n 2 point charges ....

K. Nabors, S. Kim, and J. White, "Fast capacitance extraction of general three-dimensional structures," IEEE Transactions on Microwave Theory and Techniques, vol. 40, pp. 1496--1506, November 1992.

....As VLSI circuit speeds have increased, the need for accurate interconnect models has become essential to accurate chip and system design. Recently, much work has been directed at rapidly solving for the inductance or capacitance of these structures, starting directly from Maxwell s equations [1, 2]. However inductance and capacitance are not necessarily decoupled quantities, and for higher frequencies a distributed model is necessary. In this paper, we describe an integral equation approach to modeling the impedance of interconnect structures accounting for both the charge accumulation on ....

....that with our approach, based on a modified nodal analysis formulation, it is possible to generate guaranteed passive low order models for efficient inclusion in a circuit simulator such as SPICE. Additionally, the algorithm is ripe for acceleration techniques such as the Fast Multipole Method [1, 3] or the Precorrected FFT [4] approach allowing the analysis of larger, more complex three dimensional geometries. In Section 2 we discuss the integral formulation and discretization from which we derive the large dense linear system describing the interconnect. In Section 3 we describe applying ....

K. Nabors and J. White. Fast capacitance extraction of general three-dimensional structures. IEEE Trans. on Microwave Theory and Techniques, June 1992.

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Keith Nabors, Songmin Kim and Jacob White, "Fast Capacitance Extraction of general Three-Dimensional Structures", IEEE Transactions on Microwave Theory and Technique, Vol. 40, No. 7, July 1992.

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IC Nabros and J. White "Fast capacitance extraction of general threedimensional structures" IEEE Transactions on Microwave Theory and Techniques, June 1992 vol. 40, pp. 1496-1507.

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K. Nabors, S. Kim, J. White: "Fast Capacitance Extraction of General Three-Dimensional Structures", IEEE Transactions on Microwave Theory and Techniques, Jul 1992, Vol. 40, No. 7, pp. 1496

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