Performance of a Petrov–Galerkin algebraic multilevel preconditioner for finite element modeling of the semiconductor device drift‐diffusion equations

Abstract

SUMMARYThis study compares the performance of a relatively new Petrov-Galerkin smoothed aggregation (PGSA) multilevel preconditioner with a nonsmoothed aggregation (NSA) multilevel preconditioner to accelerate the convergence of Krylov solvers on systems arising from a drift-diffusion model for semiconductor devices. PGSA is designed for nonsymmetric linear systems, Ax = b, and has two main differences with smoothed aggregation. Damping parameters for smoothing interpolation basis functions are now calculated locally and restriction is no longer the transpose of interpolation but instead corresponds to applying the interpolation algorithm to A T and then transposing the result. The drift-diffusion system consists of a Poisson equation for the electrostatic potential and two convection-diffusion-reaction-type equations for the electron and hole concentration. This system is discretized in space with a stabilized finite element method and the discrete solution is obtained by using a fully coupled preconditioned Newton-Krylov solver. The results demonstrate that the PGSA preconditioner scales significantly better than the NSA preconditioner, and can reduce the solution time by more than a factor of two for a problem with 110 million unknowns on 4000 processors. The solution of a 1B unknown problem on 24 000 processor cores of a Cray XT3/4 machine was obtained using the PGSA preconditioner.

Towards large-scale multi-socket, multicore parallel simulations: Performance of an MPI-only semiconductor device simulator

“…The numerical studies involve the steady-state solution of the two-dimensional drift-diffusion equations for a 2 Â 1.5 lm silicon bipolar junction transistor (BJT) [3,46]. The steady-state calculation is performed with a voltage bias of 0.3 V. Fig.…”

“…The two weak scaling studies consider different amount of work per core (4-8 times as many degrees of freedom (DOF) per core). All these studies (Section 3.2) use the transpose-free quasi-minimal residual (TFQMR) Krylov subspace method for the linear solver [21] with three-level ML Petrov-Galerkin smoothed aggregation (PGSA) multigrid preconditioner [9,46]. A W(1, 1) cycle was used (W-cycle with one presmoothing and one postsmoothing relaxation sweep).…”

A parallel fully coupled algebraic multilevel preconditioner applied to multiphysics PDE applications: Drift‐diffusion, flow/transport/reaction, resistive MHD

“…As well the use of the W-cycle multilevel solver, that spends more time solving coarser representations of the problem is also significantly faster than the V-cycle. These results point out that a careful choice of the various multilevel parameters such as aggregate size, the type of multilevel cycle, the number of multilevel cycles, and the number of relaxation sweeps of the smoothers can have a significant impact on the performance of multilevel preconditioners [45,67].…”

Enriched residual free bubbles for semiconductor device simulation

“…The results for potential, electron concentration and hole concentration after the first iteration of this procedure are illustrated in Figs. 6,7,8,9,10,11 for each of the numerical methods listed above.…”

“…Diffusion is introduced in such a way that it is aligned with the streamline direction and appropriate formulas are available to calculate the "right" amount of diffusion. The natural application of the method to semiconductor device simulation was carried out by Carey et al [18] and more recently, the method has been applied to large simulations using parallel preconditioning algorithms [10,11]. In addition, there exist exponential fitting techniques [4] which essentially stabilise the current continuity equations though exponential shape functions.…”