Feng-Nan Hwang

A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation

We develop a parallel Jacobi-Davidson approach for finding a partial set of eigenpairs of large sparse polynomial eigenvalue problems with application in quantum dot simulation. A Jacobi-Davidson eigenvalue solver is implemented based on the Portable, Extensible Toolkit for Scientific Computation (PETSc). The eigensolver thus inherits PETScs efficient and various parallel operations, linear solvers, preconditioning schemes, and easy usages. The parallel eigenvalue solver is then used to solve higher degree polynomial eigenvalue problems arising in numerical simulations of three dimensional quantum dots governed by Schrodingers equations. We find that the parallel restricted additive Schwarz preconditioner in conjunction with a parallel Krylov subspace method (e.g. GMRES) can solve the correction equations, the most costly step in the Jacobi-Davidson algorithm, very efficiently in parallel. Besides, the overall performance is quite satisfactory. We have observed near perfect superlinear speedup by using up to 320 processors. The parallel eigensolver can find all target interior eigenpairs of a quintic polynomial eigenvalue problem with more than 32 million variables within 12 minutes by using 272 Intel 3.0GHz processors.

Development of a parallel semi-implicit two-dimensional plasma fluid modeling code using finite-volume method

Abstract In this paper, the development of a two-dimensional plasma fluid modeling code using the cell-centered finite-volume method and its parallel implementation on distributed memory machines is reported. Simulated discharge currents agree very well with the measured data in a planar dielectric barrier discharge (DBD). Parallel performance of simulating helium DBD solved by the different degrees of overlapping of additive Schwarz method (ASM) preconditioned generalized minimal residual method (GMRES) for different modeling equations is investigated for a small and a large test problem, respectively, employing up to 128 processors. For the large test problem, almost linear speedup can be obtained by using up to 128 processors. Finally, a large-scale realistic two-dimensional DBD problem is employed to demonstrate the capability of the developed fluid modeling code for simulating the low-temperature plasma with complex chemical reactions.

Improving Robustness and Parallel Scalability of Newton Method Through Nonlinear Preconditioning

Inexact Newton method with backtracking is one of the most popular techniques for solving large sparse nonlinear systems of equations. The method is easy to implement, and converges well for many practical problems. However, the method is not robust. More precisely speaking, the convergence may stagnate for no obvious reason. In this paper, we extend the recent work of Tuminaro, Walker and Shadid [2002] on detecting the stagnation of Newton method using the angle between the Newton direction and the steepest descent direction. We also study a nonlinear additive Schwarz preconditioned inexact Newton method, and show that it is numerically more robust. Our discussion will be based on parallel numerical experiments on solving some high Reynolds numbers steady-state incompressible Navier-Stokes equations in the velocity-pressure formulation.

Nonlinear Preconditioning Techniques for Full-Space Lagrange--Newton Solution of PDE-Constrained Optimization Problems

The full-space Lagrange--Newton algorithm is one of the numerical algorithms for solving problems arising from optimization problems constrained by nonlinear partial differential equations. Newton-type methods enjoy fast convergence when the nonlinearity in the system is well-balanced; however, for some problems, such as the control of incompressible flows, even linear convergence is difficult to achieve and a long stagnation period often appears in the iteration history. In this work, we introduce a nonlinearly preconditioned inexact Newton algorithm for the boundary control of incompressible flows. The system has nine field variables, and each field variable plays a different role in the nonlinearity of the system. The nonlinear preconditioner approximately removes some of the field variables, and as a result, the nonlinearity is balanced and inexact Newton converges much faster when compared to the unpreconditioned inexact Newton method or its two-grid version. Some numerical results are presented to d...

Development of a parallel implicit solver of fluid modeling equations for gas discharges

A parallel fully implicit PETSc-based fluid modeling equations solver for simulating gas discharges is developed. Fluid modeling equations include: the neutral species continuity equation, the charged species continuity equation with drift-diffusion approximation for mass fluxes, the electron energy density equation, and Poisson’s equation for electrostatic potential. Except for Poisson’s equation, all model equations are discretized by the fully implicit backward Euler method as a time integrator, and finite differences with the Scharfetter–Gummel scheme for mass fluxes on the spatial domain. At each time step, the resulting large sparse algebraic nonlinear system is solved by the Newton–Krylov–Schwarz algorithm. A 2D-GEC RF discharge is used as a benchmark to validate our solver by comparing the numerical results with both the published experimental data and the theoretical prediction. The parallel performance of the solver is investigated.

A new paradigm for solving plasma fluid modeling equations

A new paradigm for solving plasma fluid modeling equations is proposed and verified in this paper. Model equations include continuity equations for charged species with drift-diffusion approximation, electron energy equation, and Poisson’s equation. Resulting discretized equations are solved jointly by the Newton–Krylov–Schwarz (NKS) [1] scheme by means of a parallelized toolkit called PETSc. All model equations are nondimensionalized and are discretized using fully implicit finite-difference method with the Scharfetter–Gummel scheme for the fluxes. At electrodes, thermal flux is considered for electrons, while both thermal and drift fluxes are considered for ions. A quasi-1D argon gas discharge with a radio frequency power source (13.56 MHz, Vp−p = 200 Volts), gap distance = 20 mm and 20 mm × 20 mm (100 × 100 mesh points) in size is used as the test case. Results of evolution of potential and plasma number density are shown Fig. 1, which are comparable to previous studies. Table 1 lists all the resulting timings of the present parallelized code using different combination of preconditioners (Additive

A parallel hybrid numerical algorithm for simulating gas flow and gas discharge of an atmospheric-pressure plasma jet

Abstract Development of a hybrid numerical algorithm which couples weakly with the gas flow model (GFM) and the plasma fluid model (PFM) for simulating an atmospheric-pressure plasma jet (APPJ) and its acceleration by two approaches is presented. The weak coupling between gas flow and discharge is introduced by transferring between the results obtained from the steady-state solution of the GFM and cycle-averaged solution of the PFM respectively. Approaches of reducing the overall runtime include parallel computing of the GFM and the PFM solvers, and employing a temporal multi-scale method (TMSM) for PFM. Parallel computing of both solvers is realized using the domain decomposition method with the message passing interface (MPI) on distributed-memory machines. The TMSM considers only chemical reactions by ignoring the transport terms when integrating temporally the continuity equations of heavy species at each time step, and then the transport terms are restored only at an interval of time marching steps. The total reduction of runtime is 47% by applying the TMSM to the APPJ example presented in this study. Application of the proposed hybrid algorithm is demonstrated by simulating a parallel-plate helium APPJ impinging onto a substrate, which the cycle-averaged properties of the 200th cycle are presented. The distribution patterns of species densities are strongly correlated by the background gas flow pattern, which shows that consideration of gas flow in APPJ simulations is critical.

One-dimensional simulation of nitrogen dielectric barrier discharge driven by a quasi-pulsed power source and its comparison with experiments

Parallel-plate nitrogen dielectric barrier discharge (DBD) driven by a quasi-pulsed power source (60 kHz) is simulated using a fully-implicit 1D self-consistent fluid modeling code. Simulated discharged currents of gap distances (0.5, 0.7 and 1.0 mm) agree very well with measured data; while simulated discharged currents of wider gap distance (1.2 mm) fail to reproduce the measured data. It is found that the discharge mode is homogeneous Townsend-like for the former case; while it is filamentary-like for the latter case based on the experimental observation. These findings demonstrate that previous numerical studies showing mode transition by changing the gap distance may require further investigation.

Parallel two-level domain decomposition based Jacobi–Davidson algorithms for pyramidal quantum dot simulation

Abstract We consider a quintic polynomial eigenvalue problem arising from the finite volume discretization of a quantum dot simulation problem. The problem is solved by the Jacobi–Davidson (JD) algorithm. Our focus is on how to achieve the quadratic convergence of JD in a way that is not only efficient but also scalable when the number of processor cores is large. For this purpose, we develop a projected two-level Schwarz preconditioned JD algorithm that exploits multilevel domain decomposition techniques. The pyramidal quantum dot calculation is carefully studied to illustrate the efficiency of the proposed method. Numerical experiments confirm that the proposed method has a good scalability for problems with hundreds of millions of unknowns on a parallel computer with more than 10,000 processor cores.

A combined linear and nonlinear preconditioning technique for incompressible navier-stokes equations

We propose a new two-level nonlinear additive Schwarz preconditioned inexact Newton algorithm (ASPIN). The two-level nonlinear preconditioner combines a local nonlinear additive Schwarz preconditioner and a global linear coarse preconditioner. Our parallel numerical results based on a lid-driven cavity incompressible flow problem show that the new two-level ASPIN is nearly scalable with respect to the number of processors if the coarse mesh size is fine enough.