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Register a new userCRKSPH-compatible discretization of the SUPG and SAAF transport equations
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The self-adjoint angular flux and streamline-upwind Petrov-Galerkin transport equations are discretized using reproducing kernels with the collocation method to produce a discretization that is compatible with conservative reproducing kernel smoothed particle hydrodynamics. A novel second derivative is derived for the diffusion-like term in the self-adjoint angular flux equation. The resulting equations involve only evaluations of kernels and physical data at the nodal centers.
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The time-dependent radiation transport equation is discretized using the meshless-local Petrov-Galerkin method with reproducing kernels. The integration is performed using a Voronoi tessellation, which creates a partition of unity that only depends on the position and extent of the kernels. The resolution of the integration automatically follows the particles and requires no manual adjustment. The discretization includes streamline-upwind Petrov-Galerkin stabilization to prevent oscillations and improve numerical conditioning. The angular quadrature is selectively refineable to increase angular resolution in chosen directions. The time discretization is done using backward Euler. The transport solve for each direction and the solve for the scattering source are both done using Krylov iterative methods. Results indicate first-order convergence in time and second-order convergence in space for linear reproducing kernels.
A pore network modeling (PNM) framework for the simulation of transport of charged species, such as ions, in porous media is presented. It includes the Nernst-Planck (NP) equations for each charged species in the electrolytic solution in addition to a charge conservation equation which relates the species concentration to each other. Moreover, momentum and mass conservation equations are adopted and there solution allows for the calculation of the advective contribution to the transport in the NP equations. The proposed framework is developed by first deriving the numerical model equations (NMEs) corresponding to the partial differential equations (PDEs) based on several different time and space discretization schemes, which are compared to assess solutions accuracy. The derivation also considers various charge conservation scenarios, which also have pros and cons in terms of speed and accuracy. Ion transport problems in arbitrary pore networks were considered and solved using both PNM and finite element method (FEM) solvers. Comparisons showed an average deviation, in terms of ions concentration, between PNM and FEM below $5%$ with the PNM simulations being over ${10}^{4}$ times faster than the FEM ones for a medium including about ${10}^{4}$ pores. The improved accuracy is achieved by utilizing more accurate discretization schemes for both the advective and migrative terms, adopted from the CFD literature. The NMEs were implemented within the open-source package OpenPNM based on the iterative Gummel algorithm with relaxation. This work presents a comprehensive approach to modeling charged species transport suitable for a wide range of applications from electrochemical devices to nanoparticle movement in the subsurface.
The main objective of this paper is to develop a general method of geometric discretization for infinite-dimensional systems and apply this method to the EPDiff equation. The method described below extends one developed by Pavlov et al. for incompressible Euler fluids. Here this method is presented in a general case applicable to all, not only divergence-free, vector fields. Also, a different (pseudospectral) representation of the velocity field is used. We will apply this method to the one-dimensional EPDiff equation and present numerical results.
Wereportonanewmultiscalemethodapproachforthestudyofsystemswith wide separation of short-range forces acting on short time scales and long-range forces acting on much slower scales. We consider the case of the Poisson-Boltzmann equation that describes the long-range forces using the Boltzmann formula (i.e. we assume the medium to be in quasi local thermal equilibrium). We developed a new approach where fields and particle information (mediated by the equations for their moments) are solved self-consistently. The new approach is implicit and numerically stable, providing exact energy conservation. We tested different implementations all leading to exact energy conservation. The new method requires the solution of a large set of non-linear equations. We considered three solution strategies: Jacobian Free Newton Krylov, an alternative, called field hiding, based on hiding part of the residual calculation and replacing them with direct solutions and a Direct Newton Schwarz solver that considers simplified single particle-based Jacobian. The field hiding strategy proves to be the most efficient approach.
The nonequilibrium Greens function (NEGF) method is often used to predict transport in atomistically resolved nanodevices and yields an immense numerical load when inelastic scattering on phonons is included. To ease this load, this work extends the atomistic mode space approach of Ref. [1] to include inelastic scattering on optical and acoustic phonons in silicon nanowires. This work also includes the exact calculation of the real part of retarded scattering self-energies in the reduced basis representation using the Kramers-Kronig relations. The inclusion of the Kramers-Kronig relation for the real part of the retarded scattering self-energy increases the impact of scattering. Virtually perfect agreement with results of the original representation is achieved with matrix rank reductions of more than 97%. Time-to-solution improvements of more than 200$times$ and peak memory reductions of more than 7$times$ are shown. This allows for the solution of electron transport scattered on phonons in atomically resolved nanowires with cross-sections larger than 5 nm $times$ 5 nm.
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