This paper deals with the preconditioning of the curl-curl operator. We use H（curl）- conforming finite elements for the discretization of our corresponding magnetostatic model problem. Jumps in the material paramete...This paper deals with the preconditioning of the curl-curl operator. We use H（curl）- conforming finite elements for the discretization of our corresponding magnetostatic model problem. Jumps in the material parameters influence the condition of the problem. We will demonstrate by theoretical estimates and numerical experiments that hierarchical matrices are well suited to construct efficient parallel preconditioners for the fast and robust iterative solution of such problems.展开更多
We present a novel efficient implementation of the flexible boundary condition(FBC)method,initially proposed by Sinclair et al.,for large single-periodic problems.Efficiency is primarily achieved by constructing a hie...We present a novel efficient implementation of the flexible boundary condition(FBC)method,initially proposed by Sinclair et al.,for large single-periodic problems.Efficiency is primarily achieved by constructing a hierarchical matrix(H-matrix)representation of the periodic Green matrix,reducing the complexity for updating the boundary conditions of the atomistic problem from quadratic to almost linear in the number of pad atoms.In addition,our implementation is supported by various other tools from numerical analysis,such as a residual-based transformation of the boundary conditions to accelerate the convergence.We assess the method for a comprehensive set of examples,relevant for predicting mechanical properties,such as yield strength or ductility,including dislocation bow-out,dislocation-precipitate interaction,and dislocation cross-slip.The main result of our analysis is that the FBC method is robust,easy-to-use,and up to two orders of magnitude more efficient than the current state-of-the-art method for this class of problems,the periodic array of dislocations(PAD)method,in terms of the required number of per-atom force computations when both methods give similar accuracy.This opens new prospects for large-scale atomistic simulations—without having to worry about spurious image effects that plague classical boundary conditions.展开更多
文摘This paper deals with the preconditioning of the curl-curl operator. We use H（curl）- conforming finite elements for the discretization of our corresponding magnetostatic model problem. Jumps in the material parameters influence the condition of the problem. We will demonstrate by theoretical estimates and numerical experiments that hierarchical matrices are well suited to construct efficient parallel preconditioners for the fast and robust iterative solution of such problems.
基金Financial support from the Fonds National Suisse(FNS),Switzerland,(project 191680)is highly acknowledged.
文摘We present a novel efficient implementation of the flexible boundary condition(FBC)method,initially proposed by Sinclair et al.,for large single-periodic problems.Efficiency is primarily achieved by constructing a hierarchical matrix(H-matrix)representation of the periodic Green matrix,reducing the complexity for updating the boundary conditions of the atomistic problem from quadratic to almost linear in the number of pad atoms.In addition,our implementation is supported by various other tools from numerical analysis,such as a residual-based transformation of the boundary conditions to accelerate the convergence.We assess the method for a comprehensive set of examples,relevant for predicting mechanical properties,such as yield strength or ductility,including dislocation bow-out,dislocation-precipitate interaction,and dislocation cross-slip.The main result of our analysis is that the FBC method is robust,easy-to-use,and up to two orders of magnitude more efficient than the current state-of-the-art method for this class of problems,the periodic array of dislocations(PAD)method,in terms of the required number of per-atom force computations when both methods give similar accuracy.This opens new prospects for large-scale atomistic simulations—without having to worry about spurious image effects that plague classical boundary conditions.
