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Proceedings of the 1992 Annual Meeting of JSME/MMD

Finite Element Quantum Mechanical Analysis

Nobuhiro YOSHIKAWA, Yasuyuki Mitsui, Osamu Kuwazuru, Yoshinori SHIHARA

Published Jan 1, 2003• 0 citations
Publisher

Abstract

Finite Element Method， O ( N ) Methods
DOI
10.1299/jsmezairiki.2003.0_473

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This work is concerned with the construction and analysis of structure-preserving Galerkin methods for computing the dynamics of rotating Bose-Einstein condensate (BEC) based on the Gross-Pitaevskii equation with angular momentum rotation. Due to the presence of the rotation term, constructing finite element methods (FEMs) that preserve both mass and energy remains an unresolved issue, particularly in the context of nonconforming FEMs. Furthermore, in comparison to existing works, we provide a comprehensive convergence analysis, offering a thorough demonstration of the methods' optimal and high-order convergence properties. Finally, extensive numerical results are presented to check the theoretical analysis of the structure-preserving numerical method for rotating BEC, and the quantized vortex lattice's behavior is scrutinized through a series of numerical tests.
In this paper, the two following fundamental questions in solid mechanics are answered by extending quantum-mechanical rules to macro level: 1. Why cannot both kinetic and kinematic quantities be known on a boundary point? 2. How does kinematic boundary data transfer over the continuum? For this purpose, the simple two-node bar finite element is introduced as a two-level quantum system (qubit); hence, two bar elements which are attached in series constitute an entangled two-qubit system. When a nodal or inter-nodal kinetic/kinematic boundary condition (BC) is established (i.e. a quantum measurement is performed), the system collapses in a new state. A simple bar (discretized by finite element method) might thus be considered as a set of some entangled two-element systems, which successively collapse in new states when kinetic/kinematic BCs are consecutively applied. Through such a process, boundary kinetic/kinematic information transfer along the whole bar and the static analysis is implemented.