FREEPLANE PASTE AND PRESERVE STRUCTURE CODE

FREEPLANE PASTE AND PRESERVE STRUCTURE CODE

The paper concludes with a discussion of geometric electromagnetic particle-in-cell ], a particle in cell code that incorporates Hamiltonian and geometrical structure preserving properties.Received Revised Accepted Mar 29, 2018 Throughout, applications to plasma systems are emphasized. Two kinds of simulated annealing, a relaxation technique for obtaining equilibrium states, are more » reviewed: one that uses metriplectic dynamics, which maximizes an entropy at fixed energy, and the other that uses double bracket dynamics, which preserves Casimir invariants. Metriplectic integration, which preserves the structure of conservative systems with both Hamiltonian and dissipative parts, is proposed.

Basic ideas of variational integration and Poisson integration, which can preserve the noncanonical Hamiltonian structure, are discussed. These include conservative integration, which exactly conserves invariants, symplectic integration, which exactly preserves the Hamiltonian geometric structure, and other Hamiltonian integration techniques. Geometric structures associated with and emergent from HAP formulations are discussed. Hamiltonian and action principle (HAP) formulations of plasma physics are reviewed for the purpose of explaining structure preserving numerical algorithms. Our results for a parameterized truss-structure problem demonstrate the practical importance of preserving Lagrangian structure and illustrate the proposed method's merits: it reduces computation time while maintaining high accuracy and stability, in contrast to existing nonlinear model-reduction techniques that do not preserve structure. Moreover, from the algebraic perspective, key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness: matrix gappy proper orthogonal decomposition and reduced-basis sparsification. To preserve structure, the method first approximates the system's “Lagrangian ingredients''-the Riemannian metric, the potential-energy function, the dissipation function, and the external force-and subsequently derives reduced-order equations of motion by applying the (forced) Euler-Lagrange equation with these more » quantities. We focus on parameterized simple mechanical systems subjected to Rayleigh damping and external forces, and consider an application to nonlinear structural dynamics. As such, the resulting reduced-order model retains key properties such as energy conservation and symplectic time-evolution maps. Our work proposes a model-reduction methodology that preserves Lagrangian structure and achieves computational efficiency in the presence of high-order nonlinearities and arbitrary parameter dependence. Challenging issues that arise in the presence of non-gauge invariant structure functions are also addressed. Case (ii) becomes the first example known to the author where an open algebra is handled in refined algebraic quantisation. Providing a specific parametrised family of real-valued scaling functions, the implementation of the corresponding rescaled quantum momentum-type constraints is performed using RAQ when the gauge algebra: (i) remains abelian and (ii) undergoes into an algebra of a nonunimodular group with nonconstant gauge invariant structure functions.

These rescalings induce structure functions at the level more » of the gauge algebra. The system under consideration contains two momentum-type constraints, originally abelian, where rescalings of these constraints by a non-vanishing function of the coordinates are allowed. In the present work, the first steps to generalise this analysis to cases where more constraints occur are developed. In a previous work, refined algebraic quantisation (RAQ) within a family of classically equivalent constrained Hamiltonian systems that are related to each other by rescaling one momentum-type constraint was investigated.

Freeplane paste and preserve structure