159 research outputs found
Optimal Constraint Projection for Hyperbolic Evolution Systems
Techniques are developed for projecting the solutions of symmetric hyperbolic
evolution systems onto the constraint submanifold (the constraint-satisfying
subset of the dynamical field space). These optimal projections map a field
configuration to the ``nearest'' configuration in the constraint submanifold,
where distances between configurations are measured with the natural metric on
the space of dynamical fields. The construction and use of these projections is
illustrated for a new representation of the scalar field equation that exhibits
both bulk and boundary generated constraint violations. Numerical simulations
on a black-hole background show that bulk constraint violations cannot be
controlled by constraint-preserving boundary conditions alone, but are
effectively controlled by constraint projection. Simulations also show that
constraint violations entering through boundaries cannot be controlled by
constraint projection alone, but are controlled by constraint-preserving
boundary conditions. Numerical solutions to the pathological scalar field
system are shown to converge to solutions of a standard representation of the
scalar field equation when constraint projection and constraint-preserving
boundary conditions are used together.Comment: final version with minor changes; 16 pages, 14 figure
A symplectic, symmetric algorithm for spatial evolution of particles in a time-dependent field
A symplectic, symmetric, second-order scheme is constructed for particle
evolution in a time-dependent field with a fixed spatial step. The scheme is
implemented in one space dimension and tested, showing excellent adequacy to
experiment analysis.Comment: version 2; 16 p
Strongly hyperbolic Hamiltonian systems in numerical relativity: Formulation and symplectic integration
We consider two strongly hyperbolic Hamiltonian formulations of general
relativity and their numerical integration with a free and a partially
constrained symplectic integrator. In those formulations we use hyperbolic
drivers for the shift and in one case also for the densitized lapse. A system
where the densitized lapse is an external field allows to enforce the momentum
constraints in a holonomically constrained Hamiltonian system and to turn the
Hamilton constraint function from a weak to a strong invariant.
These schemes are tested in a perturbed Minkowski and the Schwarzschild
space-time. In those examples we find advantages of the strongly hyperbolic
formulations over the ADM system presented in [arXiv:0807.0734]. Furthermore we
observe stabilizing effects of the partially constrained evolution in
Schwarzschild space-time as long as the momentum constraints are enforced.Comment: This version clarifies some points concerning the interpretation of
the result
CFD investigation of a complete floating offshore wind turbine
This chapter presents numerical computations for floating offshore wind turbines for a machine of 10-MW rated power. The rotors were computed using the Helicopter Multi-Block flow solver of the University of Glasgow that solves the Navier-Stokes equations in integral form using the arbitrary Lagrangian-Eulerian formulation for time-dependent domains with moving boundaries. Hydrodynamic loads on the support platform were computed using the Smoothed Particle Hydrodynamics method. This method is mesh-free, and represents the fluid by a set of discrete particles. The motion of the floating offshore wind turbine is computed using a Multi-Body Dynamic Model of rigid bodies and frictionless joints. Mooring cables are modelled as a set of springs and dampers. All solvers were validated separately before coupling, and the loosely coupled algorithm used is described in detail alongside the obtained results
Linearly scaling direct method for accurately inverting sparse banded matrices
In many problems in Computational Physics and Chemistry, one finds a special
kind of sparse matrices, termed "banded matrices". These matrices, which are
defined as having non-zero entries only within a given distance from the main
diagonal, need often to be inverted in order to solve the associated linear
system of equations. In this work, we introduce a new O(n) algorithm for
solving such a system, being n X n the size of the matrix. We produce the
analytical recursive expressions that allow to directly obtain the solution, as
well as the pseudocode for its computer implementation. Moreover, we review the
different options for possibly parallelizing the method, we describe the
extension to deal with matrices that are banded plus a small number of non-zero
entries outside the band, and we use the same ideas to produce a method for
obtaining the full inverse matrix. Finally, we show that the New Algorithm is
competitive, both in accuracy and in numerical efficiency, when compared to a
standard method based in Gaussian elimination. We do this using sets of large
random banded matrices, as well as the ones that appear when one tries to solve
the 1D Poisson equation by finite differences.Comment: 24 pages, 5 figures, submitted to J. Comp. Phy
Free and constrained symplectic integrators for numerical general relativity
We consider symplectic time integrators in numerical General Relativity and
discuss both free and constrained evolution schemes. For free evolution of
ADM-like equations we propose the use of the Stoermer-Verlet method, a standard
symplectic integrator which here is explicit in the computationally expensive
curvature terms. For the constrained evolution we give a formulation of the
evolution equations that enforces the momentum constraints in a holonomically
constrained Hamiltonian system and turns the Hamilton constraint function from
a weak to a strong invariant of the system. This formulation permits the use of
the constraint-preserving symplectic RATTLE integrator, a constrained version
of the Stoermer-Verlet method.
The behavior of the methods is illustrated on two effectively 1+1-dimensional
versions of Einstein's equations, that allow to investigate a perturbed
Minkowski problem and the Schwarzschild space-time. We compare symplectic and
non-symplectic integrators for free evolution, showing very different numerical
behavior for nearly-conserved quantities in the perturbed Minkowski problem.
Further we compare free and constrained evolution, demonstrating in our
examples that enforcing the momentum constraints can turn an unstable free
evolution into a stable constrained evolution. This is demonstrated in the
stabilization of a perturbed Minkowski problem with Dirac gauge, and in the
suppression of the propagation of boundary instabilities into the interior of
the domain in Schwarzschild space-time.Comment: 25 pages, 7 figures; This version contains minor clarifications and
correction
Discrete Nonholonomic Lagrangian Systems on Lie Groupoids
This paper studies the construction of geometric integrators for nonholonomic
systems. We derive the nonholonomic discrete Euler-Lagrange equations in a
setting which permits to deduce geometric integrators for continuous
nonholonomic systems (reduced or not). The formalism is given in terms of Lie
groupoids, specifying a discrete Lagrangian and a constraint submanifold on it.
Additionally, it is necessary to fix a vector subbundle of the Lie algebroid
associated to the Lie groupoid. We also discuss the existence of nonholonomic
evolution operators in terms of the discrete nonholonomic Legendre
transformations and in terms of adequate decompositions of the prolongation of
the Lie groupoid. The characterization of the reversibility of the evolution
operator and the discrete nonholonomic momentum equation are also considered.
Finally, we illustrate with several classical examples the wide range of
application of the theory (the discrete nonholonomic constrained particle, the
Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on a
rotating table and the two wheeled planar mobile robot).Comment: 45 page
Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms
This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a
discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators
From bore-soliton-splash to a new wave-to-wire wave-energy model
We explore extreme nonlinear water-wave amplification in a contraction or, analogously, wave amplification in crossing seas. The latter case can lead to extreme or rogue-wave formation at sea. First, amplification of a solitary-water-wave compound running into a contraction is disseminated experimentally in a wave tank. Maximum amplification in our bore–soliton–splash observed is circa tenfold. Subsequently, we summarise some nonlinear and numerical modelling approaches, validated for amplifying, contracting waves. These amplification phenomena observed have led us to develop a novel wave-energy device with wave amplification in a contraction used to enhance wave-activated buoy motion and magnetically induced energy generation. An experimental proof-of-principle shows that our wave-energy device works. Most importantly, we develop a novel wave-to-wire mathematical model of the combined wave hydrodynamics, wave-activated buoy motion and electric power generation by magnetic induction, from first principles, satisfying one grand variational principle in its conservative limit. Wave and buoy dynamics are coupled via a Lagrange multiplier, which boundary value at the waterline is in a subtle way solved explicitly by imposing incompressibility in a weak sense. Dissipative features, such as electrical wire resistance and nonlinear LED loads, are added a posteriori. New is also the intricate and compatible finite-element space–time discretisation of the linearised dynamics, guaranteeing numerical stability and the correct energy transfer between the three subsystems. Preliminary simulations of our simplified and linearised wave-energy model are encouraging and involve a first study of the resonant behaviour and parameter dependence of the device
Chaotic systems in complex phase space
This paper examines numerically the complex classical trajectories of the
kicked rotor and the double pendulum. Both of these systems exhibit a
transition to chaos, and this feature is studied in complex phase space.
Additionally, it is shown that the short-time and long-time behaviors of these
two PT-symmetric dynamical models in complex phase space exhibit strong
qualitative similarities.Comment: 22 page, 16 figure
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