15 research outputs found
Multisymplectic formulation of fluid dynamics using the inverse map
We construct multisymplectic formulations of fluid dynamics using the inverse of the Lagrangian path map. This inverse map, the âback-to-labelsâ map, gives the initial Lagrangian label of the fluid particle that currently occupies each Eulerian position. Explicitly enforcing the condition that the fluid particles carry their labels with the flow in Hamilton's principle leads to our multisymplectic formulation. We use the multisymplectic one-form to obtain conservation laws for energy, momentum and an infinite set of conservation laws arising from the particle relabelling symmetry and leading to Kelvin's circulation theorem. We discuss how multisymplectic numerical integrators naturally arise in this approach.</p
Continuous and discrete Clebsch variational principles
The Clebsch method provides a unifying approach for deriving variational
principles for continuous and discrete dynamical systems where elements of a
vector space are used to control dynamics on the cotangent bundle of a Lie
group \emph{via} a velocity map. This paper proves a reduction theorem which
states that the canonical variables on the Lie group can be eliminated, if and
only if the velocity map is a Lie algebra action, thereby producing the
Euler-Poincar\'e (EP) equation for the vector space variables. In this case,
the map from the canonical variables on the Lie group to the vector space is
the standard momentum map defined using the diamond operator. We apply the
Clebsch method in examples of the rotating rigid body and the incompressible
Euler equations. Along the way, we explain how singular solutions of the EP
equation for the diffeomorphism group (EPDiff) arise as momentum maps in the
Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch
variational principle is discretised to produce a variational integrator for
the dynamical system. We obtain a discrete map from which the variables on the
cotangent bundle of a Lie group may be eliminated to produce a discrete EP
equation for elements of the vector space. We give an integrator for the
rotating rigid body as an example. We also briefly discuss how to discretise
infinite-dimensional Clebsch systems, so as to produce conservative numerical
methods for fluid dynamics
Generating Functionals and Lagrangian PDEs
We introduce the concept of Type-I/II generating functionals defined on the
space of boundary data of a Lagrangian field theory. On the Lagrangian side, we
define an analogue of Jacobi's solution to the Hamilton-Jacobi equation for
field theories, and we show that by taking variational derivatives of this
functional, we obtain an isotropic submanifold of the space of Cauchy data,
described by the so-called multisymplectic form formula. We also define a
Hamiltonian analogue of Jacobi's solution, and we show that this functional is
a Type-II generating functional. We finish the paper by defining a similar
framework of generating functions for discrete field theories, and we show that
for the linear wave equation, we recover the multisymplectic conservation law
of Bridges.Comment: 31 pages; 1 figure -- v2: minor change
Geometric, Variational Discretization of Continuum Theories
This study derives geometric, variational discretizations of continuum
theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the
dynamics of complex fluids. A central role in these discretizations is played
by the geometric formulation of fluid dynamics, which views solutions to the
governing equations for perfect fluid flow as geodesics on the group of
volume-preserving diffeomorphisms of the fluid domain. Inspired by this
framework, we construct a finite-dimensional approximation to the
diffeomorphism group and its Lie algebra, thereby permitting a variational
temporal discretization of geodesics on the spatially discretized
diffeomorphism group. The extension to MHD and complex fluid flow is then made
through an appeal to the theory of Euler-Poincar\'{e} systems with advection,
which provides a generalization of the variational formulation of ideal fluid
flow to fluids with one or more advected parameters. Upon deriving a family of
structured integrators for these systems, we test their performance via a
numerical implementation of the update schemes on a cartesian grid. Among the
hallmarks of these new numerical methods are exact preservation of momenta
arising from symmetries, automatic satisfaction of solenoidal constraints on
vector fields, good long-term energy behavior, robustness with respect to the
spatial and temporal resolution of the discretization, and applicability to
irregular meshes
Discrete Differential Geometry of Thin Materials for Computational Mechanics
Instead of applying numerical methods directly to governing equations, another approach to computation is to discretize the geometric structure specific to the problem first, and then compute with the discrete geometry. This structure-respecting discrete-differential-geometric (DDG) approach often leads to new algorithms that more accurately track the physically behavior of the system with less computational effort. Thin objects, such as pieces of cloth, paper, sheet metal, freeform masonry, and steel-glass structures are particularly rich in geometric structure and so are well-suited for DDG. I show how understanding the geometry of time integration and contact leads to new algorithms, with strong correctness guarantees, for simulating thin elastic objects in contact; how the performance of these algorithms can be dramatically improved without harming the geometric structure, and thus the guarantees, of the original formulation; how the geometry of static equilibrium can be used to efficiently solve design problems related to masonry or glass buildings; and how discrete developable surfaces can be used to model thin sheets undergoing isometric deformation
Convergent Numerical Schemes for the Compressible Hyperelastic Rod Wave Equation
We propose a fully discretised numerical scheme for the hyperelastic rod wave
equation on the line. The convergence of the method is established. Moreover,
the scheme can handle the blow-up of the derivative which naturally occurs for
this equation. By using a time splitting integrator which preserves the
invariants of the problem, we can also show that the scheme preserves the
positivity of the energy density
Geometric Numerical Integration
The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods