1,215 research outputs found
Finite Element Hodge for Spline Discrete Differential Forms. Application to the Vlasov-Poisson Equations.
The notion of B-spline based discrete differential forms is recalled and along with a Finite Element Hodge operator, it is used to design new numerical methods for solving the Vlasov-Poisson equations
Robust and parallel scalable iterative solutions for large-scale finite cell analyses
The finite cell method is a highly flexible discretization technique for
numerical analysis on domains with complex geometries. By using a non-boundary
conforming computational domain that can be easily meshed, automatized
computations on a wide range of geometrical models can be performed.
Application of the finite cell method, and other immersed methods, to large
real-life and industrial problems is often limited due to the conditioning
problems associated with these methods. These conditioning problems have caused
researchers to resort to direct solution methods, which signifi- cantly limit
the maximum size of solvable systems. Iterative solvers are better suited for
large-scale computations than their direct counterparts due to their lower
memory requirements and suitability for parallel computing. These benefits can,
however, only be exploited when systems are properly conditioned. In this
contribution we present an Additive-Schwarz type preconditioner that enables
efficient and parallel scalable iterative solutions of large-scale multi-level
hp-refined finite cell analyses.Comment: 32 pages, 17 figure
GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods
We present a novel framework for Finite Element Particle-in-Cell methods
based on the discretization of the underlying Hamiltonian structure of the
Vlasov-Maxwell system. We derive a semi-discrete Poisson bracket, which retains
the defining properties of a bracket, anti-symmetry and the Jacobi identity, as
well as conservation of its Casimir invariants, implying that the semi-discrete
system is still a Hamiltonian system. In order to obtain a fully discrete
Poisson integrator, the semi-discrete bracket is used in conjunction with
Hamiltonian splitting methods for integration in time. Techniques from Finite
Element Exterior Calculus ensure conservation of the divergence of the magnetic
field and Gauss' law as well as stability of the field solver. The resulting
methods are gauge invariant, feature exact charge conservation and show
excellent long-time energy and momentum behaviour. Due to the generality of our
framework, these conservation properties are guaranteed independently of a
particular choice of the Finite Element basis, as long as the corresponding
Finite Element spaces satisfy certain compatibility conditions.Comment: 57 Page
Diagnosing numerical Cherenkov instabilities in relativistic plasma simulations based on general meshes
Numerical Cherenkov radiation (NCR) or instability is a detrimental effect
frequently found in electromagnetic particle-in-cell (EM-PIC) simulations
involving relativistic plasma beams. NCR is caused by spurious coupling between
electromagnetic-field modes and multiple beam resonances. This coupling may
result from the slow down of poorly-resolved waves due to numerical (grid)
dispersion and from aliasing mechanisms. NCR has been studied in the past for
finite-difference-based EM-PIC algorithms on regular (structured) meshes with
rectangular elements. In this work, we extend the analysis of NCR to
finite-element-based EM-PIC algorithms implemented on unstructured meshes. The
influence of different mesh element shapes and mesh layouts on NCR is studied.
Analytic predictions are compared against results from finite-element-based
EM-PIC simulations of relativistic plasma beams on various mesh types.Comment: 31 pages, 20 figure
Metriplectic Integrators for the Landau Collision Operator
We present a novel framework for addressing the nonlinear Landau collision
integral in terms of finite element and other subspace projection methods. We
employ the underlying metriplectic structure of the Landau collision integral
and, using a Galerkin discretization for the velocity space, we transform the
infinite-dimensional system into a finite-dimensional, time-continuous
metriplectic system. Temporal discretization is accomplished using the concept
of discrete gradients. The conservation of energy, momentum, and particle
densities, as well as the production of entropy is demonstrated algebraically
for the fully discrete system. Due to the generality of our approach, the
conservation properties and the monotonic behavior of entropy are guaranteed
for finite element discretizations in general, independently of the mesh
configuration.Comment: 24 pages. Comments welcom
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