36 research outputs found
Parallel preconditioners for high order discretizations arising from full system modeling for brain microwave imaging
This paper combines the use of high order finite element methods with
parallel preconditioners of domain decomposition type for solving
electromagnetic problems arising from brain microwave imaging. The numerical
algorithms involved in such complex imaging systems are computationally
expensive since they require solving the direct problem of Maxwell's equations
several times. Moreover, wave propagation problems in the high frequency regime
are challenging because a sufficiently high number of unknowns is required to
accurately represent the solution. In order to use these algorithms in practice
for brain stroke diagnosis, running time should be reasonable. The method
presented in this paper, coupling high order finite elements and parallel
preconditioners, makes it possible to reduce the overall computational cost and
simulation time while maintaining accuracy
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
On Basis Constructions in Finite Element Exterior Calculus
We give a systematic and self-contained account of the construction of
geometrically decomposed bases and degrees of freedom in finite element
exterior calculus. In particular, we elaborate upon a previously overlooked
basis for one of the families of finite element spaces, which is of interest
for implementations. Moreover, we give details for the construction of
isomorphisms and duality pairings between finite element spaces. These
structural results show, for example, how to transfer linear dependencies
between canonical spanning sets, or give a new derivation of the degrees of
freedom
Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems
We present the mixed Galerkin discretization of distributed parameter
port-Hamiltonian systems. On the prototypical example of hyperbolic systems of
two conservation laws in arbitrary spatial dimension, we derive the main
contributions: (i) A weak formulation of the underlying geometric
(Stokes-Dirac) structure with a segmented boundary according to the causality
of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac
structure by a finite-dimensional Dirac structure is realized using a mixed
Galerkin approach and power-preserving linear maps, which define minimal
discrete power variables. (iii) With a consistent approximation of the
Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models.
By the degrees of freedom in the power-preserving maps, the resulting family of
structure-preserving schemes allows for trade-offs between centered
approximations and upwinding. We illustrate the method on the example of
Whitney finite elements on a 2D simplicial triangulation and compare the
eigenvalue approximation in 1D with a related approach.Comment: Copyright 2018. This manuscript version is made available under the
CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0
Numerical Modeling and High Speed Parallel Computing: New Perspectives for Tomographic Microwave Imaging for Brain Stroke Detection and Monitoring
This paper deals with microwave tomography for brain stroke imaging using state-of-the-art numerical modeling and massively parallel computing. Microwave tomographic imaging requires the solution of an inverse problem based on a minimization algorithm (e.g. gradient based) with successive solutions of a direct problem such as the accurate modeling of a whole-microwave measurement system. Moreover, a sufficiently high number of unknowns is required to accurately represent the solution. As the system will be used for detecting the brain stroke (ischemic or hemorrhagic) as well as for monitoring during the treatment, running times for the reconstructions should be reasonable. The method used is based on high-order finite elements, parallel preconditioners from the Domain Decomposition method and Domain Specific Language with open source FreeFEM++ solver
Energy Conserving Higher Order Mixed Finite Element Discretizations of Maxwell's Equations
We study a system of Maxwell's equations that describes the time evolution of
electromagnetic fields with an additional electric scalar variable to make the
system amenable to a mixed finite element spatial discretization. We
demonstrate stability and energy conservation for the variational formulation
of this Maxwell's system. We then discuss two implicit, energy conserving
schemes for its temporal discretization: the classical Crank-Nicholson scheme
and an implicit leapfrog scheme. We next show discrete stability and discrete
energy conservation for the semi-discretization using these two time
integration methods. We complete our discussion by showing that the error for
the full discretization of the Maxwell's system with each of the two implicit
time discretization schemes and with spatial discretization through a
conforming sequence of de Rham finite element spaces converges quadratically in
the step size of the time discretization and as an appropriate polynomial power
of the mesh parameter in accordance with the choice of approximating polynomial
spaces. Our results for the Crank-Nicholson method are generally well known but
have not been demonstrated for this Maxwell's system. Our implicit leapfrog
scheme is a new method to the best of our knowledge and we provide a complete
error analysis for it. Finally, we show computational results to validate our
theoretical claims using linear and quadratic Whitney forms for the finite
element discretization for some model problems in two and three spatial
dimensions