232 research outputs found
Coupled structural, thermal, phase-change and electromagnetic analysis for superconductors, volume 1
This research program has dealt with the theoretical development and computer implementation of reliable and efficient methods for the analysis of coupled mechanical problems that involve the interaction of mechanical, thermal, phase-change and electromagnetic subproblems. The focus application has been the modeling of superconductivity and associated quantum-state phase-change phenomena. In support of this objective the work has addressed the following issues: (1) development of variational principles for finite elements; (2) finite element modeling of the electromagnetic problem; (3) coupling of thermal and mechanical effects; and (4) computer implementation and solution of the superconductivity transition problem. The research was carried out over the period September 1988 through March 1993. The main accomplishments have been: (1) the development of the theory of parametrized and gauged variational principles; (2) the application of those principled to the construction of electromagnetic, thermal and mechanical finite elements; and (3) the coupling of electromagnetic finite elements with thermal and superconducting effects; and (4) the first detailed finite element simulations of bulk superconductors, in particular the Meissner effect and the nature of the normal conducting boundary layer. The grant has fully supported the thesis work of one doctoral student (James Schuler, who started on January 1989 and completed on January 1993), and partly supported another thesis (Carmelo Militello, who started graduate work on January 1988 completing on August 1991). Twenty-three publications have acknowledged full or part support from this grant, with 16 having appeared in archival journals and 3 in edited books or proceedings
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Computational Electromagnetism and Acoustics
It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems
A semi-implicit hybrid finite volume / finite element scheme for all Mach number flows on staggered unstructured meshes
In this paper a new hybrid semi-implicit finite volume / finite element
(FV/FE) scheme is presented for the numerical solution of the compressible
Euler and Navier-Stokes equations at all Mach numbers on unstructured staggered
meshes in two and three space dimensions. The chosen grid arrangement consists
of a primal simplex mesh composed of triangles or tetrahedra, and an edge-based
/ face-based staggered dual mesh. The governing equations are discretized in
conservation form. The nonlinear convective terms of the equations, as well as
the viscous stress tensor and the heat flux, are discretized on the dual mesh
at the aid of an explicit local ADER finite volume scheme, while the implicit
pressure terms are discretized at the aid of a continuous
finite element method on the nodes of the primal mesh. In the zero Mach number
limit, the new scheme automatically reduces to the hybrid FV/FE approach
forwarded in \cite{BFTVC17} for the incompressible Navier-Stokes equations. As
such, the method is asymptotically consistent with the incompressible limit of
the governing equations and can therefore be applied to flows at all Mach
numbers. Due to the chosen semi-implicit discretization, the CFL restriction on
the time step is only based on the magnitude of the flow velocity and not on
the sound speed, hence the method is computationally efficient at low Mach
numbers. In the chosen discretization, the only unknown is the scalar pressure
field at the new time step. Furthermore, the resulting pressure system is
symmetric and positive definite and can therefore be very efficiently solved
with a matrix-free conjugate gradient method.
In order to assess the capabilities of the new scheme, we show computational
results for a large set of benchmark problems that range from the quasi
incompressible low Mach number regime to compressible flows with shock waves
Poroelasticity problem: numerical difficulties and efficient multigrid solution
This work contains some of the more relevant results obtained by the author regarding the numerical solution of the Biot’s consolidation problem. The emphasis here is on the stable discretization and the highly efficient solution of the resulting algebraic system of equations, which is of saddle point type. On the one hand, a stabilized linear finite element scheme providing oscillation-free solutions for this model is proposed and theoretically analyzed. On the other hand, a monolithic multigrid method is considered for the solution of the resulting system of equations after discretization by using the stabilized scheme. Since this system is of saddle point type, special smoothers of “Vanka”-type have to be considered. This multigrid method is designed with the help of an special local Fourier analysis that takes into account the specific characteristics of the considered block-relaxations. Results from this analysis are presented and compared with those experimentally computed
Automatic 3D modeling by combining SBFEM and transfinite element shape functions
The scaled boundary finite element method (SBFEM) has recently been employed
as an efficient means to model three-dimensional structures, in particular when
the geometry is provided as a voxel-based image. To this end, an octree
decomposition of the computational domain is deployed and each cubic cell is
treated as an SBFEM subdomain. The surfaces of each subdomain are discretized
in the finite element sense. We improve on this idea by combining the
semi-analytical concept of the SBFEM with certain transition elements on the
subdomains' surfaces. Thus, we avoid the triangulation of surfaces employed in
previous works and consequently reduce the number of surface elements and
degrees of freedom. In addition, these discretizations allow coupling elements
of arbitrary order such that local p-refinement can be achieved
straightforwardly
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