7,086 research outputs found
The hp-BEM with quasi-uniform meshes for the electric field integral equation on polyhedral surfaces: a priori error analysis
This paper presents an a priori error analysis of the hp-version of the
boundary element method for the electric field integral equation on a piecewise
plane (open or closed) Lipschitz surface. We use H(div)-conforming
discretisations with Raviart-Thomas elements on a sequence of quasi-uniform
meshes of triangles and/or parallelograms. Assuming the regularity of the
solution to the electric field integral equation in terms of Sobolev spaces of
tangential vector fields, we prove an a priori error estimate of the method in
the energy norm. This estimate proves the expected rate of convergence with
respect to the mesh parameter h and the polynomial degree p
Natural hp-BEM for the electric field integral equation with singular solutions
We apply the hp-version of the boundary element method (BEM) for the
numerical solution of the electric field integral equation (EFIE) on a
Lipschitz polyhedral surface G. The underlying meshes are supposed to be
quasi-uniform triangulations of G, and the approximations are based on either
Raviart-Thomas or Brezzi-Douglas-Marini families of surface elements.
Non-smoothness of G leads to singularities in the solution of the EFIE,
severely affecting convergence rates of the BEM. However, the singular
behaviour of the solution can be explicitly specified using a finite set of
power functions (vertex-, edge-, and vertex-edge singularities). In this paper
we use this fact to perform an a priori error analysis of the hp-BEM on
quasi-uniform meshes. We prove precise error estimates in terms of the
polynomial degree p, the mesh size h, and the singularity exponents.Comment: 17 page
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
hp-version time domain boundary elements for the wave equation on quasi-uniform meshes
Solutions to the wave equation in the exterior of a polyhedral domain or a
screen in exhibit singular behavior from the edges and corners.
We present quasi-optimal -explicit estimates for the approximation of the
Dirichlet and Neumann traces of these solutions for uniform time steps and
(globally) quasi-uniform meshes on the boundary. The results are applied to an
-version of the time domain boundary element method. Numerical examples
confirm the theoretical results for the Dirichlet problem both for screens and
polyhedral domains.Comment: 41 pages, 11 figure
Generation of curved high-order meshes with optimal quality and geometric accuracy
We present a novel methodology to generate curved high-order meshes featuring optimal mesh quality and geometric accuracy. The proposed technique combines a distortion measure and a geometric L2-disparity measure into a single objective function. While the element distortion term takes into account the mesh quality, the L2-disparity term takes into account the geometric error introduced by the mesh approximation to the target geometry. The proposed technique has several advantages. First, we are not restricted to interpolative meshes and therefore, the resulting mesh approximates the target domain in a non-interpolative way, further increasing the geometric accuracy. Second, we are able to generate a series of meshes that converge to the actual geometry with expected rate while obtaining high-quality elements. Third, we show that the proposed technique is robust enough to handle real-case geometries that contain gaps between adjacent entities.Peer ReviewedPostprint (published version
Generation of Curved High-order Meshes with Optimal Quality and Geometric Accuracy
We present a novel methodology to generate curved high-order meshes featuring optimal mesh quality and geometric accuracy. The proposed technique combines a distortion measure and a geometric Full-size image (<1 K)-disparity measure into a single objective function. While the element distortion term takes into account the mesh quality, the Full-size image (<1 K)-disparity term takes into account the geometric error introduced by the mesh approximation to the target geometry. The proposed technique has several advantages. First, we are not restricted to interpolative meshes and therefore, the resulting mesh approximates the target domain in a non-interpolative way, further increasing the geometric accuracy. Second, we are able to generate a series of meshes that converge to the actual geometry with expected rate while obtaining high-quality elements. Third, we show that the proposed technique is robust enough to handle real-case geometries that contain gaps between adjacent entities.This research was partially supported by the Spanish Ministerio de Economía y Competitividad under grand contract
CTM2014-55014-C3-3-R, and by the Government of Catalonia under grand contract 2014-SGR-1471. The work of the last author was supported by the European Commission through the Marie Sklodowska-Curie Actions
(HiPerMeGaFlows project).Peer ReviewedPostprint (published version
Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes
We introduce an -version symmetric interior penalty discontinuous
Galerkin finite element method (DGFEM) for the numerical approximation of the
biharmonic equation on general computational meshes consisting of
polygonal/polyhedral (polytopic) elements. In particular, the stability and
-version a-priori error bound are derived based on the specific choice of
the interior penalty parameters which allows for edges/faces degeneration.
Furthermore, by deriving a new inverse inequality for a special class {of}
polynomial functions (harmonic polynomials), the proposed DGFEM is proven to be
stable to incorporate very general polygonal/polyhedral elements with an
\emph{arbitrary} number of faces for polynomial basis with degree . The
key feature of the proposed method is that it employs elemental polynomial
bases of total degree , defined in the physical coordinate
system, without requiring the mapping from a given reference or canonical
frame. A series of numerical experiments are presented to demonstrate the
performance of the proposed DGFEM on general polygonal/polyhedral meshes
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