276 research outputs found
On the mesh nonsingularity of the moving mesh PDE method
The moving mesh PDE (MMPDE) method for variational mesh generation and
adaptation is studied theoretically at the discrete level, in particular the
nonsingularity of the obtained meshes. Meshing functionals are discretized
geometrically and the MMPDE is formulated as a modified gradient system of the
corresponding discrete functionals for the location of mesh vertices. It is
shown that if the meshing functional satisfies a coercivity condition, then the
mesh of the semi-discrete MMPDE is nonsingular for all time if it is
nonsingular initially. Moreover, the altitudes and volumes of its elements are
bounded below by positive numbers depending only on the number of elements, the
metric tensor, and the initial mesh. Furthermore, the value of the discrete
meshing functional is convergent as time increases, which can be used as a
stopping criterion in computation. Finally, the mesh trajectory has limiting
meshes which are critical points of the discrete functional. The convergence of
the mesh trajectory can be guaranteed when a stronger condition is placed on
the meshing functional. Two meshing functionals based on alignment and
equidistribution are known to satisfy the coercivity condition. The results
also hold for fully discrete systems of the MMPDE provided that the time step
is sufficiently small and a numerical scheme preserving the property of
monotonically decreasing energy is used for the temporal discretization of the
semi-discrete MMPDE. Numerical examples are presented.Comment: Revised and improved version of the WIAS preprin
Average Interpolation Under the Maximum Angle Condition
Interpolation error estimates needed in common finite element applications
using simplicial meshes typically impose restrictions on the both the
smoothness of the interpolated functions and the shape of the simplices. While
the simplest theory can be generalized to admit less smooth functions (e.g.,
functions in H^1(\Omega) rather than H^2(\Omega)) and more general shapes
(e.g., the maximum angle condition rather than the minimum angle condition),
existing theory does not allow these extensions to be performed simultaneously.
By localizing over a well-shaped auxiliary spatial partition, error estimates
are established under minimal function smoothness and mesh regularity. This
construction is especially important in two cases: L^p(\Omega) estimates for
data in W^{1,p}(\Omega) hold for meshes without any restrictions on simplex
shape, and W^{1,p}(\Omega) estimates for data in W^{2,p}(\Omega) hold under a
generalization of the maximum angle condition which previously required p>2 for
standard Lagrange interpolation
A comparative numerical study of meshing functionals for variational mesh adaptation
We present a comparative numerical study for three functionals used for
variational mesh adaptation. One of them is a generalisation of Winslow's
variable diffusion functional while the others are based on equidistribution
and alignment. These functionals are known to have nice theoretical properties
and work well for most mesh adaptation problems either as a stand-alone
variational method or combined within the moving mesh framework. Their
performance is investigated numerically in terms of equidistribution and
alignment mesh quality measures. Numerical results in 2D and 3D are presented.Comment: Additional example (H1), journal referenc
An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems
Heterogeneous anisotropic diffusion problems arise in the various areas of
science and engineering including plasma physics, petroleum engineering, and
image processing. Standard numerical methods can produce spurious oscillations
when they are used to solve those problems. A common approach to avoid this
difficulty is to design a proper numerical scheme and/or a proper mesh so that
the numerical solution validates the discrete counterpart (DMP) of the maximum
principle satisfied by the continuous solution. A well known mesh condition for
the DMP satisfaction by the linear finite element solution of isotropic
diffusion problems is the non-obtuse angle condition that requires the dihedral
angles of mesh elements to be non-obtuse. In this paper, a generalization of
the condition, the so-called anisotropic non-obtuse angle condition, is
developed for the finite element solution of heterogeneous anisotropic
diffusion problems. The new condition is essentially the same as the existing
one except that the dihedral angles are now measured in a metric depending on
the diffusion matrix of the underlying problem. Several variants of the new
condition are obtained. Based on one of them, two metric tensors for use in
anisotropic mesh generation are developed to account for DMP satisfaction and
the combination of DMP satisfaction and mesh adaptivity. Numerical examples are
given to demonstrate the features of the linear finite element method for
anisotropic meshes generated with the metric tensors.Comment: 34 page
Analysis and Generation of Quality Polytopal Meshes with Applications to the Virtual Element Method
This thesis explores the concept of the quality of a mesh, the latter being intended as the discretization of a two- or three- dimensional domain.
The topic is interdisciplinary in nature, as meshes are massively used in several fields from both the geometry processing and the numerical analysis communities.
The goal is to produce a mesh with good geometrical properties and the lowest possible number of elements, able to produce results in a target range of accuracy.
In other words, a good quality mesh that is also cheap to handle, overcoming the typical trade-off between quality and computational cost.
To reach this goal, we first need to answer the question:
''How, and how much, does the accuracy of a numerical simulation or a scientific computation (e.g., rendering, printing, modeling operations) depend on the particular mesh adopted to model the problem? And which geometrical features of the mesh most influence the result?''
We present a comparative study of the different mesh types, mesh generation techniques, and mesh quality measures currently available in the literature related to both engineering and computer graphics applications.
This analysis leads to the precise definition of the notion of quality for a mesh, in the particular context of numerical simulations of partial differential equations with the virtual element method, and the consequent construction of criteria to determine and optimize the quality of a given mesh.
Our main contribution consists in a new mesh quality indicator for polytopal meshes, able to predict the performance of the virtual element method over a particular mesh before running the simulation.
Strictly related to this, we also define a quality agglomeration algorithm that optimizes the quality of a mesh by wisely agglomerating groups of neighboring elements.
The accuracy and the reliability of both tools are thoroughly verified in a series of tests in different scenarios
Optimal Voronoi Tessellations with Hessian-based Anisotropy
International audienceThis paper presents a variational method to generate cell complexes with local anisotropy conforming to the Hessian of any given convex function and for any given local mesh density. Our formulation builds upon approximation theory to offer an anisotropic extension of Centroidal Voronoi Tessellations which can be seen as a dual form of Optimal Delaunay Triangulation. We thus refer to the resulting anisotropic polytopal meshes as Optimal Voronoi Tessel-lations. Our approach sharply contrasts with previous anisotropic versions of Voronoi diagrams as it employs first-type Bregman diagrams , a generalization of power diagrams where sites are augmented with not only a scalar-valued weight but also a vector-valued shift. As such, our OVT meshes contain only convex cells with straight edges, and admit an embedded dual triangulation that is combinatorially-regular. We show the effectiveness of our technique using off-the-shelf computational geometry libraries
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