2,943 research outputs found
An adaptive octree finite element method for PDEs posed on surfaces
The paper develops a finite element method for partial differential equations
posed on hypersurfaces in , . The method uses traces of
bulk finite element functions on a surface embedded in a volumetric domain. The
bulk finite element space is defined on an octree grid which is locally refined
or coarsened depending on error indicators and estimated values of the surface
curvatures. The cartesian structure of the bulk mesh leads to easy and
efficient adaptation process, while the trace finite element method makes
fitting the mesh to the surface unnecessary. The number of degrees of freedom
involved in computations is consistent with the two-dimension nature of surface
PDEs. No parametrization of the surface is required; it can be given implicitly
by a level set function. In practice, a variant of the marching cubes method is
used to recover the surface with the second order accuracy. We prove the
optimal order of accuracy for the trace finite element method in and
surface norms for a problem with smooth solution and quasi-uniform mesh
refinement. Experiments with less regular problems demonstrate optimal
convergence with respect to the number of degrees of freedom, if grid
adaptation is based on an appropriate error indicator. The paper shows results
of numerical experiments for a variety of geometries and problems, including
advection-diffusion equations on surfaces. Analysis and numerical results of
the paper suggest that combination of cartesian adaptive meshes and the
unfitted (trace) finite elements provide simple, efficient, and reliable tool
for numerical treatment of PDEs posed on surfaces
Volume-Enclosing Surface Extraction
In this paper we present a new method, which allows for the construction of
triangular isosurfaces from three-dimensional data sets, such as 3D image data
and/or numerical simulation data that are based on regularly shaped, cubic
lattices. This novel volume-enclosing surface extraction technique, which has
been named VESTA, can produce up to six different results due to the nature of
the discretized 3D space under consideration. VESTA is neither template-based
nor it is necessarily required to operate on 2x2x2 voxel cell neighborhoods
only. The surface tiles are determined with a very fast and robust construction
technique while potential ambiguities are detected and resolved. Here, we
provide an in-depth comparison between VESTA and various versions of the
well-known and very popular Marching Cubes algorithm for the very first time.
In an application section, we demonstrate the extraction of VESTA isosurfaces
for various data sets ranging from computer tomographic scan data to simulation
data of relativistic hydrodynamic fireball expansions.Comment: 24 pages, 33 figures, 4 tables, final versio
Generating Surface Geometry in Higher Dimensions using Local Cell Tilers
In two dimensions contour elements surround two dimensional objects, in three dimensions surfaces surround three dimensional objects and in four dimensions hypersurfaces surround hyperobjects. These surfaces can be represented by a collection of connected simplices, hence, continuous n dimensional surfaces can be represented by a lattice of connected n-1 dimensional simplices. The lattice of connected simplices can be calculated over a set of adjacent n-dimensional cubes, via for example the Marching Cubes Algorithm. These algorithms are often named local cell tilers. We propose that the local-cell tiling method can be usefully-applied to four dimensions and potentially to N-dimensions. We present an algorithm for the generation of major cases (cases that are topologically invariant under standard geometrical transformations) and introduce the notion of a sub-case which simplifies their representations. Each sub-case can be easily subdivided into simplices for rendering and we describe a backtracking tetrahedronization algorithm for the four dimensional case. An implementation for surfaces from the fourth dimension is presented and we describe and discuss ambiguities inherent within this and related algorithms
SurfelMeshing: Online Surfel-Based Mesh Reconstruction
We address the problem of mesh reconstruction from live RGB-D video, assuming
a calibrated camera and poses provided externally (e.g., by a SLAM system). In
contrast to most existing approaches, we do not fuse depth measurements in a
volume but in a dense surfel cloud. We asynchronously (re)triangulate the
smoothed surfels to reconstruct a surface mesh. This novel approach enables to
maintain a dense surface representation of the scene during SLAM which can
quickly adapt to loop closures. This is possible by deforming the surfel cloud
and asynchronously remeshing the surface where necessary. The surfel-based
representation also naturally supports strongly varying scan resolution. In
particular, it reconstructs colors at the input camera's resolution. Moreover,
in contrast to many volumetric approaches, ours can reconstruct thin objects
since objects do not need to enclose a volume. We demonstrate our approach in a
number of experiments, showing that it produces reconstructions that are
competitive with the state-of-the-art, and we discuss its advantages and
limitations. The algorithm (excluding loop closure functionality) is available
as open source at https://github.com/puzzlepaint/surfelmeshing .Comment: Version accepted to IEEE Transactions on Pattern Analysis and Machine
Intelligenc
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