1,473 research outputs found
A hierarchical construction of LR meshes in 2D
We describe a construction of LR-spaces whose bases are composed of locally linearly independent B-splines which also form a partition of unity. The construction conforms to given refinement requirements associated to subdomains. In contrast to the original LR-paper (Dokken et al., 2013) and similarly to the hierarchical B-spline framework (Forsey and Bartels, 1988) the construction of the mesh is based on a priori choice of a sequence of nested tensor B-spline spaces
Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes
The focus on locally refined spline spaces has grown rapidly in recent years
due to the need in Isogeoemtric analysis (IgA) of spline spaces with local
adaptivity: a property not offered by the strict regular structure of tensor
product B-spline spaces. However, this flexibility sometimes results in
collections of B-splines spanning the space that are not linearly independent.
In this paper we address the minimal number of B-splines that can form a linear
dependence relation for Minimal Support B-splines (MS B-splines) and for
Locally Refinable B-splines (LR B-splines) on LR-meshes. We show that the
minimal number is six for MS B-splines, and eight for LR B-splines. The risk of
linear dependency is consequently significantly higher for MS B-splines than
for LR B-splines. Further results are established to help detecting collections
of B-splines that are linearly independent
Convergence of discrete duality finite volume schemes for the cardiac bidomain model
We prove convergence of discrete duality finite volume (DDFV) schemes on
distorted meshes for a class of simplified macroscopic bidomain models of the
electrical activity in the heart. Both time-implicit and linearised
time-implicit schemes are treated. A short description is given of the 3D DDFV
meshes and of some of the associated discrete calculus tools. Several numerical
tests are presented
Hierarchical Surface Prediction for 3D Object Reconstruction
Recently, Convolutional Neural Networks have shown promising results for 3D
geometry prediction. They can make predictions from very little input data such
as a single color image. A major limitation of such approaches is that they
only predict a coarse resolution voxel grid, which does not capture the surface
of the objects well. We propose a general framework, called hierarchical
surface prediction (HSP), which facilitates prediction of high resolution voxel
grids. The main insight is that it is sufficient to predict high resolution
voxels around the predicted surfaces. The exterior and interior of the objects
can be represented with coarse resolution voxels. Our approach is not dependent
on a specific input type. We show results for geometry prediction from color
images, depth images and shape completion from partial voxel grids. Our
analysis shows that our high resolution predictions are more accurate than low
resolution predictions.Comment: 3DV 201
An adaptive space-time phase field formulation for dynamic fracture of brittle shells based on LR NURBS
We present an adaptive space-time phase field formulation for dynamic fracture of brittle shells. Their deformation is characterized by the Kirchhoff–Love thin shell theory using a curvilinear surface description. All kinematical objects are defined on the shell’s mid-plane. The evolution equation for the phase field is determined by the minimization of an energy functional based on Griffith’s theory of brittle fracture. Membrane and bending contributions to the fracture process are modeled separately and a thickness integration is established for the latter. The coupled system consists of two nonlinear fourth-order PDEs and all quantities are defined on an evolving two-dimensional manifold. Since the weak form requires C1-continuity, isogeometric shape functions are used. The mesh is adaptively refined based on the phase field using Locally Refinable (LR) NURBS. Time is discretized based on a generalized-α method using adaptive time-stepping, and the discretized coupled system is solved with a monolithic Newton–Raphson scheme. The interaction between surface deformation and crack evolution is demonstrated by several numerical examples showing dynamic crack propagation and branching
An isogeometric finite element formulation for phase transitions on deforming surfaces
This paper presents a general theory and isogeometric finite element
implementation for studying mass conserving phase transitions on deforming
surfaces. The mathematical problem is governed by two coupled fourth-order
nonlinear partial differential equations (PDEs) that live on an evolving
two-dimensional manifold. For the phase transitions, the PDE is the
Cahn-Hilliard equation for curved surfaces, which can be derived from surface
mass balance in the framework of irreversible thermodynamics. For the surface
deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation.
Both PDEs can be efficiently discretized using -continuous interpolations
without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured
spline spaces with pointwise -continuity are utilized for these
interpolations. The resulting finite element formulation is discretized in time
by the generalized- scheme with adaptive time-stepping, and it is fully
linearized within a monolithic Newton-Raphson approach. A curvilinear surface
parameterization is used throughout the formulation to admit general surface
shapes and deformations. The behavior of the coupled system is illustrated by
several numerical examples exhibiting phase transitions on deforming spheres,
tori and double-tori.Comment: fixed typos, extended literature review, added clarifying notes to
the text, added supplementary movie file
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