791 research outputs found
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
Anisotropic Surface Remeshing without Obtuse Angles
We present a novel anisotropic surface remeshing method that can efficiently eliminate obtuse angles. Unlike previous work that can only suppress obtuse angles with expensive resampling and Lloyd-type iterations, our method relies on a simple yet efficient connectivity and geometry refinement, which can not only remove all the obtuse angles, but also preserves the original mesh connectivity as much as possible. Our method can be directly used as a post-processing step for anisotropic meshes generated from existing algorithms to improve mesh quality. We evaluate our method by testing on a variety of meshes with different geometry and topology, and comparing with representative prior work. The results demonstrate the effectiveness and efficiency of our approach
Automatic higher order mesh generation and movement utilizing spring-field and vector-adding
In this research, an automatic, complex geometry applicable, unstructured curved 2D mesh generation method is developed. Based on this 2D method, an extension to 3D geometries is developed. The methodology of this mesh generation is to provide near-body and off-body meshes by an advancing-layer method to avoid mesh quality problems in transition areas, while using spring models to complement the advancing-layers by smoothing the mesh, especially in medial axis regions. After generating the linear mesh, a higher-order finite-element ready, curved mesh is obtained by deforming the linear mesh through “pipes” using a simple Vector-Adding approach. Different from most curved viscous mesh generation approaches, this method eschews a linear or nonlinear elasticity analogy while still providing positive-Jacobian mesh elements. In addition, the transformation from linear to curved meshes can be achieved by only deforming the necessary edges of the elements near domain boundaries while retaining the remaining non-curved edges to the benefit of numerical solvers. The CFD results of 30P30N airfoil on the curved mesh are given and are in good agreement with the experimental data. Further, a new method for moving boundary problems with viscous mesh layers is also presented. This method is an extension of the mesh generation approach above. It can be used not only for high-order mesh moving but also for high-order mesh generation. Also, based on the curved mesh deformation strategy, the method can handle high-order mesh movement or regeneration with minimal extra computational time compared with linear mesh movement. Several 2D boundary motion cases are tested including boundary translations, boundary rotations, and boundary morphing. The CFD results on the curved mesh after rotation are given and are in good agreement with the experimental data. This technique is also tested through the optimization of an airfoil to achieve a maximized lift and drag ratio. The results demonstrate that this approach can handle large boundary movements while preserving a good mesh quality
Feature Lines for Illustrating Medical Surface Models: Mathematical Background and Survey
This paper provides a tutorial and survey for a specific kind of illustrative
visualization technique: feature lines. We examine different feature line
methods. For this, we provide the differential geometry behind these concepts
and adapt this mathematical field to the discrete differential geometry. All
discrete differential geometry terms are explained for triangulated surface
meshes. These utilities serve as basis for the feature line methods. We provide
the reader with all knowledge to re-implement every feature line method.
Furthermore, we summarize the methods and suggest a guideline for which kind of
surface which feature line algorithm is best suited. Our work is motivated by,
but not restricted to, medical and biological surface models.Comment: 33 page
Anisotropic Mesh Adaptation for the Finite Element Solution of Anisotropic Diffusion Problems
Anisotropic diffusion problems arise in many fields of science and engineering and are modeled by partial differential equations (PDEs) or represented in variational formulations. Standard numerical schemes can produce spurious oscillations when they are used to solve those problems. A common approach is to design a proper numerical scheme or a proper mesh such that the numerical solution satisfies discrete maximum principle (DMP). For problems in variational formulations, numerous research has been done on isotropic mesh adaptation but little work has been done for anisotropic mesh adaptation. In this dissertation, anisotropic mesh adaptation for the finite element solution of anisotropic diffusion problems is investigated. A brief introduction for the related topics is provided. The anisotropic mesh adaptation based on DMP satisfaction is then discussed. An anisotropic non-obtuse angle condition is developed which guarantees that the linear finite element approximation of the steady state problem satisfies DMP. A metric tensor is derived for use in mesh generation based on the anisotropic non-obtuse angle condition. Then DMP satisfaction and error based mesh adaptation are combined together for the first time. For problems in variational formulations, two metric tensors for anisotropic mesh adaptation and one for isotropic mesh adaptation are developed. For anisotropic mesh adaptation, one metric tensor (based on Hessian recovery) is semi-a posterior and the other (based on hierarchical basis error estimator) is completely a posterior. The metric tensor for isotropic mesh adaptation is completely a posterior. All the metric tensors incorporate structural information of the underlying problem into their design and generate meshes that adapt to changes in the structure. The application of anisotropic diffusion filter in image processing is briefly discussed. Numerical examples demonstrate that anisotropic mesh adaptation can significantly improve computational efficiency while still providing good quality result. More research is needed to investigate DMP satisfaction for parabolic problems
Stability of explicit one-step methods for P1-finite element approximation of linear diffusion equations on anisotropic meshes
We study the stability of explicit one-step integration schemes for the
linear finite element approximation of linear parabolic equations. The derived
bound on the largest permissible time step is tight for any mesh and any
diffusion matrix within a factor of , where is the spatial
dimension. Both full mass matrix and mass lumping are considered. The bound
reveals that the stability condition is affected by two factors. The first one
depends on the number of mesh elements and corresponds to the classic bound for
the Laplace operator on a uniform mesh. The other factor reflects the effects
of the interplay of the mesh geometry and the diffusion matrix. It is shown
that it is not the mesh geometry itself but the mesh geometry in relation to
the diffusion matrix that is crucial to the stability of explicit methods. When
the mesh is uniform in the metric specified by the inverse of the diffusion
matrix, the stability condition is comparable to the situation with the Laplace
operator on a uniform mesh. Numerical results are presented to verify the
theoretical findings.Comment: Revised WIAS Preprin
Optimal Point Placement for Mesh Smoothing
We study the problem of moving a vertex in an unstructured mesh of
triangular, quadrilateral, or tetrahedral elements to optimize the shapes of
adjacent elements. We show that many such problems can be solved in linear time
using generalized linear programming. We also give efficient algorithms for
some mesh smoothing problems that do not fit into the generalized linear
programming paradigm.Comment: 12 pages, 3 figures. A preliminary version of this paper was
presented at the 8th ACM/SIAM Symp. on Discrete Algorithms (SODA '97). This
is the final version, and will appear in a special issue of J. Algorithms for
papers from SODA '9
Adaptive Mesh Refinement for Electromagnetic Simulation
We consider problems related to initial meshing and adaptive mesh refinement
for the electromagnetic simulation of various structures. The quality of the
initial mesh and the performance of the adaptive refinement are of great
importance for the finite element solution of the Maxwell equations, since they
directly affect the accuracy and the computational time. In this paper, we
describe the complete meshing workflow, which allows the simulation of
arbitrary structures. Test simulations confirm that the presented approach
allows to reach the quality of the industrial simulation software
- …