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- ItemAnisotropic finite element mesh adaptation via higher dimensional embedding(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dassi, Franco; Si, Hang; Perotto, Simona; Streckenbach, TimoIn this paper we provide a novel anisotropic mesh adaptation technique for adaptive finite element analysis. It is based on the concept of higher dimensional embedding, which was exploited in [1-4] to obtain an anisotropic curvature adapted mesh that fits a complex surface in ℝ3. In the context of adaptive finite element simulation, the solution (which is an unknown function ƒ: Ω ⊂; ℝd → ℝ) is sought by iteratively modifying a finite element mesh according to a mesh sizing field described via a (discrete) metric tensor field that is typically obtained through an error estimator. We proposed to use a higher dimensional embedding, Φf(x) := (x1, …, xd, s f (x1, …, xd), s ∇ f (x1, …, xd))t, instead of the mesh sizing field for the mesh adaption. This embedding contains both informations of the function ƒ itself and its gradient. An isotropic mesh in this embedded space will correspond to an anisotropic mesh in the actual space, where the mesh elements are stretched and aligned according to the features of the function ƒ. To better capture the anisotropy and gradation of the mesh, it is necessary to balance the contribution of the components in this embedding. We have properly adjusted Φf(x) for adaptive finite element analysis. To better understand and validate the proposed mesh adaptation strategy, we first provide a series of experimental tests for piecewise linear interpolation of known functions. We then applied this approach in an adaptive finite element solution of partial differential equations. Both tests are performed on two-dimensional domains in which adaptive triangular meshes are generated. We compared these results with the ones obtained by the software BAMG - a metric-based adaptive mesh generator. The errors measured in the L2 norm are comparable. Moreover, our meshes captured the anisotropy more accurately than the meshes of BAMG.
- ItemOn indecomposable polyhedra and the number of interior Steiner points(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Goerigk, Nadja; Si, HangThe existence of 3d indecomposable polyhedra, that is, the interior of every such polyhedron cannot be decomposed into a set of tetrahedra whose vertices are all of the given polyhedron, is well-known. While the geometry and combinatorial structure of such polyhedra are much less studied. In this article, we first investigate the geometry of some wellknown examples, the so-called Schönhardt polyhedron [Schönhardt, 1928] and the Bagemihl's generalization of it [Bagemihl, 1948], which will be called Bagemihl polyhedra. We provide a construction of an interior point, so-called Steiner point, which can be used to tetrahedralize the Schönhardt and the Bagemihl polyhedra. We then provide a construction of a larger class of three-dimensional indecomposable polyhedra which often appear in grid generation problems. We show that such polyhedra have the same combinatorial structure as the Schönhardt and Bagemihl polyhedra, but they may need more than one interior Steiner point to be tetrahedralized. Given such a polyhedron with n ≥ 6 vertices, we show that it can be tetrahedralized by adding at most ... interior Steiner points. We also show that this number is optimal in the worst case.
- ItemTetGen: A quality tetrahedral mesh generator and a 3D Delaunay triangulator (Version 1.5 — User’s Manual)(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Si, HangTetGen is a software for tetrahedral mesh generation. Its goal is to generate good quality tetrahedral meshes suitable for numerical methods and scientific computing. It can be used as either a standalone program or a library component integrated in other software. The purpose of this document is to give a brief explanation of the kind of tetrahedralizations and meshing problems handled by TetGen and to give a fairly detailed documentation about the usage of the program. Readers will learn how to create tetrahedral meshes using input files from the command line. Furthermore, the programming interface for calling TetGen from other programs is explained.
- Item3D boundary recovery by constrained Delaunay tetrahedralization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Si, Hang; Gärtner, KlausThree-dimensional boundary recovery is a fundamental problem in mesh generation. In this paper, we propose a practical algorithm for solving this problem. Our algorithm is based on the construction of a it constrained Delaunay tetrahedralization (CDT) for a set of constraints (segments and facets). The algorithm adds additional points (so-called Steiner points) on segments only. The Steiner points are chosen in such a way that the resulting subsegments are Delaunay and their lengths are not unnecessarily short. It is theoretically guaranteed that the facets can be recovered without using Steiner points. The complexity of this algorithm is analyzed. The proposed algorithm has been implemented. Its performance is reported through various application examples
- ItemA curvature-adapted anisotropic surface remeshing method(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Dassi, Franco; Si, HangWe present a new method for remeshing surfaces that respect the intrinsic anisotropy of the surfaces. In particular, we use the normal informations of the surfaces, and embed the surfaces into a higher dimensional space (here we use 6d). This allow us to form an isotropic mesh optimization problem in this embedded space. Starting from an initial mesh of a surface, we optimize the mesh by improving the mesh quality measured in the embedded space. The mesh is optimized by combining common local modifications operations, i.e., edge flip, edge contraction, vertex smoothing, and vertex insertion. All perations are applied directly on the 3d surface mesh. This method results a curvature-adapted mesh of the surface. This method can be easily adapted to mesh multi-patches surfaces, i.e., containing corner singularities and sharp features. The reliability and robustness of the proposed re-meshing technique is provided by a large number of examples including both implicit surfaces and CAD models.
- ItemTetGen, towards a quality tetrahedral mesh generator(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Si, HangTetGen is a C++ program for generating quality tetrahedral meshes aimed to support numerical methods and scientific computing. It is also a research project for studying the underlying mathematical problems and evaluating algorithms. This paper presents the essential meshing components developed in TetGen for robust and efficient software implementation. And it highlights the state-of-the-art algorithms and technologies currently implemented and developed in TetGen for automatic quality tetrahedral mesh generation.
- ItemTetrahedral mesh improvement using moving mesh smoothing and lazy searching flips(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dassi, Franco; Kamenski, Lennard; Si, HangIn this paper we combine two new smoothing and flipping techniques. The moving mesh smoothing is based on the integration of an ordinary differential coming from a given functional. The lazy flip technique is a reversible edge removal algorithm to automatically search flips for local quality improvement. On itself, these strategies already provide good mesh improvement, but their combination achieves astonishing results which have not been reported so far. Provided numerical examples show that we can obtain final tetrahedral meshes with dihedral angles between 40° and 123°. We compare the new method with other publicly available mesh improving codes.
- ItemA novel surface remeshing scheme via higher dimensional embedding and radial basis functions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dassi, Franco; Farrell, Patricio; Si, HangMany applications heavily rely on piecewise triangular meshes to describe complex surface geometries. High-quality meshes significantly improve numerical simulations. In practice, however, one often has to deal with several challenges. Some regions in the initial mesh may be overrefined, others too coarse. Additionally, the triangles may be too thin or not properly oriented. We present a novel mesh adaptation procedure which greatly improves the problematic input mesh and overcomes all of these drawbacks. By coupling surface reconstruction via radial basis functions with the higher dimensional embedding surface remeshing technique, we can automatically generate anisotropic meshes. Moreover, we are not only able to fill or coarsen certain mesh regions but also align the triangles according to the curvature of the reconstructed surface. This yields an acceptable trade-off between computational complexity and accuracy.
- ItemMesh smoothing: An MMPDE approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Huang, Weizhang; Kamenski, Lennard; Si, HangWe study a mesh smoothing algorithm based on the moving mesh PDE (MMPDE) method. For the MMPDE itself, we employ a simple and efficient direct geometric discretization of the underlying meshing functional on simplicial meshes. The nodal mesh velocities can be expressed in a simple, analytical matrix form, which makes the implementation of the method relatively easy and simple. Numerical examples are provided.
- ItemTetrahedral mesh improvement using moving mesh smoothing, lazy searching flips, and RBF surface reconstruction(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Dassi, Franco; Kamenski, Lennard; Farrell, Patricio; Si, HangGiven a tetrahedral mesh and objective functionals measuring the mesh quality which take into account the shape, size, and orientation of the mesh elements, our aim is to improve the mesh quality as much as possible. In this paper, we combine the moving mesh smoothing, based on the integration of an ordinary differential equation coming from a given functional, with the lazy flip technique, a reversible edge removal algorithm to modify the mesh connectivity. Moreover, we utilize radial basis function (RBF) surface reconstruction to improve tetrahedral meshes with curved boundary surfaces. Numerical tests show that the combination of these techniques into a mesh improvement framework achieves results which are comparable and even better than the previously reported ones.