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- ItemOn Tetrahedralisations Containing Knotted and Linked Line Segments(Amsterdam [u.a.] : Elsevier, 2017) Si, Hang; Ren, Yuxue; Lei, Na; Gu, XianfengThis paper considers a set of twisted line segments in 3d such that they form a knot (a closed curve) or a link of two closed curves. Such line segments appear on the boundary of a family of 3d indecomposable polyhedra (like the Schönhardt polyhedron) whose interior cannot be tetrahedralised without additional vertices added. On the other hand, a 3d (non-convex) polyhedron whose boundary contains such line segments may still be decomposable as long as the twist is not too large. It is therefore interesting to consider the question: when there exists a tetrahedralisation contains a given set of knotted or linked line segments? In this paper, we studied a simplified question with the assumption that all vertices of the line segments are in convex position. It is straightforward to show that no tetrahedralisation of 6 vertices (the three-line-segments case) can contain a trefoil knot. Things become interesting when the number of line segments increases. Since it is necessary to create new interior edges to form a tetrahedralisation. We provided a detailed analysis for the case of a set of 4 line segments. This leads to a crucial condition on the orientation of pairs of new interior edges which determines whether this set is decomposable or not. We then prove a new theorem about the decomposability for a set of n (n ≥ 3) knotted or linked line segments. This theorem implies that the family of polyhedra generalised from the Schonhardt polyhedron by Rambau [1] are all indecomposable.
- ItemScattering matrices and Dirichlet-to-Neumann maps(Amsterdam [u.a.] : Elsevier, 2017) Behrndt, Jussi; Malamud, Mark M.; Neidhardt, HagenA general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh–Weyl m-function is proved. This result is applied to scattering problems for different self-adjoint realizations of Schrödinger operators on unbounded domains, Schrödinger operators with singular potentials supported on hypersurfaces, and orthogonal couplings of Schrödinger operators. In these applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps.
- ItemOn complex dynamics in a Purkinje and a ventricular cardiac cell model(Amsterdam [u.a.] : Elsevier, 2020) Erhardt, André H.; Solem, SusanneCardiac muscle cells can exhibit complex patterns including irregular behaviour such as chaos or (chaotic) early afterdepolarisations (EADs), which can lead to sudden cardiac death. Suitable mathematical models and their analysis help to predict the occurrence of such phenomena and to decode their mechanisms. The focus of this paper is the investigation of dynamics of cardiac muscle cells described by systems of ordinary differential equations. This is generically performed by studying a Purkinje cell model and a modified ventricular cell model. We find chaotic dynamics with respect to the leak current in the Purkinje cell model, and EADs and chaos with respect to a reduced fast potassium current and an enhanced calcium current in the ventricular cell model — features that have been experimentally observed and are known to exist in some models, but are new to the models under present consideration. We also investigate the related monodomain models of both systems to study synchronisation and the behaviour of the cells on macro-scale in connection with the discovered features. The models show qualitatively the same behaviour to what has been experimentally observed. However, for certain parameter settings the dynamics occur within a non-physiological range.
- ItemPreface(Amsterdam [u.a.] : Elsevier, 2016) Canann, Scott; Owen, Steven; Si, Hang[No abstract available]
- ItemTetrahedral Mesh Improvement Using Moving Mesh Smoothing and Lazy Searching Flips(Amsterdam [u.a.] : Elsevier, 2016) Dassi, Franco; Kamenski, Lennard; Si, HangWe combine the new moving mesh smoothing, based on the integration of an ordinary differential equation coming from a given functional, with the new lazy flip technique, a reversible edge removal algorithm for local mesh quality improvement. These strategies already provide good mesh improvement on themselves, but their combination achieves astonishing results not reported so far. Provided numerical comparison with some publicly available mesh improving software show that we can obtain final tetrahedral meshes with dihedral angles between 40° and 123°.
- ItemOn Indecomposable Polyhedra and the Number of Steiner Points(Amsterdam [u.a.] : Elsevier, 2015) Goerigk, Nadja; Si, HangThe existence of 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. However, the geometry and combinatorial structure of such polyhedra are much less studied. In this article, we investigate the structure of some well-known examples, the so-called Schönhardt polyhedron [10] and the Bagemihl's generalization of it [1], which will be called Bagemihl's polyhedra. We provide a construction of an additional point, so-called Steiner point, which can be used to decompose the Schönhardt and the Bagemihl's 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's and Bagemihl's polyhedra, but they may need more than one Steiner point to be decomposed. Given such a polyhedron with n ≥ 6 vertices, we show that it can be decomposed by adding at most interior Steiner points. We also show that this number is optimal in theworst case.
- ItemExample dataset for the hMRI toolbox(Amsterdam [u.a.] : Elsevier, 2019) Callaghan, Martina F.; Lutti, Antoine; Ashburner, John; Balteau, Evelyne; Corbin, Nadège; Draganski, Bogdan; Helms, Gunther; Kherif, Ferath; Leutritz, Tobias; Mohammadi, Siawoosh; Phillips, Christophe; Reimer, Enrico; Ruthotto, Lars; Seif, Maryam; Tabelow, Karsten; Ziegler, Gabriel; Weiskopf, NikolausThe hMRI toolbox is an open-source toolbox for the calculation of quantitative MRI parameter maps from a series of weighted imaging data, and optionally additional calibration data. The multi-parameter mapping (MPM) protocol, incorporating calibration data to correct for spatial variation in the scanner's transmit and receive fields, is the most complete protocol that can be handled by the toolbox. Here we present a dataset acquired with such a full MPM protocol, which is made freely available to be used as a tutorial by following instructions provided on the associated toolbox wiki pages, which can be found at http://hMRI.info, and following the theory described in: hMRI – A toolbox for quantitative MRI in neuroscience and clinical research [1].
- ItemAnisotropic Finite Element Mesh Adaptation via Higher Dimensional Embedding(Amsterdam [u.a.] : Elsevier, 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], [2], [3], [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 f : Ω ⊂ d → ) 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 f 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 f. 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.
- ItemGeneralized Regular Quadrilateral Mesh Generation based on Surface Foliation(Amsterdam [u.a.] : Elsevier, 2017) Lei, Na; Zheng, Xiaopeng; Si, Hang; Luo, Zhongxuan; Gu, XianfengThis work introduces a novel algorithm for quad-mesh generation based on surface foliation theory. The algorithm is based on the equivalence among colorable quad-meshes, measure foliations and holomorphic differentials. The holomorphic differentials can be obtained by graph-valued harmonic maps. The algorithm has several merits: it can be applied for surfaces with general topologies; the resulting quad-meshes have global tensor product structure and the least number of singularities; the algorithmic pipeline is fully automatic. The experimental results demonstrate the efficiency and efficacy of the proposed method.
- ItemOn Tetrahedralisations of Reduced Chazelle Polyhedra with Interior Steiner Points(Amsterdam [u.a.] : Elsevier, 2016) Si, Hang; Goerigk, NadjaThe non-convex polyhedron constructed by Chazelle, known as the Chazelle polyhedron [4], establishes a quadratic lower bound on the minimum number of convex pieces for the 3d polyhedron partitioning problem. In this paper, we study the problem of tetrahedralising the Chazelle polyhedron without modifying its exterior boundary. It is motivated by a crucial step in tetrahedral mesh generation in which a set of arbitrary constraints (edges or faces) need to be entirely preserved. The goal of this study is to gain more knowledge about the family of 3d indecomposable polyhedra which needs additional points, so-called Steiner points, to be tetrahedralised. The requirement of only using interior Steiner points for the Chazelle polyhedron is extremely challenging. We first “cut off” the volume of the Chazelle polyhedron by removing the regions that are tetrahedralisable. This leads to a 3d non-convex polyhedron whose vertices are all in the two slightly shifted saddle surfaces which are used to construct the Chazelle polyhedron. We call it the reduced Chazelle polyhedron. It is an indecomposable polyhedron. We then give a set of (N + 1)2 interior Steiner points that ensures the existence of a tetrahedralisation of the reduced Chazelle polyhedron with 4(N + 1) vertices. The proof is done by transforming a 3d tetrahedralisation problem into a 2d edge flip problem. In particular, we design an edge splitting and flipping algorithm and prove that it gives to a tetrahedralisation of the reduced Chazelle polyhedron.