15 research outputs found

    Flipping Cubical Meshes

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    We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.Comment: 20 pages, 24 figures. Expanded journal version of paper from 10th International Meshing Roundtable. This version removes some unwanted paragraph breaks from the previous version; the text is unchange

    Finding Hexahedrizations for Small Quadrangulations of the Sphere

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    This paper tackles the challenging problem of constrained hexahedral meshing. An algorithm is introduced to build combinatorial hexahedral meshes whose boundary facets exactly match a given quadrangulation of the topological sphere. This algorithm is the first practical solution to the problem. It is able to compute small hexahedral meshes of quadrangulations for which the previously known best solutions could only be built by hand or contained thousands of hexahedra. These challenging quadrangulations include the boundaries of transition templates that are critical for the success of general hexahedral meshing algorithms. The algorithm proposed in this paper is dedicated to building combinatorial hexahedral meshes of small quadrangulations and ignores the geometrical problem. The key idea of the method is to exploit the equivalence between quad flips in the boundary and the insertion of hexahedra glued to this boundary. The tree of all sequences of flipping operations is explored, searching for a path that transforms the input quadrangulation Q into a new quadrangulation for which a hexahedral mesh is known. When a small hexahedral mesh exists, a sequence transforming Q into the boundary of a cube is found; otherwise, a set of pre-computed hexahedral meshes is used. A novel approach to deal with the large number of problem symmetries is proposed. Combined with an efficient backtracking search, it allows small shellable hexahedral meshes to be found for all even quadrangulations with up to 20 quadrangles. All 54,943 such quadrangulations were meshed using no more than 72 hexahedra. This algorithm is also used to find a construction to fill arbitrary domains, thereby proving that any ball-shaped domain bounded by n quadrangles can be meshed with no more than 78 n hexahedra. This very significantly lowers the previous upper bound of 5396 n.Comment: Accepted for SIGGRAPH 201

    A case study in hexahedral mesh generation: Simulation of the human mandible

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    We provide a case study for the generation of pure hexahedral meshes for the numerical simulation of physiological stress scenarios of the human mandible. Due to its complex and very detailed free-form geometry, the mandible model is very demanding. This test case is used as a running example to demonstrate the applicability of a combinatorial approach for the generation of hexahedral meshes by means of successive dual cycle eliminations, which has been proposed by the second author in previous work. We report on the progress and recent advances of the cycle elimination scheme. The given input data, a surface triangulation obtained from computed tomography data, requires a substantial mesh reduction and a suitable conversion into a quadrilateral surface mesh as a first step, for which we use mesh clustering and b-matching techniques. Several strategies for improved cycle elimination orders are proposed. They lead to a significant reduction in the mesh size and a better structural quality. Based on the resulting combinatorial meshes, gradient-based optimized smoothing with the condition number of the Jacobian matrix as objective together with mesh untangling techniques yielded embeddings of a satisfactory quality. To test our hexahedral meshes for the mandible model within an FEM simulation we used the scenario of a bite on a ‘hard nut.’ Our simulation results are in good agreement with observations from biomechanical experiments

    Surface cubications mod flips

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    Let Σ\Sigma be a compact surface. We prove that the set of surface cubications modulo flips, up to isotopy, is in one-to-one correspondence with Z/2Z⊕H1(Σ,Z/2Z)\Z/2\Z\oplus H_1(\Sigma,\Z/2\Z).Comment: revised version, 18

    Scheduling precedence-constrained jobs with stochastic processing times on parallel machines

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    We consider parallel machine scheduling problems where the jobs are subject to precedence constraints, and the processing times of jobs are governed by independent probability distributions. The objective is to minimize the weighted sum of job completion times &;j wjC_j in expectation, where wj&; 0. Building upon an LP-relaxation by Möhring, Schulz, and Uetz (J.ACM 46 (1999), pp.924-942) and an idle time charging scheme by Chekuri, Motwani, Natarajan, and Stein (SIAM J. Comp., to appear) we derive the first approximation algorithms for this model

    Scheduling under Uncertainty: Optimizing Against a Randomizing Adversary

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    Deterministic models for project scheduling and control suffer from the fact that they assume complete information and neglect random influences that occur during project execution. A typical consequence is the underestimation of the expected project duration and cost frequently observed in practice. To cope with these phenomena, we consider scheduling models in which processing times are random but precedence and resource constraints are fixed. Scheduling is done by policies which consist of an online process of decisions that are based on the observed past and the a priori knowledge of the distribution of processing times. We give an informal survey on different classes of policies and show that suitable combinatorial properties of such policies give insight into optimality, computational methods, and their approximation behavior. In particular, we present recent constant-factor approximation algorithms for simple policies in machine scheduling that are based on a suitable polyhedral relaxation of the performance space of policies
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