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We introduce a fully automatic algorithm which optimizes the high-level structure of a given quadrilateral mesh to achieve a coarser quadrangular base complex. Such a topological optimization is highly desirable, since stateof-the-art quadrangulation techniques lead to meshes which have an appropriate singularity distribution and an anisotropic element alignment, but usually they are still far away from the high-level structure which is typical for carefully designed meshes manually created by specialists and used e.g. in animation or simulation. In this paper we show that the quality of the high-level structure is negatively affected by helical configurations within the quadrilateral mesh. Consequently we present an algorithm which detects helices and is able to remove most of them by applying a novel grid preserving simplification operator (GP-operator) which is guaranteed to maintain an all-quadrilateral mesh. Additionally it preserves the given singularity distribution and in particular does not introduce new singularities. For each helix we construct a directed graph in which cycles through the start vertex encode operations to remove the corresponding helix. Therefore a simple graph search algorithm can be performed iteratively to remove as many helices as possible and thus improve the high-level structure in a greedy fashion. We demonstrate the usefulness of our automatic structure optimization technique by showing several examples with varying complexity. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Hierarchy and geometric transformations, Curve, surface, solid, and object representations

Summary. Mesh optimization is critical in numerical simulations involving complex or evolving geometry. Because of the geometric constraints, such as preservation of sharp features and conservation of volume, optimizing a surface mesh poses signi cant challenges, especially when a CAD model is unavailable. In this paper, we introduce a formulation of volume conservation in a local sense for surface meshes under smoothing or other types of mesh motion, and propose a simple and e-cient technique to solve it. We also present a simple and robust feature detection technique to enhance the e ectiveness of local volume conservation and mesh optimization. We present the theoretical foundation of our techniques and experimental study to demonstrate their e ectiveness. Key words: mesh optimization; mesh smoothing; surface meshes; volume conservation; feature detection 1

Hexahedral finite element meshes have historically offered some mathematical benefit over tetrahedral finite element meshes in terms of reduced error, smaller element counts, and improved reliability, especially with respect to finite element analyses within highly elastic, and plastic, structural domains. However, hexahedral finite element mesh generation continues to be extremely difficult to perform and automate, with hexahedral mesh generation taking several orders of magnitude longer in time to complete over current tetrahedral mesh generators. In this paper, I focus on delineating the known constraints associated with hexahedral meshes, formulating these constraints utilizing the dual of the hexahedral mesh. Utilizing these constraints, it will be possible to highlight areas where better knowledge and incorporation of these constraints can augment existing algorithms, predict failure of specific methods, and suggest some additional methods for extending the class of geometries which can be hexahedrally meshed.

We present a variational approach to smooth molecular (proteins, nucleic acids) surface constructions, starting from atomic coordinates, as available from the protein and nucleicacid data banks. Molecular dynamics (MD) simulations traditionally used in understanding protein and nucleic-acid folding processes, are based on molecular force fields, and require smooth models of these molecular surfaces. To accelerate MD simulations, a popular methodology is to employ coarse grained molecular models, which represent clusters of atoms with similar physical properties by psuedo- atoms, resulting in coarser resolution molecular surfaces. We consider generation of these mixed-resolution or adaptive molecular surfaces. Our approach starts from deriving a general form second order geometric partial differential equation in the level-set formulation, by minimizing a first order energy functional which additionally includes a regularization term to minimize the occurrence of chemically infeasible molecular surface pockets or tunnel-like artifacts. To achieve even higher computational efficiency, a fast cubic B-spline C 2 interpolation algorithm is also utilized. A narrow band, tri-cubic B-spline level-set method is then used to provide C 2 smooth and resolution adaptive molecular surfaces.

Building high-quality quadrilateral/hexahedral meshes directly from volumetric data is hard. Existing algorithms for generating meshes from volumetric data are based on primal and dual isocontouring algorithms, and current research focuses on improving the quality of such meshes. Most techniques are based on isocontouring techniques, and work by generating a grid of hexahedra on the interior/exterior of an isosurface, and then adjusting the elements that lie on the boundary of the grid to fit the surface. As a result of the element adjustment, many of these elements lose their convexity, as measured by the scaled Jacobian metric. Recovering the convexity of these elements is difficult since the position of boundary vertices is restricted to the domain of the isosurface. In this paper, we propose a solution to this problem using insights obtained from the structure of the dual of a hexahedral mesh. Our solution is to add a sheet of hexes along the boundary, composed of well-shaped elements. The additional degrees of freedom provided by this sheet enables the optimization of the original poorly-shaped elements that are no longer on the boundary allowing for the creation of a high-quality mesh. An extra benefit of our technique is being able to capture sharp features, which is done by inserting multiple sheets. Our experimental results demonstrate the successful removal of all bad elements (i.e. those elements of the mesh that have a scaled Jacobian measure of less than 0.2) in a number of complex examples.

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