Adaptive mesh refinement using piecewise-linear shape functions based on the blending function method

Use of quadrilateral elements for finite element mesh refinement can lead either to so-called irregular meshes or the necessity of adjustments between finer and coarser parts of the mesh necessary. In the case of irregular meshes, constraints have to be introduced in order to maintain continuity of the displacements. Introduction of finite elements based on blending function interpolation shape functions using piecewise boundary interpolation avoids these problems. This paper introduces an adaptive refinement procedure for these types of elements. The refinement is anh-method. Error estimation is performed using the Zienkiewicz-Zhu method. The refinement is controlled by a switching function representation. The method is applied to the plane stress problem. Numerical examples are given to show the efficiency of the methodology.     

Parallel adaptive mesh generation and decomposition

An important class of methodologies for the parallel processing of computational models defined on some discrete geometric data structures (i.e. meshes, grids) is the so calledgeometry decomposition or splitting approach. Compared to the sequential processing of such models, the geometry splitting parallel methodology requires an additional computational phase. It consists of the decomposition of the associated geometric data structure into a number of balancedsubdomains that satisfy a number of conditions that ensure the load balancing and minimum communication requirement of the underlying computations on a parallel hardware platform. It is well known that the implementation of the mesh decomposition phase requires the solution of a computationally intensive problem. For this reason several fast heuristics have been proposed. In this paper we explore a decomposition approach which is part of a parallel adaptive finite element mesh procedure. The proposed integrated approach consists of five steps. It starts with a coarse background mesh that isoptimally decomposed by applying well known heuristics. Then, the initial mesh is refined in each subdomain after linking the new boundaries introduced by its decomposition. Finally, the decomposition of the new refined mesh is improved so that it satisfies the objectives and conditions of the mesh decomposition problem. Extensive experimentation indicates the effectiveness and efficiency of the proposed parallel mesh and decomposition approach.     

Quasi-angle-preserving mesh deformation using the least-squares approach

We propose an angle-based mesh representation, which is invariant under translation, rotation, and uniform scaling, to encode the geometric details of a triangular mesh. Angle-based mesh representation consists of angle quantities defined on the mesh, from which the mesh can be reconstructed uniquely up to translation, rotation, and uniform scaling. The reconstruction process requires solving three sparse linear systems： the first system encodes the length of edges between vertices on the mesh, the second system encodes the relationship of local frames between two adjacent vertices on the mesh, and the third system defines the position of the vertices via the edge length and the local frames. From this angle-based mesh representation, we propose a quasi-angle-preserving mesh deformation system with the least-squares approach via detail-preserving mesh editing examples are presented to handle translation, rotation, and uniform scaling. Several demonstrate the effectiveness of the proposed method.     

Feature-preserving adaptive mesh generation for molecular shape modeling and simulation

We describe a chain of algorithms for molecular surface and volumetric mesh generation. We take as inputs the centers and radii of all atoms of a molecule and the toolchain outputs both triangular and tetrahedral meshes that can be used for molecular shape modeling and simulation. Experiments on a number of molecules are demonstrated, showing that our methods possess several desirable properties: feature-preservation, local adaptivity, high quality, and smoothness (for surface meshes). We also demonstrate an example of molecular simulation using the finite element method and the meshes generated by our method. The approaches presented and their implementations are also applicable to other types of inputs such as 3D scalar volumes and triangular surface meshes with low quality, and hence can be used for generation/improvement of meshes in a broad range of applications.     

Semi-formal design of reliable mesh generation systems

A reliable mesh generation infrastructure is designed based on software engineering principles. Formal methods, software design documents and clear modular decomposition criteria are introduced to improve the quality of mesh generation software. The design document for a simple 2D mesh generation data structure is presented using a semi-formal specification. The proposed semi-formal documentation system avoids any ambiguity during the software design process and will help in driving the software test cases. Using the proposed software, design techniques result in a consistent software design that is easy to extend and modify.     

Interactive mesh deformation using equality-constrained least squares

Mesh deformation techniques that preserve the differential properties have been intensively studied. In this paper, we propose an equality-constrained least squares approach for stably deforming mesh models while approximately preserving mean curvature normals and strictly satisfying other constraints such as positional constraints. We solve the combination of hard and soft constraints by constructing a typical least squares system using QR decomposition. A well-known problem of hard constraints is over-constraints. We show that the equality-constrained least squares approach is useful for resolving such over-constrained situations. In our framework, the rotations of mean curvature normals are treated using the logarithms of unit quaternions in . During deformation, mean curvature normals can be rotated while preserving their magnitudes. In addition, we introduce a new modeling constraints called rigidity constraints and show that rigidity constraints can effectively preserve the shapes of feature regions during deformation. Our framework achieves good performance for interactive deformation of mesh models.     

Hierarchical aggregation for efficient shape extraction

This paper presents an efficient framework which supports both automatic and interactive shape extraction from surfaces. Unlike most of the existing hierarchical shape extraction methods, which are based on computationally expensive top-down algorithms, our framework employs a fast bottom-up hierarchical method with multiscale aggregation. We introduce a geometric similarity measure, which operates at multiple scales and guarantees that a hierarchy of high-level features are automatically found through local adaptive aggregation. We also show that the aggregation process allows easy incorporation of user-specified constraints, enabling users to interactively extract features of interest. Both our automatic and the interactive shape extraction methods do not require explicit connectivity information, and thus are applicable to unorganized point sets. Additionally, with the hierarchical feature representation, we design a simple and effective method to perform partial shape matching, allowing efficient search of self-similar features across the entire surface. Experiments show that our methods robustly extract visually meaningful features and are significantly faster than related methods.     

Using a computational domain and a three-stage node location procedure for multi-sweeping algorithms

The multi-sweeping method is one of the most used algorithms to generate hexahedral meshes for extrusion volumes. In this method the geometry is decomposed in sub-volumes by means of projecting nodes along the sweep direction and imprinting faces. However, the quality of the final mesh is determined by the location of inner nodes created during the decomposition process and by the robustness of the imprinting process.In this work we present two original contributions to increase the quality of the decomposition process. On the one hand, to improve the robustness of the imprints we introduce the new concept of computational domain for extrusion geometries. Since the computational domain is a planar representation of the sweep levels, we improve several geometric operations involved in the imprinting process. On the other hand, we propose a three-stage procedure to improve the location of the inner nodes created during the decomposition process. First, inner nodes are projected towards source surfaces. Second, the nodes are projected back towards target surfaces. Third, the final position of inner nodes is computed as a weighted average of the projections from source and target surfaces.     

Skeleton-based computational method for the generation of a 3D finite element mesh sizing function

This paper focuses on the generation of a three-dimensional (3D) mesh sizing function for geometry-adaptive finite element (FE) meshing. The mesh size at a point in the domain of a solid depends on the geometric complexity of the solid. This paper proposes a set of tools that are sufficient to measure the geometric complexity of a solid. Discrete skeletons of the input solid and its surfaces are generated, which are used as tools to measure the proximity between geometric entities and feature size. The discrete skeleton and other tools, which are used to measure the geometric complexity, generate source points that determine the size and local sizing function at certain points in the domain of the solid. An octree lattice is used to store the sizing function as it reduces the meshing time. The size at every lattice-node is calculated by interpolating the size of the source points. The algorithm has been tested on many industrial models, and it can be extended to consider other non-geometric factors that influence the mesh size, such as physics, boundary conditions, etc.Sandia National Laboratory is a multiprogram laboratory operated by the Sandia Corporation, a Lockheed Martin Company, for the US Department of Energy under contract DE-AC04-94AL85000.     

Fast median filtering algorithms for mesh computers

Two fast algorithms for median filtering of images using parallel computers having 2-D mesh interconnections are given. Both algorithms assume that an n × n image is loaded onto the mesh with one processing element per pixel. One algorithm performs median filtering over d × d neighborhoods in O(d²) time and works with pixel values in an arbitrarily large range. This algorithm, while theoretically suboptimal, achieves a lower constant than a previously published asymptotically—optimal algorithm and is simpler to program. The second algorithm assumes that the range of pixel values is limited and relatively small, and it accomplishes median filtering in O(d) time.     

Video coding using geometry based block partitioning and reordering discrete cosine transform

Geometry based block partitioning(GBP) has been shown to achieve better performance than the tree structure based block partitioning(TSBP) of H.264.However,the residual blocks of GBP mode after motion compensation still present some nonvertical/non-horizontal orientations,and the conventional discrete cosine transform(DCT) may generate many high-frequency coefficients.To solve this problem,in this paper we propose a video coding approach by using GBP and reordering DCT(RDCT) techniques.In the proposed approach,GBP is first applied to partition the macroblocks.Then,before performing DCT,a reordering operation is used to adjust the pixel positions of the residual macroblocks based on the partition information.In this way,the partition information of GBP can be used to represent the reordering information of RDCT,and the bitrate can be reduced.Experimental results show that,compared to H.264/AVC,the proposed method achieves on average 6.38% and 5.69% bitrate reductions at low and high bitrates,respectively.     

Quadrilateral and tetrahedral mesh stripification using 2-factor partitioning of the dual graph

In order to find a 2-factor of a graph, there exists a O(n ^1.5) deterministic algorithm [7] and a O(n ³) randomized algorithm [14]. In this paper, we propose novel O(nlog³ nloglogn) algorithms to find a 2-factor, if one exists, of a graph in which all n vertices have degree 4 or less. Such graphs are actually dual graphs of quadrilateral and tetrahedral meshes. A 2-factor of such graphs implicitly defines a linear ordering of the mesh primitives in the form of strips. Further, by introducing a few additional primitives, we reduce the number of tetrahedral strips to represent the entire tetrahedral mesh and represent the entire quad surface using a single quad strip.     

Sorting in constant number of row and column phases on a mesh

An algorithm for sorting on a mesh by alternately transforming the rows and columns is presented. The algorithm runs in a constant number of row- and column-transformation phases (sixteen phases), an improvement over the previous best upper bound ofO(log logm) phases,m being the number of rows in the mesh. A corresponding lower bound of five phases is also shown.This research was supported by the National Science Foundation under Grant DCR-84-51396, and by IBM Corporation under Grant D8400622.     

Origin-based fault-tolerant routing in the mesh

The ability to tolerate faults is critical in multicomputer employing large numbers of processors. This paper describes a class of fault-tolerant routing algorithms for n-dimensional meshes that can tolerate large numbers of faults without using virtual channels. We show that these routing algorithms prevent livelock and deadlock while remaining highly adaptive.     

High-quality 2D mesh generation without obtuse and small angles

In this paper, we present an efficient method to eliminate the obtuse triangles for high quality 2D mesh generation. Given an initialization (e.g., from Centroidal Voronoi Tessellation—CVT), a limited number of point insertions and removals are performed to eliminate obtuse or small angle triangles. A mesh smoothing and optimization step is then applied. These steps are repeated till a desired good quality mesh is reached. We tested our algorithm on various 2D polygonal domains and verified that our algorithm always converges after inserting a few number of new points, and generates high quality triangulation with no obtuse triangles.     

Three-dimensional anisotropic geometric metrics based on local domain curvature and thickness

A three-dimensional anisotropic Riemannian metric is constructed from a triangulated CAD model to control its spatial discretization for numerical analysis. In addition to the usual curvature criterion, the present geometric metric is also based on the local thickness of the modeled domain. This local thickness is extracted from the domain skeleton while local curvature is deduced from the model triangulated boundaries. A Cartesian background octree is used as the support medium for this metric and skeletonization takes advantage of this structure through an octree extension of a digital medial axis transform. For this purpose, the octree has to be refined according to not only boundary curvature but also a local separation criterion from digital topology theory. The resulting metric can be used to geometrically adapt any mesh type as long as metric-based adaptation tools are available. To illustrate such an application, geometric adaptation of overlay meshes used in grid-based methods for unstructured hexahedral mesh generation is presented. However, beyond mesh generation, the present metric may also be useful as a shape analysis tool and such a possibility could be explored in future developments.     

Geometry-defining processors for engineering design and analysis

Critical issues of interactive three-dimensional geometry definition and high-speed parallel computation are addressed in a unified fashion by Geometry-Defining Processors (GDPs). GDPs are microprocessors housed in three-dimensional physical polyhedral packages which can be easily manually assembled or reconfigured to construct approximate scale models of physical objects or domains. An individual GDP communicates with neighboring GDPs in an assembly through optical ports associated with the faces of its package. An assembly of communicating GDPs is able to bothdefine a system geometry and, operating as an optimally connected parallel processor,solve the associated continuum partial differential equations required for design evaluation. Combining simplicity-of-use with efficient computational capabilities, the GDP design system should prove useful in numberous engineering applications.     

Network partitioning using harmony search and equivalencing for distributed computing

Power system has a highly interconnected network that requires intense computational effort and resources for centralized control. Distributed computing needs the systems to be partitioned optimally into clusters. The network partitioning is an optimization problem whose objective is to minimize the number of nodes in a cluster and the tie lines between the clusters. Harmony Search(HS) Algorithm is one of the recently developed meta heuristic algorithms that can be applied to optimization problems. In this work, the HS algorithm is applied to the network partitioning problem and power flow based equivalencing is done to represent the external system. Simulation is done on IEEE Standard Test Systems. The algorithm is found to be very effective in partitioning the system hierarchically and the equivalencing method gives accurate results in comparison to the centralized control.     

Greedy hot-potato routing on the two-dimensional mesh

Summary We propose hot-potato (or, deflection) packet routing algorithms on the two-dimensional mesh. The algorithms are strongly greedy in the sense that they attempt to send packets in good directions whenever possible. Furthermore, the routing operations are simple and independent of the time that has elapsed. The first algorithm gives the best evacuation time known for delivering all the packets to their destinations. A batch ofk packets with maximal source-to-destination distanced _max is delivered in 2(k-1)+d _max. The second algorithm improves this bound tok+d _max when all packets are destined to the same node. This also implies a new bound for the multitarget case, which is the first to take into account the number of in-edges of a node. The third algorithm is designed for routing permutations with source-to-destination distance at most three, in which case the algorithm terminates in at most seven steps. We also show a lower bound of five steps for this problem. Ishai Ben-Aroya received the B.A. and M.Sc. in computer science from the Technion (Israel Institute of Technology). He is currently working with Microsoft Israel R&D group. His main interests include Routing Algorithms, Cryptography and Computer Security. Tamar Eilam received the B.A. degree in Computer Science from the Technion IIL in 1995, and is currently studying towards her M.A. degree. Assaf Schuster received his B.A., M.A. and Ph.D. degrees in Computer Science from the Hebrew University of Jerusalem (the last one in 1991). He is currently a lecturer at the Technion IIL. His main interests include Networks and Routing Algorithms, Parallel and Distributed Computation, Optical Computation and Communication, Dynamically Reconfiguring Networks, and Greedy Hot Potato Routing.This work was supported in part by the French-Israeli grant for cooperation in Computer Science, and by a grant from the Israeli Ministry of Science. An extended abstract appeared in proc. 2nd European Symposium on Algorithms, September 1994