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Tools for fatigue life prediction of solder joints are developed. This report consists of two parts. In Part One, the creep and plastic deformations stored in a solder joint are calculated by implementing appropriate plastic and creep constitutive models in a nonlinear finite element program. The calculated damage in each cycle is then related to the life of the material using two failure criteria: strain-life and energy-partitioning. The importance of parameters affecting the fatigue life of solders are evaluated. In Part Two, the finite element analysis is combined with x-ray and laser imaging systems, from which real solder joint geometries can be constructed. The approaches presented herein provide a useful tool in the design and manufacturing of surface-mount assemblies. Solder, Solder joint, Fatigue life, Creep, Thermal stress, Finite element analysis, Mesh generation.

An overview of current unstructured mesh generation and adaptivity techniques is given. Basic building blocks taken from the field of computational geometry are first described. Various practical mesh generation techniques based on these algorithms are then constructed and illustrated with examples. Issues of adaptive meshing and stretched mesh generation for anisotropic problems arc treated in subsequent sections. The presentation is organized in an educational manner. for readers familiar with computational fluid dynamics. wishing to learn more about current unstructured mesh techniques.

An implicit, shock-capturing finite-difference code which is used to calculate two-dimensional inlet flow fields in a supersonic free stream is explained. The Euler equations are subjected to general nonorthogonal transformation and a body-fitted coordinate system is employed. The mathematical formulation of the problem is given along with the numerical algorithm. Initial and boundary conditions, numerical stability, program limitations, and accuracy is discussed. An overall program logic as well as instructions for program use and operation are also furnished.

The goal of this research is to develop computer programs that generate finite element models suitable for doing 3D contact analysis of faced milled spiral bevel gears in mesh. A pinion tooth and a gear tooth are created and put in mesh. There are two programs: Points.f and Pat.f to perform the analysis. Points.f is based on the equation of meshing for spiral bevel gears. It uses machine tool settings to solve for an N x M mesh of points on the four surfaces, pinion concave and convex, and gear concave and convex. Points.f creates the file POINTS.OUT, an ASCI file containing N x M points for each surface. (N is the number of node points along the length of the tooth, and M is nodes along the height.) Pat.f reads POINTS.OUT and creates the file tl.out. Tl.out is a series of PATRAN input commands. In addition to the mesh density on the tooth face, additional user specified variables are the number of finite elements through the thickness, and the number of finite elements along the tooth full fillet. A full fillet is assumed to exist for both the pinion and gear.

Geophysical model domains typically contain irregular, complex fractal-like boundaries and physical processes that act over a wide range of scales. Constructing geographically constrained boundary-conforming spatial discretizations of these domains with flexible use of anisotropically, fully unstructured meshes is a challenge. The problem contains a wide range of scales and a relatively large, heterogeneous constraint parameter space. Approaches are commonly ad hoc, model or application specific and insufficiently described. Development of new spatial domains is frequently time-consuming, hard to repeat, error prone and difficult to ensure consistent due to the significant human input required. As a consequence, it is difficult to reproduce simulations, ensure a provenance in model data handling and initialization, and a challenge to conduct model intercomparisons rigorously. Moreover, for flexible unstructured meshes, there is additionally a greater potential for inconsistencies in model initialization and forcing parameters. This paper introduces a consistent approach to unstructured mesh generation for geophysical models, that is automated, quick-to-draft and repeat, and provides a rigorous and robust approach that is consistent to the source data throughout. The approach is enabling further new research in complex multi-scale domains, difficult or not possible to achieve with existing methods. Examples being actively pursued in a range of geophysical modeling efforts are presented alongside the approach, together with the implementation library Shingle and a selection of its verification test cases.

We use a time-relaxed linear grid generator of Winslow type to propose a new deterministic-stochastic domain decomposition approach to the generation of adaptive moving meshes. The method uses the probabilistic form of the exact solution of the linear mesh generator to provide the sub-domain interface values that serve as boundary conditions for the domain decomposition. The meshes over the single sub-domains can then be obtained independently of each other in an entirely parallel fashion. We demonstrate numerically the capability of the method to yield smooth, time-evolving meshes.

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on a posteriori error estimates proposed in \cite{YinXie}. The new metric tensor explores more comprehensive information of anisotropy for the true solution than those existing ones. Numerical results show that this approach can be successfully applied to deal with poisson and steady convection-dominated problems. The superior accuracy and efficiency of the new metric tensor to others is illustrated on various numerical examples of complex two-dimensional simulations.

Speaker: Jonathan Shewchuk Date: October, 2003

Despite great advancements in the parallelization of numerical simulation codes over the last 20 years, it is still common to perform grid generation in serial. Generating large scale grids in serial often requires using special "grid generation" compute machines that can have more than ten times the memory of average machines. While some parallel mesh generation techniques have been proposed, generating very large meshes for LES or aeroacoustic simulations is still a challenging problem. An automated method for the parallel generation of very large scale off-body hierarchical meshes is presented here. This work enables large scale parallel generation of off-body meshes by using a novel combination of parallel grid generation techniques and a hybrid "top down" and "bottom up" oct-tree method. Meshes are generated using hardware commonly found in parallel compute clusters. The capability to generate very large meshes is demonstrated by the generation of off-body meshes surrounding complex aerospace geometries. Results are shown including a one billion cell mesh generated around a Predator Unmanned Aerial Vehicle geometry, which was generated on 64 processors in under 45 minutes.

Computational mathematics plays an increasingly important role in computational fluid dynamics (CFD). The aeronautics and aerospace re- search community is working on next generation of CFD capacity that is accurate, automatic, and fast. A key component of the next generation of CFD is a greatly enhanced capacity for mesh generation and adaptivity of the mesh according to solution and geometry. In this paper, we propose a new method that generates triangular meshes on domains of curved boundary. The method deforms a Cartesian mesh that covers the domain to generate a mesh with prescribed boundary nodes. The deformation fields are generated by a system of divergence and curl equations which are solved effectively by the least square finite element method.

This document serves as the final report for the grant titled Surface Generation and Cartesian Mesh Support . This completed work was in algorithmic research into automatically generating surface triangulations from CAD geometries. NASA's OVERFLOW and Cart3D simulation packages use surface triangulations as an underlying geometry description and the ability to automatically generate these from CAD files (without translation) substantially reduces both the wall-clock time and expertise required to get geometry out of CAD and into mesh generation. This surface meshing was exercised greatly during the Shuttle investigation during the last year with success. The secondary efforts performed in this grant involve work on a visualization system cut-cell handling for Cartesian Meshes with embedded boundaries.

An overview of current unstructured mesh generation and adaptivity techniques is given. Basic building blocks taken from the field of computational geometry are first described. Various practical mesh generation techniques based on these algorithms are then constructed and illustrated with examples. Issues of adaptive meshing and stretched mesh generation for anisotropic problems are treated in subsequent sections. The presentation is organized in an education manner, for readers familiar with computational fluid dynamics, wishing to learn more about current unstructured mesh techniques.

Fluid flow simulations through a valve often require evaluation of the valve in multiple opening positions. A mesh has to be generated for the valve for each position and compounding. The problem is the fact that the valve is typically part of a larger feedline system. In this paper, we propose to develop a system to create meshes for feedline systems with parametrically controlled valve openings. Herein we outline two approaches to generate the meshes for a valve in a feedline system at multiple positions. There are two issues that must be addressed. The first is the creation of the mesh on the valve for multiple positions. The second is the generation of the mesh for the total feedline system including the valve. For generation of the mesh on the valve, we will describe the use of topology matching and mesh generation parameter transfer. For generation of the total feedline system, we will describe two solutions that we have implemented. In both cases the valve is treated as a component in the feedline system. In the first method the geometry of the valve in the feedline system is replaced with a valve at a different opening position. Geometry is created to connect the valve to the feedline system. Then topology for the valve is created and the portion of the topology for the valve is topology matched to the standard valve in a different position. The mesh generation parameters are transferred and then the volume mesh for the whole feedline system is generated. The second method enables the user to generate the volume mesh on the valve in multiple open positions external to the feedline system, to insert it into the volume mesh of the feedline system, and to reduce the amount of computer time required for mesh generation because only two small volume meshes connecting the valve to the feedline mesh need to be updated.

We present a simple direct discretization for functionals used in the variational mesh generation and adaptation. Meshing functionals are discretized on simplicial meshes and the Jacobian matrix of the continuous coordinate transformation is approximated by the Jacobian matrices of affine mappings between elements. The advantage of this direct geometric discretization is that it preserves the basic geometric structure of the continuous functional, which is useful in preventing strong decoupling or loss of integral constraints satisfied by the functional. Moreover, the discretized functional is a function of the coordinates of mesh vertices and its derivatives have a simple analytical form, which allows a simple implementation of variational mesh generation and adaptation on computer. Since the variational mesh adaptation is the base for a number of adaptive moving mesh and mesh smoothing methods, the result in this work can be used to develop simple implementations of those methods. Numerical examples are given.

This article is from BMC Bioinformatics , volume 14 . Abstract Background: Multicellular organisms consist of cells of many different types that are established during development. Each type of cell is characterized by the unique combination of expressed gene products as a result of spatiotemporal gene regulation. Currently, a fundamental challenge in regulatory biology is to elucidate the gene expression controls that generate the complex body plans during development. Recent advances in high-throughput biotechnologies have generated spatiotemporal expression patterns for thousands of genes in the model organism fruit fly Drosophila melanogaster. Existing qualitative methods enhanced by a quantitative analysis based on computational tools we present in this paper would provide promising ways for addressing key scientific questions. Results: We develop a set of computational methods and open source tools for identifying co-expressed embryonic domains and the associated genes simultaneously. To map the expression patterns of many genes into the same coordinate space and account for the embryonic shape variations, we develop a mesh generation method to deform a meshed generic ellipse to each individual embryo. We then develop a co-clustering formulation to cluster the genes and the mesh elements, thereby identifying co-expressed embryonic domains and the associated genes simultaneously. Experimental results indicate that the gene and mesh co-clusters can be correlated to key developmental events during the stages of embryogenesis we study. The open source software tool has been made available at http://compbio.cs.odu.edu/fly/. Conclusions: Our mesh generation and machine learning methods and tools improve upon the flexibility, ease-of-use and accuracy of existing methods.

An isoparametric 3-D finite element mesh generator was developed with direct interface to an interactive geometric modeler program called POLYGON. POLYGON defines the model geometry in terms of boundaries and mesh regions for the mesh generator. The mesh generator controls the mesh flow through the 2-dimensional spans of regions by using the topological data and defines the connectivity between regions. The program is menu driven and the user has a control of element density and biasing through the spans and can also apply boundary conditions, loads interactively.

We propose a novel approach which employs random sampling to generate an accurate non-uniform mesh for numerically solving Partial Differential Equation Boundary Value Problems (PDE-BVP's). From a uniform probability distribution U over a 1D domain, we sample M discretizations of size N where M>>N. The statistical moments of the solutions to a given BVP on each of the M ultra-sparse meshes provide insight into identifying highly accurate non-uniform meshes. Essentially, we use the pointwise mean and variance of the coarse-grid solutions to construct a mapping Q(x) from uniformly to non-uniformly spaced mesh-points. The error convergence properties of the approximate solution to the PDE-BVP on the non-uniform mesh are superior to a uniform mesh for a certain class of BVP's. In particular, the method works well for BVP's with locally non-smooth solutions. We present a framework for studying the sampled sparse-mesh solutions and provide numerical evidence for the utility of this approach as applied to a set of example BVP's. We conclude with a discussion of how the near-perfect paralellizability of our approach suggests that these strategies have the potential for highly efficient utilization of massively parallel multi-core technologies such as General Purpose Graphics Processing Units (GPGPU's). We believe that the proposed algorithm is beyond embarrassingly parallel; implementing it on anything but a massively multi-core architecture would be scandalous.

A novel approach to unstructured quadrilateral mesh generation for planar domains is presented. Away from irregular vertices, the resulting meshes have the properties of nearly conformal grids. The technique is based on a theoretical relation between the present problem, and the inverse Poisson (IP) problem with point sources. An IP algorithm is described, which constructs a point-source distribution, whose sources correspond to the irregular vertices of the mesh. Both the background theory and the IP algorithm address the global nature of the mesh generation problem. The IP algorithm is incorporated in a complete mesh generation scheme, which also includes an algorithm for creating the final mesh. Example results are presented and discussed.

Many important real world problems require meshing, that is the approximation of a given geometry by a set of simpler elements such as triangles or quadrilaterals in two dimensions, and tetrahedra or hexahedra in three dimensions. Applications include finite element analysis and computer graphics. This work focuses on the former. A physically based model of interacting 'particles' is introduced to uniformly spread points over a 2-dimensional polygonal domain. The set of points is triangulated to form a triangle mesh. Delaunay triangulation is used because it guarantees a low computational cost and reasonably well shaped elements. Several particle interaction (repulsion and attraction) models are investigated ranging from Gaussian energy potentials to Laplacian smoothing. Particle population control mechanisms are introduced to make the size of the mesh elements converge to the desired size. In most applications spatial mesh adaptivity is desirable. Triangles should not only adapt in size but also in shape, to better fit the function to approximate. Computational fluid dynamics simulations typically require triangles stretched in the direction of the flow. A metric tensor is introduced to quantify the stretching. The triangulation procedure is changed to generate 'Delaunay' meshes in the Riemannian space defined by the metric. This new approach to mesh generation appears quite promising.

This report provides a comprehensive implementation of an unstructured mesh generation method that re nes a triangulated grid by locally bisecting triangles on their longest edge, until they satisfy a given local condition. The method is relatively simple to implement, has the capacity to quickly generate a re ned mesh with triangles that rapidly change size over a short distance, and does not create triangles with small or large angles.

The paper introduces an automation process of generating a tetrahedral mesh for computational fluid dynamics (CFD) analysis of aircraft configurations in early conceptual design. The method was developed for CFD-based sonic boom analysis of supersonic configurations, but can be applied to aerodynamic analysis of aircraft configurations in any flight regime.

The theory of elliptic mesh generation is reviewed and the fundamental problem of constructing computational space is discussed. It is argued that the construction of computational space is an NP-Complete problem and therefore requires a nonstandard approach for its solution. This leads to the development of graph-theoretic, combinatorial optimization and integer programming algorithms. Methods for the construction of two dimensional computational space are presented.

The goal of this research was to further develop zonal methods and three dimensional procedures for application of finite difference methods to solve complex aircraft configurations. For the task of three dimensional grid generation both elliptic and hyperbolic methods were developed. A chimera grid scheme, that is, the use of overset multiple grid systems, was also tested in two dimensions. In zonal methods several new algorithms were developed. These included combining transonic potential codes with thin layer Navier-Stokes equations, unsteady transonic potential, Euler and vector potential codes. This report summarizes the various numerical algorithms that were studied. (Author)

We present a framework to generate watertight mesh representations in an unsupervised manner from noisy point clouds of complex, heterogeneous objects with free-form surfaces. The resulting meshes are ready to use in applications like kinematics and dynamics simulation where watertightness and fast processing are the main quality criteria. This works with no necessity of user interaction, mainly by utilizing a modified Growing Neural Gas technique for surface reconstruction combined with several post-processing steps. In contrast to existing methods, the proposed framework is able to cope with input point clouds generated by consumer-grade RGBD sensors and works even if the input data features large holes, e.g. a missing bottom which was not covered by the sensor. Additionally, we explain a method to unsupervisedly optimize the parameters of our framework in order to improve generalization quality and, at the same time, keep the resulting meshes as coherent as possible to the original object regarding visual and geometric properties.

Recent developments for mathematical modeling and numerical simulation of biomolecular systems raise new demands for qualified, stable, and efficient surface meshing, especially in implicit-solvent modeling. In our former work, we have developed an algorithm for manifold triangular meshing for large Gaussian molecular surfaces, TMSmesh. In this paper, we present new algorithms to greatly improve the meshing efficiency and qualities, and implement into a new program version, TMSmesh 2.0. In TMSmesh 2.0, in the first step, a new adaptive partition and estimation algorithm is proposed to locate the cubes in which the surface are approximated by piecewise trilinear surface with controllable precision. Then, the piecewise trilinear surface is divided into single valued pieces by tracing along the fold curves, which ensures that the generated surface meshes are manifolds. Numerical test results show that TMSmesh 2.0 is capable of handling arbitrary sizes of molecules and achieves ten to hundreds of times speedup over the previous algorithm. The result surface meshes are manifolds and can be directly used in boundary element method (BEM) and finite element method (FEM) simulation. The binary version of TMSmesh 2.0 is downloadable at the web page http://lsec.cc.ac.cn/~lubz/Meshing.html.

Under the first period of the grant support, mathematical developments were performed on algebraic, adaptive, surface, and orthogonal transformations. Algebraic transformations were extended by forming Boolen sums of multisurface transformations and, by appropriate compositions, lifting the results to curved 2-D surfaces. Orthogonal trajectories were a major part of the general development of orthogonal transformations under the grant support. Additional parts included field methods. This report summarizes progress made in these areas and lists publications resulting from the research. (Author)

A continuum description of unstructured meshes in two dimensions, both for planar and curved surface domains, is proposed. The meshes described are those which, in the limit of an increasingly finer mesh (smaller cells), and away from irregular vertices, have ideally-shaped cells (squares or equilateral triangles), and can therefore be completely described by two local properties: local cell size and local edge directions. The connection between the two properties is derived by defining a Riemannian manifold whose geodesics trace the edges of the mesh. A function $\phi$, proportional to the logarithm of the cell size, is shown to obey the Poisson equation, with localized charges corresponding to irregular vertices. The problem of finding a suitable manifold for a given domain is thus shown to exactly reduce to an Inverse Poisson problem on $\phi$, of finding a distribution of localized charges adhering to the conditions derived for boundary alignment. Possible applications to mesh generation are discussed.

Viewgraphs on Cartesian-cell based grid generation and adaptive mesh refinement are presented. Topics covered include: grid generation; cell cutting; data structures; flow solver formulation; adaptive mesh refinement; and viscous flow.

This report documents the process to convert right and left elevation versus distance course profiles into triangular meshes for use in Road Property Files (.rdf) used in RecurDyn multibody dynamics software. Firstly, MATLAB was used to combine the left and right elevation vs. distance course profile curves into one Importing Blend File. This file was imported into Pro/Engineer and used to create road geometry. The geometry was imported into HyperMesh and meshed, creating nodes and elements. The meshed geometry was exported as an ASCII mesh database file. MATLAB was used to parse the file and extract node and element data. Finally, the data was printed in the Road Property File (.rdf) format to be used in the RecurDyn multibody dynamics program.

We propose a new two-dimensional meshing algorithm called PINW able to generate meshes that accurately approximate the distance between any two domain points by paths composed only of cell edges. This technique is based on an extension of pinwheel tilings proposed by Radin and Conway. We prove that the algorithm produces triangles of bounded aspect ratio. This kind of mesh would be useful in cohesive interface finite element modeling when the crack propagation pathis an outcome of a simulation process.

A method for generating an unstructured triangular mesh in two dimensions, suitable for computing high Reynolds number flows over arbitrary configurations is presented. The method is based on a Delaunay triangulation, which is performed in a locally stretched space, in order to obtain very high aspect ratio triangles in the boundary layer and the wake regions. It is shown how the method can be coupled with an unstructured Navier-Stokes solver to produce a solution adaptive mesh generation procedure for viscous flows.

No abstract available

In a recent note [High Energy Density Phys. 7, 161 (2011)], B.G. Wilson and V. Sonnad proposed a very useful closed form expression for the efficient generation of analytic log-linear radial meshes. The central point of the note is an implicit equation for the parameter h, involving Lambert's function W[x]. The authors mention that they are unaware of any direct proof of this equation (they obtained it by re-summing the Taylor expansion of h using high-order coefficients obtained by analytic differentiation of the implicit definition using symbolic manipulation). In the present comment, we present a direct proof of that equation.

A Frontal-Delaunay refinement algorithm for mesh generation in piecewise smooth domains is described. Built using a restricted Delaunay framework, this new algorithm combines a number of novel features, including: (i) an unweighted, conforming restricted Delaunay representation for domains specified as a (non-manifold) collection of piecewise smooth surface patches and curve segments, (ii) a protection strategy for domains containing curve segments that subtend sharply acute angles, and (iii) a new class of off-centre refinement rules designed to achieve high-quality point-placement along embedded curve features. Experimental comparisons show that the new Frontal-Delaunay algorithm outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.

Due to the fractal nature of the domain geometry in geophysical flow simulations, a completely accurate description of the domain in terms of a computational mesh is frequently deemed infeasible. Shoreline and bathymetry simplification methods are used to remove small scale details in the geometry, particularly in areas away from the region of interest. To that end, a novel method for shoreline and bathymetry simplification is presented. Existing shoreline simplification methods typically remove points if the resultant geometry satisfies particular geometric criteria. Bathymetry is usually simplified using traditional filtering techniques, that remove unwanted Fourier modes. Principal Component Analysis (PCA) has been used in other fields to isolate small-scale structures from larger scale coherent features in a robust way, underpinned by a rigorous but simple mathematical framework. Here we present a method based on principal component analysis aimed towards simplification of shorelines and bathymetry. We present the algorithm in detail and show simplified shorelines and bathymetry in the wider region around the North Sea. Finally, the methods are used in the context of unstructured mesh generation aimed at tidal resource assessment simulations in the coastal regions around the UK.

This paper presents a fast an robust mesh generation procedure that is able to generate meshes of the earth system (ocean and continent) in matters of seconds. Our algorithm takes as input a standard shape-file i.e. geospatial vector data format for geographic information system (GIS) software. The input is initially coarsened in order to automatically remove unwanted channels that are under a desired resolution. A valid non-overlapping 1D mesh is then created on the sphere using the Euclidian coordinates system $x,y,z$. A modified Delaunay kernel is then proposed that enables to generate meshes on the sphere in a straightforward manner without parametrization. One of the main difficulty in dealing with geographical data is the over-sampled nature of coastline representations. We propose here an algorithm that automatically unrefines coastline data. Small features are automatically removed while always keeping a valid (non-overlapping) geometrical representation of the domain. A Delaunay refinement procedure is subsequently applied to the domain. The refinement scheme is also multi-threaded at a fine grain level, allowing to generate about a million points per second on 8 threads. Examples of meshes of the Baltic sea as well as of the global ocean are presented.

Hybrid tools have been developed which greatly reduce the time required to generate three-dimensional structured CFD meshes for turbomachinery blade passages. RAGGS, an existing Rockwell proprietary, general purpose mesh generation and visualization system, provides the starting point and framework for tool development. Utilities which manipulate and interface with RAGGS tools have been developed to (1) facilitate blade geometry inputs from point or CAD representations, (2) automate auxiliary surface creation, and (3) streamline and automate edge, surface, and subsequent volume mesh generation from minimal inputs. The emphasis of this approach has been to maintain all the functionality of the general purpose mesh generator while simultaneously eliminating the bulk of the repetitive and tediuos manual steps in the mesh generation process. Using this approach, mesh generation cycle times have been reduced from the order of days down to the order of hours.

AUTOMESH-2D is a computer program specifically designed as a preprocessor for the scattering analysis of two dimensional bodies by the finite element method. This program was developed due to a need for reproducing the effort required to define and check the geometry data, element topology, and material properties. There are six modules in the program: (1) Parameter Specification; (2) Data Input; (3) Node Generation; (4) Element Generation; (5) Mesh Smoothing; and (5) Data File Generation.

NAS 1.26:3222 Digitized from IA1177305-07-0071 .

In the grant, a number of accomplishments were made in a variety of ways and in a variety of topics. The ways in which this was achieved were in oral communication with others, in the organization of conferences, in the journal publications, in the direction of graduate studies, and in the computer demonstration of theoretical developments. The topics include a study of shock- vortex interaction and a number of studies in grid generation. Those studies covered algebraic and interactive aspects here converged with the establishment of a powerful control point formulation for arbitrary grid generation.

The finite element method for discreitizing differential equations is introduced and briefly compared to finite difference approaches. Advantages of finite element approximations include ease of localized mesh refinement and maximization of computational resources through unstructured, graded meshes and element-based computations. The use of triangular elements lead to realistic representations of shoreline and bathymetric complexity; boundary conditions are naturally incorporated. Presentation of a theoretical framework and state-of-the-art research for determining required mesh resolution in relation to the modeled physics is followed by practical concerns of mesh generation itself. Known tools for mesh generation and promising develop. mental software are discussed. Detailed descriptions of mesh creation, editing, and model diagnostics using the ACE/gredit software, the most versatile and advanced mesh creation and modification tool available, comprises the remainder of this report. Step-by-step instructions for the semi-automatic generation of a finite element mesh are included, which has the potential to advance the rapid relocatability of finite element models.

This research study considered the technologies needed to realize the promise of optimum rates of convergence of adaptive high-order methods based on hp-discretization. The study focused on the development of the technologies needed to support robust p-version mesh generation of general non-manifold geometric domains of interest to the NAVY. To meet the goal of providing appropriate meshes for p-version finite element analysis for general domains, the authors developed an automatic mesh generation procedure to create curved finite element meshes with sufficient order of geometric approximation based on the order of polynomial used in the finite element basis. Although the initial straight-sided mesh generated by automatic mesh generators has all valid elements, the process of curving the mesh edges and faces to the appropriate model boundaries often yields elements with invalid shapes. Therefore, the developed procedures perform mesh modifications to produce a set of valid curved elements. A key ingredient is the geometric representation of the mesh entities. The standard method of Lagrangian interpolation does not lend itself to effective procedures as the order of the elements increases. Therefore, an alternative geometric form based on Bezier approximating geometry was developed. Many of the physical domains of interest to the Navy are structures dominated by thin sections for which the desired finite element discretizations employ thin volume elements. For the most effective application of p-version finite elements it is desirable to generate elements in these thin volumes that do not have long diagonals going through the thick- ness direction. This creates problems for current automatic mesh generators that use only tetrahedra. Therefore, procedures were developed to eliminate, to the greatest possible extent, such diagonals. (3 figures, 10 refs.)

A method for generating and adaptively refining a highly stretched unstructured mesh suitable for the computation of high-Reynolds-number viscous flows about arbitrary two-dimensional geometries was developed. The method is based on the Delaunay triangulation of a predetermined set of points and employs a local mapping in order to achieve the high stretching rates required in the boundary-layer and wake regions. The initial mesh-point distribution is determined in a geometry-adaptive manner which clusters points in regions of high curvature and sharp corners. Adaptive mesh refinement is achieved by adding new points in regions of large flow gradients, and locally retriangulating; thus, obviating the need for global mesh regeneration. Initial and adapted meshes about complex multi-element airfoil geometries are shown and compressible flow solutions are computed on these meshes.

The topics are presented in viewgraph form and include the following: 3D combinatorial edge swapping; 3D incremental triangulation via local transformations; a new approach to multigrid for unstructured meshes; surface mesh generation using local transforms; volume triangulations; viscous mesh generation; and future directions.

The topics are presented in viewgraph form and include the following: 3D combinatorial edge swapping; 3D incremental triangulation via local transformations; a new approach to multigrid for unstructured meshes; surface mesh generation using local transforms; volume triangulations; viscous mesh generation; and future directions.

During this period, the development of algebraic and adaptive grid techniques were continued and brought into a more refined state. Included were the topics of multisurface transformations together with Boolean operations, pointwise distributions on curves, two-dimensional surface grids embedded in three-dimensions, three-dimensional volume grids and the adaptive strategies of mean value relaxation, alternating direction, and center of mass relaxation for triangular meshes. Of paramount importance in the various adaptive strategies was the utilization of weighting functions, which was studied and still requires further development. (Author)

We show that any polyhedron forming a topological ball with an even number of quadrilateral sides can be partitioned into O(n) topological cubes, meeting face to face. The result generalizes to non-simply-connected polyhedra satisfying an additional bipartiteness condition. The same techniques can also be used to reduce the geometric version of the hexahedral mesh generation problem to a finite case analysis amenable to machine solution.

A technique is presented for rapidly producing unstructured finite element meshes in support of large-scale remote sensing simulations. These tetrahedral meshes typically have more than one million elements and more than 250 thousand nodes, and allow for arbitrary placement of objects into the domain. Open-source mesh generation packages are used in conjunction with a tetrahedra element smoothing operation to achieve the desired final meshes. Meshes can be reproduced in less than 30 minutes on a Cray XT3 architecture.

Progress was made in a number of topics including algebraic, adaptive, surface and interactive grid generation. Seventeen papers and one Ph.D. resulted, including such titles as Alternating direction adaptive grid generation, The local redistribution of points along curves for numerical grid generation, Automatic algebraic coordinate generation and The geometric construction of pointwise distributions on curves. In the first paper, an improvement was made for alternating direction adaptive grid generation method. In the second paper, a basic tool for interactive grid generation was developed from an earlier theory of weights. In the third paper, the basic formulations of algebraic transformation were extended by forming Boolean sums of multisurface transformation and, by appropriate caupositiars, lifting the results to curved 2-D surfaces. In the final referenced paper, the capability to accurately partition the available grid points among various properties was established along with the simultaneous specification of grid spacing from the boundaries.

Progress was made in a number of topics including algebraic, adaptive, surface and interactive grid generation. Seventeen papers and one Ph.D. resulted, including such titles as Alternating direction adaptive grid generation, The local redistribution of points along curves for numerical grid generation, Automatic algebraic coordinate generation and The geometric construction of pointwise distributions on curves. In the first paper, an improvement was made for alternating direction adaptive grid generation method. In the second paper, a basic tool for interactive grid generation was developed from an earlier theory of weights. In the third paper, the basic formulations of algebraic transformation were extended by forming Boolean sums of multisurface transformation and, by appropriate caupositiars, lifting the results to curved 2-D surfaces. In the final referenced paper, the capability to accurately partition the available grid points among various properties was established along with the simultaneous specification of grid spacing from the boundaries.