The IMA Volumes in Mathematics

and its Applications

Volume 75

Series Editors Avner Friedman Willard Miller, Jr.

Institute for Mathematics and its Applications

IMA

The Institute for Mathematics and its Applications was established by a grant from the National Science Foundation to the University of Minnesota in 1982. The IMA seeks to encourage the development and study of fresh mathematical concepts and questions of concern to the other sciences by bringing together mathematicians and scientists from diverse fields in an atmosphere that will stimulate discussion and collaboration.

The IMA Volumes are intended to involve the broader scientific community in this process.

1982-1983 1983-1984

1984-1985 1985-1986 1986-1987 1987-1988 1988-1989 1989-1990 1990-1991 1991-1992 1992-1993 1993-1994 1994-1995 1995-1996

Avner Friedman, Director Willard Miller, Jr., Associate Director

********** IMA ANNUAL PROGRAMS

Statistical and Continuum Approaches to Phase Transition Mathematical Models for the Economics of Decentralizp.d Resource Allocation Continuum Physics and Partial Differential Equations Stochastic Differential Equations and Their Applications Scientific Computation Applied Combinatorics Nonlinear Waves Dynamical Systems and Their Applications Phase Transitions and Free Boundaries Applied Linear Algebra Control Theory and its Applications Emerging Applications of Probability Waves and Scattering Mathematical Methods in Material Science

IMA SUMMER PROGRAMS

1987 Robotics 1988 Signal Processing 1989 Robustness, Diagnostics, Computing and Graphics in Statistics 1990 Radar and Sonar (June 18 - June 29)

New Directions in Time Series Analysis (July 2 - July 27) 1991 Semiconductors 1992 Environmental Studies: Mathematical, Computational, and Statistical Analysis 1993 Modeling, Mesh Generation, and Adaptive Numerical Methods

for Partial Differential Equations 1994 Molecular Biology

********** SPRINGER LECTURE NOTES FROM THE IMA:

The Mathematics and Physics of Disordered Media

Editors: Barry Hughes and Barry Ninham (Lecture Notes in Math., Volume 1035, 1983)

Orienting Polymers

Editor: J.L. Ericksen (Lecture Notes in Math., Volume 1063, 1984)

New Perspectives in Thermodynamics

Editor: James Serrin (Springer-Verlag, 1986)

Models of Economic Dynamics

Editor: Hugo Sonnenschein (Lecture Notes in Econ., Volume 264, 1986)

Ivo Babuska William D. Henshaw Joseph E. Oliger

Editors

Joseph E. Flaherty John E. Hopcroft Tayfun Tezduyar

Modeling, Mesh Generation, and Adaptive Numerical Methods for

Partial Differential Equations

With 201 Illustrations

Springer-Verlag New York Berlin Heidelberg London Paris

Tokyo Hong Kong Barcelona Budapest

Ivo Babuska Institute for Physical Science

and Technology University of Maryland College Park, MD 20742 USA

Joseph E. Aaherty Department of Computer Science and

Scientific Computation Research Ctr. Rensselaer Polytechnic Institute 110-8th St. Troy, NY 12180 USA

Joseph E. Hopcroft Joseph Oliger Joseph Silbert Dean of Engineering Research Institute for Cornell University Advanced Computer Science College of Engineering Mail Stop T20G-5 242 Carpenter Hall NASA Ames Research Center Ithaca, NY 14853 USA Moffet Field, CA 94035 USA

Series Editors Avner Friedman Willard Miller, Jr. Institute for Mathematics and its Applications University of Minnesota Minneapolis, MN 55455 USA

William D. Henshaw CIC-3 MS K987 Los Alamos National Laboratory Los Alamos, NM 87545 USA

Tayfun Tezduyar Army High Performance

Computing Research Center 1100 South Washington Ave. Suite 101 Minneapolis, MN 55415 USA

Mathematics Subject Classifications (1991): 65M06, 65M12, 65M15, 65M20, 65M50, 65M55, 65M60, 65M70, 65N06, 65N12, 65N15, 65N22, 65N30, 65N35, 65N50, 65N55, 65Y05, 65YIO, 68Q22, 68T05, 68U05, 68U07, 73B40, 73E99, 73K20, 76D05, 76RIO, 76Z05, 76S05,92C35

Library of Congress Cataloging-in-Publication Data Modeling, mesh generation, and adaptive numerical methods for partial

differential equations I [edited by) Ivo Babuska ... let al.). p. cm. - (The IMA volumes in mathematics and its

applications; v. 75) "Based on the proceedings of the 1993 IMA summer program"-

-Foreword. Includes bibliographical references. ISBN-13: 978-1-4612-8707-0 1. Differential equations, Partial-Numerical solutions­

-Congresses. 2. Numerical grid generation (Numerical analysis)--Congresses. I. Babuska, Ivo. II. Series. QA377.M578 1995 95-17342 515'.353-<1c20

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Xl

FOREWORD

This IMA Volume in Mathematics and its Applications

MODELING, MESH GENERATION, AND ADAPTIVE

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS

is based on the proceedings of the 1993 IMA Summer Program "Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations." We thank Ivo Babuska, Joseph E. Flaherty, William D. Hen­shaw, John E. Hopcroft, Joseph E. Oliger, and Tayfun Tezduyar for orga­nizing the workshop and editing the proceedings. We also take this oppor­tunity to thank those agencies whose financial support made the summer program possible: the National Science Foundation (NSF), the Army Re­search Office (ARO) the Department of Energy (DOE), the Minnesota Su­percomputer Institute (MSI), and the Army High Performance Computing Research Center (AHPCRC).

A vner Friedman

Willard Miller, Jr.

xiii

PREFACE

Mesh generation is one of the most time consuming aspects of com­putational solutions of problems involving partial differential equations. It is, furthermore, no longer acceptable to compute solutions without proper verification that specified accuracy criteria are being satisfied. Mesh gen­eration must be related to the solution through computable estimates of discretization errors. Thus, an iterative process of alternate mesh and so­lution generation evolves in an adaptive manner with the end result that the solution is computed to prescribed specifications in an optimal, or at least efficient, manner. While mesh generation and adaptive strategies are becoming available, major computational challenges remain. One, in particular, involves moving boundaries and interfaces, such as free-surface flows and fluid-structure interactions.

We held a three-week summer program from July 5 to July 23, 1993 entitled "Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations" at the Institute for Mathematics and its Applications (IMA) which emphasized geometric modeling, mesh gen­eration, adaptive strategies, and a posteriori error estimation. Our goals were (i) an exchange of information, (ii) a stimulation of interdisciplinary research, and (iii) a unification of similar principles used in diverse disci­plines. The workshop brought mathematicians, numerical analysts, com­puter scientists, and engineers together in a timely and appropriate means of accelerating achievement of our aims.

The first two weeks of the workshop emphasized geometric modeling and mesh generation and the last week emphasized error estimation. The first two-week period included a one-day workshop on "Unstructured Grid Methods for Fluid Mechanics Problems," hosted by the Minnesota Super­computer Institute (MSI) and jointly sponsored by the Army HPC Re­search Center, IMA, and MSI. This workshop, with speakers from United States, Japan, and Belgium, attracted more than seventy five participants. Adaptive strategies and selected applications were discussed throughout the three-week period and served as a catalyst to stimulate interaction between the various groups. Keynote lectures by Christoph Hoffmann of Purdue University, Mark Shephard of the Rensselaer Polytechnic Institute, Joseph Oliger of Stanford University and RIACS, and Randolph Bank of the University of California at San Diego described the state of the art in geometric modeling, mesh generation, adaptivity using component meth­ods, and error estimation. There were 173 participants in the workshop and 66 keynote, invited, and contributed presentations. This volume represents written versions of 21 of these lectures.

These proceedings are organized roughly in order of their presentation at the workshop. Thus, the initial papers are concerned with geometry and mesh generation and discuss the representation of physical objects and sur­faces on a computer and techniques to use this data to generate, principally,

xv

XVI PREFACE

unstructured meshes oftetrahedral or hexahedral elements. The remainder of the papers cover adaptive strategies, error estimation, and applications. Several submissions deal with high-order p- and hp-refinement methods where mesh refinement/coarsening (h-refinement) is combined with local variation of method order (p-refinement). Combinations of mathemati­cally verified and physically motivated approaches to error estimation are represented. Applications center on fluid mechanics.

The workshop and these proceedings represent an accurate picture of the state of the art of the automatic solution of problems involving partial differential equations. One area that appears to be under-represented con­cerns parallel adaptive procedures. We would expect this area to grow in future symposia on these subjects.

We would like to thank the IMA for giving us the opportunity to hold this workshop. We would also like to recognize and acknowledge the sup­port of the Minnesota Supercomputer Institute and the Army HPC Re­search Center. Individual thanks are extended to Avner Friedman and Willard Miller, Jr. of the IMA for coordinating, scheduling, and provid­ing logistic support for the workshop and to Patricia V. Brick, Stephan J. Skogerboe, and Kaye Smith of the IMA and Pamela Paslow of the Rensse­laer Polytechnic Institute for providing editorial support.

Ivo Babuska, College Park, Maryland Joseph E. Flaherty, Troy, New York William D. Henshaw, Los Alamos, New Mexico John E. Hopcroft, Ithaca, New York Joseph E. Oliger, Palo Alto, California Tayfun Tezduyar, Minneapolis, Minnesota

CONTENTS

Foreword ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. XUi

Preface ............................................................. xv

Workshop schedule ................................................. XXi

List of participants ................................................ xlix

WEEK 1

NURBS and grid generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 Robert E. Barnhill, Gerald Farin, and Bernd Hamann

Coping with degeneracies in Delaunay triangulation. . . . . . . . . . . . . . . . .. 23 Isabel Beichl and Francis Sullivan

Geometric approaches to mesh generation ............................ 31 Christoph M. Hoffmann

Refining quadrilateral and brick element meshes. . . . . . . . . . . . . . . . . . . . .. 53 Robert Schneiders and Jiirgen Debye

Automatic meshing of curved three-dimensional domains: Curving finite elements and curvature-based mesh control ..................... 67

Mark S. Shephard, Saikat Dey, and Marcel K. Georges

WEEK 2

Optimization of tetrahedral meshes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97 Eric Briere de l'Isle and Paul Louis George

A class of error estimators based on interpolating the finite element solutions for reaction-diffusion equations. . . . . . . . . . . . . . . . . . . . . . . . . . .. 129

Tao Lin and Hong Wang

Accuracy-based time step criteria for solving parabolic equations .... 153 Rabi Mohtar and Larry Segerlind

WEEK 3

Adaptive domain decomposition methods for advection-diffusion problems .......................................................... 165

Claudio Carlenzoli and Alfio Quarteroni

xvii

XVlll CONTENTS

WEEK 3 CONTINUED

LP-posteriori error analysis of mixed methods for linear and quasilinear elliptic problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 187

Zhangxin Chen

A characteristic-Galer kin method for the N avier-Stokes equations in thin domains with free boundaries ............................... 201

Giovanni M. Cometti

Parallel partitioning strategies for the adaptive solution of conservation laws .................................................. 215

Karen D. Devine, Joseph E. Flaherty, Raymond M. Loy, and Stephen R. Wheat

Adaptive multi-grid method for a periodic heterogeneous medium in 1 - D .................................................. 243

Jacob Fish and Vladimir Belsky

A knowledge-based approach to the adaptive finite element analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 267

Kamyar Haghighi and Eun Kang

An asymptotically exact, pointwise, a posteriori error estimator for the finite element method with super convergence properties. . .. 277

Jens Hugger

A mesh-adaptive collocation technique for the simulation of advection-dominated single- and multiphase transport phenomena in porous media. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 307

Manfred Koch

Three-step H-P adaptive strategy for the incompressible Navier-Stokes equations ............................................ 347

J. Tinsley Oden, Weihan Wu, and Mark Ainsworth

Applications of automatic mesh generation and adaptive methods in computational medicine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 367

J.A. Schmidt, C.R. Johnson, J.C. Eason, and R.S. MacLeod

Solution of elastic-plastic stress analysis problems by the p-version of the finite element method .............................. 395

Barna A. Szabo, Ricardo L. Actis, and Stefan M. Holzer

CONTENTS XIX

WEEK 3 CONTINUED

Adaptive finite volume methods for time-dependent P.D.E.S .......... 417 J. Ware and M. Berzins

Superconvergence of the derivative patch recovery technique and a posteriori error estimation ..................................... 431

Zhimin Zhang and 1.Z. Zhu

IMA SUMMER PROGRAM MODELING, MESH GENERATION,

AND ADAPTIVE NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS

July 6 - 23, 1993

Coordinators: Joseph E. Flaherty (Chairman), Ivo Babuska, John E. Hopcroft, William D. Henshaw, Joseph E. Oliger, and Tayfun Tezduyar.

WORKSHOP SCHEDULE

WEEK 1: GEOMETRIC MODELING AND MESH GENERATION

Modeling is concerned with three areas:

i Representation of physical paris. Objects are typically represented by solid models for use in conjunction with the analysis of, e.g., stress or heat conduction.

ii Representation of surfaces. Surfaces are common in the automo­bile and aircraft industry for use with, e.g., fluid flow analysis. Emphasis will be placed on surfaces with special features such as dimples, holes, and nipples rather than on those with more general or aesthetic features.

iii Approximation of physical data. The task is to generate a contin­uous representation of samples of discrete experimental data.

Once the physical boundaries of the problem domain have been described/ modeled, it is necessary to create a surface or volume grid to be used in the solution of the partial differential system. Mesh generation involves a number of approaches. The solution of partial differential equations can often be facilitated by transforming the equations to a different domain where they are easier to solve. The simplest examples of this arise in the use of conformal mappings to solve the Laplace equation, but such trans­formations are also used in a much broader context. Many systems may be solved on a single grid; however, others benefit when a set of component grids are used to generate a composite grid where the component grids can be used to effectively match the boundaries of the domain. While struc­tured grids provide efficiency, the flexibility of unstructured grids is often desired. Other important issues concern the shape of computational cells in relation to the two- or three-dimensional domain. Lectures on the ap-

plication of adaptive methods to problems in science and engineering will complement the talks on modeling and mesh generation. Problems that emphasize geometrical complexity will be selected from computational me­chanics, electro-magnetics, and thermodynamics and heat conduction.

xxi

XXll WORKSHOP SCHEDULE

Tuesday, July 6

Louis J. Billera, Cornell University Fiber polytopes and the set of subdivisions of a fixed point set

Abstract: Given a set of n points A C Rd , we wish to study the set of all possible triangulations of the polytope Q = conv A using only vertices from the set A. This set turns out to be parametrized by an (n - d - 1)­dimensional polytope, ~(A), the secondary polytope of A, introduced in recent work of Gel'fand, Kapranov and Zelevinsky studying generalized discriminants. The vertices of the secondary polytope correspond to trian­gulations of A, while edges correspond to familiar transformations between triangulations. The notion of secondary polytope opens up the possibility of studying prop­erties of triangulations of A by means of properties of the associated poly­tope ~(A). It is relatively straightforward, for example, to determine which vertex of ~(A) corresponds to the Delaunay triangulation ofthe set A, dual to the Voronoi diagram of A. We describe a polytope, introduced in joint work with B. Sturmfels, which is more general than the secondary polytope and which allows one to study subdivisions more general than triangulations. For a projection of convex polytopes 7r : P -+ Q, we consider the Minkowski average over all fibers 7r- 1 (x), x E Q. This average ~(P, Q) is a convex polytope, called the fiber polytope, whose face lattice mirrors the combinatorial structure of all poly­hedral subdivisions of Q induced by the faces of P. We study ~(P, Q) for a number of important cases of 7r and P. For example, when Q = conv A, the canonical projection of the vertices of the (n - 1 )-simplex .6.n - 1 onto A leads to the secondary polytope ~(A). On the other hand, if P = en, the n-cube, then ~(P, Q) has vertices corresponding to tilings of the zonotope Q = 7r( P) by zonotopes.

KEYNOTE LECTURE

Christoph M. Hoffmann, Purdue University Geometric approaches to mesh generation. I

Abstract: We describe three different geometric approaches to mesh gener­ation. In the first approach, due to Anderson and Cox, a shape is thought to be composed from simpler primitives such as cylinder, block, and sphere. Each primitive is separately meshed, and the final object is then meshed with overlapping meshes. Based on the overlapped meshes, the finite ele­ment problem is formulated. In the second approach, mesh overlap is ge­ometrically resolved by moving nodes and remeshing overlapping volumes. The third approach, due to Armstrong, is based on the medial-axis trans­form and generates hexahedral elements. Here, the medial-axis transform serves to subdivide the volume into smaller regions, and the interaction of adjacent elements across regions is analyzed resulting in a suitable meshing strategy. There are a number of approaches related to the third approach,

WORKSHOP SCHEDULE XXlll

especially in two dimensions, based on the medial-axis or on a Delaunay triangulation. For each approach, we will describe the fundamental ideas rather than the full technical details in the hope that this co-exposition of different techniques stimulates debate among the participants.

Itzhak Levit, Lockheed Missiles & Space Co. Adaptive mesh refine­ment for large scale aerospace shell structures

Abstract: Adaptive Mesh Refinement, AMR, in the context of general pur­pose, large scale, aerospace structural analysis involves three different as­pects: refinement strategies, error estimation techniques and large scale equation solvers - all of which will be addressed in this talk. A variety of adaptive refinement strategies have been implemented by the Lockheed team in the NASA sponsored COmputational MEchanics Testbed (COMET) and in Lockheed's Structural Analysis of General Shells (STAGS) programs over the last four years. These methods include an array of mesh transition-based techniques (hrrefinement), inter-element constraint-based techniques (he-refinement), polynomial-enrichment tech­niques (p-refinement), and a superposition based technique (h.-refinement) each of which will be briefly described. A new error estimation technique based on a "Look Ahead" approach is currently being developed and tested in the context of AMR. The basic theory and implementation aspects of the method will be presented as well as some preliminary results. A fully vectorized iterative equation solver based on the PCG method and a COMPACT storage format will be presented together with the results of a numerical performance study. Finally, the results of applying the above tools to the analysis of a NASA High Speed Civil Transport (HSCT) aircraft model (mach 2,3 version) will be presented and discussed. This research is sponsored in part by NASA/Langley CSM and ASIP pro­grams and in part by Lockheed Independent Research Computational Me­chanics and Multi-Disciplinary Analysis programs.

Francis Sullivan, Supercomputing Research Center Coping with dege­neracies in mesh generation

Abstract: The fundamental geometrical operation in most 3-d mesh gen­eration codes is construction of tetrahedra of the 3-d Delaunay triangula­tion. We describe an extremely efficient 3-d triangulation algorithm and report on the performance of a CM-5 implementation. The code triangu­lates 16,000 points giving more than 100,000 tetrahedra in less than 50 seconds.) An interesting aspect of the parallel implementation is a new method of checking for and handling of degeneracy. In their primitive form, geomet-

XXIV WORKSHOP SCHEDULE

rical algorithms simply assume that there is no degeneracy. If primitive methods are used on data that is nearly degenerate, they either fail com­pletely or else give nonsensical results. It turns out that the best existing methods for degeneracy checking depend very strongly on the strict se­quential ordering of the construction of the tetrahedra. Joint work with Isabel Beichl.

Wednesday, July 7

Doug Moore, Rice University Limits on balanced quad tree growth

Robert Schneiders, RWTH Aachen Generation of hexahedral element meshes for metal forming simula­tion

Abstract: The three-dimensional simulation of large plastic deformations is often hampered by fast mesh distortion. Meshes must be recreated very often (remeshing) which makes simulations practically impossible if mesh generation is not performed automatically. In our talk we present an algorithm for the generation of hexahedral element meshes. We use a grid-based approach: the interior of the geometry is meshed with a regular mesh, the boundary region is filled by a projection technique. The quality of the mesh generator is demonstrated by the example of the simulation of a complex forging process.

KEYNOTE LECTURE

Christoph M. Hoffmann, Purdue University Geometric approaches to mesh generation. II

Peter R. Eiseman, Program Development Corporation GridPro/ az3000 - Multiblock grid generation

William Henshaw, IBM T.J. Watson Research Centre Software for the solution of PDEs on overlapping grids

Abstract: Overlapping grids can be used to create smooth grids on compli­cated geometries. An overlapping grid consists of a set of logically rectan­gular curvilinear grids that cover a domain and overlap where they meet. The grid construction program CMPGRD has been developed by Geoff Chesshire and myself for the creation of overlapping grids in two and three space dimensions. Efficient and accurate finite-difference methods can be implemented on overlapping grids. In this talk I will describe some programs that have been written to solve a variety of PDEs on overlapping grids. These pro­grams include

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(1) a solver for a general class of boundary value problems (e.g. Laplace's equation, steady Navier-Stokes) in 2D/3D to 2nd/4th order accu­racy using a variety of iterative and direct solvers such as conjugate­gradient or multigrid.

(2) a solver for the incompressible Navier-Stokes (2D/3D, 2nd/4th or­der accurate).

(3) a solver for hyperbolic-parabolic systems (e.g. the compressible N avier-Stokes).

Min-Yee Jiang, Mississippi State University From EAGLE to NGP, I Abstract: In 1991 the NSF Engineering Research Center at Mississippi State formed a national coalition of industrial and federal laboratory participants to develop a comprehensive grid generation system defined by users and de­signed by experts. This NSF /ERC provides a natural mechanism to form such a coalition of interest, funding, and talent, and is a natural and endur­ing agency to support, maintain, and enhance the system. The objective of this National Grid Project is to increase the efficiency and productivity of the grid generation process to the high level required for high performance computational simulation of real-world systems for design and analysis in engineering applications. The Project is achieving this objective by de­veloping a comprehensive interactive/automated grid generation system which will provide a uniform environment for the construction of compu­tational grids using state-of-the-art technology. This comprehensive grid generation system is a necessary element to meet the grand challenges of the nation's High Performance Computing and Communications Program. The system incorporates a surface definition system including an interface with IGES files and CAD systems; B-splines and NURBS; surface-surface intersection; surface quality enhancement; a surface grid generation sys­tem based on surface parametric coordinates; a volume grid generation system including transfinite interpolation, elliptic and hyperbolic genera­tion systems; both block-structured and unstructured grids with hybrid and chimera (overset) combinations; and orthogonal, reflective, and peri­odic boundaries. Block interfaces and connectivity are automated, and the system includes automated fault diagnostics and corrective measures.

Wayne Mastin, Mississippi State University From EAGLE to NGP, II

Stephen Vavasis, Cornell University 3-D mesh generation with prova­ble quality bounds

Joint work with S. Mitchell, Sandia.

Thursday, July 8

Geoffrey Chesshire, IBM The CMPGRD system for overlapping grid generation

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Abstract: We discuss some issues in the generation and use of overlapping curvilinear grids, including:

• Generation of smooth grids for high-order accurate discretizations • Reparametrization of surfaces to eliminate grid singularities • Overlapping fillet and collar grids • Algorithms for computing the connections between component grids • Estimation and control of grid quality • Optimizations for time-dependent geometry • Support for adaptive mesh refinement

We (Bill Henshaw and I) have implemented these ideas in CMPGRD, a system for overlapping grid generation in two and three dimensions.

Robert Barnhill, Arizona State University NURBS and grids Abstract: We present some areas where NURB-based tools may aid nu­merical grid generation. We discuss curve and surface smoothing, and also accelerated grid generation. We also address the problem of reparametriz­ing NURB surfaces in order to achieve more uniform grids. Joint work with Gerald Farin.

L. Paul Chew, Cornell University Mesh generation, curved surfaces, and guaranteed-quality triangles

Abstract: We present a mesh generation technique that creates high-quality triangular meshes for both flat regions and regions on curved surfaces. The resulting meshes are guaranteed to exhibit the following properties: (1) internal and external boundaries are respected, (2) triangle shape is guar­anteed - all triangles have angles between 30 and 120 degrees (with the possible exception of some badly shaped triangles that may be required by the specified boundaries), and (3) density can be controlled, producing small triangles in "interesting" areas and large triangles elsewhere. This technique can be extended to provide control of triangle shape across re­gions, producing, for instance, elongated triangles along boundaries. This work is based on a type of constrained Delaunay triangulation generalized in such a way that Delaunay-like properties hold even for triangulations on curved-surfaces.

KEYNOTE LECTURE

Mark S. Shephard, Rensselaer Polytechnic Institute The automatic generation of valid finite element mesh­es for general 3-D geometric domains

Abstract: Topics covered in this lecture will include:

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1. The definition of a valid finite element mesh, a geometric triangu­lation

2. A mesh validity assurance algorithm 3. An overview of the algorithmic approaches to automatic 3-D mesh

generation 4. A description of the Finite Octree mesh generator

I. Schmelzer, Inst. fur Ang. Analy. und Stochastik A I-3D grid genera­tor with functional geometry interface

Abstract: It is a Delaunay grid generator based on an anisotropical oc­tree approach. The geometry interfaced is based on a set of dimension­independent query functions. It allows grid generation without the cre­ation of a boundary grid, for example using characteristic functions. It was developed for complex, time dependent geometries, for example in semi­conductor technology simulation.

Chrisochoides Nikos, Syracuse University Parallel structured grid generation for 2 and 3-dimensional general domains

Abstract: Data-parallel PDE solvers for distributed memory MIMD sys­tems require the mapping of a grid or a coefficient matrix (discrete PDE operator) into the local memory of the processors. A traditional approach in solving this problem is the sequential grid generation or discretization of the PDE operator and the use of graph partitioning/allocation/coloring methods for the solution of the data-mapping problem. There are two disadvantages in this approach: (1) the sequential grid generation (or dis­cretization of the PD E operator) and sequential loading of the grid (or the coefficient matrix) to the processors, and (2) the partitioning of a grid (or coefficient matrix) graph with very large number of vertices (or edges) -for real problems the number of vertices (or edges) is of the order of 106 .

In this talk we address these problems and solve them by presenting a data-parallel grid generation method based on composite block structures.

To implement efficiently the data-parallel grid methods we are developing a Mapping Environment for Numerical Unstructure & Structure - Compu­tations (MENUS-C). We will outline MENUS-C software architecture and describe in more detail the Data Mapper, a library of Templates for Do­main Decomposition Methods, and the Scheduler, a machine-independent, multithread based library of communication and synchronization routines required for the implementation of the grid generation on large MIMD systems with distributed and shared address space.

Ramana G. Venkata, Stanford University 3-D composite grids for fluid flow applications

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Friday, July 9

Timothy J. Baker, Princeton University Generation of surface and volume meshes based on the Delaunay triangulation

Abstract: Algorithms based on the Delaunay triangulation have proved to be very effective and robust for the generation of triangular and tetra­hedral meshes around complex shapes. This paper will describe an imple­mentation that achieves O(N log N) time complexity generating tetrahedral meshes of N points. This will be followed by a discussion of issues which determine mesh quality and ways of improving mesh quality by the selective addition of extra mesh points.

KEYNOTE LECTURE

Mark S. Shephard, Rensselaer Polytechnic Institute The integration of automatic mesh generation with geometric modeling

Abstract: Topics covered in this lecture will include:

1. The integration of automatic mesh generation and general non­manifold geometric modeling through functional interfaces

2. Example procedures for the automatic generation of finite element models for non-manifold geometric models

3. Consistent handling of geometric tolerances during mesh genera­tion

4. Mapping functions to support element stiffness integration account­ing for the model geometry

5. Support of evolving geometry problems 6. A non-manifold mesh data structure and multiple meshes within

a single tree

Marshall Bern, Xerox Corporation Optimal triangulation problems

WEEK 2: ERROR ESTIMATION AND ADAPTIVE STRATEGIES

Week 2 will be concerned with error estimation for finite difference solu­tions and component methods of adaptive analysis. A single mathematical model can rarely be used over the entire computational domain or over an entire range of accuracy specifications. Significant gains in efficiency can be achieved by using appropriate simpler models over the majority of the domain while reserving the more complex models for those regions where accuracy or physical considerations dictate their use. The necessary form of stability requirements will be derived. Having such stability estimates,

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techniques of obtaining error estimates by Richardson's extrapolation and deferred correction will be described with an emphasis on hyperbolic and parabolic problems.

A posteriori error estimates for high-order finite difference and finite ele­ment methods as well as for spectral methods will also be discussed. Lec­turers will illustrate adaptive techniques for component, high-order, and spectral methods. These adaptive strategies involve automatic mesh re­finement / coarsening (h-refinement), automatic local variation of the order of the numerical methods (p-refinement), and mesh motion (r-refinement). Techniques that combine these basic approaches will also be represented.

Applications will be chosen from incompressible and compressible fluid me­chanics.

Monday, July 12

KEYNOTE LECTURE

Mark S. Shephard, Rensselaer Polytechnic Institute Mesh control functions and automated, adaptive ana­lysis with auto­matic mesh gene­rators

Abstract: Topics covered in this lecture will include:

1. Curvature-based mesh refinement 2. Mesh modification operators to refine, coarsen and improve ele­

ment shapes 3. An explicit node point smoothing algorithm 4. Examples of Finite Quadtree and Finite Octree based automated,

adaptive analysis procedures for thermomechanical analysis, form­ing simulations, CVD and unsteady aerodynamics

5. A generalized analysis attribute structure 6. Description of a general analysis idealization control framework

KEYNOTE LECTURE

Joseph Oliger, Stanford University Adaptive component grid methods for time dependent PDE's. I

Abstract: These two lectures will outline the development of adaptive com­ponent grid methods for time dependent partial differential equations and discuss local error estimation, convergence of these methods, and the form of stability estimate required to obtain global error control. The first lec­ture will discuss the use of component grids for an adequate description of

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the problem domain and the algorithm for generating adaptive refinements to obtain an adequate representation of the solution.

J .E. Aiken, Rice University rp-Adaptive finite element analysis for incompressible flows

Abstract: We present the solution ofthe incompressible Navier-Stokes equa­tion by adaptive finite element solutions. This is done first by a classic r-method adaption, and then by a new rp-method. We begin by briefly reviewing the common adaptive methods. They all assume the availability of some type of error indicator or error estimator. Adaptive methods are usually compared in terms of the reduction in error as a function of the increase in the total number of degrees of freedom to be computed. The p-method [8] keeps the mesh topology unchanged, but does vary the el­ement interpolation order as indicated by the error indicators. Some codes upgrade to the maximum required degree and use it in uniformly in all elements. However, it is more common to vary the order on each element edge or face. When two elements on a common interface have different polynomial orders, one must enrich the edge interpolation functions of the lower order element [6]. This can be done with Lagrangian or Hierarchial functions, although the latter are preferred by most analysts. Of course, the functions used to define the edge order of an element propagate into its interior. The hp-method also splits some elements into smaller elements as it changes the local order. The hp solutions are often optimal because the error continues to curve down as N increases (as more degrees of freedom are added). That is, there is an exponential decrease in the error as the number of unknowns is increased. In a classic r-method, or optimal grid adaptivity, one holds the number of degrees of freedom fixed while seeking the best approximate solution. It relocates the nodes to find the best finite element approximation for a fixed polynomial interpolation order. Often one may solve an elliptic system for the node coordinates. We have utilized a Min-Max procedure for equal ele­ment error, and a direct relocation calculation that treats the error density as a mass like quantity [7]. Typically one obtains a 30 to 40% reduction in error. The new rp-method allows additional error reduction through the addition and removal of unknowns on each element edge. Thus, it brings together proven technologies through the combination of local p- and r-methods. The work takes advantage of the most recent advances in error estimators and variable degree element technology to develop an adaptive rp-method that relocates nodes on a mesh of fixed connectivity while locally varying the degree p of each element edge (and interior). By assuring that elements on a common edge or face are the same size and polynomial degree we avoid any constraint equations of the type that occur in the h- and hp-methods. We present a rp-method version for solving the 2-D Navier-Stokes equa­tions via a pressure segregation algorithm using variable degree Serendipity

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quadrilateral elements with different interpolation orders on the edges of an element. We refer to this as anisotropic interpolation. Detailed procedures for two-dimensional elements have been given by EI-Zafrany and Cookson [4], while a large family of two- and three-dimensional elements has been given by Attenbach and Scholz. Here the mesh locations (and thus the element Jacobian) change with each iteration. Therefore, all the element matrices are computed in each mesh relocation so there is not a disadvan­tage in a Lagrangian approach. Also, the Lagrangian interpolation leads to a mass matrix that can be easily diagonalized, which is not true of a hierarchial mass matrix used in the hp-methods. We employ the element patch error estimators [1,9] and concentrate on how best to relocate the corner vertices (and possibly side nodes) of the mesh. The grid relocation seeks to produce equal error in each element. For an element with a larger than average error we want its nodes to move so as to reduce its area. Since a node is connected to several elements it should move in the direction of its neighbor that has the current largest error. The movement magnitude depends on the relative error size in the elements surrounding the node. This suggests using the element error den­sity (error per unit area) as a "mass-like" quantity assigned to each element centroid. Each node is moved to the center of "mass" of the elements in the current relocation iteration. Nodes initially on the boundary are moved back to the boundary via an orthogonal projection from their new location. Since a p-enrichment of an element is more expensive than a mesh reloca­tion we establish heuristic rules to relocate the mesh first. If a relocation would caused the elements to become badly distorted then a p-enrichment is attempted instead. A small expert system controls the choice of element relocation and/or enrichment or de-enrichment. We present results for a flow over a square step, the driven cavity, and the sudden expansion flow problem. The rp-adaptive method is shown to be relatively simple yet effective in producing improved solutions.

References

1 Ainsworth, M. and Oden, J .T., "A Procedure for A Posteriori Error Estimation for h-p Finite Element Methods," Compo Math. Appl. Mech. Eng., VoLl01, pp. 73-96, 1992.

2 Akin, J.E., Finite Elements for Analysis and Design. London: Academic Press, 1993.

3 Diaz, A. R., Kikuchi, N., and Taylor, J. E., "A Method of Grid Optimization for Finite Element Methods," Compo Meth. Appl. Mech. Eng., Vo1.41, pp. 29-45, 1983.

4 EI-Zafrany, A. and Cookson, R.A., "Derivation of Lagran- gian and Hermite Shape Functions for Quadrilateral Elements," Int. J. Num. Meth. Eng., Vo1.23, pp. 1939-1958,1986.

5 Oden, J.T., Demkowicz, L., Rochowicz, W., Westermann, T.A.,

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and Hardy, 0., "Toward a Universal h-p Adaptive Finite Element Strategy - Parts I, II, III," Compo Meth. Appl. Mech. Eng., Vol.77, pp. 79-212,1989.

6 Oden, J .T., "The Best FEM," Finite Elements in Analysis and Design, Vol.7, pp. 103-114,1990.

7 Ramaswamy, B. and Akin, J. E., "Design of an Optimal Grid for Finite Element Methods Incompressible Fluid Flow Problems," Engineering Computations, Vol.7, pp. 311-326, Dec. 1990.

8 Szabo, B. and Babuska, I., Finite Element Analysis. John Wiley & Sons, 1991.

9 Zienkiewicz, O.C. and Zhu, J .Z., "Superconvergent Patch Recov­ery Techniques and Adaptive Finite Element Refinement," Compo Meth. Appl. Mech. Engr., Vo1.101, pp. 207-224,1992. Joint work with B. Ramaswamy.

T. Belytschko, Northwestern University Adaptive methods for fracture problems

Abstract: The modeling of fracture is of great importance in industry be­cause it is crucial to the life and performance of a component. One of the great difficulties in modeling fracture is that the crack path is usually not rectilinear and is therefore difficult to accommodate within the structure of a finite element mesh. Although some success has been achieved by interac­tive remeshing, in which the user guides the path of the crack such methods are awkward and also introduce errors when projections are made between different meshes. In this presentation, several hybrid methods involving element-free-Galerkin methods and boundary element methods are studied and compared to adaptive finite element mesh procedures. Some of the topics which are studied are: the accuracy with which the stress intensity factor can be computed in its relationship to errors in the element stresses, the implications of this on error indicators, and the cost alternative pro­cedures. Several examples of an academic and also of industrial interest, such as the modeling of fracture problems in a setting aging aircraft, are reported. Joint work with Y.Y. Lu, T. Blacker, and M. Tabbara.

Joseph Maubach, University of Pittsburgh n-Dimensional n-simplex adaptive refinement

Rabi Mohtar, Michigan State University Accuracy-based time-step criteria for solving parabolic equations

WORKSHOP SCHEDULE

Tuesday, July 13

Workshop on Unstructured Grid Methods for Fluid Mechanics Problems

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Cosponsored by the Minnesota Supercomputer Institute and the Army High Performance Computing Research Center

Tayfun Tezduyar, organizer

Tayfun Tezduyar, University of Minnesota/ AHPCRC Welcoming Remarks

L.E. Scriven, University of Minnesota Generating grid in order to solve viscous free surface flow problems in coating theory

Thomas Hughes, Stanford University A potpourri of recent developments in stabilized methods for computa­tional fluid dynamics

T. Tezduyar, University of Minnesota Mesh update strategies for mass­ively parallel finite element com­putation of flow problems in­volving moving boundaries and interfaces

Joint work with A. Johnson, M. Behr, and S. Mittal.

Herman Deconinck, von Karman Institute, Brussels Multidimensional upwinding: From scalar advection to systems of con­servation laws

Timothy Barth, NASA Ames Research Center Unstructured grids and finite-volume solvers for the Euler and N avier­Stokes equations

Abstract: The one hour lecture will discuss recent advances in three-dimen­sional tetrahedral mesh generation and flow simulation. The first portion of the lecture will discuss an incremental triangulation algorithm for con­structing Delaunay triangulations and several variants of this algorithm which locally optimize other grid quality measures. Nontetrahedralizabil­ity and other issues related to the construction of constrained and/or con­forming triangulations in three space dimensions will be discussed. The suitability of these algorithms for general mesh adaptation will also be dis­cussed and demonstrated for fluid flow simulations arising in computational aerodynamics.

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The second portion of the lecture will briefly review an upwind finite­volume algorithm for solving the Euler and Navier-Stokes equations. The lecture will focus on the following recent improvements to this algorithm: implicit solution strategies using domain-decomposed and preconditioned minimum (quasi-minimum) residual methods, edge data structures, and parallel implementations. Examples will be shown for aerodynamic prob­lems of interest.

Kazuo Kashiyama, Chuo University, Tokyo An unstructured grid met­hod for water environmen­tal flow problems

Abstract: Mesh generation is one of the most time-consuming aspect of the finite element analysis of water environmental flow, such as shallow water flow, since the geographical configuration of the analytical region is gener­ally complicated and the data of water depth is needed for computations. For this type of environmental flow problems, an efficient automatic mesh generator is developed based on Delauny triangulation method. Using this method, finite element mesh can be generated with the Courant number or the ratio of wavelength and element size to be nearly constant in accor­dance with the geographical configuration. Consequently, the numerical accuracy and stability can be improved considerably. The automatic mesh generation method is applied to some flow modeling with complicated geo­graphical features to check the ability of the method. The present method is more economical in computational time than the conventional methods.

Wednesday, July 14

KEYNOTE LECTURE

Joseph Oliger, Stanford University Adaptive component grid methods for time dependent PDE's. II

Abstract: The second lecture will discuss the error estimation issues which are required to control the algorithm and to guarantee an error bound for the computed results.

Raymond C.Y. Chin, ,Indiana U. - Purdue U. at Indianapolis Preconditioning, domain decomposition adaptivity and all that

Abstract: Solutions to problems of technological interests in science and en­gineering usually have many scales of variations which underscore the var­ious locally competing physical, chemical, and biological processes. Prob­lems of this nature, called multiple-scales problems, are difficult to solve accurately and efficiently using standard numerical techniques because of the appearance of local regions of large gradients and/or temporal varia­tions. This talk is concerned with how to incorporate analytic information in the construction and development of high resolution numerical methods for multiple-scales problems of science and engineering. Analytic inform a-

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tion is used to suggest strategies in performing domain decomposition, in develop non-standard discretization techniques, in the selection of meshes, and in deriving local coordinate transformations to expose the underly­ing interacting processes. Examples will be given to illustrate the use of analytic information. For scientists and engineers, computers are used to study and gain insight into the inherent physical, chemical, and biological processes. The resulting solutions to problems, given as tables of numbers or charts of solution curves, may not be easily interpreted or understood. Advanced visualization techniques with interrogative capabilities together with local analysis must be employed to sort-out the details. In looking at the process, then, it would seem a good idea to suggest that, if analysis is to be a part of the course of investigation, why don't we use it from the outset. Great benefits can be accrued from a rudimentary analysis as it can guide us in the development of accurate and efficient methods for solving the problem as well as in the interpretation of the solution.

Roger Strawn, NASA Dynamic mesh adaption for 3-D unstructured grids

M.D. Smooke, Yale University Computational and experimental study of axisymmetric laminar flames

Abstract: The ability to predict the coupled effects of complex transport phenomena with detailed chemical kinetics is critical in the modeling of turbulent reacting flows, in improving engine efficiency and in understand­ing the processes by which pollutants are formed. While the majority of multistep kinetic flame studies in the literature have been one-dimensional in nature, recent advances in the development of computational algorithms, reaction kinetics and mainframe supercomputers have enabled the combus­tion scientist to investigate chemically reacting systems that were compu­tationally infeasible only a few years ago. In this talk we focus our atten­tion on one such system-the axisymmetric laminar flame. We investigate experimentally and computationally the structure of two-dimensional, ax­isymmetric, laminar, methane-air flames in which a cylindrical fuel stream is surrounded by a coflowing oxidizer jet. Experimentally, spontaneous Ra­man spectroscopy is used to generate profiles of the major species with laser induced fluorescence for the OH and CH radical fields. The temperature is determined from the number density of the species. Computationally, we develop a detailed chemistry, complex transport combustion model to predict the temperature, velocity and species concentrations as a function of the two coordinate directions. Results of the study include 1) a detailed quantitative description of the fluid dynamic-thermochemistry structure of the flame 2) an illustration of the applicability of partial equilibrium/full equilibrium chemistry approximations in diffusion flames 3) an assessment of mole fraction versus mixture fraction correlations for flamelet models of turbulent combustion and 4) a discussion of issues related to the modeling

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of the fluid dynamic and thermochemistry solution fields on both serial and parallel computers.

Thursday, July 15

Frederic Hecht, INRIA 2D and 3D automatic mesh generation and adaptation for finite element methods

P.L. George, INRIA Shape optimization for tetrahedral meshes

James Quirk, NASA Simulating shock wave phenomena

Sukumar Chakravarthy, Rockwell International Science Center Unification of exterior and interior boundary conditions for inviscid computational fluid dynamics

Abstract: A novel and versatile unification of "exterior" and "interior" boundary conditions has been developed. This enables boundary conditions to be imposed at the edges of the computational domain or in its interior. This leads to very easy and unified treatment of multizonal, overlapped, nonoverlapped, and sliding grids as well as moving objects. This unification is part of a larger comprehensive numerical framework that includes mul­tidimensional Essentially NonOscillatory (ENO) interpolation, general cell shapes, generalized neighbors, etc. These aspects will be described along with a particular application of the approach to computationally simulate the flow past separating stores. The talk will end by summarizing some implications of this approach to modeling, mesh generation and adaptive numerical methods.

M.B. Bieterman, The Boeing Company An adaptive grid method for aerodynamic analysis and design

Abstract: One gridding approach for aerodynamic analysis and design prob­lems involving complex configurations is the use of locally refined rectan­gular grids that are automatically adapted to the solution. This is the approach implemented in Boeing's TRANAIR code, which incorporates a steady full potential flow model. TRANAIR can be applied to perform inviscid and boundary layer coupled CFD analyses in two and three dimen­sions. An inverse design and optimization capability has been implemented as well. The discretization methods used for these problems include finite element approximations of potential on the rectangular grid elements and interelement upwinding of density in supersonic flow regions. Grids are lo­cally refined and coarsened during the solution process according to values of computed local error indicators. In this lecture, aspects of the adaptive gridding strategy employed for various problems will be described. This will be followed by comparison of results for several examples. The examples

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will be chosen to illustrate the authors' recent work, which includes im­plementation of second order upwinding for higher resolution in supersonic flow regions and the development of the inverse design and optimization capability. Joint work with J.E. Bussoletti, C.L. Hilmes, W.P. Huffman, F.T. Johnson, R.G. Melvin, and D.P. Young.

Frederic Hecht, INRIA 2D mesh generator and boundary constructor (EMC2)

Friday, July 16

Ron Trompert, CWI-Netherlands A refinement strategy for the local uniform grid refinement method

Abstract: Local uniform grid refinement is an adaptive grid technique for computing solutions of partial differential equations which are locally steep. The main feature of local uniform grid refinement is that the PDEs are solved on a series of nested local uniform increasingly finer sub grids which are created up to a level of refinement where sufficient spatial accuracy is reached. On each local subgrid a new initial boundary value problem is solved for one time step in a consecutive order, from coarse to fine. The sub grids are automatically adjusted at discrete times in order to follow the movement of the steep parts of the solution. This lecture discusses the results of an error analysis. From this a refine­ment strategy which controls the grid refinement process is derived based on estimating and controlling the global space error. The refinement strat­egy aims at domination of the global space error by the global space error at the finest subgrid. This strategy does not only take the inevitable trun­cation errors into account but also the additional errors introduced by the grid refinement process itself. The philosophy behind the refinement strategy is very general which im­plies that it is applicable to various systems of PDEs. The performance of the local uniform grid refinement method together with this refinement strategy will be illustrated with some numerical example problems.

Tao Lin, VPI & SU Estimating gradients of solutions of partial differ­ential equations by finite elements

WEEK 3: ERROR ESTIMATION AND ADAPTIVE STRATEGIES

In Week 3, we continue the discussion of error estimation procedures with a concentration on techniques for finite element methods. Strategies for both initial and boundary value problems will be considered. Combinations

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of mathematically verified and physically motivated approaches will be described with a goal of stimulating interaction between groups working on each strategy. Optimal adaptive strategies including combinations of mesh refinement and order enrichment (hp-refinement) will be addressed. Applications will emphasize solid mechanics and chemical reactions.

J. Tinsley Oden, University of Texas-Austin A unified approach to a­posteriori error estimati­on for hp-adaptive finite element methods

Abstract: A general approach to a-posteriori error estimation for non­uniform hp-adaptive finite element approximations of various classes of boundary-value problems is described. Rigorous a-posteriori error bounds are derived for elliptic systems, Stokes' problem, and the Navier-Stokes equation. In addition, a parallel hp-adaptivity scheme is described together with sev­eral applications. The presentation summarizes work by the author, Mark Ainsworth, Wei­han Wu, and Vincent Legat on adaptive methods for linear and nonlinear boundary-value problems.

Claes Johnson, Chalmers University Adaptive finite element methods for incompressible and compress­ible flow.

Abstract: We present theoretical and numerical results indicating that it may be possible to realize reliable and efficient error control in Computa­tional Fluid Mechanics, for both incompressible and compressible flow. As far as we know these are the first results to this effect. We present adaptive algorithms for finite element methods based on a posteriori error estimates. The proofs of the a posteriori error estimates are based on a combination of (i) strong hydrodynamic stability, and (ii) Galerkin orthogonality. The critical issue is the evaluation of the strong stability which measures the accumulation in time of certain derivatives of the solution of an associated linearized perturbation equation in terms of the initial data. We present some model cases, where the evaluation of strong stability can be made by analytical means, and discuss evaluation by computational means for more complex cases.

M. Berzens, The University, Leeds Reliable finite volume methods for time-dependent p.d.e.s

Abstract: In the numerical solution of time-dependent p.d.e.s the accuracy is influenced by the spatial discretisation method used, the spatial mesh and the method of time integration. In particular the spatial discretisation method and positioning of the spatial mesh points should ensure that the spatial error is controlled to meet the user's requirements. It is then desir-

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able to integrate the o.d.e. system in time with sufficient accuracy so that the temporal error does not corrupt the spatial accuracy or the reliability of the spatial error estimates. However, in most existing methods for time dependent p.d.e.s either a stability control and a specialized time integra­tion method is employed or an an o.d.e. solver is used which makes use of local time error control that is unrelated to the spatial error. This paper is concerned with combining the spatial and temporal error balancing approach of Berzins [5] with the adaptive mesh algorithms of Berzins et al. [1], [3]. This combination of error control strategies provides an algorithm that adapts the space mesh in order that the spatial error is controlled and then, accordingly, adjusts the local error tolerance used in the o.d.e. integrator so that the two errors are related in some way. This not only ensures that the main error present is that due to spatial discreti­sation but offers the tantalizing possibility of overall error control. In the error balancing procedure the local time error is controlled so that it is a fraction of the growth in the spatial error over the timestep. An analysis is presented to show that although the method attempts to con­trol accuracy a Courant-like stability condition is also satisfied. Numerical experiments show that the error control strategy appears to offer an effec­tive method of balancing the spatial and temporal errors. The unstructured triangular mesh spatial discretisation method, see [1] and [4], is a cell-centered, finite volume scheme that is shown to achieve second-order accuracy by use of a six triangle stencil around each edge. The growth of the spatial error is measured locally in time by using the computed solution as the high-order solution and constructing an error estimate by calculating a computationally inexpensive low-order solution. Thus local extrapolation in space is effectively being used. These spatial error estimates are shown experimentally to mimic the behavior of the global spatial error. Adaptive unstructured mesh techniques e.g. [1] or [3], are used to rezone the mesh at times chosen by the time integration algorithm. The selection of appropriate times is made by using a combi­nation of present estimated errors and predicted future errors. The time integration technique used in this paper is a new adaptive Theta Method, [2], suitably modified to use the new error control approach. The implementation of the algorithm raises a number of issues. For in­stance, since the accuracy tolerance for the time integration over the next time step after spatial remeshing depends partially on the error incurred prior to spatial remeshing, this tolerance must be modified according to the expected reduction in the spatial discretization error. The present algorithm thus goes some way towards providing a reliable solution method for two-dimensional time-dependent flows. The numerical results shows that this can be achieved for both simple convection type problems and for Euler flows in two space dimensions.

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References.

[1] M.Berzins, P.Baehmann, J.E. Flaherty and J. Lawson, Towards Reli­able Software for Time-Dependent Problems in CFD. Proceedings of 1990 Mafelap Conference, Ed J .R. Whiteman Academic Press.

[2] M. Berzins, and R.M. Furzeland, An Adaptive Theta Method for the Solution of Stiff and Non-Stiff Differential Equations, Applied Nu­merical Mathematics 7 (1992) pp. 1-19.

[3] M.Berzins, J.Lawson and J.Ware Spatial and Temporal Error Con­trol in the Adaptive Solution of Systems of Conservations Laws. Proceedings of 1992 IMACS PDE Conference, New Jersey, USA.

[4] M.Berzins and J.Ware Finite Volume Techniques for Time-Dependent Fluid-Flow Problems, Proceedings of 1992 IMACS PDE Confer­ence , New Jersey, USA.

[5] M.Berzins, Temporal Error Control in the Method of Lines for Con­vection Dominated Equations. Paper submitted for publication, 1992. Joint work with J. Ware.

Joseph E. Flaherty, Rensselaer Polytechnic Institute Adaptive and parallel high-order methods for conservation laws

Abstract: We consider the adaptive and parallel solution of vector systems of hyperbolic conservation laws by a local finite element method. High­order spatial approximations are constructed using Galerkin's method with the discontinuous solution approximation expressed in terms of a basis of piecewise continuous Legendre polynomials. Temporal integration is per­formed using explicit Runge-Kutta methods. Adaptivity involves a combi­nation of local (h-type) mesh refinement or coarsening in regions of high or low solution activity and (p-type) variation of method order. This hp­refinement strategy produces methods having very high convergence rates. Local and global a posteriori estimates of spatial errors guide adaptive enrichment and are obtained by p-refinement. Thus, an error estimate is obtained by using a piecewise polynomial approximation of one degree higher than that used for the finite element solution. Superconvergence simplifies computation of the error estimate and adds to the efficiency of the procedure. This local finite element method is well suited to parallel computation on distributed-memory computers. The local projections only involve nearest­neighbor communications regardless of the degree of the piecewise polyno­mial approximation. Upwinding, projection limiting, and the discontinuous basis result in sharp resolution at shocks and other discontinuities without spurious oscillations or excessive diffusion. Load balancing strategies for adaptive procedures are quite complex. Some preliminary notions will be presented for h- and p-refinement procedures. Computational results using these techniques on a ten-dimensional NCube-

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2 hypercube exhibited excellent scaled parallel performance.

Zhangxin Chen, University of Minnesota LP-posteriori error analysis of mixed methods for linear and quasilinear elliptic problems

Abstract: We consider mixed finite element methods for the approximation of linear and quasilinear second-order elliptic problems. A class of postpro­cessing methods for improving mixed finite element solutions is analyzed. In particular, error estimates in LP, 1 :s: p :s: 00, are given. These postpro­cessing methods are applicable to all the existing mixed methods, and can be easily implemented.

Chris Johnson, University of Utah Modeling, mesh generation, and adaptive methods for large scale bioelectric field problems

E. Bansch, University of Freiburg Adaptive finite-element techniques for transient problems

Tuesday, July 20

Richard E. Ewing, Texas A&M University Adaptive grid refinements for transient flow problems

Abstract: In the multidisciplinary numerical simulation of certain multi­phase fluid flow processes, many phenomena are sufficiently localized and transient that self-adaptive local grid refinement techniques are necessary to resolve the local physical behavior. For large-scale simulation problems, efficiency is the key to the choice of specific adaptive strategies. A priori and a posteriori error estimates can be effectively used to help determine the adaptive strategy. Purely local refinement techniques require complex data tree structures and associated specialized solution techniques. Techniques which involve a relatively coarse macro-mesh with potential local refine­ment in each separate mesh will be discussed. The macro-mesh will be the basis for domain decomposition techniques and parallel solution algorithms. Uniform meshing in the sub domains will allow efficient vectorization as well as parallelization of the algorithms. Similarly, different solution processes can be applied to different sub domains. A preconditioner, based upon the domain-decomposition techniques of Bramble, Pasciak, and Schatz, is uti­lized to efficiently solve the combination domain-decomposition and local grid refinement problem. Techniques for applying this concept to resolve sharp, moving fluid interfaces in large-scale simulation problems are dis­cussed.

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KEYNOTE LECTURE

Randolph E. Bank, UC San Diego Analysis of a posteriori error esti­mations, based on hierarchical bases (with R.K. Smith)

Slimane Adjerid, University Houari-Algeria A posteriori error estima­tion for elliptic and para­bolic problems

James M. Hyman, Los Alamos National Laboratory Static and dynamic adaptive methods for solving partial different­ial equations

Abstract: Adaptive mesh methods can improve the accuracy and efficiency of the numerical approximations to evolutionary and steady state systems of PDEs. These gradients can occur, for example, in boundary layers, shock waves or combustion fronts. To approximate the solution accurately in these regions it is often necessary to generate a mesh that is dense where the solution is rapidly changing. Also, for reasons of efficiency, the mesh should be sparse where the solution is smooth. These adaptive algorithms are costly, but without any local refinement many numerical calculations would be wasteful, or even worse, not resolve some important aspects of the solution satisfactorily. In an evolutionary PDE as the solution changes the mesh must also change to adaptively refine regions where the solution is developing sharp gradients and to remove redundant points from regions where the solution is becoming smoother. Thus, the mesh must have a dynamic behavior much in the same way the solution does. In this talk, I will illustrate several effective adaptive grid methods and describe some new approaches to implement the methods in a modular efficient way.

Kamyar Haghighi, Purdue University A knowledge-based approach to adaptive finite element analysis

Wednesday, July 21

Ivo Babuska, University of Maryland, College Park Adaptive modelling for partial differential equations on thin domains

Abstract: The outstanding problem in the engineering sciences is the mod­elling of plate and shells, especially laminated materials. The talk will present an approach of adaptive hierarchical modelling with an a-posterior error estimation technique which lead to the approximate solution of the original problem in a 3-dimensional setting. Numerical examples will be presented.

WORKSHOP SCHEDULE xliii

KEYNOTE LECTURE

Randolph E. Bank, UC San Diego Some nuts and bolts of adaptive method

T. Strouboulis, Texas A&M University Validation of error estimators by numerical approach

Joint work with Ivo Babuska.

Barna Szabo, Washington University Reliability in numerical simula­tion

Manfred Koch, Florida State University A mesh-adaptive collocation technique for the simulation of linear and nonlinear ad­vection-dominated single-and multiphase transport pheno­mena in porous media

Abstract: The development and implementation of a new mesh-adaptive ID collocation technique is presented that allows the efficient numerical solu­tion of transient advection-dominated transport problems in porous media that are governed by a hyperbolic/parabolic PDE or a system of PDE's. The method employs a collocation discretization of up to order six in space, in conjunction with an adaptive refining and remeshing of the grid at lo­cations, where the solution and/or its derivatives vary significantly. Local error estimates based on the theoretical convergence theorems of the collo­cation method are computed through comparisons of two solutions obtained on two different meshes. The mesh-adaptation strategy consists then in re­distributing and/or refining the mesh locally, such that the local error is approximately equal over the whole domain. This strategy avoids the ac­cumulation of an excessive number of grid points when a transient sharp front is moving across the mesh. For the time integration of the PDE several implicit methods such as a first-order backward Euler and a second-order Taylor-Donea technique are employed. Numerical simulations on a variety of high Peclet-number trans­port problems that are characterized by the presence of sharp fronts are carried out. Examples include classical linear advection-diffusion, non­linear adsorption, propagation of wetting fronts through the unsaturated zone, two-phase Buckley-Leverett flow without and with capillary forces (Rapoport-Leas equation), and three-phase flow pertaining to petroleum reservoir modeling. The results of computations show that the adaptive mesh/Euler method is unconditional stable and, compared with classical FD or FE methods is dispersive free, however, shows some amount of numerical dissipation that can be reduced through the use of sufficiently small time steps. The

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adaptive mesh/Taylor-Donea method, on the other hand, exhibits much less diffusion and allows the use of larger time steps. However, the more delicate balancing of spatial- and time derivatives makes this method more prone to numerical dispersion. This shows that for the proper solution of a transient PDE the spatial mesh adaption alone does not guarantee error­free solutions but that time-integration is equally important. The application the adaptive collocation method to the various physical transport phenomena cited above demonstrates the power of this technique method of capturing the physical processes that exhibit themselves in the form of the sharp front, such as the nature of the saturation-permeability flow function, of the adsorption isotherm, of the capillary forces and of the multi phase wetting functions.

Kunibert Siebert, University of Freiburg A-posteriori error estimators on anisotropically refined meshes and local refinement of rectangular and primatic grids

Abstract: Additionally to non-adaptive finite element methods there are two basic tools needed in adaptive methods. The first tool is an algorithm for local refinement of some given triangulation. We present algorithms for the local refinement of rectangular and prismatic grids by repeated hori­zontal and vertical bisection. The second tool we need is an a-posteriori error estimator. Using the in­formation generated by such an error estimator the refinement of the grid is controlled. For 2nd order elliptic problems we present error estimators on anisotropically refined n-D rectangular and 3-D primatic grids that will give valid information about the size of the error; additionally it generates information about the direction in which some element has to be refined (for 2-D rectangular and 3-D primatic grids we will get the information to refine an element horizontally or vertically or isotropic ally ).

Peter K. Moore, Tulane University Adaptive hp-refinement in space and time for parabolic systems

Thursday, July 22

Ken Morgan, University of Wales Unstructured grid methods for aero­dynamic flows

KEYNOTE LECTURE

Randolph E. Bank, UC San Diego Analysis of some space-time moving finite element methods (with R. Santos)

WORKSHOP SCHEDULE xlv

Jacob Fish, Rensselaer Polytechnic Institute Multiscale computational techniques for heteroge­neous media

J .Z. Zhu, UES, Inc. Accurate and effective a posteriori error estima-tion in the finite element method

Abstract: This paper addresses the issue of accurate and effective estima­tion of the discretization error of the finite element solutions. In particular, we investigate the relation between Zienkiewicz-Zhu error estimator and the error estimator of the element residual type. We demonstrate that if a derivative recovery technique is properly constructed, the element resid­ual type of error estimator can be derived from the Zienkiewicz-Zhu error estimator. We also show that asymptotically exact a posteriori error esti­mators can be achieved providing that the recovered solution used in the error estimator is superconvergent. The recognition of the relation between different types of error estimators enables us to develop accurate and com­putationally cost effective error estimators. It also provides a different view point on the theory of the a posterior error estimation. Joint work with O.C. Zienkiewicz.

Masayuki Kaiho, University of Minnesota Overlapping grid methods for unsteady viscous incom­pressible flows

Abstract: A new FEM/FDM overlapping scheme for unsteady incompress­ible flow problem is presented. As is well known, flow analysis can be clas­sified into the finite difference method and the finite element method. Each analysis has advantages and disadvantages. The finite difference method al­lows the use of efficient numerical solution procedures and is advantageous in terms of computing time and storage. However, boundary representa­tion requires special logic and is generally inaccurate. On the other hand, the finite element method is best suited to analyzing flows within arbitrar­ily shaped flow geometries. However, it requires greater computing time and storage than the finite difference method. Neither the finite difference method nor the finite element method alone seems suited to applying to the large-scale and high-resolution flow analysis in complex flow geome­tries. The present scheme presents a new numerical technique to solve viscous incompressible flow problems which combines the finite element method and the finite difference method. Computations are calculated us­ing an overlapping grid system, where finite element meshes are generated around arbitrarily shaped bodies and a finite difference grid is partially super-imposed on finite element meshes. The grid has been employed to cover the remaining region of interest. Values of unknown variables on this grid system are exchanged through interconnected boundaries to join subregions together as a single region. The modified ABMAC method pro­posed by Vie celli has been used as the basic algorithm. This composite

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grid scheme not only provides an efficient procedure to solve the large­scale flow problem, but also enables flow analysis around moving bodies with rotation or translation. Numerical results including moving boundary problems and three-dimensional problems indicate good agreement with experimental results.

Gang Bao, IMA Finite element approximation of time harmonic waves in periodic structures

Abstract: Consider the diffraction of a time harmonic wave incident on a periodic surface of some inhomogeneous material. It is shown that the scat­tering (diffraction) problem may be modeled by a Helmholtz equation with transparent boundary conditions. The diffraction problem may be solved by a finite element method. In this work, optimal error estimates for the finite element method are established. The speaker will also discuss some related problems.

Ray Spiteri, University of British Columbia Collocation software for the solution of boundary value differential-alge­braic equations

Giovanni Cornetti, Paris VI Convection in arbitrary Lagrangian Eule­rian free surface problem

Friday, July 23

Kapil Mathur, Thinking Machines Corp. Massively parallel computing: Unstructured finite element simulations

Joint work with Z. Johan, S. Lennart Johnson and T. J. R. Hughs.

A. Quarteroni, Polytechnic of Milan Numerical modelling of shallow water equations

Abstract: Shallow water equations (briefly, SWE) provide a model to de­scribe fluid dynamical processes of various nature, and find therefore wide­spread application in science and engineering. A rigorous mathematical analysis is not available, unless for few specific cases under strict assump­tions on the problem's data. In particular, the issue of which kind of boundary conditions are allowed is not completely understood yet. In this presentation several sets of boundary conditions of physical interest that are admissible from the mathematical viewpoint are considered. Another critical issue is which formulation (either conservative or non con­servative) is better to describe numerically the physical problem at hand. An ultimate answer is not available for general situations. However, a few remarks can be done in this respect.

WORKSHOP SCHEDULE xlvii

The space-discretization of SWE considered in this presentation is based upon finite element spaces: the stability analysis reveals that the primary physical variables, velocity and elevation, need to be discretized in a con­sistent manner. For what concerns the temporal discretization a couple of new methods are discussed. The former is based on a semi-implicit time advancing scheme, the latter on a fractional step procedure. Several numerical test cases are discussed.

Jens Hugger, University of Copenhagen An asymptotically exact, pointwise, a posteriori error estimator for the finite ele­ment method with super con­vergence properties

Abstract: When the finite element solution of a variational problem pos­sesses certain super convergence properties, it is possible very inexpensively to obtain a correction term providing an additional order of approximation of the solution. The correction can be used for error estimation locally or globally in whatever norm is preferred, or if no error estimation is wanted it can be used for postprocessing of the solution to improve the quality. The correction term will be derived for linear or nonlinear problems in any space dimension with possible singularities and zeros of derivatives in the exact solution.

LIST OF PARTICIPANTS

Adair, Roberto University of Minnesota

Adjerid, Slimane University Houari-Algeria

Ainsworth, Mark Leister U ni versi ty

Akin, J.E. Rice University

Babuska, Ivo University of Maryland

Baensch, Eberhard University of Freiburg

Baines, M.J. University of Reading

Baker, Tim Princeton University

Bank, Randolph E. U of California-San Diego

Bao, Gang IMA

Barnhill, Robert Arizona State University

Barrera, Pablo National University of Mexico

Barth, Timothy NASA

Bedrosian, Gary General Electric Company

Beichl, Isabel NIST

Belytschko, Ted Northwestern University

Benbourenane, Mohamed Northern Illinois University

Bern, Marshall Xerox Corporation

Berzins, Martin University of Leeds-England

Bieterman, Michael B. Boeing Computer Services

Billera, Louis J. Cornell University

Bogar, Kristina University of Utah

Brenner, Susanne C. Clarkson University

Brown, David L. Los Alamos National Laboratories

Burns, Tim NIST

Castro, Gustavo 3M

Chakravarthy, S.R. Rockwell International

xlix

Chapman, Andrew University of Minnesota

Chen, Zhangxin Texas A&M University

Chesshire, Geoffrey IBM

Chew, Paul Cornell University

Chin, Ray C.Y. Purdue University-Indianapolis

Chrisochoides, Nikos P. Syracuse University

Copps, Kevin Texas A&M

Cornetti, Giovanni U.' Pierre et Marie Curie (Paris VI)

Deconinck, Herman Von Karmen Inst. for Fluid Dyn.

Devine, Karen Sandia National Laboratory

Edelman, Paul University of Minnesota

Eiseman, Peter R. Program Development Corporation

Ellwood, Kevin R.J. Ford Motor Company

Ewing, Richard E. Texas A&M University

Fabes, Eugene University of Minnesota

Ferguson, Warren Southern Methodist University

Field, David A. General Motors

Fish, Jacob Rensselaer Polytechnic Institute

Flaherty, Joseph E. Rensselaer Polytechnic Institute

Franca, Adriana Purdue University

Gangaraj, Srihari K. Texas A&M University

Gates, Ian University of Minnesota

George, Paul-Luois INRIA

Goenaga, Alberto University of Minnesota

Golik, Wojciech University of Missouri

Gropp, William D. Argonne National Laboratory

Gutierrez-Miravete, E. The Hartford Graduate Center

LIST OF PARTICIPANTS

Haghighi, Kamyar Purdue University

Hakula, Harri Helsinki University of Technology

Hecht, Frederic INRIA

Heisler, Joseph L. U. of Massachusetts-Amherst

Henriquez, Craig Duke University

Henshaw, William D. Los Alamos National Laboratory

Hoffmann, Christoph Purdue University

Hugger, Jens University of Copenhagen

Hughes, Thomas J .R. Stanford University

Hulbert, Gregory M. University of Michigan

Hwang, Hyun-Cheol SUNY-Stony Brook

Hyman, James M. Los Alamos National Laboratory

Jiang, Min-Yee Mississippi State University

Johnson, Christopher R. University of Utah

Johnson, Claes Chalmers University of Technology

Kaiho, Masayuki University of Minnesota

Kang, Eun Purdue University

Kashiyama, Kazuo Chuo University

Kloucek, Petr University of Minnesota

Koch, Manfred Florida State University

Kozak, Darryn J. Cray Research

Kwok, Wa Purdue University

Lanucara, Piero CASPUR!U. of Rome

Latypov, Azat University of Windsor

Lawrence, Peter James University of Greenwich

Levit, Itzhak Lockheed Research Laboratory

Lin, Tao VPI & SU

Liu, Bo University of Minnesota

Liu, Shaobo 3M

Luskin, Mitchell University of Minnesota

Ma,X.Z. Texas A&M University

Mastin, Wayne Mississippi State University

Mathur, Kapil Thinking Machines Corp.

Maubach, Joseph University of Pittsburgh

Micchelli, Charles IBM

Min, Shirley Medtronic

Mohtar, Rabi Michigan State University

Moore, Doug Rice University

Moore, Peter K. Tulane University

Morgan, Ken University of Wales

Mortazavi, Hamid 3M

Mukherjee, Arup Penn State University

Nguyen, Hung Viet University of Minnesota

Nieber, John University of Minnesota

Nijhawan, Sandeep University of Minnesota

Oden, J. Tinsley University of Texas-Austin

Olejniczak, Debra University of Minnesota

Olejniczak, Joseph University of Minnesota

Oliger, Joseph E. NASA Ames Research Center

Oliveira, Leandro Purdue University

Olver, Peter University of Minnesota

Petzold, Linda University of Minnesota

Quarteroni, Alfio Politecnico di Milano

Quirk, James NASA

LIST OF PARTICIPANTS

Ren, Yuhe University of Minnesota

Rosen, J.B. University of Minnesota

Ruppert, James NASA

Saltzman, Jeff Los Alamos National Laboratory

Samuelsson, Klas Royal Inst. of Technology-Sweden

Saunders, Bonita NIST

Schmelzer, Ilj a Inst.fur Ang. Analy. und Stach.

Schneiders, Robert Inst. Informatik-Germany

Scriven, L.E. "Skip" University of Minnesota

Sheikh, Qasim M. Cray Research

Shephard, Mark Rensselear Polytechnic Institute

Siebert, Kunibert Inst. fuer Ange. Mathematik

Silveiro, Mauncio University of Minnesota

Singh, Kumud Lora'nd Eotvos University-Hungary

Smooke, Mitchell Yale University

Spiteri, Raymond University of British Columbia

Stein, Keith US Army-Natick RD&E Center

Strawn, Roger C. NASA

Strouboulis, Theofanis Texas A&M University

Sullivan, Francis SuperComputing Research Center

Svobodny, Thomas P. Wright State University

Szabo, Barna Washington University

Taghavi, Reza Cray Research

Tezduyar, Tayfun Army High Performance Computing Research Center

Thune, Michael Uppsala University

Trompert, Ron CWI-Netherlands

Upadhyay, Chandra S. Texas A&M University

Vavasis, Steve Cornell University

Vazquez, Juan UAM, Spain

Venkata, Ramana G. Stanford University

Wang, Xiaodi Michigan State University

Warren, Gary NASA

Williams, Glenn University of North Carolina

Xia, Xu University of Minnesota

Zegeling, Paul Utrecht University

Zhang, Shangqian Michigan State University

Zhang, Yongmin University of Chicago

Zhu, J.Z. UES, Inc.

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