**November 9, 2003**

**Organizers: **Ulrich Reif, Technische Universität
Darmstadt, Germany

Vadim Shapiro, University of Wisconsin-Madison

Igor Tsukanov, University of Wisconsin-Madison

Many problems in geometric design and engineering analysis are dominated by spatial discretization of the geometric domain into a conforming grid of simple elements (mesh). This tutorial will demonstrate that many such problems may be formulated and solved without explicit meshing, using approximate distance fields and splines.

**Ulrich Reif** (http://www.mathematik.tu-darmstadt.de/~reif/)
is a Professor at the Department of Mathematics at Darmstadt University
of Technoloy since 2000. He studied mathematics and electrical engineering
at Stuttgart University, where received his PhD in 1993 and his Habilitation
in 1999. His fields of interest include Geometric Modeling, Subdivision
algorithms, and spline approximation. Recent work addresses finite element
simulations with weighted extended B-splines.

**Vadim Shapiro** (http://www.engr.wisc.edu/me/faculty/shapiro_vadim.html)
is a Professor of Mechanical Engineering and Computer Sciences at the University
of Wisconsin-Madison where he has been on faculty since 1994. Prior to that
he was a member of research staff at the General Motors R&D Center working
in the areas of geometric modeling and design automation. He received BA
degrees in mathematics and in computer science from New York University,
MS in computer science from UCLA, and MS and PhD in mechanical engineering
from Cornell University. In addition to solid modeling, his research interests
include unification of physical and geometric representations and computational
design.

**Igor Tsukanov** (http://sal-cnc.me.wisc.edu/~igor/igor.html)
is a research scientist at the Department of Mechanical Engineering of University
of Wisconsin-Madison working on automation of the R-function meshfree method.
He received his Ph.D. degree in 1997 from the Ukrainian National Academy
of Sciences. From 1992 till 1998 he worked in the research group headed
by Prof. Rvachev at Institute for Problems in Machinery of Ukrainian National
Academy of Sciences. In 1998 he joined Spatial Automation Laboratory at
UW-Madison. Research interests include meshfree methods of engineering analysis,
numerical methods for PDE's, automation.

The approach to modeling relies on the observation that any field problem may be modeled as a function of weighted distances to a given collection of geometric features. In addition to various geometric applications, field-modeling applications include (heterogeneous) material modeling and meshfree solution of engineering analysis problems. The tutorial will consists of three parts:

1. Fields as functions of approximate distances. Every field function satisfying some given boundary conditions on a collection of geometric features may be put into a canonical form in terms of powers of distances to the feature. Once a distance field is known, the canonical form allows systematic construction, comparison, and approximation of fields satisfying boundary conditions without relying on meshing. Many different approximations of distance fields are known, but we will explore in details smooth approximations of distance fields using theory of R-functions originally developed by V. L. Rvachev.

2. Formulating and solving boundary value problems. Once the boundary conditions are satisfied in terms of distance fields, a solution to boundary value problem maybe constructed by a linear combination of appropriately weighted basis functions. In particular, weighted extended B-splines (web-splines) allow constructing efficient and numerically stable approximations on arbitrary domains.

3. Computational tools and applications. Effective applications of the above techniques require accurate and efficient computational procedures for sampling, differentiation, integration, and visualization of fields over solids and domains. We will discuss the required utilities, feasible solutions, and demonstrate a generic system for field modeling using approximate distance fields. Example applications include modeling of solids with heterogeneous materials and meshfree solution of various engineering analysis problems including heat transfer, vibration, stress analysis, and thermo-elasticity.

Course Objectives

Attendees will be able to model and solve a variety of field problems without depending on meshing.

Introductory: 20% Intermediate: 60% Advanced: 20%

Graduate students, researchers, industrial scientists

Calculus, differential equations, numerical methods, splines.

The course will consist of 3 parts as outline above. Detailed outline
will be available at a later date.