An implicit representation typically describes a set of points as the zero set of some real valued function. Such implicit representations of curves and surfaces have been used widely in geometric and solid modeling, and often take on a form of approximate distance fields. This tutorial will explore how such implicit representations may be extended to represent not only local geometric information but also to model and analyze globally defined fields satisfying a collection of prescribed boundary conditions. Briefly, the main premise of the presented material is 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 cover a number of topics:
Distance fields and their smooth approximations. While many different constructions are known, this tutorial will explore in details smooth approximations of distance fields using theory of R-functions originally developed by V. L. Rvachev. Brief introduction to the theory of R-functions will be followed by comparison of various systems of R-functions and their properties.
Distance canonical form. Every field function satisfying some given boundary conditions on a geometric feature 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.
Distance-based interpolation techniques. Fields associated with various geometric features may be combined into single fields using weighted inverse distance-based interpolation techniques. We will show that the Shepard’s method of interpolation of scattered data is a special case of a more general approach relying on approximate distance fields to known geometric features.
Computational tools. 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, as well as computational challenges.
Applications. We will describe and demonstrate a generic system for field modeling using approximate distance fields. Specific applications include modeling of solids with heterogeneous materials and meshfree solution of numerous engineering analysis problems including heat transfer, vibration, stress analysis, and thermoelasticity.
Vadim Shapiro is Associate 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 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.