Theory of R-functions [12] provides the methodology for constructing exact implicit functions for any semianalytic set. This paper systematically explores and compares the known constructions in terms of their differential properties and explains how such functions may be constructed automatically from CSG and boundary representations of solids. The constructed functions may be automatically differentiated and integrated and have many important applications in meshfree engineering analysis, motion planning, and scientific visualization.

In a general setting, the transfinite interpolation problem requires constructing a single function #### that takes on the prescribed values and/or derivatives on some collection of point sets. The sets of points may contain isolated points, bounded or unbounded curves, as well as surfaces and regions of arbitrary topology. All such closed semi-analytic sets may be represented implicitly by real valued functions with guaranteed differential properties.

We propose a universal approach to the problem of computer modeling of shapes with continuously varying material properties satisfying prescribed material conditions on a finite collection of material features and global constraints. The central notion is a parameterization of space by distances from the material features -- either exactly or approximately. Functions of such distances provide a systematic and intuitive means for modeling of desired material distributions as they arise in design, manufacturing, analysis and optimization of components with varying material properties.

The paper presents an approach to modeling heterogeneous objects as multidimensional point sets with multiple attributes (hypervolumes). A theoretical framework is based on a hybrid model of hypervolumes combining a cellular representation and a constructive representation using real-valued functions. This model allows for independent but unifying representation of geometry and attributes, and makes it possible to represent dimensionally non-homogeneous entities and their cellular decompositions. Hypervolume model components such as objects, operations and relations are introduced and outlined. The framework's inherent multidimensionality allowing, in particular, to deal naturally with time dependence promises to model complex dynamic objects composed of different materials with constructive building of their geometry and attributes. Attributes given at each point can represent properties of arbitrary nature (material, photometric, physical, statistical, etc.). To demonstrate a particular application of the proposed framework, we present an example of multimaterial modeling -- a multilayer geological structure with cavities and wells. Another example illustrating the treatment of attributes other than material distributions is concerned with time-dependent adaptive mesh generation where function representation is used to describe object geometry and density of elements in the cellular model of the mesh. The examples have been implemented by using a specialized modeling language and software tools being developed by the authors.

This article revisits the main ideas and foundations of solid modeling in engineering, summarizes recent progress and bottlenecks, and speculates on possible future directions

In a meshfree system, a geometric model of a domain neither conforms to, nor is restricted by a spatial discretization.

The presented approach to discretization of functionally defined heterogeneous objects is oriented towards applications associated with numerical simulation procedures, for example, finite element analysis (FEA). Such applications impose specific constraints upon the resulting surface and volume meshes in terms of their topology and metric characteristics, exactness of the geometry approximation, and conformity with initial attributes. The function representation of the initial object is converted into the resulting cellular representation described by a simplicial complex. We consider in detail all phases of the discretization algorithm from initial surface polygonization to final tetrahedral mesh generation and its adaptation to special FEA needs. The initial object attributes are used at all steps both for controlling geometry and topology of the resulting object and for calculating new attributes for the resulting cellular representation.

In this document we discuss the data structure and algorithms for direct application of recursive chain rules to numerical computations of partial derivatives in forward mode. The proposed data structure providing constant time access to the partial derivatives accelerates the automatic differentiation computations. We implemented the presented algorithms in a software package, which simplifies automatic differentiation of functions represented by a computer program. The library is available for public use and can be downloaded from http://sal-cnc.me.wisc.edu. This manual contains detailed descriptions of all library functions and several examples illustrating the applications and usage of the proposed automatic differentiation software.

The presented approach to discretization of functionally defined heterogeneous objects is oriented towards applications associated with numerical simulation procedures, for example, finite element analysis (FEA). Such applications impose specific constraints upon the resulting surface and volume meshes in terms of their topology and metric characteristics, exactness of the geometry approximation, and conformity with initial attributes. The function representation of the initial object is converted into the resulting cellular representation described by a simplicial complex. We consider in detail all phases of the discretization algorithm from initial surface polygonization to final tetrahedral mesh generation and its adaptation to special FEA needs. The initial object attributes are used at all steps both for controlling geometry and topology of the resulting object and for calculating new attributes for the resulting cellular representation. Categories and Subject Descriptors

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