Vadim Shapiro

The long-standing goal of computer aided design (CAD)/computer aided engineering (CAE) integration demands seamless interfaces between geometric design and engineering analysis/simulation tasks. The key challenge to this integration stems from the distinct and often incompatible roles geometric representations play, respectively, in design and analysis. This paper critically examines and compares known mesh-based and meshfree approaches to CAD/CAE integration, focusing on the basic tasks and components required for building fully integrated engineering applications. For each task, we identify the fundamental requirements and challenges and discuss how they may be met by known techniques and proposed solutions.

V. L. Rvachev called R-functions ‘logically charged functions’ because they encode complete logical information within the standard setting of real analysis. He invented them in the 1960s as a means for unifying logic, geometry, and analysis within a common computational framework – in an effort to develop a new computationally effective language for modelling and solving boundary value problems. Over the last forty years, R-functions have been accepted as a valuable tool in computer graphics, geometric modelling, computational physics, and in many areas of engineering design, analysis, and optimization. Yet, many elements of the theory of R-functions continue to be rediscovered in different application areas and special situations. The purpose of this survey is to expose the key ideas and concepts behind the theory of R-functions, explain the utility of R-functions in a broad range of applications, and to discuss selected algorithmic issues arising in connection with their use.

Recent advances in shape optimization rely on free‐form implicit representations, such as level sets, to support boundary deformations and topological changes. By contrast, parametric shape optimization is formulated directly in terms of meaningful geometric design variables, but usually does not support free‐form boundary and topological changes. We propose a novel approach to shape optimization that combines and retains the advantages of the earlier optimization techniques. The shapes in the design space are represented implicitly as level sets of a higher‐dimensional function that is constructed using B‐splines (to allow free‐form deformations), and parameterized primitives combined with R‐functions (to support desired parametric changes).Our approach to shape design and optimization offers great flexibility because it provides explicit parametric control of geometry and topology within a large space of free‐form shapes. The resulting method is also general in that it subsumes most other types of shape optimization as special cases. We describe an implementation of the proposed technique with attractive numerical properties. The explicit construction of an implicit representation supports straightforward sensitivity analysis that can be used with most gradient‐based optimization methods. Furthermore, our implementation does not require any error‐prone polygonization or approximation of level sets (isocurves and isosurfaces). The effectiveness of the method is demonstrated by several numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.

Geometric data interoperability is critical in industrial applications where geometric data are transferred (translated) among multiple modeling systems for data sharing and reuse. A big obstacle in data translation lies in that geometric data are usually imprecise and geometric algorithm precisions vary from system to system. In the absence of common formal principles, both industry and academia embraced ad hoc solutions, costing billions of dollars in lost time and productivity. This paper explains how the problem of interoperability, and data translation in particular, may be formulated and studied in terms of a recently developed theory of ε-solidity. Furthermore, a systematic classification of problems in data translation shows that in most cases ε-solids can be maintained without expensive and arbitrary geometric repairs.