David A. Field

Qualitative measures for initial meshes

This paper reviews geometric measures used to assess the shape of finite elements in two- and three-dimensional meshes. Measures have been normalized and made scale invariant whenever possible. This paper also introduces a Universal Similarity Region that enhances comparisons of triangles and their measures. As a byproduct, the USR provides a dynamic way to compare improved triangular meshes. Copyright

Education and training for CAD in the auto industry

Abstract CAD-systems envisioned and remarkably well specified in the 1950s have powered themselves into the central role they enjoy in todays automotive industry through continuous improvements and technological breakthroughs. This paper emphasizes the parallel and continuing evolution in the training and educational needs of users of CAD-systems. In the context of early historical developments of CAD at General Motors, this paper categorizes CAD-users in the automobile industry and presents their current and future needs. The variance in their educational and training needs poses an ongoing challenge for educational and industrial institutions to meet.

Designing Bézier conic segments with monotone curvature

Abstract For some applications of computer-aided geometric design it is important to maintain strictly monotone curvature along a curve segment. Here we analyze the curvature distributions of segments of conic sections represented as rational quadratic Bezier curves in standard form. We show that if the end points and the weight are fixed, then the curvature of the conic segment will be strictly monotone if and only if the other control point lies inside well-defined regions bounded by circular arcs. We also show that if the turning angle of the curve is less than or equal to 90°, then there are always values of the weight that ensure strict monotonicity of the curvature distribution. Furthermore, bounds on such values of the weight are easily computed.

The legacy of automatic mesh generation from solid modeling

This paper briefly reviews early efforts to automatically create finite element meshes from solid models. Although parametrically defined blocks of finite elements initially dominated mesh generation, the creation of complex solids forced mesh generation away from the direct use of parametrizations. This review emphasizes the earliest and most important techniques for mesh generation that relied on solid modeling. In addition to including a brief summary of parametrically defined meshes, this review also includes the more recent advancing front technique. This paper concludes with brief remarks that incorporate current directions and developments in automating mesh generation.

Delaunay criteria for triangulating surfaces

Computer aided design and manufacturing processes frequently map planar triangulations onto surfaces. Due to their properties relevant to finite element analysis, Delaunay triangulations have become popular for triangulating arbitrary planar domains that are subsequently mapped onto surfaces. Although surface triangulations cannot enjoy all the properties of planar triangulations, utilization of as many of the properties of Delaunay triangulations as possible in algorithms for triangulating surfaces can have advantages as well as disadvantages. This paper focuses on the advantages and disadvantages of using Delaunay criteria in triangulating surfaces.© (1992) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

The focal point method for solving systems of linear equations

The Focal Point method of solving systems of linear equations is based on the linearity of matrix transformations and the fact that each equation in the system defines a hyperplane of solutions. As with Gaussian elimination the number of multiplications is a cubic polynomial inn with leading coefficientn3/3. Some of the benefits of this new method are that it needs approximatelyn2/4 storage registers and that the zeroes of the system are preserved. Thus for sparse systems the operation count can be significantly reduced. Numerical examples using initial segments of the Hilbert matrix demonstrate that this new method can be very accurate.ZusammenfassungDie Vektor Projektions Methode zur Lösung linearer Gleichungssysteme ist darauf begründet, daß jede Gleichung eine Hyperebene definiert und daß die Berechnung der Projektionspunkte linear ist. Wie bei der Gaussschen Eliminierung wird die Anzahl der Multiplikationen durch ein kubisches Polynom mit höchstem Koeffizientenn3/3 beschrieben. Zwei Vorteile der neuen Methode sind, daß nur etwan2/4 Speicherplätze benötigt werden, und daß Null-Koeffizienten im Gleichungsystem erhalten bleiben. Damit kann die Anzahl der Rechenoperationen bei dünnbesetzten Matrizen erheblich reduziert werden. Die gute Genauigkeit des neuen Verfahrens wird am Beispiel der numerischen Inversion von Hilbert-Matrizen verschiedener Größe gezeigt.

Convergence rates for Pade´-based iterative solutions of equations

Abstract Convergence rates for iterative solutions of equations are given when the iterates are Pade approximants. A new theorem for the univariate case is extended to the use of multivariate Pade approximants for systems of nonlinear equations.

Three dimensional Delaunay triangulation on a Cray X-MP

A software package for generation of tetrahedral finite-element mesh, whose kernel is a robust three-dimensional Delaunay triangulation algorithm, was ported to a CRAY X-MP for vector processing. The total execution time for the critical subroutines of the kernel decreased by a factor of six over scalar mode on the CRAY X-MP. The kernel is characterized by simple data structures and O(N/sup 2/) arithmetic operation counts, N being the number of finite-element nodes. Although the kernel is essentially a sequential algorithm, its simple data structures allow for key uses of vector processing and for streamlining sequential processing.<<ETX>>

From solid modeling to finite element analysis

Solid modeling, finite element mesh generation and analysis of finite element solutions are obviously tightly connected in the design redesign cycle. However, practical realizations of each of these aspects of mathematical analysis of solid objects requires a significant amount of internal independence. Each aspect has to be justified on its own merits and their actual development reflects this independence. Together they form a powerful system and separately they perform useful functions which can interface with other computer codes and systems.

Determining invertible two and three dimensional quadratic isoparametric finite element transformations

This paper presents an algorithm determining the invertibility of any planar, triangular quadratic isoparametric finite element transformation. Extensions of the algorithm to three-dimensional isoparametric finite element transformations yield conditions which guarantee invertibility of 10-node tetrahedra and 8-node bricks.