The Construction of Curves and Surfaces Using Numerical Optimization Techniques

Numerical optimization techniques are playing an increasing role in curve and surface construction. Often difficult problems in curve and surface construction, especially when some aspect of shape control is involved, can be phrased as a constrained optimization problem. Four such classes of problems are explored: parametric curve fitting with non-linear shape constraints; explicit surface fitting with linear shape constraints; surface fitting to scattered data giving rise to ill-posed problems; finally, variable knot problems. In each of these problems there is a nonlinear aspect: either the shape of the curve or surface is important for manufacturing or engineering reasons or the shape affects the convergence of numerical algorithms which use the curve or surface or the placement of knots affects the accuracy of the fits. In all cases the class of functions used is that of parametric spline curves and tensor or direct product spline surfaces. The reason for choosing this class is that splines provide flexible models that are easily evaluated and stored. Furthermore, the B-spline representation of splines leads to convenient expressions for shape control over regions

Applications and identification of surface correlations

We compare theoretical, experimental, and computational approaches to random rough surfaces. The aim is to produce rough surfaces with desirable correlations and to analyze the correlation functions extracted from the surface profiles. Physical applications include ultracold neutrons in a rough waveguide, lateral electronic transport, and scattering of longwave particles and waves. Results provide guidance on how to deal with experimental and computational data on rough surfaces. A supplemental goal is to optimize the neutron waveguide for GRANIT experiments. The measured correlators are identified by fitting functions or by direct spectral analysis. The results are used to compare the calculated observables with theoretical values. Because of fluctuations, the fitting procedures lead to inaccurate physical results even if the quality of the fit is very good unless one guesses the right shape of the fitting function. Reliable extraction of the correlation function from the measured surface profile seems virtually impossible without independent information on the structure of the correlation function. Direct spectral analysis of raw data rarely works better than the use of a "wrong" fitting function. Analysis of surfaces with a large correlation radius is hindered by the presence of domains and interdomain correlations

Reconstruction of freeform surfaces for metrology

The application of freeform surfaces has increased since their complex shapes closely express a product's functional specifications and their machining is obtained with higher accuracy. In particular, optical surfaces exhibit enhanced performance especially when they take aspheric forms or more complex forms with multi-undulations. This study is mainly focused on the reconstruction of complex shapes such as freeform optical surfaces, and on the characterization of their form. The computer graphics community has proposed various algorithms for constructing a mesh based on the cloud of sample points. The mesh is a piecewise linear approximation of the surface and an interpolation of the point set. The mesh can further be processed for fitting parametric surfaces (Polyworks® or Geomagic®). The metrology community investigates direct fitting approaches. If the surface mathematical model is given, fitting is a straight forward task. Nonetheless, if the surface model is unknown, fitting is only possible through the association of polynomial Spline parametric surfaces. In this paper, a comparative study carried out on methods proposed by the computer graphics community will be presented to elucidate the advantages of these approaches. We stress the importance of the pre-processing phase as well as the significance of initial conditions. We further emphasize the importance of the meshing phase by stating that a proper mesh has two major advantages. First, it organizes the initially unstructured point set and it provides an insight of orientation, neighbourhood and curvature, and infers information on both its geometry and topology. Second, it conveys a better segmentation of the space, leading to a correct patching and association of parametric surfaces.EMR

On accurate estimation of transverse stresses in multilayer laminates

New numerical algorithms are proposed for the accurate evaluation of transverse stresses in general composite and sandwich laminates. A set of higher-order theories with C0 isoparameteric finite elements and exact three-dimensional equilibrium equations are used. The integration of the equilibrium equations is carried out through exact surface fitting method, direct integration method and forward and central direct finite difference methods. Sixteen- and nine-noded quadrilateral Lagrangian elements with selective numerical integration techniques based on Gauss-Legendre product rules are used in the analysis. Validity of the present numerical techniques and the higher-order theories are demonstrated by comparing the present results with the available elasticity and other closed-form solutions for cross-ply, angle-ply and sandwich laminates. The exact surface fitting method is seen to give accurate estimate of the transverse stresses compared to other methods