Panagiotis D. Kaklis

An isogeometric BEM for exterior potential-flow problems in the plane

In this paper, the isogeometric concept introduced by Hughes, in the context of Finite Element Method, is applied to Boundary Element Method (BEM), for solving an exterior planar Neumann problem. The developed isogeometric-BEM concept is based on NURBS, for representing the exact body geometry and employs the same basis for representing the potential and/or the density of the single layer. In order to examine the accuracy of the scheme, numerical results for the case of a circle and a free-form body are presented and compared against analytical solutions. This enables performing a numerical error analysis, verifying the superior convergence rate of the isogeometric BEM versus low-order BEM. When starting from the initial NURBS representation of the geometry and then using knot insertion for refinement of the NURBS basis, the achieved rate of convergence is O(DoF-4). This rate may be further improved by using a degree-elevated initial NURBS representation of the geometry (kh-refinement).

Convexity-preserving interpolatory parametric splines of non-uniform polynomial degree

Abstract This note presents an O(n2) algorithm for evaluating point and tangents of a rational tensor product Bezier surface patch.

Advanced course on FAIRSHAPE

The task of fairing a curve interpolating a given point data set and potentially given end conditions by minimizing an explicit fairness measure is discussed from the viewpoint of the resulting curve quality. Results are compared for different choices of a fairness criteria applied to a variety of data sets. The improvements achievable by going from integer to rational cubic B-spline curves are examined in particular. Fairness quality can be raised both by lessening the constraints and by increasing the freedoms in curve representation.

Convexity-preserving Fairing

Abstract This paper develops a two-stage automatic algorithm for fairing C2-continuous cubic parametric B-splines under convexity, tolerance and end constraints. The first stage is a global procedure, yielding a C2 cubic B-spline which satisfies the local-convexity, local-tolerance and end constraints imposed by the designer. The second stage is a local finefairing procedure employing an iterative knot-removal knotreinsertion technique, which adopts the curvature-slope discontinuity as the fairness measure of a C2 spline. This procedure preserves the convexity and end properties of the output of the first stage and, moreover, it embodies a globaltolerance constraint. The performance of the algorithm is discussed for four data sets.

Sectional-curvature preserving skinning surfaces

Abstract In this work we develop a method for constructing sectional-curvature preserving (scp) C 2 -continuous surfaces, which interpolate point-sets lying on parallel planes. The working function space consists of skinning surfaces, whose skeletal lines and blending functions are polynomial splines of nonuniform degree. The asymptotic behaviour of these surfaces and their sectional curvature, as the segment degrees tend to infinity globally or semilocally, is thoroughly studied. This study provides sufficient geometrical conditions on the given data ensuring that, it the segment degrees increase semilocally then the surface will eventually become scp in the corresponding parameter subdomain. Based on the obtained asymptotic results, an automatic algorithm for constructing scp interpolatory surfaces is developed and numerically tested.

Bounding the distance between 2D parametric Bézier curves and their control polygon

Employing the techniques presented by Nairn, Peters and Lutterkort in [1], sharp bounds are firstly derived for the distance between a planar parametric Bézier curve and a parameterization of its control polygon based on the Greville abscissae. Several of the norms appearing in these bounds are orientation dependent. We next present algorithms for finding the optimal orientation angle for which two of these norms become minimal. The use of these bounds and algorithms for constructing polygonal envelopes of planar polynomial curves, is illustrated for an open and a closed composite Bézier curve.

A BEM-ISOGEOMETRIC METHOD WITH APPLICATION TO THE WAVEMAKING RESISTANCE PROBLEM OF SHIPS AT CONSTANT SPEED

In the present work IsoGeometric Analysis (IGA), initially proposed by Hughes et al (2005), is applied to the solution of the boundary integral equation associated with the Neumann-Kelvin (NK) problem and the calculation of the wave resistance of ships, following the formulation by Brard (1972) and Baar & Price (1988). As opposed to low-order panel methods, where the body is represented by a large number of quadrilateral panels and the velocity potential is assumed to be piecewise constant (or approximated by low degree polynomials) on each panel, the isogeometric concept is based on exploiting the NURBS basis, which is used for representing exactly the body geometry and adopts the very same basis functions for approximating the singularity distribution (or in general the dependent physical quantities). In order to examine the accuracy of the present method, in a previous paper Belibassakis et al (2009), numerical results obtained in the case of submerged bodies are compared against analytical and benchmark solutions and low-order panel method predictions, illustrating the superior efficiency of the isogeometric approach. In the present paper we extent previous analysis to the case of wavemaking resistance problem of surface piercing bodies. The present approach, although focusing on the linear NK problem which is more appropriate for thin ship hulls, it carries the IGA novelty of integrating CAD systems for ship-hull design with computational hydrodynamics solvers.Copyright

Polynomial cubic splines with tension properties

In this paper we present a new class of spline functions with tension properties. These splines are composed by polynomial cubic pieces and therefore are conformal to the standard, NURBS based CAD/CAM systems.

A method for computing the tension parameters in convexity-preserving spline-in-tension interpolation

SummaryThis paper is concerned with the problem of convexity-preservng (orc-preserving) interpolation by using Exponential Splines in Tension (or ESTs). For this purpose the notion of ac-preserving interpolant, which is usually employed in spline-in-tension interpolation, is refined and the existence ofc-preserving ESTs is established for the so-calledc-admissible data sets. The problem of constructing ac-preserving and visually pleasing EST is then treated by combining a generalized Newton-Raphson method, due to Ben-Israel, with a step-length technique which serves the need for “visual pleasantness”. The numerical performance of the so formed iterative scheme is discussed for several examples.

Spatial shape‐preserving interpolation using ν‐splines

We present a global iterative algorithm for constructing spatial G2‐continuous interpolating ν‐splines, which preserve the shape of the polygonal line that interpolates the given points. Furthermore, the algorithm can handle data exhibiting two kinds of degeneracy, namely, coplanar quadruples and collinear triplets of points. The convergence of the algorithm stems from the asymptotic properties of the curvature, torsion and Frénet frame of ν‐splines for large values of the tension parameters, which are thoroughly investigated and presented. The performance of our approach is tested on two data sets, one of synthetic nature and the other of industrial interest.