This paper proposes two, geodesic-curvature based, criteria for shape-preserving interpolation on smooth surfaces, the first criterion being of non-local nature, while the second criterion is a local (weaker) version of the first one. These criteria are tested against a family of on-surface C2 splines obtained by composing the parametric representation of the supporting surface with variable-degree (≥3) splines amended with the preimages of the shortest-path geodesic arcs connecting each pair of consecutive interpolation points. After securing that the interpolation problem is well posed, we proceed to investigate the asymptotic behaviour of the proposed on-surface splines as degrees increase. Firstly, it is shown that the local-convexity sub-criterion of the local criterion is satisfied. Second, moving to non-local asymptotics, we prove that, as degrees increase, the interpolant tends uniformly to the spline curve consisting of the shortest-path geodesic arcs. Then, focusing on isometrically parametrized developable surfaces, sufficient conditions are derived, which secure that all criteria of the first (strong) criterion for shape-preserving interpolation are met. Finally, it is proved that, for adequately large degrees, the aforementioned sufficient conditions are satisfied. This permits to build an algorithm that, after a finite number of iterations, provides a C2 shape-preserving interpolant for a given data set on a developable surface.

In this work, we propose and test a method to expedite Global Sensitivity Analysis (GSA) in the context of shape optimisation of free-form shapes. To leverage the computational burden that is likely to occur in engineering problems, we construct a Shape-Signature-Vector (SSV) and propose to use it as a substitute for physics. SSV is composed of shapes’ integral properties, in our case geometric moments and their invariants of varying order, and is used as quantity-of-interest (QoI) for prior estimation of parametric sensitivities. Opting for geometric moments is motivated by the fact that they are intrinsic properties of shapes’ underlying geometry, and their evaluation is essential in many physical computations as they act as a medium for interoperability between geometry and physics. The proposed approach has been validated in the area of computer-aided ship design with regard to the capability of global- and composite-SSV to reveal parametric sensitivities of different ship hulls for the wave-making resistance coefficient (Cw), which is a critical QoI towards improving ship’s efficiency and thus decreasing emissions. More importantly, the longitudinal distribution of the volume below the ship’s floating waterline, which is measurable via geometric moments, has an impact on Cw. Through extensive experimentation, we show a strong correlation between the sensitive parameters obtained with respect to SSV and those based on Cw. Consequently, we can estimate parameters’ sensitivity with considerably reduced computational cost compared to when sensitivity analysis is performed with respect to Cw. Finally, two design spaces are constructed with sensitive parameters evaluated from SSV and Cw, and spaces’ quality and richness are analysed in terms of their capability to provide an optimised solution.

In shape optimisation problems, subspaces generated with conventional dimension reduction approaches often fail to extract the intrinsic geometric features of the shape that would allow the exploration of diverse but valid candidate solutions. More importantly, they also lack incorporation of any notion of physics against which shape is optimised. This work proposes a shape-supervised dimension reduction approach. To simultaneously tackle these deficiencies, it uses higher-level information about the shape in terms of its geometric integral properties, such as geometric moments and their invariants. Their usage is based on the fact that moments of a shape are intrinsic features of its geometry, and they provide a unifying medium between geometry and physics. To enrich the subspace with latent features associated with shape’s geometrical features and physics, we also evaluate a set of composite geometric moments, using the divergence theorem, for appropriate shape decomposition. These moments are combined with the shape modification function to form a Shape Signature Vector (SSV) uniquely representing a shape. Afterwards, the generalised Karhunen–Loève expansion is applied to SSV, embedded in a generalised (disjoint) Hilbert space, which results in a basis of the shape-supervised subspace retaining the highest geometric and physical variance. Validation experiments are performed for a three-dimensional wing and a ship hull model. Our results demonstrate a significant reduction of the original design space’s dimensionality for both test cases while maintaining a high representation capacity and a large percentage of valid geometries that facilitate fast convergence to the optimal solution. The code developed to implement this approach is available at https://github.com/shahrozkhan66/SSDR.git.

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).

This paper proposes a framework for constructing G1 surfaces that interpolate data points on parallel cross sections, consisting of simple disjoined and non-nested contours, the number of which may vary from plane to plane. Using appropriately estimated cross tangent vectors at the given points, we split the problem into a sequence of local Hermite problems, each of which can be one of the following three types: "one-to-one", "one-to-many" or "many-to-many". The solution of the "one-to-many" branching problem, where one contour on the i-plane is to be connected to script M sign-contours on the (i+1)-plane, is based on combining skinning with trimming and hole filling. More specifically, we firstly construct a G1 surrounding curve of all script M sign-contours on the (i+1)-plane, consisting of contour portions connected with linear segments, the so-called bridges. Next, we build a surface that skins the i-plane contour with the (i+1)-plane surrounding curve and trim suitably along the bridges. The resulting multi-sided hole is covered with quadrilateral Gordon-Coons patches that possess G1 continuity. For this purpose, we develop a hole-filling technique that employs shape-preserving guide curves and is able to preserve data symmetries. The "many-to-many "problem is handled by combining the "one-to-many" methodology with a zone-separation technique, that achieves to split the configuration into two "one-to-many" problems. The methodology, implemented as a C++ Rhino v3.0 plug-in, is illustrated via a synthetic example. © 2006 ACM.

Transformation of hardcopy ship drawings to electronic ones is usually accomplished through scanning and raster-to-vector conversions. Such conversions are, however, limited to produce low-degree vector entities, such as line segments, poly-lines and circular arcs. As a consequence, free-form curves, appearing in the original hardcopy, are usually disintegrated to a significant number of overlapping line and/or arc segments. The algorithm presented in this paper, consists of a scan-line processing of line segments that are grouped (clustered) with the aid of a moving scan-line and an appropriately defined distance to previously grouped entities. The performance of the algorithm is illustrated for the body-plan of a bulk carrier.