American Society of Mechanical Engineers

A survey is provided of shape parameterization techniques formultidisciplinary optimization, and some emerging ideas are highlighted. The survey focuses on the suitability of available techniques for multidisciplinary applications of complex cone gurations using high-e delity analysis tools such as computational e uid dynamics and computational structural mechanics. The suitability criteria are based on the efe ciency, effectiveness, ease of implementation, and availability of analytical sensitivities for geometry and grids. A section on sensitivity analysis, grid regeneration, and grid deformation techniques is also provided.

Survey of Shape Parameterization Techniques for High-Fidelity Multidisciplinary Shape Optimization

We present a new method to convert the geometry of a mesh into frequency space. The eigenfunctions of the Laplace-Beltrami operator are used to define Fourier-like function basis and transform. Since this generalizes the classical Spherical Harmonics to arbitrary manifolds, the basis functions will be called Manifold Harmonics. It is well known that the eigenvectors of the discrete Laplacian define such a function basis. However, important theoretical and practical problems hinder us from using this idea directly. From the theoretical point of view, the combinatorial graph Laplacian does not take the geometry into account. The discrete Laplacian (cotan weights) does not have this limitation, but its eigenvectors are not orthogonal. From the practical point of view, computing even just a few eigenvectors is currently impossible for meshes with more than a few thousand vertices. In this paper, we address both issues. On the theoretical side, we show how the FEM (Finite Element Modeling) formulation defines a function basis which is both geometry-aware and orthogonal. On the practical side, we propose a band-by-band spectrum computation algorithm and an out-of-core implementation that can compute thousands of eigenvectors for meshes with up to a million vertices. Finally, we demonstrate some applications of our method to interactive convolution geometry filtering and interactive shading design.

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Spectral Geometry Processing with Manifold Harmonics

Implicit and Nonparametric Shape Reconstruction from Unorganized Data Using a Variational Level Set Method

This article develops a dynamic generalization of the nonuniform rational B-spline (NURBS) model. NURBS have become a defacto standard in commercial modeling systems because of their power to represent free-form shapes as well as common analytic shapes. To date, however, they have been viewed as purely geometric primitives that require the user to manually adjust multiple control points and associated weights in order to design shapes. Dynamic NURBS, or D-NURBS, are physics-based models that incorporate mass distributions, internal deformation energies, and other physical quantities into the popular NURBS geometric substrate. Using D-NURBS, a modeler can interactively sculpt curves and surfaces and design complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. D-NURBS move and deform in a physically intuitive manner in response to the user's direct manipulations. Their dynamic behavior results from the numerical integration of a set of nonlinear differential equations that automatically evolve the control points and weights in response to the applied forces and constraints. To derive these equations, we employ Lagrangian mechanics and a finite-element-like discretization. Our approach supports the trimming of D-NURBS surfaces using D-NURBS curves. We demonstrate D-NURBS models and constraints in applications including the rounding of solids, optimal surface fitting to unstructured data, surface design from cross sections, and free-form deformation. We also introduce a new technique for 2D shape metamorphosis using constrained D-NURBS surfaces.

/pdf/dynamic-nurbs-with-geometric-constraints-for-interactive-3h332bw0kg.pdf

Dynamic NURBS with geometric constraints for interactive sculpting

A method of generating free-form surfaces using solutions to a suitably chosen partial differential equation is discussed. By varying the boundary conditions and a parameter in the partial differential equation, it is demonstrated that a wide variety of surface shapes are accessible to the method.

Using partial differential equations to generate free-form surfaces: 91787

A method is proposed for representing surfaces as solutions to partial differential equations. It is shown, by examples from the field of blend generation, that the method can easily achieve the required degree of continuity between the blend and the surfaces to which it attaches. The surfaces also have the property of being geometrically ‘well-behaved’.

Generating blend surfaces using partial differential equations

Interactive design of practical surfaces using the partial differential equation (PDE) method is considered. The PDE method treats surface design as a boundary value problem (ensuring that surfaces can be defined using a small set of design parameters). Owing to the elliptic nature of the PDE operator, the boundary conditions imposed around the edges of the surface control the internal shape of the surface. Moreover, surfaces obtained in this manner tend to be smooth and fair. The PDE chosen has a closed form solution allowing the interactive manipulation of the surfaces in real time. Thus we present efficient techniques by which we show how surfaces of practical significance can be constructed interactively in real time.

Techniques for interactive design using the PDE method

A method is presented for approximating surfaces, which are the solutions of partial differential equations, in terms of B-splines so that they can be represented in a form which is compatible with more established techniques of surface design. This allows surface data generated in this way to be incorporated into other design systems. Furthermore, the properties of B-splines can be used to good effect to enhance the PDE method, and this is illustrated by exercising local control of the surface shape in one of the examples given in the results.

Representing PDE surfaces in terms of B-splines

A new method is presented for the parametrization of aircraft geometry; it is efficient in the sense that a relatively small number of "design" parameters are required to describe a complex surface geometry. The method views surface generation as a boundary-value problem and produces surfaces as the solutions to elliptic partial differential equations, hence its name, the PDE method. The use of the PDE method will be illustrated in this article by the parametrization of double delta geometries; it will be shown that it is possible to capture the basic features of the large-scale geometry of the aircraft in terms of a small set of design variables.

Malcolm I. G. Bloor

Papers

Using partial differential equations to generate free-form surfaces: 91787

Generating blend surfaces using partial differential equations

Techniques for interactive design using the PDE method

Representing PDE surfaces in terms of B-splines

Efficient parametrization of generic aircraft geometry