Jens Gravesen

The Geometry of the Scroll Compressor

The scroll compressor is an ingenious machine used for compressing air or refrigerant; it was originally invented in 1905 by Leon Creux. The classical design consists of two nested identical scrolls given by circle involutes, one of which is rotated through

Curves and surfaces represented by polynomial support functions

\text{180}^{\circ}

Schrödinger problems for surfaces of revolution—the finite cylinder as a test example

with respect to the other. By specifying not a parametrization of the curve, but instead the radius of curvature as a function of tangent direction and using the intrinsic equation of a planar curve, the design can be changed in a way that allows all relevant geometrical quantities to be calculated in closed analytical form.

Constructing invariant fairness measures for surfaces

This paper studies shapes (curves and surfaces) which can be described by (piecewise) polynomial support functions. The class of these shapes is closed under convolutions, offsetting, rotations and translations. We give a geometric discussion of these shapes and present methods for the approximation of general curves and surfaces by them. Based on the rich theory of spherical spline functions, this leads to computational techniques for rational curves and surfaces with rational offsets, which can deal with shapes without inflections/parabolic points.

Adaptive subdivision and the length and energy of Be´zier curves

A set of ordinary differential equations is derived employing the method of differentiable forms so as to describe the quantum mechanics of a particle constrained to move on a general two-dimensional surface of revolution. Eigenvalues and eigenstates are calculated quasianalytically in the case of a finite cylinder (finite along the axis) and compared with the eigenvalues and eigenstates of a full three-dimensional Schrodinger problem corresponding to a hollow cylinder in the limit where the inner and outer radii approach each other. Good agreement between the two models is obtained for a relative difference less than 20% in inner and outer radii.

On rationally supported surfaces

The paper proposes a rational method to derive fairness measures for surfaces. It works in cases where isophotes, reflection lines, planar intersection curves, or other curves are used to judge the fairness of the surface. The surface fairness measure is derived by demanding that all the given curves should be fair with respect to an appropriate curve fairness measure. The method is applied to the field of ship hull design where the curves are plane intersections. The method is extended to the case where one considers, not the fairness of one curve, but the fairness of a one parameter family of curves. Six basic third order invariants by which the fairing measures can be expressed are defined. Furthermore, the geometry of a plane intersection curve is studied, and the variation of the total, the normal, and the geodesic curvature and the geodesic torsion is determined.

Planar Parametrization in Isogeometric Analysis

Abstract It is an often used fact that the control polygon of a Bezier curve approximates the curve and that the approximation gets better when the curve is subdivided. In particular, if a Bezier curve is subdivided into some number of pieces, then the arc-length of the original curve is greater than the sum of the chord-lengths of the pieces, and less than the sum of the polygon-lengths of the pieces. Under repeated subdivisions, the difference between this lower and upper bound gets arbitrarily small. If Lc denotes the total chord-length of the pieces and Lp denotes the total polygon-length of the pieces, the best estimate of the true arc-length is (2L c + (n − 1)L p ) (n + 1) , where n is the degree of the Bezier curve. This convex combination of Lc and Lp is best in the sense that the error goes to zero under repeated subdivision asymptotically faster than the error of any other convex combination, and it forms the basis for a fast adaptive algorithm, which determines the arc-length of a Bezier curve. The energy of a curve is half the square of the curvature integrated with respect to arc-length. Like in the case of the arc-length, it is possible to use the chord-length and polygon-length of the pieces of a subdivided Bezier curve to estimate the energy of the Bezier curve.

ISOGEOMETRIC SHAPE OPTIMIZATION FOR ELECTROMAGNETIC SCATTERING PROBLEMS

We analyze the class of surfaces which are equipped with rational support functions. Any rational support function can be decomposed into a symmetric (even) and an antisymmetric (odd) part. We analyze certain geometric properties of surfaces with odd and even rational support functions. In particular it is shown that odd rational support functions correspond to those rational surfaces which can be equipped with a linear field of normal vectors, which were discussed by Sampoli et al. (Sampoli, M.L., Peternell, M., Juttler, B., 2006. Rational surfaces with linear normals and their convolutions with rational surfaces. Comput. Aided Geom. Design 23, 179-192). As shown recently, this class of surfaces includes non-developable quadratic triangular Bezier surface patches (Lavicka, M., Bastl, B., 2007. Rational hypersurfaces with rational convolutions. Comput. Aided Geom. Design 24, 410-426; Peternell, M., Odehnal, B., 2008. Convolution surfaces of quadratic triangular Bezier surfaces. Comput. Aided Geom. Design 25, 116-129).

Surfaces parametrized by the normals

Before isogeometric analysis can be applied to solving a partial differential equation posed over some physical domain, one needs to construct a valid parametrization of the geometry. The accuracy of the analysis is affected by the quality of the parametrization. The challenge of computing and maintaining a valid geometry parametrization is particularly relevant in applications of isogemetric analysis to shape optimization, where the geometry varies from one optimization iteration to another. We propose a general framework for handling the geometry parametrization in isogeometric analysis and shape optimization. It utilizes an expensive non-linear method for constructing/updating a high quality reference parametrization, and an inexpensive linear method for maintaining the parametrization in the vicinity of the reference one. We describe several linear and non-linear parametrization methods, which are suitable for our framework. The non-linear methods we consider are based on solving a constrained optimization problem numerically, and are divided into two classes, geometry-oriented methods and analysis-oriented methods. Their performance is illustrated through a few numerical examples.

Surfaces with piecewise linear support functions over spherical triangulations

We consider the benchmark problem of magnetic energy density enhancement in a small spatial region by varying the shape of two symmetric conducting scatterers. We view this problem as a prototype for a wide variety of geometric design problems in electromagnetic applications. Our approach for solving this problem is based on shape optimization and isogeometric analysis. One of the major di-culties we face to make these methods work together is the need to maintain a valid parametrization of the computational domain during the optimization. Our approach to generating a domain parametrization is based on minimizing a second order approximation to the Winslow functional in the vicinity of a reference parametrization. Furthermore, we enforce the validity of the parametrization by ensuring the non-negativity of the coe-cients of a B-spline expansion of the Jacobian. The shape found by this approach outperforms earlier design computed using topology optimization by a factor of one billion.