Michael Pauley

Isogeometric Segmentation: The case of contractible solids without non-convex edges

Abstract We present a novel technique for segmenting a three-dimensional solid with a 3-vertex-connected edge graph consisting of only convex edges into a collection of topological hexahedra. Our method is based on the edge graph, which is defined by the sharp edges between the boundary surfaces of the solid. We repeatedly decompose the solid into smaller solids until all of them belong to a certain class of predefined base solids. The splitting step of the algorithm is based on simple combinatorial and geometric criteria. The segmentation technique described in the paper is part of a process pipeline for solving the isogeometric segmentation problem that we outline in the paper.

Isogeometric segmentation. Part II: On the segmentability of contractible solids with non-convex edges

Motivated by the discretization problem in isogeometric analysis, we consider the challenge of segmenting a contractible boundary-represented solid into a small number of topological hexahedra. A satisfactory segmentation of a solid must eliminate non-convex edges because they prevent regular parameterizations. Our method works by searching a sufficiently connected edge graph of the solid for a cycle of vertices, called a cutting loop, which can be used to decompose the solid into two new solids with fewer non-convex edges. This can require the addition of auxiliary vertices to the edge graph. We provide theoretical justification for our approach by characterizing the cutting loops that can be used to segment the solid, and proving that the algorithm terminates. We select the cutting loop using a cost function. For this cost function we propose terms which help to select geometrically and combinatorially favorable cutting loops. We demonstrate the effects of these terms using a suite of examples.

The Isogeometric Segmentation Pipeline

We present a pipeline for the conversion of 3D models into a form suitable for isogeometric analysis (IGA). The input into our pipeline is a boundary represented 3D model, either as a triangulation or as a collection of trimmed non-uniform rational B-spline (NURBS) surfaces. The pipeline consists of three stages: computer aided design (CAD) model reconstruction from a triangulation (if necessary); segmentation of the boundary-represented solid into topological hexahedra; and volume parameterization. The result is a collection of volumetric NURBS patches. In this paper we discuss our methods for the three stages, and demonstrate the suitability of the result for IGA by performing stress simulations with examples of the output.

Isogeometric segmentation

In the context of segmenting a boundary represented solid into topological hexahedra suitable for isogeometric analysis, it is often necessary to split an existing face by constructing auxiliary curves. We consider solids represented as a collection of trimmed spline surfaces, and design a curve which can split the domain of a trimmed surface into two pieces satisfying the following criteria: the curve must not intersect the boundary of the original domain, it must not intersect itself, the two resulting pieces should have good shape, and the endpoints and the tangents of the curve at the endpoints must be equal to specified values. A method is proposed for splitting a trimmed surface into two with a curve.The curve is required to have specified endpoints and tangents at the endpoints.The splitting is central to an algorithm for isogeometric segmentation of 3D models.The curve optimizes a penalty function that measures the quality of the shapes.We study regularity properties and methods for computing the penalty function.

Parabolic curves in Lie groups

To interpolate a sequence of points in Euclidean space, parabolic splines can be used. These are curves which are piecewise quadratic. To interpolate between points in a (semi-)Riemannian manifold, we could look for curves such that the second covariant derivative of the velocity is zero. We call such curves Jupp and Kent quadratics or JK-quadratics because they are a special case of the cubic curves advocated by Jupp and Kent. When the manifold is a Lie group with bi-invariant metric, we can relate JK-quadratics to null Lie quadratics which arise from another interpolation problem. We solve JK-quadratics in the Lie groups SO(3) and SO(1,2) and in the sphere and hyperbolic plane, by relating them to the differential equation for a quantum harmonic oscillator.

Jupp and Kent's cubics in Lie groups

We study a generalization of the cubic polynomial to Riemannian manifolds and other affine connection spaces, modifying the differential equation x(4)=0 by replacing higher derivatives with covariant derivatives. In matrix groups, with two particular choices of a left-invariant connection, we can convert the equation into a system of first order linear differential equations. We give asymptotics in a generic case in GL(n) and in the n-sphere.

Numerical filtering of linear state-space models with Markov switching

A class of discrete-time random processes that have seen a wide variety of applications consists of a linear state-space model whose parameters are modulated by the state of a finite-state Markov chain. A typical way to filter such processes is with collapsing methods, which approximate the underlying distribution by a mixture of Gaussians indexed by the recent history of the Markov chain. The computational cost of such methods increases rapidly as the error decreases to zero. This paper presents an alternative approach to filtering these processes based on keeping track of the values of the underlying probability density function and characteristic function on grids. It has favourable convergence properties under certain assumptions.