Isogeometric Analysis (IgA)

In this thesis a recently proposed numerical method for solving partial differential equations,
isogeometric analysis (IGA), is utilized for the purpose of shape optimization, with a particular
emphasis on applications to two-dimensional design problems arising in electromagnetic applications.
The study is motivated by the fact that in contrast with most commonly utilized finite
element approximations, IGA allows one to exactly represent geometries arising in computer
aided design applications with relatively few variables using splines.
The following problems coming from theoretical considerations or engineering applications
are solved in the thesis utilizing IGA:
 finding a shape having a few prescribed eigenvalues of the Laplace operator;
 shape optimization of sub-wavelength micro-antennas for energy concentration;
 shape optimization of nano-antennas for field enhancement;
 economical design of magnetic density separators.
From the point of view of method development, several heuristic approaches for extending a
valid parametrization of the boundary onto the domain’s interior are examined in the thesis. The
parametrization approaches and a method for validating a spline parametrization are combined
into an iterative algorithm for shape optimization of two dimensional electromagnetic problems.
The algorithm may also be relevant for problems in other engineering disciplines.
Using the methods developed in this thesis, remarkably we have obtained antennas that
perform one million times better than an earlier topology optimization result. This shows a
great potential of shape optimization using IGA in the area of electromagnetic antenna design
in particular, and for electromagnetic problems in general. Our conclusion is that IGA is well
suited for shape optimization.

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.

—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 difficulties 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 non-linear Winslow functional in the vicinity of a reference parametrization. Furthermore, we enforce the validity of the parametrization by ensuring the non-negativity of the coefficients 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.

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 hexa-hedra. 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.

The goal of dust astronomy is to uncover the information contained in individual cosmic dust grains. A series of previous dust instruments lead to novel in-situ instrumentation suitable for determining the origin and chemical and elemental composition of dust particles. A new instrument was developed that combines large target area, high mass-resolution, and accurate trajectory determination. The Hyperdust instrument is a combination of a Dust Trajectory Sensor (DTS) with an in-situ chemical analyzer. Dust particles' trajectories are determined by the measurement of induced charge signals, when a charged grain flies through a position-sensitive electrode system. A modern DTS can measure dust particles as small as 0.3 µm in radius and dust speeds up to 100 km/s. The chemical analyzer with ~0.1 m 2 sensitive target area has a mass resolution > 200. The Hyperdust instrument is capable of distinguishing interstellar and interplanetary grains based on their trajectory information. The Hyperdust instrument is currently being developed to Technical Readiness Level (TRL) 6 funded by NASA's MatISSE program to be a low-mass, high performance instrument for future in-situ exploration. This paper describes its current state of development.

Background: Thispaperpresentsasimpleandeffectiveformulationbasedona rotation-free isogeometric approach for the assessment of collapse limit loads of plastic thin plates in bending. Methods: Theformulationreliesonthekinematic(orupperbound)theoremand namely B-splines or non-uniform rational B-splines (NURBS), resulting in both exactly geometric representation and high-order approximations. Only one deflection variable (without rotational degrees of freedom) is used for each control point. This allows us to design the resulting optimization problem with a minimum size that is very useful to solve large-scale plate problems. The optimization formulation of limit analysis is transformed into the form of a second-order cone programming problem so that it can be solved using highly efficient interior-point solvers. Resultsandconclusions: Severalnumericalexamplesaregiventodemonstrate reliability and effectiveness of the present method in comparison with other published methods. Keywords: P...

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.

Purpose-The waste recycling industry increasingly relies on magnetic density separators. These devices generate an upward magnetic force in ferro-fluids allowing to separate the immersed particles according to their mass density. Recently, a new separator design has been proposed that significantly reduces the required amount of permanent magnet material. The purpose of this paper is to reduce the undesired end-effects in the upward force that this design generates by altering the shape of the fer-romagnetic covers of the individual poles. Design/methodology/approach-We represent the shape of the fer-romagnetic pole covers with B-splines and define a cost functional that measures the non-uniformity in the magnetic force in an area above the poles. We apply an isogeometric shape optimization procedure, which allows us to accurately represent, analyze and optimize the geometry using only a few design variables. The design problem is regularized by imposing constraints that enforce the convexity of the pole cover shapes. It is solved by a non-linear optimization procedure. We validate the implementation of our algorithm using a simplified variant of our design problem with a known analytical solution. The algorithm is subsequently applied to the problem posed. Research limitations/implications-The shape optimization attains its target and yields pole cover shapes that give rise to a magnetic field that is uniform over a larger domain. This increased uniformity is obtained at the cost of a pole cover shape that differs per pole. This limitation has negligible impact on the manufacturing of the separator. The new pole cover shapes, therefore, lead to improved performance of the density separation. Originality/value-This paper treats the shapes optimization of magnetic density separators systematically and presents new shapes for the ferromagnetic pole covers. Due to the larger uniformity of the generated field, these shapes should enable larger amounts of waste to be processed than the previous design.

We present a novel approach to modeling and simulating blood cells, including RBCs, using isogeometric analysis. The particulates are modeled as deformable capsules with a membrane of finite thickness containing fluid in the exterior and interior. The deformation of the capsule results from the fluid flow in the capillary, and this in turn gives rise to membrane forces which are communicated to the fluid in the vicinity of the capsule.
Through NURBS-based modeling we can represent a geometrically accurate model and perform stress analysis using the same geometric framework. Additionally, due to the inherent geometric exactness and smoothness of the NURBS representation, models can be simulated using relatively few elements as compared with traditional FEM-based methods.