Dang Manh Nguyen

Planar Parametrization in Isogeometric Analysis

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.

ISOGEOMETRIC SHAPE OPTIMIZATION FOR ELECTROMAGNETIC SCATTERING PROBLEMS

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.

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.

Nonuniform Discontinuous Galerkin Filters via Shift and Scale

Convolving the output of Discontinuous Galerkin (DG) computations with symmetric Smoothness-Increasing Accuracy-Conserving (SIAC) filters can improve both smoothness and accuracy. To extend convolution to the boundaries, several one-sided spline filters have recently been developed. We interpret these filters as instances of a general class of position-dependent spline filters that we abbreviate as PSIAC filters. These filters may have a non-uniform knot sequence and may leave out some B-splines of the sequence. For general position-dependent filters, we prove that rational knot sequences result in rational filter coefficients. We derive symbolic expressions for prototype knot sequences, typically integer sequences that may include repeated entries and corresponding B-splines, some of which may be skipped. Filters for shifted or scaled knot sequences are easily derived from these prototype filters so that a single filter can be re-used in different locations and at different scales. Moreover, the convolution itself reduces to executing a single dot product making it more stable and efficient than the existing approaches based on numerical integration. The construction is demonstrated for several established and one new boundary filter.

Explicit Least-Degree Boundary Filters for Discontinuous Galerkin

Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improve both smoothness and accuracy of the output. At domain boundaries, these filters have to be one-sided for non-periodic boundary conditions. Recently, position-dependent smoothness-increasing accuracy-preserving (PSIAC) filters were shown to be a superset of the well-known one-sided RLKV and SRV filters. Since PSIAC filters can be formulated symbolically, PSIAC filtering amounts to forming linear products with local DG output and so offers a more stable and efficient implementation. The paper introduces a new class of PSIAC filters NP0 that have small support and are piecewise constant. Extensive numerical experiments for the canonical hyperbolic test equation show NP0 filters outperform the more complex known boundary filters. NP0 filters typically reduce the L∞ error in the boundary region below that of the interior where optimally superconvergent symmetric filters of the same support are applied. NP0 filtering can be implemented as forming linear combinations of the data with short rational weights. Exact derivatives of the convolved output are easy to compute.