G. Sangalli

ISOGEOMETRIC COLLOCATION METHODS

We initiate the study of collocation methods for NURBS-based isogeometric analysis. The idea is to connect the superior accuracy and smoothness of NURBS basis functions with the low computational cost of collocation. We develop a one-dimensional theoretical analysis, and perform numerical tests in one, two and three dimensions. The numerical results obtained confirm theoretical results and illustrate the potential of the methodology.

Mathematical analysis of variational isogeometric methods

This review paper collects several results that form part of the theoretical foundation of isogeometric methods. We analyse variational techniques for the numerical resolution of PDEs based on splines or NURBS and we provide optimal approximation and error estimates in several cases of interest. The theory presented also includes estimates for T-splines, which are an extension of splines allowing for local refinement. In particular, we focus our attention on elliptic and saddle point problems, and we define spline edge and face elements. Our theoretical results are demonstrated by a rich set of numerical examples. Finally, we discuss implementation and efficiency together with preconditioning issues for the final linear system.

ANALYSIS-SUITABLE T-SPLINES OF ARBITRARY DEGREE: DEFINITION, LINEAR INDEPENDENCE AND APPROXIMATION PROPERTIES

T-splines are an important tool in IGA since they allow local refinement. In this paper we define analysis-suitable T-splines of arbitrary degree and prove fundamental properties: Linear independence of the blending functions and optimal approximation properties of the associated T-spline space. These are corollaries of our main result: A T-mesh is analysis-suitable if and only if it is dual-compatible. Indeed, dual compatibility is a concept already defined and used in L. Beirao da Veiga et al.5 Analysis-suitable T-splines are dual-compatible which allows for a straightforward construction of a dual basis.

An isogeometric method for the Reissner–Mindlin plate bending problem

We present a new isogeometric method for the discretization of the Reissner–Mindlin plate bending problem. The proposed scheme follows a recent theoretical framework that makes possible the construction of a space of smooth discrete deflections Wh and a space of smooth discrete rotations Θh such that the Kirchhoff constraint is exactly satisfied at the limit. Therefore we obtain a formulation which is natural from the theoretical/mechanical viewpoint and locking-free by construction. We prove that the method is uniformly stable and satisfies optimal convergence estimates. Finally, the theoretical results are fully supported by numerical tests.

Efficient matrix computation for tensor-product isogeometric analysis: The use of sum factorization

The geometrically rigorous nonlinear analysis of elastic shells is considered in the context of finite, but small, strain theory. The research is focused on the introduction of the full shell metric and examination of its influence on the nonlinear structural response. The exact relation between the reference and equidistant strains is employed and the complete analytic elastic constitutive relation between energetically conjugated forces and strains is derived via the reciprocal shift tensor. Utilizing these strict relations, the geometric stiffness matrix is derived explicitly by the variation of the unknown metric. Moreover, a compact form of this matrix is presented. Despite the linear displacement distribution due to the Kirchhoff-Love hypothesis, a nonlinear strain distribution arises along the shell thickness. This fact is sometimes disregarded for the nonlinear analysis of thin shells based on the initial geometry, thereby ignoring the strong curviness of a shell at some subsequent configuration. We show that the curviness of a shell at each configuration determines the appropriate shell formulation. For shells that become strongly curved at some configurations during deformation, the nonlinear distribution of strain throughout the thickness must be considered in order to obtain accurate results. We investigate four computational models: one based on the full analytical constitutive relation, and three simplified ones. Robustness, efficiency and accuracy of the presented formulation are examined via selected numerical experiments. Our main finding is that the employment of the full metric is often required when the complete response of the shells is sought, even for the initially thin shells. Finally, the simplified model that provided the best balance between efficiency and accuracy is suggested for the nonlinear analysis of strongly curved shells.

Efficient assembly based on B-spline tailored quadrature rules for the IgA-SGBEM

Abstract This paper deals with the discrete counterpart of 2D elliptic model problems rewritten in terms of Boundary Integral Equations. The study is done within the framework of Isogeometric Analysis based on B-splines. In such a context, the problem of constructing appropriate, accurate and efficient quadrature rules for the Symmetric Galerkin Boundary Element Method is here investigated. The new integration schemes, together with row assembly and sum factorization, are used to build a more efficient strategy to derive the final linear system of equations. Key ingredients are weighted quadrature rules tailored for B-splines, that are constructed to be exact in the whole test space, also with respect to the singular kernel. Several simulations are presented and discussed, showing accurate evaluation of the involved integrals and outlining the superiority of the new approach in terms of computational cost and elapsed time with respect to the standard element-by-element assembly.

Dual Compatible Splines on Nontensor Product Meshes

In this paper we introduce the concept of dual compatible (DC) splines on nontensor product meshes, study the properties of this class, and discuss their possible use within the isogeometric framework. We show that DC splines are linear independent and that they also enjoy good approximation properties.

A LAGRANGE MULTIPLIER METHOD FOR ELLIPTIC INTERFACE PROBLEMS USING NON-MATCHING MESHES

When considering multi-domain problems with non-matching meshes using Lagrange multiplier techniques, two basic problems occur. First and foremost, the discrete spaces for the discretization of the primal variable and the multipliers have to fulfill the inf-sup condition (see [7]) in order the resulting numerical scheme to be stable; it turns out that for many natural choices of approximations this is not the case. Fortunately, this problem can be alleviated by using stabilized multiplier methods (cf. [14, 17, 1, 4], for instance) or by using mesh-dependent penalty methods (see e.g. [2, 3]). The second problem is that products of traces of the primal variable and the multipliers have to be integrated on the interfaces. For methods known