Hierarchical Riesz bases for Hs(Omega), 1 < s < 5/2

On arbitrary polygonal domains $Omega subset RR^2$, we construct $C^1$ hierarchical Riesz bases for Sobolev spaces $H^s(Omega)$. In contrast to an earlier construction by Dahmen, Oswald, and Shi (1994), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from $s in (2,frac{5}{2})$ to $s in (1,frac{5}{2})$. Since the latter range includes $s=2$, with respect to the present basis, the stiffness matrices of fourth-order elliptic problems are uniformly well-conditioned

A Hermite interpolatory subdivision scheme for C2C^2-quintics on the Powell-Sabin 12-split

In order to construct a $C^1$-quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell-Sabin 12-split. It has been shown previously that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme. In this paper we introduce a nodal macro-element on the 12-split for the space of quintic splines that are locally $C^3$ and globally $C^2$. For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage. Using the available first derivatives for Phong shading, visually appealing plots can be generated after just a couple of refinements.Comment: 17 pages, 7 figure

Computational Gradient Elasticity and Gradient Plasticity with Adaptive Splines

Classical continuum mechanics theories are largely insufficient in capturing size effects observed in many engineering materials: metals, composites, rocks etc. This is attributed to the absence of a length scale that accounts for microstructural effects inherent in these materials. Enriching the classical theories with an internal length scale solves this problem. One way of doing this, in a theoretically sound manner, is introducing higher order gradient terms in the constitutive relations. In elasticity, introducing a length scale removes the singularity observed at crack tips using the classical theory. In plasticity, it eliminates the spurious mesh sensitivity observed in softening and localisation problems by defining the width of the localisation zone thereby maintaining a well-posed boundary value problem. However, this comes at the cost of more demanding solution techniques. Higher-order continuity is usually required for solving gradient-enhanced continuum theories, a requirement difficult to meet using traditional finite elements. Hermitian finite elements, mixed methods and meshless methods have been developed to meet this requirement, however these methods have their drawbacks in terms of efficiency, robustness or implementational convenience. Isogeometric analysis, which exploits spline-based shape functions, naturally incorporates higher-order continuity, in addition to capturing the exact geometry and expediting the design-through-analysis process. Despite its potentials, it is yet to be fully explored for gradient-enhanced continua. Hence, this thesis develops an isogeometric analysis framework for gradient elasticity and gradient plasticity. The linearity of the gradient elasticity formulation has enabled an operator-split approach so that instead of solving the fourth-order partial differential equation monolithically, a set of two partial differential equations is solved in a staggered manner. A detailed convergence analysis is carried out for the original system and the split set using NURBS and T-splines. Suboptimal convergence rates in the monolithic approach and the limitations of the staggered approach are substantiated. Another advantage of the spline-based approach adopted in this work is the ease with which different orders of interpolation can be achieved. This is useful for consistency, and relevant in gradient plasticity where the local (explicit formulation) or nonlocal (implicit formulation) effective plastic strain needs to be discretised in addition to the displacements. Using different orders of interpolation, both formulations are explored in the second-order and a fourth-order implicit gradient formulation is proposed. Results, corroborated by dispersion analysis, show that all considered models give good regularisation with mesh-independent results. Compared with finite element approaches that use Hermitian shape functions for the plastic multiplier or mixed finite element approaches, isogeometric analysis has the distinct advantage that no interpolation of derivatives is required. In localisation problems, numerical accuracy requires the finite element size employed in simulations to be smaller than the internal length scale. Fine meshes are also needed close to regions of geometrical singularities or high gradients. Maintaining a fine mesh globally can incur high computational cost especially when considering large structures. To achieve this efficiently, selective refinement of the mesh is therefore required. In this context, splines need to be adapted to make them analysis-suitable. Thus, an adaptive isogeometric analysis framework is also developed for gradient elasticity and gradient plasticity. The proposed scheme does not require the mesh size to be smaller than the length scale, even during analysis, until a localisation band develops upon which adaptive refinement is performed. Refinement is based on a multi-level mesh with truncated hierarchical basis functions interacting through an inter-level subdivision operator. Through Bezier extraction, truncation of the bases is simplified by way of matrix multiplication, and an element-wise standard finite element data structure is maintained. In sum, a robust computational framework for engineering analysis is established, combining the flexibility, exact geometry representation, and expedited design-through analysis of isogeometric analysis, size-effect capabilities and mesh-objective results of gradient-enhanced continua, the standard convenient data structure of finite element analysis and the improved efficiency of adaptive hierarchical refinement

Isogeometric analysis and hierarchical refinement for multi-field contact problems

The present work deals with multi-field contact problems in the context of IGA. In particular, a thermomechanical as well as a fracture mechanical system is considered, where novel formulations are introduced for both. The corresponding discrete contact formulations are based on a variationally consistent mortar approach adapted for NURBS discretized and hierarchical refined surfaces. Finally, the capabilities of the proposed framework are demonstrated within numerous numerical examples

An immersed boundary hierarchical B-spline method for flexoelectricity

© 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper develops a computational framework with unfitted meshes to solve linear piezoelectricity and flexoelectricity electromechanical boundary value problems including strain gradient elasticity at infinitesimal strains. The high-order nature of the coupled PDE system is addressed by a sufficiently smooth hierarchical B-spline approximation on a background Cartesian mesh. The domain of interest is embedded into the background mesh and discretized in an unfitted fashion. The immersed boundary approach allows us to use B-splines on arbitrary domain shapes, regardless of their geometrical complexity, and could be directly extended, for instance, to shape and topology optimization. The domain boundary is represented by NURBS, and exactly integrated by means of the NEFEM mapping. Local adaptivity is achieved by hierarchical refinement of B-spline basis, which are efficiently evaluated and integrated thanks to their piecewise polynomial definition. Nitsche's formulation is derived to weakly enforce essential boundary conditions, accounting also for the non-local conditions on the non-smooth portions of the domain boundary (i.e. edges in 3D or corners in 2D) arising from Mindlin's strain gradient elasticity theory. Boundary conditions modeling sensing electrodes are formulated and enforced following the same approach. Optimal error convergence rates are reported using high-order B-spline approximations. The method is verified against available analytical solutions and well-known benchmarks from the literature.Peer ReviewedPostprint (author's final draft

Nitsche’s method for two and three dimensional NURBS patch coupling

We present a Nitche’s method to couple non-conforming two and three-dimensional NURBS (Non Uniform Rational B-splines) patches in the context of isogeometric analysis (IGA). We present results for linear elastostatics in two and and three-dimensions. The method can deal with surface-surface or volume-volume coupling, and we show how it can be used to handle heterogeneities such as inclusions. We also present preliminary results on modal analysis. This simple coupling method has the potential to increase the applicability of NURBS-based isogeometric analysis for practical applications

Smooth representation of thin shells and volume structures for isogeometric analysis

The purpose of this study is to develop self-contained methods for obtaining smooth meshes which are compatible with isogeometric analysis (IGA). The study contains three main parts. We start by developing a better understanding of shapes and splines through the study of an image-related problem. Then we proceed towards obtaining smooth volumetric meshes of the given voxel-based images. Finally, we treat the smoothness issue on the multi-patch domains with C1 coupling. Following are the highlights of each part. First, we present a B-spline convolution method for boundary representation of voxel-based images. We adopt the filtering technique to compute the B-spline coefficients and gradients of the images effectively. We then implement the B-spline convolution for developing a non-rigid images registration method. The proposed method is in some sense of “isoparametric”, for which all the computation is done within the B-splines framework. Particularly, updating the images by using B-spline composition promote smooth transformation map between the images. We show the possible medical applications of our method by applying it for registration of brain images. Secondly, we develop a self-contained volumetric parametrization method based on the B-splines boundary representation. We aim to convert a given voxel-based data to a matching C1 representation with hierarchical cubic splines. The concept of the osculating circle is employed to enhance the geometric approximation, where it is done by a single template and linear transformations (scaling, translations, and rotations) without the need for solving an optimization problem. Moreover, we use the Laplacian smoothing and refinement techniques to avoid irregular meshes and to improve mesh quality. We show with several examples that the method is capable of handling complex 2D and 3D configurations. In particular, we parametrize the 3D Stanford bunny which contains irregular shapes and voids. Finally, we propose the B´ezier ordinates approach and splines approach for C1 coupling. In the first approach, the new basis functions are defined in terms of the B´ezier Bernstein polynomials. For the second approach, the new basis is defined as a linear combination of C0 basis functions. The methods are not limited to planar or bilinear mappings. They allow the modeling of solutions to fourth order partial differential equations (PDEs) on complex geometric domains, provided that the given patches are G1 continuous. Both methods have their advantages. In particular, the B´ezier approach offer more degree of freedoms, while the spline approach is more computationally efficient. In addition, we proposed partial degree elevation to overcome the C1-locking issue caused by the over constraining of the solution space. We demonstrate the potential of the resulting C1 basis functions for application in IGA which involve fourth order PDEs such as those appearing in Kirchhoff-Love shell models, Cahn-Hilliard phase field application, and biharmonic problems

Isogeometric treatment of large deformation contact and debonding problems with T-splines: a review

AbstractWithin a setting where the isogeometric analysis (IGA) has been successful at bringing two different research fields together, i.e. Computer Aided Design (CAD) and numerical analysis, T-spline IGA is applied in this work to frictionless contact and mode-I debonding problems between deformable bodies in the context of large deformations. Based on the concept of IGA, the smooth basis functions are adopted to describe surface geometries and approximate the numerical solutions, leading to higher accuracy in the contact integral evaluation. The isogeometric discretizations are here incorporated into an existing finite element framework by using Bézier extraction, i.e. a linear operator which maps the Bernstein polynomial basis on Bézier elements to the global isogeometric basis. A recently released commercial T-spline plugin for Rhino is herein used to build the analysis models adopted in this study.In such context, the continuum is discretized with cubic T-splines, as well as with Non Uniform Rational B-Splines (NURBS) and Lagrange polynomial elements for comparison purposes, and a Gauss-point-to-surface (GPTS) formulation is combined with the penalty method to treat the contact constraints. The purely geometric enforcement of the non-penetration condition in compression is generalized to encompass both contact and mode-I debonding of interfaces which is approached by means of cohesive zone (CZ) modeling, as commonly done by the scientific community to analyse the progressive damage of materials and interfaces. Based on these models, non-linear relationships between tractions and relative displacements are assumed. These relationships dictate both the work of separation per unit fracture surface and the peak stress that has to be reached for the crack formation. In the generalized GPTS formulation an automatic switching procedure is used to choose between cohesive and contact models, depending on the contact status. Some numerical results are first presented and compared in 2D for varying resolutions of the contact and/or cohesive zone, including frictionless sliding and cohesive debonding, all featuring the competitive accuracy and performance of T-spline IGA. The superior accuracy of T-spline interpolations with respect to NURBS and Lagrange interpolations for a given number of degrees of freedom (Dofs) is always verified. The isogeometric formulation is also extended to 3D bodies, where some examples in large deformations based on T-spline discretizations show an high smoothness of the reaction history curves