Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS as a basis for the approximation of the solution of PDEs. In this work, we investigate to which extent IGA solvers can benefit from the so-called sparse-grids construction in its combination technique form, which was first introduced in the early 90s in the context of the approximation of high-dimensional PDEs. The tests that we report show that, in accordance to the literature, a sparse-grid construction can indeed be useful if the solution of the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the case of non-smooth solutions when some a-priori knowledge on the location of the singularities of the solution can be exploited to devise suitable non-equispaced meshes. Finally, we remark that sparse grids can be seen as a simple way to parallelize pre-existing serial IGA solvers in a straightforward fashion, which can be beneficial in many practical situations.

Questa esperienza si inquadrata nel più ampio contesto del restauro del Villino Florio a Palermo parzialmente distrutto nel 1962 in seguito ad un incendio. Costruito per volere dalla ricca famiglia Florio dall'architetto Ernesto Basile e realizzato tra il 1899 e il 1902 è questa una delle prime opere architettoniche in stile Liberty d'Italia e viene considerato uno dei capolavori dell'Art Nouveau anche a livello europeo. Il restauro, condotto dalla Soprintendenza di Palermo ha interessato anche lo scalone monumentale, con il complesso ramage floreale che ne decorava il soffitto. La mancanza di riferimenti progettuali relativi alla sezione del ramage e di cui esiste solo una documentazione fotografica in bianco e nero, ha portato alla necessità di approfondire la problematica della sua ricostruzione seguendo metodi scientifici nuovi ed avanzati. Il modello 3D del ramage, realizzato dall'ITLab del CNR IBAM di Lecce, costituisce la base informativa necessaria alla successiva realizzazione dell'oggetto reale, eseguita con l'ausilio di macchine a controllo numerico su moduli in legno di rovere, assemblati e rifiniti così come apparivano nelle foto d'epoca. Grazie all'approccio numerico è stato possibile controllare l'intero processo di lavorazione e predisporre adeguate opere di rinforzo strutturale atte a sostenere il notevole peso del soffitto.

In this paper we investigate numerically the order of convergence of an isogeometric collocation method that builds upon the least-squares collocation method presented in Anitescu et al. (2015) and the variational collocation method presented in Gomez and De Lorenzis (2016). The focus is on smoothest B-splines/NURBS approximations, i.e, having global . Cp-1 continuity for polynomial degree . p. Within the framework of Gomez and De Lorenzis (2016), we select as collocation points a subset of those considered in Anitescu et al. (2015), which are related to the Galerkin superconvergence theory. With our choice, that features local symmetry of the collocation stencil, we improve the convergence behavior with respect to Gomez and De Lorenzis (2016), achieving optimal . L2-convergence for odd degree B-splines/NURBS approximations. The same optimal order of convergence is seen in Anitescu et al. (2015), where, however a least-squares formulation is adopted. Further careful study is needed, since the robustness of the method and its mathematical foundation are still unclear.

GeoPDEs (http://rafavzqz.github.io/geopdes) is an Octave/Matlab package for the solution of partial differential equations with isogeometric analysis, first released in 2010. In this work we present in detail the new design of the package, based on the use of Octave and Matlab classes. Compared to previous versions the new design is much clearer, and it is also more efficient in terms of memory consumption and computational time.

We present the design of an object oriented general purpose library for isogeometric analysis, where the mathematical concepts of the isogeometric method and their relationships are directly mapped into classes and their interactions. The encapsulation of mathematical concepts into interacting building blocks gives flexibility to use the library in a wide range of scientific areas and applications. We provide a precise framework for a lot of loose, available information regarding the implementation of the isogeometric method, and also discuss the similarities and differences between this and the finite element method. We also describe how to implement this proposed design in a C++11 open source library, \textttigatools (http://www.igatools.org). The library uses advanced object oriented and generic programming techniques to ensure reusability, reliability, and maintainability of the source code. It includes isogeometric elements of the h-div and h-curl type, and supports the development of dimension independent code (including manifolds and tensor-valued spaces). We finally present a number of code examples to illustrate the flexibility and power of the library, including surface domains, nonlinear elasticity, and Navier--Stokes computations on nontrivial geometries.

In this paper we discuss the use of the sum-factorization for the calculation of the integrals arising in Galerkin isogeometric analysis. While introducing very little change in an isogeometric code based on element-by-element quadrature and assembling, the sum-factorization approach, taking advantage of the tensor-product structure of splines or NURBS shape functions, significantly reduces the quadrature computational cost.

In this paper, IGA collocation methods are for the first time introduced for the solution of thin structural problems described by the Bernoulli-Euler beam and Kirchhoff plate models. In particular, a precise description of the proposed methods, of the relevant implementation details, and of the strategy to efficiently deal with different combinations of boundary conditions is given. Finally, several numerical experiments confirm that the proposed formulations represent an efficient and geometrically flexible tool for the simulation of thin structures.

Within the general framework of isogeometric methods, collocation schemes have been recently proposed as a viable and promising low-cost alternative to standard isogeometric Galerkin approaches. In this paper, isogeometric collocation methods for the numerical approximation of Reissner-Mindlin plate problems are proposed for the first time. Locking-free primal and mixed formulations are herein considered, and the potential of isogeometric collocation as a geometrically flexible and computationally efficient simulation tool for shear deformable plates is shown through the solution of several numerical tests.

We propose new collocation methods for phase-field models. Our algorithms are based on isogeometric analysis, a new technology that makes use of functions from computational geometry, such as, for example, Non-Uniform Rational B-Splines (NURBS). NURBS exhibit excellent approximability and controllable global smoothness, and can represent exactly most geometries encapsulated in Computer Aided Design (CAD) models. These attributes permitted us to derive accurate, efficient, and geometrically flexible collocation methods for phase-field models. The performance of our method is demonstrated by several numerical examples of phase separation modeled by the Cahn-Hilliard equation. We feel that our method successfully combines the geometrical flexibility of finite elements with the accuracy and simplicity of pseudo-spectral collocation methods, and is a viable alternative to classical collocation methods.

We propose and study high-regularity isogeometric discretizations of the Stokes problem. We address the Taylor-Hood isogeometric element, already known in this context, and a new Subgrid element which allows highest regularity velocity and pressure fields. Our stability analysis grounds on a characterization of full-rank scalar products for splines, which is the key theoretical result of this paper. We include numerical testing on two- and three-dimensional benchmarks.

In this work we present the application of isogeometric collocation techniques to the solution of spatial Timoshenko rods. The strong form equations of the problem are presented in both displacement-based and mixed formulations and are discretized via NURBS-based isogeometric collocation. Several numerical experiments are reported to test the accuracy and efficiency of the considered methods, as well as their applicability to problems of practical interest. In particular, it is shown that mixed collocation schemes are locking-free independently of the choice of the polynomial degrees for the unknown fields. Such an important property is also analytically proven.

We develop new quadrature rules for isogeometric analysis based on the solution of a local nonlinear problem. A simple and robust algorithm is developed to determine the rules which are exact for important B-spline spaces of uniform and geometrically stretched knot spacings. We consider both periodic and open knot vector configurations and illustrate the efficiency of the rules on selected boundary value problems. We find that the rules are almost optimally efficient, but much easier to obtain than optimal rules, which require the solution of global nonlinear problems that are often ill-posed.

GeoPDEs (http://geopdes.sourceforge.net) is a suite of free software tools for applications on Isogeometric Analysis (IGA). Its main focus is on providing a common framework for the implementation of the many IGA methods for the discretization of partial differential equations currently studied, mainly based on B-Splines and Non-Uniform Rational B-Splines (NURBS), while being flexible enough to allow users to implement new and more general methods with a relatively small effort. This paper presents the philosophy at the basis of the design of GeoPDEs and its relation to a quite comprehensive, abstract definition of IGA.

In this paper, we propose a theoretical study of the approximation properties of NURBS spaces, which are used in Isogeometric Analysis. We obtain error estimates that are explicit in terms of the mesh-size h, the degree p and the global regularity, measured by the parameter k. Our approach covers the approximation with global regularity from C (0) up to C (k-1), with 2k - 1 a parts per thousand currency sign p. Notice that the interesting case of higher regularity, up to k = p, is still open. However, our results give an indication of the role of the smoothness k in the approximation properties, and offer a first mathematical justification of the potential of Isogeometric Analysis based on globally smooth NURBS.

n this paper, we discuss the application of IsoGeometric Analysis to incompressible viscous flow problems. We consider, as a prototype problem, the Stokes system and we propose various choices of compatible spline spaces for the approximations to the velocity and the pressure fields. The proposed choices can be viewed as extensions of the Taylor-Hood, Nédélec and Raviart-Thomas pairs of finite element spaces, respectively. We study the stability and convergence properties of each method and discuss the conservation properties of the discrete velocity field in each case.

We initiate the study of efficient quadrature rules for NURBS-based isogeometric analysis. A rule of thumb emerges, the "half-point rule", indicating that optimal rules involve a number of points roughly equal to half the number of degrees-of-freedom, or equivalently half the number of basis functions of the space under consideration. The half-point rule is independent of the polynomial order of the basis. Efficient rules require taking into account the precise smoothness of basis functions across element boundaries. Several rules of practical interest are obtained, and a numerical procedure for determining efficient rules is presented. We compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics. The new rules represent improvements over those used previously in isogeometric analysis.

In this paper, an approach for shape tuning and predictable surface deformation is proposed. It pertains to the development of Fully Free Form Deformation Features sd-F4d which have been proposed to avoid low-level manipulations of free form surfaces. In our approach, d-F4 are applied through the specification of higher level parameters and constraints such as curves and points to be interpolated by the resulting surfaces. From the system perspective, the deformation is performed through the modification of the static equilibrium of bar networks coupled to the control polyhedra of the trimmed patches composing the free form surfaces on which the d-F4 are defined. The equations system coming from the constraints specification is often underconstrained, the selection of one among the whole set of possible solutions requires the definition of an optimization problem where an objective function has to be minimized. In this paper we propose a formulation of this optimization problem where the objective function can be defined as a multiple combination of various local quantities related either to the geometry of the bar network (e.g., the length of a bar or the displacement of a node), or to its mechanical characteristics (e.g. the external force applied at a node or a bar deformation energy). Different types of combinations are also proposed and analyzed according to the induced shape behaviors. In this way the shape of a d-F4 can be controlled globally, with a unique minimization, or locally with different minimizations applied to subdomains of the surface.