RESULTS FROM 1 TO 12 OF 12

2018, Articolo in rivista, ENG

A sparse-grid isogeometric solver

J. Beck, G. Sangalli, and L. Tamellini

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.

Computer methods in applied mechanics and engineering 335, pp. 128–151

DOI: 10.1016/j.cma.2018.02.017

2017, Articolo in rivista, ENG

Non-prismatic Timoshenko-like beam model: Numerical solution via isogeometric collocation

G. Balduzzi, S. Morganti, F. Auricchio, and A. Reali

The present paper combines an effective beam theory with a simple and accurate numerical technique opening the door to the prediction of the structural behavior of planar beams characterized by a continuous variation of the cross-section geometry, that in general deeply influences the stress distribution and, therefore, leads to non-trivial constitutive relations. Accounting for these peculiar aspects, the beam theory is described by a mixed formulation of the problem represented by six linear Ordinary Differential Equations (ODEs) with non-constant coefficients depending on both the cross-section displacements and the internal forces. Due to the ODEs' complexity, the solution can be typically computed only numerically also for relatively simple geometries, loads, and boundary conditions; however, the use of classical numerical tools for this problem, like a (six-field) mixed finite element approach, might entail several issues (e.g., shear locking, ill-conditioned matrices, etc.). Conversely, the recently proposed isogeometric collocation method, consisting of the direct discretization of the ODEs in strong form and using the higher-continuity properties typical of spline shape functions, allows an equal order approximation of all unknown fields, without affecting the stability of the solution. This makes such an approach simple, robust, efficient, and particularly suitable for solving the system of ODEs governing the non-prismatic beam problem. Several numerical experiments confirm that the proposed mixed isogeometric collocation method is actually cost-effective and able to attain high accuracy.

Computers & mathematics with applications (1987) 74 (7), pp. 1531–1541

DOI: 10.1016/j.camwa.2017.04.025

2017, Articolo in rivista, ENG

Optimal-order isogeometric collocation at Galerkin superconvergent points

M. Montardini, G. Sangalli, and L. Tamellini

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.

Computer methods in applied mechanics and engineering 316, pp. 741–757

DOI: 10.1016/j.cma.2016.09.043

2016, Articolo in rivista, ENG

A new design for the implementation of isogeometric analysis in Octave and Matlab: GeoPDEs 3.0

R. Vázquez

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.

Computers & mathematics with applications (1987) 72 (3), pp. 523–554

DOI: 10.1016/j.camwa.2016.05.010

2015, Articolo in rivista, ENG

Igatools: An isogeometric analysis library

M.S. Pauletti, M. Martinelli, N. Cavallini, and P. Antolín

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.

SIAM journal on scientific computing (Online) 37 (4), pp. 465–496

DOI: 10.1137/140955252

2013, Articolo in rivista, ENG

BPX-preconditioning for isogeometric analysis

A. Buffa, H. Harbrecht, A. Kunoth, and G. Sangalli

We consider elliptic PDEs (partial differential equations) in the framework of isogeometric analysis, i.e., we treat the physical domain by means of a B-spline or NURBS mapping which we assume to be regular. The numerical solution of the PDE is computed by means of tensor product B-splines mapped onto the physical domain. We construct additive multilevel preconditioners and show that they are asymptotically optimal, i.e., the spectral condition number of the resulting preconditioned stiffness matrix is independent of h. Together with a nested iteration scheme, this enables an iterative solution scheme of optimal linear complexity. The theoretical results are substantiated by numerical examples in two and three space dimensions.

Computer methods in applied mechanics and engineering 265, pp. 63–70

DOI: 10.1016/j.cma.2013.05.014

2012, Articolo in rivista, ENG

Avoiding shear locking for the Timoshenko beam problem via isogeometric collocation methods

L. Beirao da Veiga, C. Lovadina, and A. Reali

In this work we study isogeometric collocation methods for the Timoshenko beam problem, considering both mixed and displacement-based formulations. In particular, we show that locking-free solutions are obtained for mixed methods independently on the approximation degrees selected for the unknown fields. Moreover, several numerical tests are provided in order to support our theoretical results and to show the good behavior and the flexibility of isogeometric collocation methods in this context.

Computer methods in applied mechanics and engineering 241-244, pp. 38–51

DOI: 10.1016/j.cma.2012.05.020

2012, Articolo in rivista, ENG

A simple algorithm for obtaining nearly optimal quadrature rules for NURBS-based isogeometric analysis

F. Auricchio, F. Calabro', T.J.R. Hughes, A. Reali, and G. Sangalli

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.

Computer methods in applied mechanics and engineering 249-252, pp. 15–27

DOI: 10.1016/j.cma.2012.04.014

2012, Articolo in rivista, ENG

Isogeometric collocation for elastostatics and explicit dynamics

F. Auricchio, L. Beirao da Veiga, T.J.R. Hughes, A. Reali, and G. Sangalli

We extend the development of collocation methods within the framework of Isogeometric Analysis (IGA) to multi-patch NURBS configurations, various boundary and patch interface conditions, and explicit dynamic analysis. The methods developed are higher-order accurate, stable with no hourglass modes, and efficient in that they require a minimum number of quadrature evaluations. The combination of these attributes has not been obtained previously within standard finite element analysis.

Computer methods in applied mechanics and engineering 249-252, pp. 2–14

DOI: 10.1016/j.cma.2012.03.026

2012, Articolo in rivista, ENG

Multivariate Additive PLS Spline Boosting in Agro-Chemistry Studies.

Lombardo R., Durand J.F., Leone A.P.

Routinely, the multi-response Partial Least-Squares (PLS) is used in regression and classification problems showing good performances in many applied studies. In this paper, we aim to present PLS via spline functions focusing on supervised classification studies and showing how PLS methods historically belong to L2 boosting family. The theory of the PLS boost models is presented and used in classification studies. As a natural enrichment of linear PLS boost, we present its multi-response non-linear version by univariate and bivariate spline functions to transform the predictors. Three case studies of different complexities concerning soils and its products will be discussed, showing the gain in diagnostic provided by the non-linear additive PLS boost discriminant analysis compared to the linear one.

Current analytical chemistry (Print) 8 (2), pp. 236–253

2010, Articolo in rivista

Efficient quadrature for NURBS-based isogeometric analysis

T.J.R. Hughes, A. Reali, and G. Sangalli

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.

Computer methods in applied mechanics and engineering 199, pp. 301–313

DOI: 10.1016/j.cma.2008.12.004

2006, Articolo in rivista, ENG

Kernel B-splines and interpolation

Bozzini M.; Lenarduzzi L; and Schaback R.

This paper applies difference operators to conditionally positive definite kernels in order to generate {\em kernel $B$--splines} that have fast decay towards infinity. Interpolation by these new kernels provides better condition of the linear system, while the kernel $B$--spline inherits the approximation orders from its native kernel. We proceed in two different ways: either the kernel $B$--spline is constructed adaptively on the data knot set $X$, or we use a fixed difference scheme and shift its associated kernel $B$--spline around. In the latter case, the kernel $B$--spline so obtained is strictly positive in general. Furthermore, special kernel $B$--splines obtained by hexagonal second finite differences of multiquadrics are studied in more detail. We give suggestions in order to get a consistent improvement of the condition of the interpolation matrix in applications.

Numerical algorithms 41, pp. 1–16

DOI: 10.1007/s11075-005-9000-8

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Keyword

B-splines

RESULTS FROM 1 TO 12 OF 12