arXiv subjects

Annalisa Buffa

Publications and source records attributed to Annalisa Buffa.

49 records · Page 3Linked to original sources

arXiv2019

Adaptive isogeometric analysis on two-dimensional trimmed domains based on a hierarchical approach

The focus of this work is on the development of an error-driven isogeometric framework, capable of automatically performing an adaptive simulation in the context of second- and fourth-order, elliptic partial differential equations defined on two-dimensional trimmed domains. The method is steered by an a posteriori error estimator, which is computed with the aid of an auxiliary residual-like problem formulated onto a space spanned by splines with single element support. The local refinement of the basis is achieved thanks to the use of truncated hierarchical B-splines. We prove numerically the applicability of the proposed estimator to various engineering-relevant problems, namely the Poisson problem, linear elasticity and Kirchhoff-Love shells, formulated on trimmed geometries. In particular, we study several benchmark problems which exhibit both smooth and singular solutions, where we recover optimal asymptotic rates of convergence for the error measured in the energy norm and we observe a substantial increase in accuracy per-degree-of-freedom compared to uniform refinement. Lastly, we show the applicability of our framework to the adaptive shell analysis of an industrial-like trimmed geometry modeled in the commercial software Rhinoceros, which represents the B-pillar of a car.

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arXiv2019

Isogeometric Analysis on V-reps: first results

Inspired by the introduction of Volumetric Modeling via volumetric representations (V-reps) by Massarwi and Elber in 2016, in this paper we present a novel approach for the construction of isogeometric numerical methods for elliptic PDEs on trimmed geometries, seen as a special class of more general V-reps. We develop tools for approximation and local re-parametrization of trimmed elements for three dimensional problems, and we provide a theoretical framework that fully justify our algorithmic choices. We validate our approach both on two and three dimensional problems, for diffusion and linear elasticity.

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arXiv2019

Optimizing micro-tiles in micro-structures as a design paradigm

In recent years, new methods have been developed to synthesize complex porous and micro-structured geometry in a variety of ways. In this work, we take these approaches one step further and present these methods as an efficacious design paradigm. Specifically, complex micro-structure geometry can be synthesized while optimizing certain properties such as maximal heat exchange in heat exchangers, or minimal weight under stress specifications. By being able to adjust the geometry, the topology and/or the material properties of individual tiles in the micro-structure, possibly in a gradual way, a porous object can be synthesized that is optimal with respect to the design specifications. As part of this work, we exemplify this paradigm on a variety of diverse applications.

cs.CG↗

arXiv2019

Multipatch Approximation of the de Rham Sequence and its Traces in Isogeometric Analysis

We define a conforming B-spline discretisation of the de Rham complex on multipatch geometries. We introduce and analyse the properties of interpolation operators onto these spaces which commute w.r.t. the surface differential operators. Using these results as a basis, we derive new convergence results of optimal order w.r.t. the respective energy spaces and provide approximation properties of the spline discretisations of trace spaces for application in the theory of isogeometric boundary element methods. Our analysis allows for a straightforward generalisation to finite element methods.

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arXiv2019

A hierarchical approach to the a posteriori error estimation of isogeometric Kirchhoff plates and Kirchhoff-Love shells

This work focuses on the development of a posteriori error estimates for fourth-order, elliptic, partial differential equations. In particular, we propose a novel algorithm to steer an adaptive simulation in the context of Kirchhoff plates and Kirchhoff-Love shells by exploiting the local refinement capabilities of hierarchical B-splines. The method is based on the solution of an auxiliary residual-like variational problem, formulated by means of a space of localized spline functions. This space is characterized by $C^1$ continuous B-splines with compact support on each active element of the hierarchical mesh. We demonstrate the applicability of the proposed estimator to Kirchhoff plates and Kirchhoff-Love shells by studying several benchmark problems which exhibit both smooth and singular solutions. In all cases, we obtain optimal asymptotic rates of convergence for the error measured in the energy norm and an excellent approximation of the true error.

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arXiv2018

Isogeometric Mortar Coupling for Electromagnetic Problems

This paper discusses and analyses two domain decomposition approaches for electromagnetic problems that allow the combination of domains discretised by either Nédélec-type polynomial finite elements or spline-based isogeometric analysis. The first approach is a new isogeometric mortar method and the second one is based on a modal basis for the Lagrange multiplier space, called state-space concatenation in the engineering literature. Spectral correctness and in particular inf-sup stability of both approaches are analytically and numerically investigated. The new mortar method is shown to be unconditionally stable. Its construction of the discrete Lagrange multiplier space takes advantage of the high continuity of splines, and does not have an analogue for Nédélec finite elements. On the other hand, the approach with modal basis is easier to implement but relies on application knowledge to ensure stability and correctness.

cs.CE↗

arXiv2018

A priori error for unilateral contact problems with Lagrange multiplier and IsoGeometric Analysis

In this paper, we consider unilateral contact problem without friction between a rigid body and deformable one in the framework of isogeometric analysis. We present the theoretical analysis of the mixed problem using an active-set strategy and for a primal space of NURBS of degree $p$ and $p-2$ for a dual space of B-Spline. A inf-sup stability is proved to ensure a good property of the method. An optimal a priori error estimate is demonstrated without assumption on the unknown contact set. Several numerical examples in two- and three-dimensional and in small and large deformation demonstrate the accuracy of the proposed method.

math.NA↗

arXiv2016

A posteriori error estimators for hierarchical B-spline discretizations

In this article we develop function-based a posteriori error estimators for the solution of linear second order elliptic problems considering hierarchical spline spaces for the Galerkin discretization. We prove a global upper bound for the energy error. The theory hinges on some weighted Poincaré type inequalities, where the B-spline basis functions are the weights appearing in the norms. Such inequalities are derived following the lines in [Veeser and Verfürth, 2009], where the case of standard finite elements is considered. Additionally, we present numerical experiments that show the efficiency of the error estimators independently of the degree of the splines used for discretization, together with an adaptive algorithm guided by these local estimators that yields optimal meshes and rates of convergence, exhibiting an excellent performance.

math.NA↗

arXiv2015

Complexity of hierarchical refinement for a class of admissible mesh configurations

An adaptive isogeometric method based on $d$-variate hierarchical spline constructions can be derived by considering a refine module that preserves a certain class of admissibility between two consecutive steps of the adaptive loop [6]. In this paper we provide a complexity estimate, i.e., an estimate on how the number of mesh elements grows with respect to the number of elements that are marked for refinement by the adaptive strategy. Our estimate is in the line of the similar ones proved in the finite element context, [3,24].

math.NA↗

arXiv2015

New refinable spaces and local approximation estimates for hierarchical splines

We study the local approximation properties in hierarchical spline spaces through multiscale quasi-interpolation operators. This construction suggests the analysis of a subspace of the classical hierarchical spline space (Vuong et al., 2011) which still satisfies the essential properties of the full space. The B-spline basis of such a subspace can be constructed using parent-children relations only, making it well adapted to local refinement algorithms.

math.NA↗

arXiv2015

The Influence of Quadrature Errors on Isogeometric Mortar Methods

Mortar methods have recently been shown to be well suited for isogeometric analysis. We review the recent mathematical analysis and then investigate the variational crime introduced by quadrature formulas for the coupling integrals. Motivated by finite element observations, we consider a quadrature rule purely based on the slave mesh as well as a method using quadrature rules based on the slave mesh and on the master mesh, resulting in a non-symmetric saddle point problem. While in the first case reduced convergence rates can be observed, in the second case the influence of the variational crime is less significant.

math.NA↗

arXiv2015

Adaptive isogeometric methods with hierarchical splines: error estimator and convergence

The problem of developing an adaptive isogeometric method (AIGM) for solving elliptic second-order partial differential equations with truncated hierarchical B-splines of arbitrary degree and different order of continuity is addressed. The adaptivity analysis holds in any space dimensions. We consider a simple residual-type error estimator for which we provide a posteriori upper and lower bound in terms of local error indicators, taking also into account the critical role of oscillations as in a standard adaptive finite element setting. The error estimates are properly combined with a simple marking strategy to define a sequence of admissible locally refined meshes and corresponding approximate solutions. The design of a refine module that preserves the admissibility of the hierarchical mesh configuration between two consectutive steps of the adaptive loop is presented. The contraction property of the quasi-error, given by the sum of the energy error and the scaled error estimator, leads to the convergence proof of the AIGM.

math.NA↗

arXiv2014

Isogeometric mortar methods

The application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically. For the Lagrange multiplier two choices of uniformly stable spaces are presented, both of them are spline spaces but of a different degree. In one case, we consider an equal order pairing for which a cross point modification based on a local degree reduction is required. In the other case, the degree of the dual space is reduced by two compared to the primal. This pairing is proven to be inf-sup stable without any necessary cross point modification. Several numerical examples confirm the theoretical results and illustrate additional aspects. Keywords: isogeometric analysis, mortar methods, inf-sup stability, cross point modification.

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