Isogeometric analysis based on Geometry Independent Field approximaTion (GIFT) and Polynomial Splines over Hierarchical T-meshes (original) (raw)
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Computer Methods in Applied Mechanics and Engineering, 2016
Local refinement with hierarchical B-spline structures is an active topic of research in the context of geometric modeling and isogeometric analysis. By exploiting a multilevel control structure, we show that truncated hierarchical B-spline (THB-spline) representations support interactive modeling tools, while simultaneously providing effective approximation schemes for the manipulation of complex data sets and the solution of partial differential equations via isogeometric analysis. A selection of illustrative 2D and 3D numerical examples demonstrates the potential of the hierarchical framework.
2011
Abstract Isogeometric analysis has become a powerful alternative to standard finite elements due to its flexibility in handling complex geometries. One of the major drawbacks of NURBS-based isogeometric finite elements is the inefficiency of local refinement. In this study, we present an alternative to NURBS-based isogeometric analysis that allows for local refinement. The idea is based on polynomial splines and exploits the flexibility of T-meshes for local refinement.
Refinement Algorithms for Adaptive Isogeometric Methods with Hierarchical Splines
Axioms
The construction of suitable mesh configurations for spline models that provide local refinement capabilities is one of the fundamental components for the analysis and development of adaptive isogeometric methods. We investigate the design and implementation of refinement algorithms for hierarchical B-spline spaces that enable the construction of locally graded meshes. The refinement rules properly control the interaction of basis functions at different refinement levels. This guarantees a bounded number of nonvanishing (truncated) hierarchical B-splines on any mesh element. The performances of the algorithms are validated with standard benchmark problems.
Adaptive isogeometric analysis with hierarchical box splines
Computer Methods in Applied Mechanics and Engineering, 2016
Isogeometric analysis is a recently developed framework based on finite element analysis, where the simple building blocks in geometry and solution space are replaced by more complex and geometricallyoriented compounds. Box splines are an established tool to model complex geometry, and form an intermediate approach between classical tensor-product B-splines and splines over triangulations. Local refinement can be achieved by considering hierarchically nested sequences of box spline spaces. Since box splines do not offer special elements to impose boundary conditions for the numerical solution of partial differential equations (PDEs), we discuss a weak treatment of such boundary conditions. Along the domain boundary, an appropriate domain strip is introduced to enforce the boundary conditions in a weak sense. The thickness of the strip is adaptively defined in order to avoid unnecessary computations. Numerical examples show the optimal convergence rate of box splines and their hierarchical variants for the solution of PDEs.
Geometry-Independent Field approximaTion ( GIFT ) for Adaptive Spline-Based Finite Element Analysis
2014
In isogeometric analysis (IGA), the same spline representation is employed for both the geometry of the domain and approximation of the unknown fields over this domain.This identity of the geometry and field approximation spaces was put forward in the now classic 2005 paper [20] as a key advantage on the way to the integration of Computer Aided Design (CAD) and subsequent analysis in Computer Aided Engineering (CAE). [20] claims indeed that any change to the geometry of the domain is automatically inherited by the approximation of the field variables, without requiring the regeneration of the mesh at each change of the domain geometry. Yet, in Finite Element versions of IGA, a parameterization of the interior of the domain must still be constructed, since CAD only provides information about the boundary. The identity of the boundary and field representation decreases the flexibility of the modeling process, because an approximation able to represent the domain geometry accurately ne...
Isogeometric boundary element analysis using unstructured T-splines
We couple collocated isogeometric boundary element methods and unstructured analysis-suitable T-spline surfaces for linear elastostatic problems. We extend the definition of analysis-suitable T-splines to encompass unstructured control grids (unstructured meshes) and develop basis functions which are smooth (rational) polynomials defined in terms of the Be ́zier extraction framework and which pass standard patch tests. We then develop a collocation procedure which correctly accounts for sharp edges and corners, extraordinary points, and T-junctions. This approach is applied to several demanding three-dimensional problems, including a T-spline model of a propeller. Remarkable accuracy is obtained in all cases.
Quasi-interpolation in isogeometric analysis based on generalized B-splines
Computer Aided Geometric Design, 2010
Isogeometric analysis is a new method for the numerical simulation of problems governed by partial differential equations. It possesses many features in common with finite element methods (FEM) but takes some inspiration from Computer Aided Design tools. We illustrate how quasi-interpolation methods can be suitably used to set Dirichlet boundary conditions in isogeometric analysis. In particular, we focus on quasi-interpolant projectors for generalized B-splines, which have been recently proposed as a possible alternative to NURBS in isogeometric analysis.
Isogeometric finite element data structures based on Bézier extraction of T-splines
International Journal for Numerical Methods in Engineering, 2011
We develop finite element data structures for T-splines based on Bézier extraction generalizing our previous work for NURBS. As in traditional finite element analysis, the extracted Bézier elements are defined in terms of a fixed set of polynomial basis functions, the so-called Bernstein basis. The Bézier elements may be processed in the same way as in a standard finite element computer program, utilizing exactly the same data processing arrays. In fact, only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same. A byproduct of the extraction process is the element extraction operator. This operator localizes the topological and global smoothness information to the element level, and represents a canonical treatment of T-junctions, referred to as 'hanging nodes' in finite element analysis and a fundamental feature of T-splines. A detailed example is presented to illustrate the ideas. Figure 2. A schematic diagram illustrating the central role played by Bézier extraction in unifying CAGD and FEA.
Adaptive mesh refinement strategies in isogeometric analysis— A computational comparison
Computer Methods in Applied Mechanics and Engineering, 2017
We explain four variants of an adaptive finite element method with cubic splines and compare their performance in simple elliptic model problems. The methods in comparison are Truncated Hierarchical B-splines with two different refinement strategies, T-splines with the refinement strategy introduced by Scott et al. in 2012, and T-splines with an alternative refinement strategy introduced by some of the authors. In four examples, including singular and non-singular problems of linear elasticity and the Poisson problem, the H1-errors of the discrete solutions, the number of degrees of freedom as well as sparsity patterns and condition numbers of the discretized problem are compared. * The authors gratefully acknowledge support by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 "Reliable simulation techniques in solid mechanics: Development of non-standard discretization methods, mechanical and mathematical analysis" under the project "Adaptive isogeometric modeling of propagating strong discontinuities in heterogeneous materials" (KA3309/3-1 and PE2143/2-1).