Subdivision-stabilised immersed b-spline finite elements for moving boundary flows

Rüberg, Thomas; Cirak, Fehmi

doi:10.1016/j.cma.2011.10.007

Cited by 111 publications

(133 citation statements)

References 37 publications

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“…This can either be done by combining them with geometrically nearby functions [23,24] (as is also done in Web-splines [13,12]) or by simply excluding them from the approximation space, e.g., [7,29,33].…”

Section: Introductionmentioning

confidence: 99%

Condition number analysis and preconditioning of the finite cell method

Prenter

Verhoosel

Zwieten

et al. 2017

Computer Methods in Applied Mechanics and Engineering

139

179

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The (Isogeometric) Finite Cell Method -in which a domain is immersed in a structured background mesh -suffers from conditioning problems when cells with small volume fractions occur. In this contribution, we establish a rigorous scaling relation between the condition number of (I)FCM system matrices and the smallest cell volume fraction. Ill-conditioning stems either from basis functions being small on cells with small volume fractions, or from basis functions being nearly linearly dependent on such cells. Based on these two sources of ill-conditioning, an algebraic preconditioning technique is developed, which is referred to as Symmetric Incomplete Permuted Inverse Cholesky (SIPIC). A detailed numerical investigation of the effectivity of the SIPIC preconditioner in improving (I)FCM condition numbers and in improving the convergence speed and accuracy of iterative solvers is presented for the Poisson problem and for two-and three-dimensional problems in linear elasticity, in which Nitche's method is applied in either the normal or tangential direction. The accuracy of the preconditioned iterative solver enables mesh convergence studies of the finite cell method.

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Section: Introductionmentioning

confidence: 99%

Condition number analysis and preconditioning of the finite cell method

Prenter

Verhoosel

Zwieten

et al. 2017

Computer Methods in Applied Mechanics and Engineering

139

179

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“…[6,23,25]). In addition, a simplified formula can be derived for uniform parameter spaces [22,24]. In the bivariate case, the extrapolation weights are simply determined by the tensor product of univariate values.…”

Section: Stable Basis For Trimmed Geometriesmentioning

confidence: 99%

“…Hence, the current section focuses on this aspect. In particular, it is proposed to stabilize trimmed parameter spaces by using so-called extended B-splines which have been originally introduced in the context of fictitious domain -finite element methods [21][22][23][24]. The basic idea of extended B-splines is to replace basis functions which may cause instabilities by extrapolations of neighboring ones which have a sufficient large support.…”

Section: Stable Basis For Trimmed Geometriesmentioning

confidence: 99%

Integration of Design and Analysis Through Boundary Integral Equations

Marussig

Zechner

Beer

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. The direct integration of Computer Aided Geometric Design (CAGD) models into a numerical simulation improves the accuracy of the geometrical representation of the problem as well as the efficiency of the overall analysis process.In this work, the complementary features of isogeometric analysis and boundary integral equations are combined to obtain a coalescence of design and analysis which is based on a boundary-only discretization. Following the isogeometric concept, the functions used by CAGD are employed for the simulation. An independent field approximation is applied to obtain a more flexible and efficient formulation. In addition, a procedure is presented which allows a stable analysis of trimmed geometries and a straightforward positioning of collocation points.Several numerical examples demonstrate the characteristics and benefits of the proposed approach. In particular, the independent field approximation improves the computational efficiency and reduces the storage requirements without any loss of accuracy. The proposed methodology permits a seamless integration of the most common design models into an analysis of linear elasticity problems. 6526Benjamin Marussig, Jürgen Zechner, Gernot Beer, and Thomas-Peter Fries INTRODUCTIONIsogeometric analysis aims to close the existing gap between the design process and analysis such that a simulation can be performed without generating a mesh. Consequently, the accuracy and efficiency of the overall simulation process is improved, since meshing is timeconsuming [1, 2] and introduces additional (geometrical) approximation errors. In addition, the basis functions used by design models, i.e. NURBS, provide further benefits such as high continuity [3,4].However, during the last years, it has become clear that a true integration of design and analysis is far from trivial due to several reasons: first of all, most engineering design models are based on a boundary representation (B-Rep) rather than a volume description. Secondly, three dimensional B-Rep models are usually defined by a non-conforming partition of NURBS surfaces, i.e. their mathematical parametrizations have no explicit relation to each other. Thirdly, each boundary surface is based on a tensor product structure which is a very efficient representation but has limitations due to its four sided nature. As a result, almost all NURBS based design models use trimming procedures to increase the flexibility of tensor product surfaces. This means that only a certain area of a surface is visualized while the underlying mathematical parametrization remains unchanged.In this work, a coherent framework is presented which allows a seamless integration of trimmed NURBS models into an analysis. In general, the governing equations of the problem are expressed by means of boundary integral equations which are discretized by a numerical approximation method. Here, the boundary element method (BEM) is used since it is the most versatile approach. However, it should be pointed out that other schemes like the Nyst...

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“…This issue has been successfully addressed in the context of b-spline finite elements [6,12,15]. In this work, we build up on this concept of constraining critical degrees of freedom and apply it to Lagrangian basis functions on unstructured meshes.…”

mentioning

confidence: 99%

“…The method we present here is based on our previous works [6,[17][18][19] and related to [2,3,16,20,21]. Although, as shown in the cited works, the method can be transferred to many physical applications, we focus on the problem class of nonlinear elasticity.…”

mentioning

confidence: 99%

An unstructured immersed finite element method for nonlinear solid mechanics

Rüberg

Cirak

Aznar

2016

Adv. Model. and Simul. in Eng. Sci.

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We present an immersed finite element technique for boundary-value and interface problems from nonlinear solid mechanics. Its key features are the implicit representation of domain boundaries and interfaces, the use of Nitsche's method for the incorporation of boundary conditions, accurate numerical integration based on marching tetrahedrons and cut-element stabilisation by means of extrapolation. For discretisation structured and unstructured background meshes with Lagrange basis functions are considered. We show numerically and analytically that the introduced cut-element stabilisation technique provides an effective bound on the size of the Nitsche parameters and, in turn, leads to well-conditioned system matrices. In addition, we introduce a novel approach for representing and analysing geometries with sharp features (edges and corners) using an implicit geometry representation. This allows the computation of typical engineering parts composed of solid primitives without the need of boundary-fitted meshes.

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Subdivision-stabilised immersed b-spline finite elements for moving boundary flows

Cited by 111 publications

References 37 publications

Condition number analysis and preconditioning of the finite cell method

Condition number analysis and preconditioning of the finite cell method

Integration of Design and Analysis Through Boundary Integral Equations

An unstructured immersed finite element method for nonlinear solid mechanics

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