Mapping techniques for isogeometric analysis of elliptic boundary value problems containing singularities

Jeong, Jae Woo; Oh, Hae‐Soo; Kang, Sunbu; Kim, Hyun-Ju

doi:10.1016/j.cma.2012.09.009

Cited by 18 publications

(26 citation statements)

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“…In this section, we construct a NURBS geometrical mapping to deal with monotone singularities of type r .Â /, where is a rational number with 0 < < 1, .Â / is a piecewise smooth function, and .r, Â/ are the polar coordinates. The novel geometrical mapping constructed in this section is a generalization of that in [1].…”

Section: Novel Geometrical Mappings By Which Push-forwards Of Basis Smentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Recently, Jeong et al [1] introduced a novel geometrical mapping to deal with various types of singularities in the framework of IGA. The mapping method (coined as the mapping technique in [1]) was very successful in handling two dimensional elliptic boundary value problems containing singularities, whenever B-spline functions are used for approximations. The mapping method (or the mapping technique) is concerned with construction of a novel geometrical mapping, by which 151 For example, the piecewise quadratic polynomial B-spline functions N i, 3 .…”

Section: Introductionmentioning

confidence: 99%

“…In section 4, we introduce enrichment approaches, which do not change the given design, for IGA of the boundary value problems containing singularities. In section 5, we present numerical tests showing that the proposed enrichment methods in IGA are as effective as the mapping method [1]. Finally, concluding remarks are stated in section 6.…”

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confidence: 99%

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Enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and/or corners

Kim

Jeong

2013

Numerical Meth Engineering

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The mapping method was introduced in Jeong et al. (2013) for highly accurate isogeometric analysis (IGA) of elliptic boundary value problems containing singularities. The mapping method is concerned with constructions of novel geometrical mappings by which push-forwards of B-splines from the parameter space into the physical space generate singular functions that resemble the singularities. In other words, the pullback of the singularity into the parameter space by the novel geometrical mapping (a non-uniform rational basis spline (NURBS) surface mapping) becomes highly smooth. One of the merits of IGA is that it uses NURBS functions employed in designs for the finite element analysis. However, push-forwards of rational NURBS may not be able to generate singular functions. Moreover, the mapping method is effective for neither the k-refinement nor the h-refinement. In this paper, highly accurate stress analysis of elastic domains with cracks and=or corners are achieved by enriched IGA, in which push-forwards of NURBS via the design mapping are combined with push-forwards of B-splines via the novel geometrical mapping (the mapping technique). In a similar spirit of X-FEM (or GFEM), we propose three enrichment approaches: enriched IGA for corners, enriched IGA for cracks, and partition of unity IGA for cracks. push-forwards of B-spline functions generate singular functions in the physical space that resemble the singularities.For example, suppose a one dimensional geometrical mapping is constructed to be x D F.Á/ D Á p , where p is a positive integer. If an approximation space O S h Á , spanned by B-spline functions (Bernstein polynomials on each subinterval [knot span] of the parameter space), contains polynomials 1, Á, Á 2 , Á 3 , , then the push-forward of this approximation space onto the physical space by the mapping F contains singular functions 1, x 1=p , x 2=p , x 3=p , that can exactly capture the singularity of type O.x 1=p /.Actually, the mapping method is similar to the method of auxiliary mapping (MAM), introduced by Babuška and Oh [9,10], that can effectively handle singularity problems in the framework of conventional finite element analysis [11][12][13][14][15]. However, the mapping method is more flexible than MAM because the mapping method makes it possible to independently control the radial and the angular directions of the function to be approximated. Similar to MAM, the mapping method is not effective in the h-refinement of IGA. Moreover, the k-refinement in IGA results in a significant reduction of the degrees of freedom. However, it does not generate a complete polynomial and, hence, the mapping method is not effective in the k-refinement.Most importantly, NURBS basis functions are employed for IGA. However, our mapping method to deal with singularities is not effective for NURBS basis functions. Thus, in this paper, we enrich NURBS basis functions with B-spline functions via a novel geometrical mapping so that the enriched IGA can effectively handle singularities without altering the design...

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Section: Novel Geometrical Mappings By Which Push-forwards Of Basis Smentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and/or corners

Kim

Jeong

2013

Numerical Meth Engineering

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“…Some researchers also tried to implement IGA directly on irregular meshes. Jeong et al [47] presented some mapping techniques for isogeometric analysis of elliptic boundary value problems containing singularities. It works with proper selection of control vertices, but the continuity at extraordinary vertices was not discussed.…”

Section: Introductionmentioning

confidence: 99%

Parametric mesh regularization for interpolatory shape design and isogeometric analysis over a mesh of arbitrary topology

Yuan

2015

Computer Methods in Applied Mechanics and Engineering

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Mapped finite element methods: High‐order approximations of problems on domains with cracks and corners

Chiaramonte

Shen

Lew

2017

Numerical Meth Engineering

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Summary Linear elasticity problems posed on cracked domains, or domains with re‐entrant corners, yield singular solutions that deteriorate the optimality of convergence of finite element methods. In this work, we propose an optimally convergent finite element method for this class of problems. The method is based on approximating a much smoother function obtained by locally reparameterizing the solution around the singularities. This reparameterized solution can be approximated using standard finite element procedures yielding optimal convergence rates for any order of interpolating polynomials, without additional degrees of freedom or special shape functions. Hence, the method provides optimally convergent solutions for the same computational complexity of standard finite element methods. Furthermore, the sparsity and the conditioning of the resulting system is preserved. The method handles body forces and crack‐face tractions, as well as multiple crack tips and re‐entrant corners. The advantages of the method are showcased for four different problems: a straight crack with loaded faces, a circular arc crack, an L‐shaped domain undergoing anti‐plane deformation, and lastly a crack along a bimaterial interface. Optimality in convergence is observed for all the examples. A proof of optimal convergence is accomplished mainly by proving the regularity of the reparameterized solution. Copyright © 2016 John Wiley & Sons, Ltd.

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Mapping techniques for isogeometric analysis of elliptic boundary value problems containing singularities

Cited by 18 publications

References 18 publications

Enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and/or corners

Enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and/or corners

Parametric mesh regularization for interpolatory shape design and isogeometric analysis over a mesh of arbitrary topology

Mapped finite element methods: High‐order approximations of problems on domains with cracks and corners

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