Accurate mode-separated energy release rates for delamination cracks

Oh, Hae‐Soo

doi:10.1016/j.jcp.2003.07.025

Cited by 7 publications

(4 citation statements)

References 29 publications

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“…In this paper, the physical domains of the elliptic boundary value problems containing multiple singularities, were re-designed by the patchwise mapping methods. In the patchwise mapping method, the construction of the approximation space is different from that in the conventional mapping method [16] due to the use of multiple singular functions.…”

Section: Discussionmentioning

confidence: 99%

Patchwise Mapping Method for Solving Elliptic Boundary Value Problems Containing Multiple Singularities

Kim¹

2019

JAMP

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In the paper [1], the geometrical mapping techniques based on Non-Uniform Rational B-Spline (NURBS) were introduced to solve an elliptic boundary value problem containing a singularity. In the mapping techniques, the inverse function of the NURBS geometrical mapping generates singular functions as well as smooth functions by an unconventional choice of control points. It means that the push-forward of the NURBS geometrical mapping that generates singular functions, becomes a piecewise smooth function. However, the mapping method proposed is not able to catch singularities emerging at multiple locations in a domain. Thus, we design the geometrical mapping that generates singular functions for each singular zone in the physical domain. In the design of the geometrical mapping, we should consider the design of control points on the interface between/among patches so that global basis functions are in 0 C space. Also, we modify the B-spline functions whose supports include the interface between/among them. We put the idea in practice by solving elliptic boundary value problems containing multiple singularities.

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Section: Discussionmentioning

confidence: 99%

Patchwise Mapping Method for Solving Elliptic Boundary Value Problems Containing Multiple Singularities

Kim¹

2019

JAMP

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“…and the relative error in energy norm in percent defined by (13). For the construction of NURBS and the related k-refinements, one can use existing softwares and toolboxes such as GeoPDEs and NURBS Toolbox in MATLAB.…”

Section: Numerical Testsmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…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.…”

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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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The Numerical Methods for Oscillating Singularities in Elliptic Boundary Value Problems

Kim

Lee

2001

Journal of Computational Physics

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Accurate mode-separated energy release rates for delamination cracks

Cited by 7 publications

References 29 publications

Patchwise Mapping Method for Solving Elliptic Boundary Value Problems Containing Multiple Singularities

Patchwise Mapping Method for Solving Elliptic Boundary Value Problems Containing Multiple Singularities

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

The Numerical Methods for Oscillating Singularities in Elliptic Boundary Value Problems

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