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

Abstract

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...

Isogeometric analysis enhanced by the scaled boundary finite element method

“…By substituting Eq. (14) in the definition of strain-displacement relations, the strains ε(ξ, η) are expressed as:…”

“…A key feature of this framework is that the geometry is represented exactly by NURBS and the isoparametric concept is invoked to define the field variables. Since its inception, the method has been applied to a variety of problems such as plates and shells [7][8][9], as cohesive elements [10], for shape optimization [11], fluid-structure interaction problems [12], problems with strong discontinuities and singularities [13][14][15], optimization problems [16] to name a few. Jia et al [17] by incorporating reproducing kernel approximation methods, alleviated the instabilities of the conventional triangular B-spline element.…”

Graded parametric CutFEM and CutIGA for elliptic boundary value problems in domains with corners

“…A method belonging to the first category was proposed in [13] where the parametric mappings describing the geometry were modified to grade the approximation space appropriately towards the singularity. In another contribution [20] a method related to the first and second category was presented in which the approximation space was enriched by certain basis functions constructed using push forward operations with a mapping similar to the one in [13]. The present work belongs to the first category and is based on a mapping in the form of simple radial scaling, which has the effect of a graded mesh.…”

Implicitly enriched Galerkin methods for numerical solutions of fourth‐order partial differential equations containing singularities

“…NURBS used for design usually do not satisfy given boundary conditions and hence they should be modified to be used for analysis. Various methods constructing PU functions with flat‐top and their applications are shown in . PU functions with flat‐top were also applied for analysis of Kirchhoff plates . To handle singularities arising in fourth‐order equations, modified B‐spline functions are enriched by implicitly generating or explicitly adding singular functions that resemble the singularities of the given differential equations. Original enrichment methods (such as PUFEM, GFEM, and X‐FEM ), in which singular functions are explicitly added for approximation functions, face increased degrees of freedom (DOF), elevated condition numbers, and integration of singular enrichment functions for which the standard quadrature rules do not give accurate calculations.…”

“…NURBS used for design usually do not satisfy given boundary conditions and hence they should be modified to be used for analysis. Various methods constructing PU functions with flat-top and their applications are shown in [12][13][14][15]. PU functions with flat-top were also applied for analysis of Kirchhoff plates [16].…”