Stable isogeometric analysis of trimmed geometries

Abstract

We explore extended B-splines as a stable basis for isogeometric analysis with trimmed parameter spaces. The stabilization is accomplished by an appropriate substitution of Bsplines that may lead to ill-conditioned system matrices. The construction for non-uniform knot vectors is presented. The properties of extended B-splines are examined in the context of interpolation, potential, and linear elasticity problems and excellent results are attained. The analysis is performed by an isogeometric boundary element formulation using collocation. It is argued that extended B-splines provide a flexible and simple stabilization scheme which ideally suits the isogeometric paradigm.

Isogeometric Analysis on V-reps: First results

“…The design of robust techniques to deal with this issue in the framework of trimmed isogeometric analysis is beyond the scope of the present paper. Contributions in this direction can be found in [23,28,41].…”

Isogeometric Analysis with Trimmed CAD Models

“…In case of trimmed parameter spaces, however, this support may be arbitrarily cut and thus, its size within the domain of interest, i.e., supp {B i,p } ∩ A v , can become arbitrarily small. B-splines with small support are indeed troublesome, since they can lead to ill-conditioned system matrices when a trimmed basis is used in an analysis [5]. In the following these basis functions are labeled as degenerate B-splines.…”

“…The former is the power basis form of the extended polynomial segment, which may be derived exactly by Taylor expansion, and the Newton polynomials ψ j,p result from a quasi interpolation procedure, i.e., the de Boor-Fix functional [8]. A detailed discussion on the conversion of the polynomials to power basis form and the role of quasi interpolation is given in [5]. By taking the trivial extrapolations weights (5) and (6) into account, the final extended B-spline is defined as Fig.…”

“…1, it becomes apparent that the proper substitution of degenerate B-splines by extended polynomial segments resolves this stability issues of a trimmed basis. For more information regarding the properties of extended B-splines, the interested reader is referred to [5,7].…”

Realization of CAD-integrated shell simulation based on isogeometric B-Rep analysis

“…Further literature on trimming and the achievable quality of the solutions by employing alternative approaches to the presented AGIP method by [1] in "Numerical integration of surfaces" section, can be found in [19][20][21][22][23][24][25][26] (Fig. 7).…”