Adaptive hierarchical refinement of NURBS in cohesive fracture analysis

Num Anal Meth Geomechanics

2018

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Summary Powell‐Sabin B‐splines, which are based on triangles, are employed to model cohesive crack propagation without a predefined interface. The method removes limitations that adhere to isogeometric analysis regarding discrete crack analysis. Isogeometric analysis requires that the initial mesh be aligned a priori with the final crack path to a certain extent. These restrictions are partly related to the fact that in isogeometric analysis, the crack is introduced in the parameter domain by meshline insertions. Herein, the crack is introduced directly in the physical domain. Because of the use of triangles, remeshing and tracking the real crack path in the physical domain is relatively standard. The method can be implemented in existing finite element programmes in a straightforward manner through the use of Bézier extraction. The accuracy of the approach to model free crack propagation is demonstrated by several numerical examples, including discrete crack modelling in an L‐shaped beam and the Nooru‐Mohamed tension‐shear test.

Section: Introductionmentioning

confidence: 99%

Cohesive fracture analysis using Powell‐Sabin B‐splines

Num Anal Meth Geomechanics

2018

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“…Consequently, the LR T‐splines

scriptN

associated with

scriptT

can now be described by the LR T‐splines

N_{r}

associated with

T_{r}

boldΓ boldN ((), ξ^{1}, ξ^{2}) = boldΓ boldS N_{r} ((), ξ^{1}, ξ^{2})

with N and N r being the LR T‐spline blending functions associated with the LR T‐meshes

scriptT

and

T_{r}

, respectively. S is the refinement operator,() and Γ is a diagonal matrix with the scaling weight γ of N along the diagonal. This makes it possible that the LR T‐splines

scriptN

satisfy the partition of unity property.…”

Section: Bézier Extraction Of Lr T‐splines After Meshline Insertionsmentioning

confidence: 99%

“…The mesh represents a discontinuous interface at ξ 2 =1/2 in elements e 1 and e 2. To produce such interface,

C^{- 1}

continuous blending functions have to be created, which is achieved by using knot lines of multiplicity m = p +1 . In Figure A, the number of knot lines at ξ 2 =1/2 is m =2+1=3.…”

Section: Bézier Extraction Of Lr T‐splines After Meshline Insertionsmentioning

confidence: 99%

Locally Refined T‐splines

Numerical Meth Engineering

2018

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Summary We extend Locally Refined (LR) B‐splines to LR T‐splines within the Bézier extraction framework. This discretization technique combines the advantages of T‐splines to model the geometry of engineering objects exactly with the ability to flexibly carry out local mesh refinement. In contrast to LR B‐splines, LR T‐splines take a T‐mesh as input instead of a tensor‐product mesh. The LR T‐mesh is defined, and examples are given how to construct it from an initial T‐mesh by repeated meshline insertions. The properties of LR T‐splines are investigated by exploiting the Bézier extraction operator, including the nested nature, linear independence, and the partition of unity property. A technique is presented to remove possible linear dependencies between LR T‐splines. Like for other spline technologies, the Bézier extraction framework enables to fully use existing finite element data structures.

“…Inserting a series of single meshlines,

{({}, ε_{i})}_{i = 1}^{n}

, in

scriptT

results in

T_{r}

with n r anchors. The T‐splines

scriptN

that are associated with

scriptT

are now described by the T‐splines

N_{r}

associated with

T_{r}

normalΓ boldN ((), ξ^{1}, ξ^{2}) = normalΓ boldS N_{r} ((), ξ^{1}, ξ^{2}),

where S is the refinement operator,() N and N r are the blending functions associated with

scriptT

and

T_{r}

, respectively, and Γ is a diagonal matrix with the scaling weights γ of N . Using Equation , we can now solve for S :

boldN = boldC B_{r} = boldS C_{r} B_{r},

where C is the Bézier extraction operator of the anchors on

scriptT

over the elements on T r , C r denotes the Bézier extraction operator of the anchors on T r over the elements on T r , and B r contains the Bernstein polynomials of the elements on T r .…”

Section: Bézier Extraction Of Refined T‐splinesmentioning

confidence: 99%

The role of Bézier extraction in adaptive isogeometric analysis: Local refinement and hierarchical refinement

Numerical Meth Engineering

2017

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SummaryWe present 2 adaptive refinement techniques, namely, adaptive local refinement and adaptive hierarchical refinement, for isogeometric analysis. An element-wise point of view is adopted, exploiting Bézier extraction, which facilitates the implementation of adaptive refinement in isogeometric analysis. Locally refined and hierarchical T-splines are used for the description of the geometry as well as for the approximation of the solution space in the analysis. The refinement is conducted with the aid of a subdivision operator, which is computed by again exploiting the Bézier extraction operator. The concept and algorithm of an element-based adaptive isogeometric analysis are illustrated.Numerical examples are given to examine the accuracy, the convergence, and the condition number.