Locally Refined T‐splines

Num Anal Meth Geomechanics

2018

Self Cite

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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“…A local knot vector

{boldΞ}_{i} ((), i = 1, ⋯, n)

is prescribed for each anchor to construct N , with n as the number of anchors on

scriptT

. A T‐spline surface

scriptS ((), ξ^{1}, ξ^{2})

is described by anchors and blending functions:

scriptS ((), ξ^{1}, ξ^{2}) = \sum_{α \in scriptA} P_{α} N_{α} ((), ξ^{1}, ξ^{2}) γ_{α},

where

scriptA

is the index set of anchors, P α denotes the coordinates of anchors, and γ α the scaling weight, which enables the T‐splines to satisfy the partition of unity property . The construction of the local knot vectors and the blending functions has been described in May et al…”

Section: Bézier Extraction Of Refined T‐splinesmentioning

confidence: 99%

“…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%

“…Herein, we will use Bézier extraction. First, we review some basic aspects of a recent LR T‐spline technology . Then, different refinement strategies will be given for adaptive local refinement on the basis of T‐splines, including implementation aspects.…”

Section: Local Refinement Using Bézier Extractionmentioning

confidence: 99%

“…Locally refined T‐splines are an extension of LR B‐splines. () We consider an initial T‐mesh with n anchors. Each anchor is associated with a local knot vector

{boldΞ}_{i} ((), i = 1, ⋯, n)

and a blending function

N_{i} ((), ξ^{1}, ξ^{2})

.…”

Section: Local Refinement Using Bézier Extractionmentioning

confidence: 99%

See 2 more Smart Citations

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

Numerical Meth Engineering

2017

Self Cite

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.

Discrete fracture analysis using locally refined T‐splines

Verhoosel

Numerical Meth Engineering

2018

Self Cite

Summary Locally refined T‐splines are used to model discrete crack propagation without a predefined interface. The crack is introduced by meshline insertions in the locally refined T‐mesh, which yields discontinuous basis functions. To implement the method in existing finite element programs, Bézier extraction is used. A detailed description is given as to how the crack path is inserted and how the domain is reparameterized after insertion. The versatility and accuracy of the approach to model discrete crack propagation without the crack path being predefined is demonstrated by two examples, namely, an L‐shaped beam and a single‐edge notched beam. When the crack approaches the physical boundaries, limitations to reparameterization arise, as will be discussed at the hand of a double‐edge notched specimen.