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

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: Numerical Examplesmentioning

confidence: 99%

Cohesive fracture analysis using Powell‐Sabin B‐splines

Num Anal Meth Geomechanics

2018

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“…However, quantitatively differences arise. To quantify the error, we therefore compute the relative error over the domain, which is defined by the L 2 error norm

ε = \frac{{(‖‖, σ_{1} - {\overset{σ}{true}}_{1})}_{L^{2} ((), {normalΩ}_{c}^{r})}}{\sqrt{\int_{{normalΩ}_{c}} σ_{1} \cdot σ_{1} normald S}} = \frac{\sqrt{\int_{{normalΩ}_{c}^{r}} ((), σ_{1} - {\overset{σ}{true}}_{1}) \cdot ((), σ_{1} - {\overset{σ}{true}}_{1}) normald S}}{\sqrt{\int_{{normalΩ}_{c}} σ_{1} \cdot σ_{1} normald S}},

where σ 1 stands for the exact solution referred to the stress on the mesh Ω c before the crack insertion, and

{\overset{σ}{true}}_{1}

denotes the solution after the crack insertion and remeshing.…”

Section: State Vector Update After Crack Insertionmentioning

confidence: 99%

Energy conservation during remeshing in the analysis of dynamic fracture

Numerical Meth Engineering

2019

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Summary The analysis of (dynamic) fracture often requires multiple changes to the discretisation during crack propagation. The state vector from the previous time step must then be transferred to provide the initial values of the next time step. A novel methodology based on a least‐squares fit is proposed for this mapping. The energy balance is taken as a constraint in the mapping, which results in a complete energy preservation. Apart from capturing the physics better, this also has advantages for numerical stability. To further improve the accuracy, Powell‐Sabin B‐splines, which are based on triangles, have been used for the discretisation. Since scriptC1 continuity of the displacement field holds at crack tips for Powell‐Sabin B‐splines, the stresses at and around crack tips are captured much more accurately than when using elements with a standard Lagrangian interpolation, or with NURBS and T‐splines. The versatility and accuracy of the approach to simulate dynamic crack propagation are assessed in two case studies, featuring mode‐I and mixed‐mode crack propagation.

“…The last 2 examples directly illustrate LR T‐splines after meshline insertions. An example of error‐guided refinement in isogeometric analysis can be found in the work of de Borst and Chen …”

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.