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Article Contents

# Adaptive isogeometric methods with hierarchical splines: An overview

• * Corresponding author: Annalisa Buffa
• We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and present the study of its numerical properties. By following [10,12,11], optimal convergence rates of the AIGM can be proved when suitable approximation classes are considered. This is in line with the theory of adaptive methods developed for finite elements, recently well reviewed in [43]. The important output of our analysis is the definition of classes of admissibility for meshes underlying hierarchical splines and the design of an optimal adaptive strategy based on these classes of meshes. The adaptivity analysis is validated on a selection of numerical tests. We also compare the results obtained with suitably graded meshes related to different classes of admissibility for 2D and 3D configurations.

Mathematics Subject Classification: Primary: 41A15, 65N30; Secondary: 65D07, 65N12.

 Citation:

• Figure 1.  A univariate quadratic B-spline of level $\ell$ (left) and its truncation obtained by considering $\Omega^{\ell+1} = [1,3.0]$ (right)

Figure 2.  The REFINE and REFINE_RECURSIVE modules

Figure 3.  Exact solutions for the smooth function with a peak (left) and the singular function (right)

Figure 4.  Numerical error and estimator (left) and effectivity index (right) for the smooth function with a peak. On the left, the error (solid lines) and the estimator (dashed lines) are plotted for different degrees. The red, green, blue and black lines (star, diamond, cross and circle markers, respectively) represent admissibility classes $m = 2,3,4,\infty$, respectively, while the magenta line (square marker) corresponds to uniform refinement. The same coloring and marking is used for the effectivity index (divided by 10) on the right figure

Figure 5.  Hierarchical meshes obtained for the smooth solution with a peak

Figure 6.  Numerical error and estimator (left) and effectivity index (right) for the example with singular solution. On the left, the error (solid lines) and the estimator (dashed lines) are plotted for different degrees. The red, green, blue and black lines (star, diamond, cross and circle markers, respectively) represent admissibility classes $m = 2,3,4,\infty$, respectively, while the magenta line (square marker) corresponds to uniform refinement. The same coloring and marking is used for the effectivity index (divided by 10) on the right figure

Figure 7.  Hierarchical meshes obtained for the example with singular solution

Figure 8.  Geometry and solution for the 3D example

Figure 9.  Convergence of the estimator for the 3D example. The red, green, blue and black lines (star, diamond, cross and circle markers, respectively) represent admissibility classes $m = 2,3,4,\infty$, respectively, while the magenta line (square marker) corresponds to uniform refinement

Figure 10.  Meshes obtained for the 3D example degree $\mathbf{p} = (4,4)$ and different values of the admissibility class $m$

Table 1.  Number of elements ($\# {\cal Q}$) and degrees of freedom ($\# {\cal T}({\cal Q})$) for Example 1

 $\mathbf{p}=(2,2)$ $\mathbf{p}=(3,3)$ $\mathbf{p}=(4,4)$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $m=2$ $3232$ $2876$ $3880$ $3313$ $3736$ $3400$ $m=3$ $2500$ $2244$ $3868$ $3317$ $2824$ $2224$ $m=4$ $3628$ $3272$ $3064$ $2614$ $2224$ $1672$ $m=\infty$ $3616$ $3244$ $2980$ $2446$ $2284$ $1756$

Table 2.  Number of elements ($\# {\cal Q}$) and degrees of freedom ($\# {\cal T}({\cal Q})$) for Example 2

 $\mathbf{p}=(2,2)$ $\mathbf{p}=(3,3)$ $\mathbf{p}=(4,4)$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $\# {\cal Q}$ $\# {\cal T}({\cal Q})$ $m=2$ $68677$ $68860$ $8776$ $8938$ $3700$ $3822$ $m=3$ $68014$ $68201$ $6346$ $6631$ $2875$ $2966$ $m=4$ $68008$ $68195$ $6208$ $6484$ $3010$ $3138$ $m=\infty$ $68008$ $68195$ $6208$ $6484$ $2827$ $2895$

Figures(10)

Tables(2)