Adaptive isogeometric analysis of high-order phase-field fracture based on THB-splines

Abstract

In recent decades, the study of fracture propagation in solids has increasingly relied on phase-field models. Several recent contributions have highlighted the potential of this approach in both static and dynamic frameworks. However, a major limitation remains the high computational cost. Two main strategies have been identified to mitigate this issue: the use of locally refined meshes and the adoption of higher-order models. In this work, leveraging Truncated Hierarchical B-splines (THB-splines), we introduce adaptive simulations of higher-order phase-field formulations (AT1 and AT2), focusing primarily on two-dimensional fracture problems.

Figures (12)

• Figure 1: The concept of Truncated Hierarchical B-spline refinement in one dimension. The THB-spline basis in the top row is locally refined on the interval $[\frac{3}{8},\frac{7}{8}]$ (shaded area) and the functions influenced by the local refinement are color-coded as follows: (yellow) non-truncated functions overlapping the refined area; (green and blue) truncated function of the coarse level, (orange) activated functions of the fine level. The second up to the last rows decompose three functions of the coarse level which are affected by the refinement: left and right the truncated coarse functions and in the middle the eliminated coarse function. In the second row, the original functions of the coarse level are highlighted. In the third row, the representation of the function in the finer level is provided, with the original function represented by a dotted line. In the last row, the representation coefficients of the functions which are fully contained in the marked (shaded) interval are set to zero, yielding the truncated functions (light) in the left and right columns. Since the function in the middle row is fully eliminated, the middle plot in the bottom row provides the functions from the fine level inserted instead.
• Figure 2: Step-wise illustration of admissible refinement on a mesh corresponding to a THB-spline basis of degree $p=2$. Firstly, (\ref{['fig:admissibility1']}) illustrates the marking of two elements (in blue and green) of the finest level $\ell$, based on the damage field (in red). Secondly, (\ref{['fig:admissibility2']}) shows the support extensions of the marked elements as shaded areas. Thirdly, (c) (\ref{['fig:admissibility3']}) shows the element of level $\ell-2$ intersecting with the support extensions of the marked elements in yellow. Lastly, (d) (\ref{['fig:admissibility4']}) shows the refined mesh.
• Figure 3: The effect of cross-talk and the remedy of local refinement on a 1D bar. The top row represents a 1D B-spline basis (top left) and a 1D THB-spline basis (top right) locally refined in the interval $[\frac{1}{5},\frac{4}{5}]$ (blue) such that the middle function of the B-spline basis is eliminated. The bottom row represents the solution of a linear elasticity problem with a degraded material according to \ref{['eq:elastic_strain_en', 'eq:free_en_split']} using the red area between $[\frac{2}{5},\frac{3}{5}]$ as damage field, and end displacements of $-1$ and $1$ at $\xi=0$ and $\xi=1$, respectively. On the B-spline basis (bottom left), a gradual distribution of the displacement over the length coordinate $\xi$ is observed, whereas the result obtained on the THB-spline basis shows a discontinuity between the left ($[0,\frac{2}{5}]$) and right ($[\frac{3}{5},1]$) part of the damaged region.
• Figure 4: Illustration of cross-talk elimination through refinement of support extensions, provided a damage field (in red). The shaded cells represent elements marked for refinement. Firstly, (\ref{['fig:adaptivity_cross-talk_a']}) shows the damage on the coarsest (initial) mesh, where every element with sufficiently high damage value is marked for refinement. Secondly, (\ref{['fig:adaptivity_cross-talk_b']}) shows the mesh after refinement to level 1, where elements with sufficiently high damage value are again marked for refinement. The light shaded region of elements of level $\ell=0$ represents element marked by the admissibility algorithm. Thirdly, (\ref{['fig:adaptivity_cross-talk_c']}) shows the mesh after refinement to the finest level $\ell=2$, where now elements corresponding to the support extension of the previously marked elements are marked for refinement (in this case there is no need for additional refinement through admissibility). Finally, (\ref{['fig:adaptivity_cross-talk_d']}) shows the final mesh after refinement to the finest level $\ell=2$, where all basis functions of level $\ell=1$ have been eliminated over the damaged region, hence cross-talk is eliminated.
• Figure 5: Schematic representation of sudden phase-field propagation combined with mesh adaptivity. From left to right: (\ref{['fig:propagation_step_km2']}) phase-field at step $k-2$, (\ref{['fig:propagation_step_km1']}) phase-field at step $k-1$; (\ref{['fig:propagation_step_km1']}) slightly propagated phase-field with elements marked for refinement (gray); (\ref{['fig:propagation_step_k_it_i']}) significantly propagated phase-field, partially distorted due to representation on coarse elements with elements marked for refinement in gray; (\ref{['fig:propagation_step_k_it_j']}) phase-field represented on a locally refined mesh according to the phase-field in iteration $i<j$ as presented in \ref{['fig:propagation_step_k_it_i']}.

Theorems & Definitions (5)

• Example 3.1: THB-spline representation and truncation
• Example 3.2: Admissible meshing
• Example 4.1: Cross-talk
• Example 4.2: Cross-talk elimination by local refinement
• Example 4.3: Adaptive phase-field refinement