Adaptive Mesh Refinement for Phase-Field Fracture

Learning objectives

By the end of this lesson, you should be able to:

  • Explain why AMR is useful in phase-field fracture.
  • Choose refinement indicators based on damage, energy, and mesh size.
  • Understand the four-scheme AMR strategy used in stress-based phase-field fracture.
  • Write a compact AMR workflow.

1. Why AMR is needed

Phase-field fracture requires fine mesh resolution near the diffused crack zone.

A common requirement is

\[ h\le\frac{\ell_c}{5} \]

or at least

\[ h\le\frac{\ell_c}{3}. \]

Uniformly applying this mesh size everywhere is expensive, especially in 3D. Adaptive mesh refinement reduces cost by refining only where fine resolution is needed.

2. Uniform versus adaptive refinement

Uniform refinement:

entire domain → fine mesh everywhere

This is simple but wasteful.

Adaptive refinement:

crack zone / high-energy region → fine mesh
far elastic region              → coarse mesh

This is efficient because the crack usually occupies only a small part of the domain.

3. Basic AMR loop

A standard AMR loop is:

solve on current mesh
compute refinement indicators
mark elements
refine marked elements
transfer fields to new mesh
continue solving

For phase-field fracture, the most important transferred fields are usually:

\[ D, \qquad \mathcal{H}, \]

where \(D\) is damage and \(\mathcal{H}\) is the history or crack-driving field.

4. Useful refinement indicators

Common indicators include:

Indicator Purpose
Damage field, \(D\) Identifies and refines the existing damaged or cracked zone
Damage gradient, \(\lvert \nabla D \rvert\) Identifies and refines the diffuse crack interface, where damage changes rapidly
Tensile elastic energy, \(\psi_e^+\) Identifies regions with high tensile driving energy and likely crack initiation
Incremental energy change, \(\Delta \psi_e^+\) or \(\Delta \mathcal{H}\) Identifies zones of active crack growth between load or time steps
History field, \(\mathcal{H}\) Identifies regions with high accumulated crack-driving force
Element size, \(h\) Imposes a lower refinement limit to avoid excessive mesh refinement

Damage alone is not enough before crack initiation, because \(D\) may still be zero everywhere.

5. Stress-based phase-field model in the reference paper

The reference AMR paper uses a stress-based phase-field fracture model with damage

\[ D\in[0,1]. \]

The degradation function is

\[ g_d(D)=(1-D)^2+\kappa. \]

Damage is driven by an effective tensile stress measure. The critical strain energy density is

\[ \psi_c=\frac12\frac{\sigma_c^2}{E}, \]

where \(\sigma_c\) is the critical stress for damage initiation.

The crack surface density is

\[ \gamma(D,\nabla D)= \frac{D^2}{2\ell_c}+\frac{\ell_c}{2}|\nabla D|^2. \]

This formulation makes damage initiation stress-threshold-based rather than purely energy-release-based.

6. Coupled equations

The mechanical equilibrium equation is

\[ \nabla\cdot\sigma+b=0. \]

The damage equation has the form

\[ \ell_c^2\Delta D-D+(1-D)\mathcal{H}=0, \]

with natural boundary condition

\[ \nabla D\cdot n=0. \]

The gradient term is the reason the mesh must resolve \(\ell_c\).

7. Multi-level conditional AMR

The reference strategy uses four schemes:

\[ S_1,\quad S_2,\quad S_3,\quad S_4. \]

They are applied as

\[ S_4\circ S_3\circ S_2\circ S_1. \]

The meaning is:

S1: mark likely crack-growth regions
S2: remove unnecessary elastic-region refinement
S3: mark damaged regions
S4: stop refinement below h_min

8. Scheme S1: energy-change marking

Before damage appears, a damage-based indicator cannot identify future cracks. So \(S_1\) marks elements where elastic energy changes most.

For element \(e\), define

\[ \mathcal{E}_1^e=\tilde\psi_e^{j+1}-\tilde\psi_e^j. \]

Mark if

\[ \mathcal{E}_1^e>\alpha_1\max(\mathcal{E}_1). \]

This targets crack-initiation and crack-tip regions.

9. Scheme S2: remove safe elastic regions

Some regions may have energy change but still be far below damage initiation.

Unmark elements if

\[ \tilde\psi_e^+\le \alpha_2\psi_c. \]

This avoids wasting refinement in safe elastic regions.

10. Scheme S3: damage marking

Once damage exists, refine damaged elements.

Mark if

\[ D_e>\alpha_3. \]

A small threshold such as \(\alpha_3=0.01\) captures even weakly damaged regions.

11. Scheme S4: stop over-refinement

Refinement should stop once the target size is reached.

Use an element-size indicator such as

\[ \mathcal{E}_4^e=2\times\text{circumradius}(e). \]

Unmark elements if

\[ \mathcal{E}_4^e<h_{\min}. \]

Usually,

\[ h_{\min}\approx\frac{\ell_c}{5}. \]

12. Summary of the four schemes

Scheme Indicator Action Purpose
\(S_1\) change in elastic energy mark find active/future crack zone
\(S_2\) tensile energy threshold unmark avoid safe elastic zones
\(S_3\) damage \(D\) mark resolve cracked zone
\(S_4\) element size \(h\) unmark prevent over-refinement

This combination is more reliable than using damage alone.

13. Compact AMR algorithm

initialize mesh, u, D, H

for each load step:

    solve coupled/staggered phase-field problem

    compute energy-change indicator
    apply S1 marking

    remove safe elastic elements using S2

    add damaged elements using S3

    remove elements already smaller than h_min using S4

    if elements are marked:
        refine mesh
        transfer D and H to new mesh
        continue solve

14. Practical cautions

AMR can introduce errors if not handled carefully:

  • field transfer may violate irreversibility,
  • repeated refinement can create too many elements,
  • poor element quality can bias crack paths,
  • refinement must maintain \(h/\ell_c\) near the crack,
  • refinement indicators should anticipate crack growth, not only follow existing damage.

A good AMR strategy refines both the current damaged zone and the region where damage is likely to grow next.