Regularization Length Scale, Mesh Size, and Irreversibility

Learning objectives

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

  • Explain the role of the phase-field length scale \(\ell\).
  • Choose mesh size \(h\) relative to \(\ell\).
  • Distinguish mesh convergence from length-scale convergence.
  • Understand why irreversibility is required.

1. Meaning of the length scale

Phase-field fracture replaces a sharp crack by a smooth damage field

\[ d:\Omega\rightarrow[0,1]. \]

The regularized crack surface density for AT2 is

\[ \gamma_\ell(d,\nabla d) = \frac{d^2}{2\ell}+\frac{\ell}{2}|\nabla d|^2. \]

The parameter \(\ell\) controls the width of the diffused crack.

Small \(\ell\):

  • sharper crack,
  • closer to Griffith fracture,
  • harder nonlinear problem,
  • requires finer mesh.

Large \(\ell\):

  • wider damage band,
  • easier to resolve,
  • may smear the crack,
  • may alter peak load and crack path.

2. AT2 crack profile

For AT2, a typical one-dimensional crack profile is

\[ d(x)=e^{-|x|/\ell}. \]

Thus,

\[ d(\ell)=e^{-1}\approx0.37, \qquad d(2\ell)=e^{-2}\approx0.14, \qquad d(3\ell)=e^{-3}\approx0.05. \]

Most visible damage lies within a few multiples of \(\ell\). Therefore, the mesh must resolve not only the crack center but also the surrounding damage band.

3. Mesh requirement

Let \(h\) be the local element size. The mesh should satisfy

\[ h\ll \ell. \]

Common practical rules are

\[ h\le\frac{\ell}{3} \]

or more conservatively

\[ h\le\frac{\ell}{5}. \]

If \(h\gtrsim\ell\), the damage zone is under-resolved. This can produce:

  • wrong fracture energy,
  • mesh-biased crack paths,
  • incorrect peak load,
  • unstable damage growth,
  • poor convergence.

4. Mesh convergence versus length-scale convergence

There are two different studies.

Mesh convergence

Fix \(\ell\), then refine \(h\):

\[ h\rightarrow0\quad\text{with fixed }\ell. \]

This checks whether the finite element solution resolves the chosen regularized model.

Length-scale convergence

Reduce \(\ell\) toward the sharp-crack limit:

\[ \ell\rightarrow0. \]

But every time \(\ell\) is reduced, \(h\) must also be reduced so that \(h/\ell\) remains small.

The common mistake is reducing \(\ell\) without refining the mesh.

5. Computational cost

If \(\ell\) is halved and the rule \(h\le\ell/5\) is maintained, the mesh must also become about twice as fine.

In 2D, halving \(h\) increases element count by roughly

\[ 2^2=4. \]

In 3D, it increases element count by roughly

\[ 2^3=8. \]

So small \(\ell\) becomes expensive very quickly, especially in 3D.

6. Effect of \(\ell\) on apparent strength

In classical phase-field fracture, the apparent nucleation stress often scales like

\[ \sigma_c^{\text{app}}\sim\sqrt{\frac{E G_c}{\ell}}. \]

Therefore, \(\ell\) may affect not only crack width but also apparent strength.

This matters because:

  • if \(\ell\) is treated as numerical, it should be chosen small and resolved;
  • if \(\ell\) is treated as physical, it should be calibrated carefully.

This is one motivation for generalized models that prescribe strength independently from \(G_c\) and \(\ell\).

7. AT1 versus AT2

AT2 uses

\[ \gamma_\ell^{AT2}=\frac{d^2}{2\ell}+\frac{\ell}{2}|\nabla d|^2. \]

It is smooth but can show gradual damage initiation.

AT1 uses

\[ \gamma_\ell^{AT1}=\frac{3}{8}\left(\frac{d}{\ell}+\ell|\nabla d|^2\right). \]

It tends to produce a more threshold-like onset of damage.

Both require proper mesh resolution.

8. Irreversibility

Damage should not heal:

\[ d_n\ge d_{n-1}. \]

Without irreversibility, unloading could artificially reduce damage and close cracks in an unphysical way.

Common methods:

Method Use
History field Simple approximate irreversibility
Nodal clipping Easy post-processing bound enforcement
Penalty method Penalizes healing
Bound-constrained solve Directly enforces \(d_{n-1}\le d_n\le1\)

The clean mathematical condition is

\[ d_{n-1}\le d_n\le1. \]

9. Practical checklist

Before trusting a phase-field fracture simulation, check:

  • Is \(h/\ell\) small enough near the crack?
  • Does the crack path change under mesh refinement?
  • Does the load-displacement curve converge for fixed \(\ell\)?
  • Does reducing \(\ell\) require further mesh refinement?
  • Is irreversibility enforced?
  • Is \(\ell\) being used as a numerical parameter or calibrated material length?

These checks prevent many common phase-field fracture errors.