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Discrete approximation of the Griffith functional by adaptive finite elements
Numerical Analysis · Submitted 24 Feb 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim It is proved to be approximated in the sense of $\Gamma$-convergence by a sequence of discrete integral functionals defined on continuous piecewise affine functions.
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The abstract, sentence by sentence
This paper is devoted to show a discrete adaptive finite element approximation result for the isotropic two-dimensional Griffith energy arising in fracture mechanics. The problem is addressed in the geometric measure theoretic framework of generalized special functions of bounded deformation which corresponds to the natural energy space for this functional. It is proved to be approximated in the sense of $\Gamma$-convergence by a sequence of discrete integral functionals defined on continuous piecewise affine functions. The main feature of this result is that the mesh is part of the unknown of the problem, and it gives enough flexibility to recover isotropic surface energies.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This paper is devoted to show a discrete adaptive finite element approximation result for the isotropic two-dimensional Griffith energy arising in fracture mechanics. | contribution 0.97 | 0.85 | 0.17 | asserted 1.00 |
| s1 | The problem is addressed in the geometric measure theoretic framework of generalized special functions of bounded deformation which corresponds to the natural energy space for this functional. | method 0.52 | 0.25 | 0.04 | asserted 1.00 |
| s2 | It is proved to be approximated in the sense of $\Gamma$-convergence by a sequence of discrete integral functionals defined on continuous piecewise affine functions. | result 0.98 | 0.91 | 0.59 | asserted 1.00 |
| s3 | The main feature of this result is that the mesh is part of the unknown of the problem, and it gives enough flexibility to recover isotropic surface energies. | contribution 0.55 | 0.86 | 0.35 | asserted 0.99 |
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The paper
Original abstract
This paper is devoted to show a discrete adaptive finite element approximation result for the isotropic two-dimensional Griffith energy arising in fracture mechanics. The problem is addressed in the geometric measure theoretic framework of generalized special functions of bounded deformation which corresponds to the natural energy space for this functional. It is proved to be approximated in the sense of $\Gamma$-convergence by a sequence of discrete integral functionals defined on continuous piecewise affine functions. The main feature of this result is that the mesh is part of the unknown of the problem, and it gives enough flexibility to recover isotropic surface energies.
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References
How the literature uses it
1 arXiv paper cite this in 11 sentences
Papers citing it · 9 sentences without a confident label
Uses its method or tool 1 paper
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Looking at the constructive proof for the recovery sequence in [ref] , one can check that there exists $u_∈V_h_(Ø)$ and a triangulation $𝐓_h_ ∈_h_(Ø)$ such that $u_$ is affine on each of its triangle, [equation] (Beware that this requires that $θ_0 ≤Θ_0 := 45^∘- arctan(1/2)$ as the triangulation $𝐓_h_$ might have triangles with angles equal to $Θ_0$.) Since $u_$ is affine on each triangle $T ∈𝐓_h_$, the fact that the length of every edge of $T$ is larger than $h_$ together with [ref] imply that there exists a constant $C(θ_0)> 0$ such that [equation] Introducing the characteristic functions $χ_= 1_ { 𝐀_1 e(u_) : e(u_) ≥2 κβ/ h_}$, one can check that $(u_, χ_) ∈X_h_(Ø)$ and [equation] On the one hand, [ref] entails that [equation] is bounded, so that $(u_, χ_) →(u,0)$ strongly in $L^1(Ø;^2) ×L^1(Ø)$ when $↘0$.
Effects of oscillation scales in discrete brittle damage models · Brittle fracture · 0.84
What it cites · 3 sentences without a confident label
Uses the method or tool of 1 paper
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Given a sequence ${u_}_>0$ with uniformly bounded energy, one can apply [ref] to the modified function $v_:=u_1_{|e(u_)|^2 ≤κ/}$ which consists in putting the value zero inside each triangle $T$ where the (symmetric) gradient of $u_$ is “large”.
Equilibrium configurations for nonhomogeneous linearly elastic materials with surface discontinuities · Strategy of proof · 0.88
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