Abstract
Variational phase-field models of fracture are widely used to simulate nucleation and propagation of cracks in brittle materials. They are based on the approximation of the solutions of a free-discontinuity fracture energy by two smooth function: a displacement and a damage field. Their numerical implementation is typically based on the discretization of both fields by nodal \(\mathbb {P}^1\) Lagrange finite elements. In this article, we propose a nonconforming approximation by discontinuous elements for the displacement and nonconforming elements, whose gradient is more isotropic, for the damage. The handling of the nonconformity is derived from that of heterogeneous diffusion problems. We illustrate the robustness and versatility of the proposed method through series of examples.
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The source code and data files for all examples are available at https://github.com/marazzaf/DG_CR.git.
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Acknowledgements
FM would like to thank A. Chambolle for stimulating discussions. FM’s work was supported by the US National Science Foundation under Grant Number OIA-1946231 and the Louisiana Board of Regents for the Louisiana Materials Design Alliance (LAMDA). Part of this work was performed while BB was the A.K. & Shirley Barton Professor of Mathematics at Louisiana State University. BB acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2022-04536.
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Frédéric Marazzato and Blaise Bourdin authors contributed equally to this work.
Appendix: A Proof of theorem 1
Appendix: A Proof of theorem 1
Note that the initial damage field is such that \(\alpha _0 = 0\). The following proof works by recurrence. Let us consider that at each time-step \(t_i\), the discrete damage field at the previous time-step \(t_{i-1}\), written \(\alpha _{i- 1,h}\), converges strongly in \(H^1 (\Omega )\) towards \(\alpha _{i-1} \in H^1(\Omega )\).
Lemma 1
(Compactness) Let independently \(u_{i,h} \in U_{i,h}\) be solution of (10) and \(\alpha _{i,h}\in A_h\) be a solution of (11). There exists \(v_{i} \in U_i\) and \(\beta _i \in A\) such that, up to a subsequence, \(u_{i,h} \rightarrow v_i\) strongly in \(\left( L^2(\Omega )\right) ^d\) and \(\alpha _{i,h} \rightarrow \beta _i\) strongly in \(L^2(\Omega )\), and \(\nabla _h u_{i,h} \rightharpoonup \nabla v_i\) weakly in \(\left( L^2(\Omega )\right) ^{d \times d}\) and \(\nabla _h \alpha _{i,h} \rightharpoonup \nabla \beta _i\) weakly in \(L^2(\Omega )^d\).
Proof
Since \(w_i \in \left( H^{1/2}(\partial \Omega )\right) ^d\), there exists \(f_i \in \left( H^1(\Omega )\right) ^d\) such that \({f_i}_{|\partial \Omega _D} = w_i\) on \(\partial \Omega _D\). Therefore, one has
where \([f_i]_F = 0, \forall F \in {\mathcal {F}}^i_h\) because \(f_i \in \left( H^1(\Omega )\right) ^d\). As in the proof of Proposition 1, one has
Therefore,
where \(C_1\) and \(C_2\) are non-negative constants.
Thus \(\Vert u_{i,h} \Vert _{ip}\) is bounded from above. We can apply Kolmogorov compactness criterion [28, p. 194]. Thus, there exists \(v_i \in U_i\) such that, up to a subsequence, \(u_{i,h} \rightarrow v_i\) strongly in \(\left( L^2(\Omega )\right) ^d\) and \(\nabla _h u_{i,h} \rightharpoonup \nabla v_i\) weakly in \(\left( L^2(\Omega )\right) ^{d\times d}\).
Now let us get a bound on the damage. Testing (11) with \(\alpha _{i-1,h}\), one has
Thus, using a Cauchy–Schwarz inequality and the fact that \(\alpha _{h}\le 1\) and \(\alpha _{i-1,h} \le 1\), one has
where \(C>0\) and \(C'>0\) are generic non-negative constants.
The second order polynomial in the variable \(\Vert \nabla _h \alpha _{i,h}\Vert _{L^2(\Omega )}\) is negative between its two real roots and thus \(\Vert \nabla _h \alpha _{i,h}\Vert _{L^2(\Omega )}\) is bounded from above. Using the compactness of the Crouzeix–Raviart FE [47, p. 297], there exists \(\beta _i \in A\) such that, up to a subsequence, \(\alpha _{i,h} \rightarrow \beta _i\) strongly in \(L^2(\Omega )\) and \(\nabla _h \alpha _{i,h} \rightharpoonup \nabla \beta _i\) weakly in \(\left( L^2(\Omega )\right) ^d\).
\(\square \)
Proposition 3
(Existence of solution to the discretized problem) There exists \((u_{i,h},\alpha _{i,h}) \in V_{i,h}\) solving (10) and (11) simultaneously.
Proof
Let \(T:(v_h,\beta _h) \mapsto (u_h,\alpha _h)\), where \(u_h\) is the solution of \({\mathcal {U}}_h(\beta _h;u_h,\bullet ) = 0\) over \(U_{i,h}\) and \(\alpha _h\) is the solution of \(\mathcal {A}_h (v_h;\alpha _h,\bullet -\alpha _{h}) \ge f(v_h; \bullet -\alpha _{h})\) over \(K_{i,h}\). Assuming \(v_h\) and \(\beta _h\) verify the bounds proved in the proof of Lemma 1, then \(u_h\) and \(\alpha _h\) verify these same bounds. Thus T is a mapping of a nonempty compact convex subset of \(V_{i,h}\) into itself. As, \({\mathcal {U}}_h(\beta _h)\) and \({\mathcal {A}}_h(v_h)\) are continuous bilinear forms, T is a continuous map. As a consequence, using Brouwer fixed point theorem [48, p. 179], there exists a fixed point \((u_{i,h},\alpha _{i,h})\) solving (10) and (11) simultaneously. \(\square \)
Lemma 2
\((v_i,\beta _i)\) is a solution of (3a).
Proof
Let \(\varphi \in \left( {\mathcal {C}}_c^\infty (\Omega )\right) ^d\) be a function with compact support in \(\Omega \). Testing (10) with \(\pi _h \varphi \), one has
The last two terms in the left-hand side vanish when \(h \rightarrow 0\) because \(\varphi ,v_i \in \left( H^1(\Omega )\right) ^d\). Regarding the first term in the left-hand side, one has
Passing to the limit in (I), one obtains the expected term
Let us now prove that (II) and (III) vanish as \(h \rightarrow 0\). Using a Cauchy–Schwarz inequality, one has
We focus on the second term in the right-hand side.
since \(\Pi _h\) is a projection in \(L^2(\Omega )\). Using the strong convergence \(\alpha _h \rightarrow \beta _i\) in \(L^2(\Omega )\) and the fact that a is continuous gives the desired result. Regarding (III), using a Cauchy–Schwarz inequality, one has
where \(C>0\) is a generic non-negative constant. Using a classical local approximation result (see [29, Proposition 1.135] for instance), one has:
where \(\nabla (a(\beta _i)) = a'(\beta _i)\nabla \beta _i \in \left( L^2(\Omega )\right) ^d\) because \(\beta _i \in L^\infty (\Omega ) \cap H^1(\Omega )\) and a is \({\mathcal {C}}^1\) and thus (III) vanishes as \(h \rightarrow 0\). \(\square \)
Lemma 3
\(\textrm{e}_h(u_{i,h}) \rightarrow \textrm{e}(v_i)\) strongly in \(\left( L^2(\Omega ) \right) ^{d \times d}\), where \(v_i\) is a solution of (3a).
Proof
We consider again \(f_i \in \left( H^1(\Omega )\right) ^d\) such that \(f_i=w_i\) on \(\partial \Omega _D\). We are going to test (10) with \({\tilde{v}}_h = u_{i,h} - \pi _h f_i\) so that \({\tilde{v}}_h \in U_{0,h}\). One thus has
Using the strong convergence in \(u_{i,h}\) and \(\alpha _{i,h}\) in the right-hand side gives 0 but it also ensures that the quadratic term in \(\textrm{e}_h(u_{i,h})\) in the left-hand side has a limit when \(h \rightarrow 0\). Thus \(e_h(u_{i,h}) \rightarrow e(v_i)\) strongly in \(\left( L^2(\Omega )\right) ^{d\times d}\), when \(h \rightarrow 0\). \(\square \)
Lemma 4
\((v_i,\beta _i)\) is a solution of (3b).
Proof
Let \(\varphi \in \mathcal {C} ^\infty _c(\Omega )\), \(\varphi \ge 0 \). We are going to test (11) with \(\beta _h = \pi _h + \alpha _{i,h}\). One thus has
Owing to the weak convergence of \(\nabla _h \alpha _{i,h}\), one has
Therefore, for any \(\beta \in K_i\),
Lemma 5
\(\nabla _h \alpha _{i,h} \rightarrow \nabla \beta _i\) strongly in \(\left( L^2(\Omega )\right) ^d\), where \(\beta _i\) is the solution of (3b).
Proof
Because of the weak convergence of \(\nabla _h \alpha _{i,h}\) towards \(\nabla \beta _i\), one has
Let us test (11) with \(\beta _h = \pi _h \beta _i\). One has
Because of the weak convergence of \(\nabla _h \alpha _{i,h}\),
Due to the strong convergence of \(\alpha _{i,h}\) towards \(\beta _i\), one finally gets
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Marazzato, F., Bourdin, B. A DG/CR discretization for the variational phase-field approach to fracture. Comput Mech 72, 693–705 (2023). https://doi.org/10.1007/s00466-023-02294-y
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DOI: https://doi.org/10.1007/s00466-023-02294-y