A DG/CR discretization for the variational phase-field approach to fracture

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.

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

$$\begin{aligned} {\mathcal {U}}_h(\alpha _{i,h};u_{i,h},u_{i,h} - f_i) = 0. \end{aligned}$$

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

$$\begin{aligned} \frac{2 \mu \eta _\ell }{K^2} \Vert \textrm{e}_h(u_{i,h}) \Vert ^2_{L^2} + \frac{\zeta \eta _\ell ^2}{1+\eta _\ell } |u_{i,h}|^2_J \le {\mathcal {U}}_h(\alpha _{i,h};u_{i,h},u_{i,h}). \end{aligned}$$

Therefore,

$$\begin{aligned} C_1 \Vert u_{i,h} \Vert _{ip}^2 \le {\mathcal {U}}_h(\alpha _{i,h};u_{i,h},u_{i,h}) = {\mathcal {U}}_h(\alpha _{i,h};u_{i,h},f_i) \\ \le C_2 \Vert u_{i,h} \Vert _{ip}, \end{aligned}$$

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

$$\begin{aligned} \mathcal {A}_{h}(u_{i,h}; \alpha _ {i,h}, \alpha _{i,h} - \alpha _{i- 1,h}) \le f(u_h; \alpha _{i,h} - \alpha _{i-1,h}). \end{aligned}$$

Thus, using a Cauchy–Schwarz inequality and the fact that $\alpha _{h}\le 1$ and $\alpha _{i-1,h} \le 1$, one has

$$\begin{aligned} \frac{2G_c}{c_w}&\int _ {\Omega } \ell |\nabla _h \alpha _ {i,h}|^2 dx \le \mathcal {A}_{h} (u_{i,h}; \alpha _{i,h}, \alpha _ {i,h})\\&\le \mathcal {A}_{h}(u_ {i,h}; \alpha _{i,h}, \alpha _{i- 1,h}) + f(u_{i,h}; \alpha _{i,h} - \alpha _{i-1,h}) \\&\le 2 \int _ {\Omega } \mathbb {C}\textrm{e}_h(u_ {i,h}) \cdot \textrm{e}_h(u_{i,h}) dx \\&\quad + \frac{2G_c\ell }{c_w} \Vert \nabla _h \alpha _{i,h} \Vert _ {L^2(\Omega )}\Vert \nabla _h \alpha _{i-1,h}\Vert _{L^2 (\Omega )} \\&\le C\Vert u_{i,h} \Vert _{ip}^2 + C'\Vert \nabla _h \alpha _{i,h}\Vert _{L^2(\Omega )} \end{aligned}$$

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

$$\begin{aligned}{} & {} \int _{\Omega } (a_h(\alpha _{i,h}) + \eta _\ell ) \mathbb {C}\textrm{e}_h(u_{i,h}) \cdot \textrm{e}_h(\pi _h \varphi )dx \\{} & {} \quad - \sum _{F \in \mathcal {F}^i_h} \int _F n \cdot \left( \{(a_h(\alpha _ {i,h}) + \eta _\ell ) \sigma _h(u_ {i,h}) \}_F \cdot [\pi _h \varphi ]_F\right. \\{} & {} \quad \left. -\{(a_h(\alpha _{i,h}) + \eta _\ell ) \sigma _h(\pi _h \varphi ) \}_F \cdot [u_{i,h}]_F \right) dS\\{} & {} \quad +\, \sum _{F \in {\mathcal {F}}^i_h} \frac{\zeta \gamma _F}{h_F} \int _F [u_{i,h}]_F \cdot [\pi _h \varphi ]_F dS = 0. \end{aligned}$$

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

$$\begin{aligned} \begin{aligned}&\int _{\Omega } (a_h(\alpha _{i,h}) + \eta _\ell ) \mathbb {C}\textrm{e}_h(u_{i,h}) \cdot \textrm{e}_h(\pi _h \varphi )dx \\&\quad = \int _ {\Omega } (a(\beta _i) + \eta _\ell ) \mathbb {C}\textrm{e}_h(u_{i,h}) \cdot \textrm{e}_h(\pi _h \varphi )dx \\&\qquad + \int _ {\Omega } (a_h(\alpha _{i,h}) - \Pi _h a(\beta _i)) \mathbb {C}\textrm{e}_h (u_{i,h}) \cdot \textrm{e}_h(\pi _h \varphi ) dx \\&\qquad + \int _{\Omega } (\Pi _h a (\beta _i) - a(\beta _i)) \mathbb {C} \textrm{e}_h(u_{i,h}) \cdot \textrm{e}_h(\pi _h \varphi )dx \\&\quad = (I) + (II) + (III) \end{aligned} \end{aligned}$$

Passing to the limit in (I), one obtains the expected term

$$\begin{aligned} \int _{\Omega } (a(\beta _i) + \eta _\ell ) \mathbb {C}\textrm{e}(v_i) \cdot \textrm{e}(\varphi )dx. \end{aligned}$$

Let us now prove that (II) and (III) vanish as $h \rightarrow 0$. Using a Cauchy–Schwarz inequality, one has

$$\begin{aligned} \begin{aligned} (II)&\le \left( \int _{\Omega } (\mathbb {C}\textrm{e}_h (u_{{i,}h}) \cdot \textrm{e}_h(\pi _h \varphi ))^2dx \right) ^{1/2} \\&\quad \Vert \Pi _h a(\alpha _{i,h}) - \Pi _h a (\beta _i) \Vert _{L^2(\Omega )} \\&\le C \Vert \varphi \Vert _{W^ {1,\infty }(\Omega )} \Vert u_{i,h} \Vert _{ip} \Vert \Pi _h a(\alpha _{i,h}) - \Pi _h a(\beta _i) \Vert _ {L^2(\Omega )}. \end{aligned} \end{aligned}$$

We focus on the second term in the right-hand side.

$$\begin{aligned} \Vert \Pi _h a(\alpha _{i,h}) - \Pi _h a(\beta _i) \Vert _{L^2(\Omega )} \le \Vert a(\alpha _{i,h}) - a(\beta _i) \Vert _{L^2(\Omega )}, \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} (III)&\le \left( \int _{\Omega } (\mathbb {C}\textrm{e}_h(u_{i,h}) \cdot \textrm{e}_h(\pi _h \varphi ))^2dx \right) ^{1/2} \Vert a(\beta _i) \\&\quad - \Pi _h a(\beta _i) \Vert _{L^2(\Omega )} \\&\le C \Vert \varphi \Vert _{W^{1,\infty }(\Omega )} \Vert u_{i,h}\Vert _{ip} \Vert a(\beta _i) - \Pi _h a(\beta _i) \Vert _{L^2(\Omega )} \end{aligned}, \end{aligned}$$

where $C>0$ is a generic non-negative constant. Using a classical local approximation result (see [29, Proposition 1.135] for instance), one has:

$$\begin{aligned} \Vert a(\beta _i) - \Pi _h a(\beta _i) \Vert _{L^2(\Omega )} \le Ch \Vert \nabla (a(\beta _i))\Vert _{L^2(\Omega )}, \end{aligned}$$

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

$$\begin{aligned} \begin{aligned}&\int _ {\Omega } (a_h(\alpha _{i,h}) + \eta _\ell ) \mathbb {C}\textrm{e}_h(u_h) \cdot \nabla _h \tilde{v}_h dx \\&= \sum _{F \in \mathcal {F}^i_h} \int _F n \cdot \left( \{(a_h(\alpha _ {i,h}) + \eta _\ell )\sigma _h(u_ {i,h}) \}_F \cdot [\tilde{v}_h]_F\right. \\&\quad \left. - \{(a_h(\alpha _{i,h}) + \eta _\ell ) \sigma _h(\tilde{v}_h) \}_F \cdot [u_{i,h}]_F \right) dS \\&\quad - \sum _ {F \in \mathcal {F}^i_h} \frac{\zeta \gamma _F}{h_F} \int _F [u_{i,h}]_F \cdot [\tilde{v}_h]_F dS, \end{aligned} \end{aligned}$$

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

$$\begin{aligned} \mathcal {A}_h(u_{i,h}; \alpha _ {i,h}, \pi _h \varphi ) \ge f(u_{i,h}; \pi _h \varphi _h). \end{aligned}$$

(13)

Owing to the weak convergence of $\nabla _h \alpha _{i,h}$, one has

$$\begin{aligned}{} & {} \int _{\Omega } \textrm{e}(v_i) : \mathbb {C} : \textrm{e}(v_i) (\beta _i-1) {\varphi } dx \nonumber \\{} & {} \quad + \frac{G_c}{c_w} \int _{\Omega }{ \left( 2\ell \nabla \beta _i \cdot {\nabla \varphi } + \frac{\varphi }{\ell }\right) dx \ge } 0. \end{aligned}$$

(14)

Therefore, for any $\beta \in K_i$,

$$\begin{aligned}{} & {} \int _{\Omega } \textrm{e}(v_i) : \mathbb {C} : \textrm{e}(v_i) (\beta _i-1) (\beta - \beta _i) dx \nonumber \\{} & {} \quad + \frac{G_c}{c_w} \int _ {\Omega } { \left( 2\ell \nabla \beta _i \cdot (\beta - \beta _i) + \frac{\beta - \beta _i}{\ell }\right) dx \ge } 0.\nonumber \\ \end{aligned}$$

(15)

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

$$\begin{aligned} \Vert \nabla \beta _i \Vert _{L^2 (\Omega )} \le \liminf _{h \rightarrow 0} \Vert \nabla _h \alpha _{i,h} \Vert _ {L^2(\Omega )}. \end{aligned}$$

(16)

Let us test (11) with $\beta _h = \pi _h \beta _i$. One has

$$\begin{aligned}{} & {} f(u_{i,h}; \beta _h - \alpha _{i,h}) + \mathcal {A}_h(u_ {i,h}; \alpha _{i,h}, \alpha _{i,h})\nonumber \\{} & {} \quad \le \mathcal {A}_h(u_{i,h}; \alpha _{i,h},\beta _h). \end{aligned}$$

(17)

Because of the weak convergence of $\nabla _h \alpha _{i,h}$,

$$\begin{aligned} 0 + \limsup _{h \rightarrow 0} \mathcal {A}_h (u_{i,h}; \alpha _{i,h}, \alpha _ {i,h})\le & {} \int _\Omega \mathbb {C} \textrm{e}(v_i) \cdot \textrm{e}(v_i) \beta _i^2 dx \nonumber \\{} & {} + \frac{2G_c\ell }{c_w} \Vert \nabla \beta _i \Vert _{L^2(\Omega )}^2, \nonumber \\ \end{aligned}$$

(18)

Due to the strong convergence of $\alpha _{i,h}$ towards $\beta _i$, one finally gets

$$\begin{aligned} \limsup _{h \rightarrow 0} \frac{2G_c\ell }{c_w} \Vert \nabla _h \alpha _{i,h} \Vert ^2_ {L^2(\Omega )} \le \frac{2G_c\ell }{c_w} \Vert \nabla \beta _i \Vert ^2_{L^2(\Omega )}. \end{aligned}$$

(19)