# Crouzeix-Raviart finite element approximation for elastoplastic torsion

Franz Chouly, Patrick Hild

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Franz Chouly, Patrick Hild. Crouzeix-Raviart finite element approximation for elastoplastic torsion.
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# Crouzeix-Raviart finite element approximation for elastoplastic torsion

Franz Chouly<sup>a,*</sup>, Patrick Hild<sup>b</sup>

<sup>a</sup>Centro de Matemática (CMAT), IRL-2030 CNRS IFUMI, Facultad de Ciencias, Universidad de la República, 11400 Montevideo, Uruguay

<sup>b</sup>Institut de Mathématiques de Toulouse - UMR CNRS 5219, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France

## Abstract

This study is concerned with the elastoplastic torsion problem written in its weak form as a variational inequality in two space dimensions. We consider a Crouzeix-Raviart finite element approximation. We establish an optimal error bound of order $h$ in the natural norm, for a regular enough continuous solution.

**Keywords:** variational inequalities; elastoplastic torsion problem; finite elements; Crouzeix-Raviart finite elements; error estimates.

**2020 MSC:** 65N15, 65N30.

## 1. Introduction

Problems written with weak formulations involving variational inequalities represent various nonlinear phenomena which occur in mechanics and physics [12]. We focus on the elastoplastic torsion problem, as presented in, e.g., [17]. Very few works address the numerical analysis of this problem: see [7, 16, 17, 18] and more recent works [1, 5, 8, 9, 10]. We study a direct approximation of the variational inequality, with the restriction on the norm of the gradient, making use of Crouzeix-Raviart finite elements [11]. We prove error estimates in dimension two, with an optimal error bound in $O(h)$, for a regular enough continuous solution. For this purpose, we make use of the special properties of the Crouzeix-Raviart interpolation operator and appropriate bounds for the nonconforming terms.

As usual, we denote by $H^s(\cdot), s \in \mathbb{R}$, the Sobolev spaces. For an open subset $D$ of $\mathbb{R}^n$, the natural norm of $H^s(D)$ is denoted by $\|\cdot\|_{s,D}$. The space $H_0^1(D)$ is the subspace of functions in $H^1(D)$ with vanishing trace on $\partial D$. The letter $C$ stands for a generic constant, independent of the discretization parameters.

<footnote>*Corresponding author</footnote>

<footnote>Email addresses: fchouly@cmat.edu.uy (Franz Chouly), patrick.hild@math.univ-toulouse.fr (Patrick Hild)</footnote>
## 2. The elastoplastic torsion problem

Let $\Omega \subset \mathbb{R}^2$ be an open bounded polytope, connected and with Lipschitz boundary. We consider the variational inequality modelling the torsion of an infinitely long elastoplastic cylinder of cross section $\Omega$ and plasticity yield $r > 0$. To simplify we assume that $r = 1$. The problem is to find the stress potential $u$ such that

$$ u \in K: \quad a(u, v - u) \geq L(v - u), \quad \forall v \in K, \qquad (1) $$

where $a: H^1(\Omega) \times H^1(\Omega) \to \mathbb{R}$ is the bilinear form given by: $a(u, v) := \int_{\Omega} \nabla u \cdot \nabla v$, $\forall u, v \in H^1(\Omega)$, and $L(v) := \int_{\Omega} fv$, $\forall v \in H^1(\Omega)$, $f \in L^2(\Omega)$. The notation $K$ represents the nonempty closed convex set of admissible stress potentials:

$$ K := \{v \in H_0^1(\Omega) \mid |\nabla v| \leq 1 \text{ a.e. in } \Omega\}, $$

where $|\cdot|$ denotes the Euclidean norm in $\mathbb{R}^2$. From Stampacchia's Theorem we deduce that Problem (1) admits a unique solution. Moreover, we know from [2] and references therein that there exists a Radon measure $\lambda \in (L^\infty(\Omega))'$ that verifies

$$ f = -\operatorname{div}(\lambda \nabla u), \quad \lambda \geq 1, \quad (1-\lambda)(|\nabla u| - 1) = 0. \qquad (2) $$

## 3. Crouzeix-Raviart finite element approximation

We follow the presentation in [13, 15] with slight changes of notations. Let $(\mathcal{T}_h)_{h>0}$ be a family of regular triangulations of the domain $\Omega$ ($h := \max_{T \in \mathcal{T}_h} h_T$ where $h_T$ is the diameter of $T$, a generic triangle of $\mathcal{T}_h$). We denote by $\mathcal{F}_h$ the set of all (interior and boundary) edges of $\mathcal{T}_h$, by $F$ a generic edge of $\mathcal{F}_h$. For $F$ an interior edge, we denote by $T_1$ and $T_2$ the two triangles that share $F$, and $n_i$ the outward exterior normal to the boundary of $T_i$, $i = 1, 2$. If $F$ is a boundary edge, we denote by $T_1$ the triangle that shares $F$ and $n_1$ the outward exterior normal to the boundary of $T$ (that is in fact the normal exterior $n$ to $\partial\Omega$). For every function $v$ that is piecewise continuous (i.e., continuous on every triangle $T$) we define its jump on the edge $F$ as $[v] = v_{|_{T_1}} - v_{|_{T_2}}$ if $F$ is an interior facet and $[v] = v_{|_{T_1}}$ if $F$ is a boundary edge. We introduce the broken scalar product

$$ (w, z)_h := \sum_{T \in \mathcal{T}_h} \int_T w \cdot z, $$

for $w, z \in L^2(\Omega; \mathbb{R}^2)$. We define the Crouzeix-Raviart finite element space as

$$ V_h^{\mathrm{CR}} := \left\{ v_h \in L^\infty(\Omega) \mid \begin{array}{l} v_{h|_T} \in \mathbb{P}_1(T), \forall T \in \mathcal{T}_h; \\ \displaystyle \int_F [v_h] = 0, \forall F \in \mathcal{F}_h \end{array} \right\}. $$

We set

$$ K_h^{\mathrm{CR}} := \{v_h \in V_h^{\mathrm{CR}} \mid |\nabla v_h| \le 1 \text{ a.e. in } \Omega\}. $$

Note that $K_h^{\mathrm{CR}} \not\subset K$ since $V_h^{\mathrm{CR}} \not\subset H_0^1(\Omega)$. The discrete problem is as follows:

$$ u_h \in K_h^{\mathrm{CR}}: \quad (\nabla u_h, \nabla (v_h - u_h))_h \geq L(v_h - u_h) \quad \forall v_h \in K_h^{\mathrm{CR}}. \qquad (3) $$
Let us introduce the broken semi-norm

$$|v|_h := \left( \sum_{T \in \mathcal{T}_h} \|\nabla v\|_{0,T}^2 \right)^{\frac{1}{2}} = (\nabla v, \nabla v)_h^{\frac{1}{2}},$$

and the norm $\|v\|_h := (\|v\|_{0,\Omega}^2 + |v|_h^2)^{\frac{1}{2}}$ for $v \in V_h^{CR} + H^1(\Omega)$. Thanks to the extended Poincaré inequality [13, Lemma 3.31, Chapter 3], $| \cdot |_h$ is a norm on $V_h^{CR}$. As a result, we apply Stampacchia's Theorem and Problem (3) admits a unique solution $u_h \in K_h^{CR}$.

## 4. A priori error estimate

We derive an a priori error bound for $u - u_h$, the main tool is a Falk-type lemma, an abstract error estimate adapted to variational inequalities which also takes into account the use of nonconforming methods such as the Crouzeix-Raviart approximation. For Crouzeix-Raviart finite element approximation of the obstacle problem, similar difficulties arise because of the non-conformity, see [3, 4, 6, 19]. Remark that optimal a priori and a posteriori error analysis for the elastoplastic torsion has been carried out recently in [1] with Raviart-Thomas and Crouzeix-Raviart elements but with a different approach (Fenchel duality theory).

### 4.1. Crouzeix-Raviart interpolation operator

For simplicity, we will use the same notation for operators over scalar and vector fields. Let us first denote by $\mathcal{I}_h^{CR}$ the global Crouzeix-Raviart interpolation operator mapping onto $V_h^{CR}$ and $\mathcal{I}_T^{CR}$ its local restriction to a generic triangle $T$, defined in, e.g., [15, Chapter 36]. We denote by $\Pi_T$ the local $L^2(T)$-projection onto constants, for a generic triangle $T$. The following remarkable commuting property will play an important role in the analysis (see, e.g., [15, Exercise 36.1, page 188] or also [1, 11]):

$$\nabla(\mathcal{I}_T^{CR}v) = \Pi_T(\nabla v) \quad (4)$$

for every $v \in H^1(T)$. As a first consequence, the Crouzeix-Raviart interpolation operator $\mathcal{I}_h^{CR}$ preserves the $L^\infty$-norm of the euclidian norm of the gradient.

**Lemma 4.1.** For every $v \in H^1(\Omega)$, there holds

$$|\nabla \mathcal{I}_h^{CR} v| \le \| |\nabla v| \|_{L^\infty(\Omega)} \quad \text{in } \Omega. \quad (5)$$

**Proof:** We take $v \in H^1(\Omega)$ and $T \in \mathcal{T}_h$. We denote $w := \nabla v|_T$ with $w = (w_1, w_2)$ its scalar components. Take $i = 1, 2$, use the definition of $\Pi_T$ ($\Pi_T w_i = (\int_T w_i)/|T|$) and then the Cauchy-Schwarz inequality to bound

$$|\Pi_T w_i|^2 \le \frac{1}{|T|^2} \left(\int_T 1^2\right) \left(\int_T w_i^2\right) = \frac{1}{|T|} \left(\int_T w_i^2\right).$$

Therefore, we bound the whole vector $w$ as follows:

$$|\Pi_T w|^2 = \sum_{i=1}^2 |\Pi_T w_i|^2 \le \frac{1}{|T|} \sum_{i=1}^2 \left(\int_T w_i^2\right) = \frac{1}{|T|} \int_T \left(\sum_{i=1}^2 w_i^2\right) = \frac{1}{|T|} \int_T |w|^2 \le \|w\|_{L^\infty(T)}^2.$$

We conclude by applying (4) to obtain first $|\nabla \mathcal{I}_T^{CR} v| \le \| |\nabla v| \|_{L^\infty(T)}$ and then (5). $\square$
## 4.2. Our main result

Our main result is:

**Theorem 4.1.** Let $\Omega \subset \mathbb{R}^2$ be an open subset, bounded connected and with Lipschitz boundary. Suppose that the solution $u$ to Problem (1) belongs to $H^2(\Omega)$ and that the Lagrange multiplier $\lambda$ belongs to $L^\infty(\Omega) \cap H^1(\Omega)$. Let $u_h$ be the solution to Problem (3). There holds

$$ \|u - u_h\|_h \leq Ch (\|u\|_{2,\Omega} + \|f\|_{0,\Omega} + \|\lambda\|_{L^\infty(\Omega)} \|u\|_{2,\Omega} + \|\lambda\|_{1,\Omega}), \quad (6) $$

where the constant $C$ does not depend on $h$.

**Proof:** We start from the extended Poincaré inequality that implies

$$ c\|u - u_h\|_h^2 \leq |u - u_h|_h^2 = (\nabla u, \nabla u)_h + (\nabla u_h, \nabla u_h)_h - (\nabla u, \nabla u_h)_h - (\nabla u_h, \nabla u)_h $$

where $c > 0$ is the ellipticity constant. We take $v \in K$, $v_h \in K_h^{CR}$, then we use (1) and (3):

$$ (\nabla u, \nabla u)_h + (\nabla u_h, \nabla u_h)_h \leq (\nabla u, \nabla v)_h - L(v - u) + (\nabla u_h, \nabla v_h)_h - L(v_h - u_h). $$

We rearrange the terms and get: $\forall v \in K, \forall v_h \in K_h^{CR}$

$$ c\|u - u_h\|_h^2 \leq \underbrace{(\nabla(u - u_h), \nabla(u - v_h))}_\infty + \underbrace{(\nabla u, \nabla(v_h - u))}_\infty - L(v_h - u) + \underbrace{(\nabla u, \nabla(v - u_h))}_\infty - L(v - u_h). \quad (7) $$

For the first expression we use Cauchy-Schwarz and Young inequalities:

$$ \mathcal{E}_1 \leq \frac{c}{4} \|u - u_h\|_h^2 + C \|u - v_h\|_h^2, \quad (8) $$

with $C > 0$. Now for the expression $\mathcal{E}_2$, thanks to Lemma 4.1, we can take $v_h = \mathcal{I}_h^{CR}u \in K_h^{CR}$ in (7). First we bound

$$ \begin{aligned} & (\nabla u, \nabla (\mathcal{I}_h^{CR} u - u))_h \\ & = \sum_{T \in \mathcal{T}_h} \int_T \nabla u \cdot (\Pi_T \nabla u - \nabla u) = \sum_{T \in \mathcal{T}_h} \int_T (\nabla u - \Pi_T \nabla u) \cdot (\Pi_T \nabla u - \nabla u) \leq Ch^2 \|u\|_{2,\Omega}^2, \end{aligned} $$

where we used again the property (4), the projection property of $\Pi_T$ followed by its approximation properties [14, Theorem 18.16]. The term $\int_{\Omega} f(\mathcal{I}_h^{CR}u - u)$ is bounded similarly thanks to Cauchy-Schwarz inequality and the $L^2$-approximation property of the Crouzeix-Raviart interpolation operator [15, Lemma 36.1]. This yields

$$ \mathcal{E}_2 \leq Ch^2 (\|f\|_{0,\Omega} \|u\|_{2,\Omega} + \|u\|_{2,\Omega}^2). \quad (9) $$

Finally, for the expression $\mathcal{E}_3$, we take $v = u \in K$ in (7), we use (2) followed by integration by parts on each triangle $T$ and obtain:

$$ \begin{aligned} \mathcal{E}_3 &= (\nabla u, \nabla(u - u_h))_h + \int_{\Omega} \operatorname{div}(\lambda \nabla u)(u - u_h) \\ &= ((1 - \lambda)\nabla u, \nabla(u - u_h))_h + \sum_{T \in \mathcal{T}_h} \int_{\partial T} \lambda(\nabla u \cdot n)(u - u_h). \end{aligned} $$
The first term of $\mathcal{E}_3$ is treated as follows, using $0 = -1 + 1$ followed by (2):

$$((1 - \lambda)\nabla u, \nabla(u - u_h))_h = \int_{\Omega} \underbrace{(1 - \lambda)(|\nabla u|^2 - 1)}_{=0} + \int_{\Omega} (1 - \lambda)(1 - \nabla u \cdot \nabla u_h).$$

Since $u \in K$, $u_h \in K_h^{CR}$ we get from the Cauchy-Schwarz inequality (here the same notation holds for the absolute value and for the euclidian norm in $\mathbb{R}^2$) : $|\nabla u \cdot \nabla u_h| \le |\nabla u||\nabla u_h| \le 1 \times 1 = 1$. Hence $(1 - \lambda)(1 - \nabla u \cdot \nabla u_h) \le 0$ on $\Omega$. We conclude that

$$\mathcal{E}_3 \le \sum_{T \in \mathcal{T}_h} \int_{\partial T} \lambda(\nabla u \cdot n)(u - u_h) = \underbrace{\sum_{F \in \mathcal{F}_h} \int_F \lambda(\nabla u \cdot n)[[u - u_h]]}_{\mathcal{E}_J}.$$

We now consider expression $\mathcal{E}_J$: let $F$ be an edge and let $T$ be a triangle having $F$ as an edge. The notation $n$ stands for the unit outward normal to $T$. Let $\bar{\lambda}$ be the average of $\lambda$ on $T$. We rewrite $\mathcal{E}_J$ as a sum of two terms:

$$\mathcal{E}_J := \underbrace{\sum_{F \in \mathcal{F}_h} \int_F \bar{\lambda}(\nabla u \cdot n)[[u - u_h]]}_{\mathcal{E}_J^1} + \underbrace{\sum_{F \in \mathcal{F}_h} \int_F (\lambda - \bar{\lambda})(\nabla u \cdot n)[[u - u_h]]}_{\mathcal{E}_J^2}.$$

We denote $\mathcal{I}_h$ the standard continuous elementwise $\mathbb{P}_1$ Lagrange interpolation operator. Remark that $\nabla(\mathcal{I}_h u)|_T \cdot n$ is constant on $F$. It follows from the definition of $V_h^{CR}$ and $[[u - u_h]] = [[u_h]]$ the following relationship: $\int_F (\nabla(\mathcal{I}_h u)|_T \cdot n)[[u - u_h]] = 0$. So there holds

$$|\mathcal{E}_J^1| = \left| \sum_{F \in \mathcal{F}_h} \int_F \bar{\lambda}(\nabla u - \nabla(\mathcal{I}_h u)) \cdot n[[u - u_h]] \right| \le \|\lambda\|_{L^\infty(\Omega)} \sum_{F \in \mathcal{F}_h} \|\nabla(u - \mathcal{I}_h u)\|_{0,F} \|[[u - u_h]]\|_{0,F}.$$

From the multiplicative trace inequality (see, e.g., [14, section 12.3]) we get the scaled trace inequality which gives :

$$\|\nabla(u - \mathcal{I}_h u)\|_{0,F} \le C(h_T^{-1/2}\|\nabla(u - \mathcal{I}_h u)\|_{0,T} + h_T^{1/2}\|u\|_{2,T}) \le Ch_T^{1/2}\|u\|_{2,T},$$

where we use that $\nabla(\mathcal{I}_h u)$ is constant on $T$. We then sum on all the edges ; from the approximation bound on the jump term [15, Lemma 36.9] and the Cauchy-Schwarz and Young inequalities, we conclude

$$|\mathcal{E}_J^1| \le \frac{c}{8} \|\nabla(u - u_h)\|_{0,\Omega}^2 + Ch^2 \|\lambda\|_{L^\infty(\Omega)}^2 \|u\|_{2,\Omega}^2.$$

The term $\mathcal{E}_J^2$ is less usual. We use the inequality $|\nabla u| \le 1$ and again the scaled trace inequality $\|\lambda - \bar{\lambda}\|_{0,F} \le C(h_T^{-1/2}\|\lambda - \bar{\lambda}\|_{0,T} + h_T^{1/2}\|\nabla\lambda\|_{0,T}) \le Ch_T^{1/2}\|\lambda\|_{1,T}$ (here we need the assumption $\lambda \in H^1(\Omega)$ to obtain an optimal estimate). We get $|\nabla u \cdot n||\lambda - \bar{\lambda}\|_{0,F} \le Ch_T^{1/2}\|\lambda\|_{1,T}$. We apply again the approximation bound on the jump term [15, Lemma 36.9] and the Cauchy-Schwarz and Young inequalities. This yields

$$|\mathcal{E}_J^2| \le \frac{c}{8} \|\nabla(u - u_h)\|_{0,\Omega}^2 + Ch^2 \|\lambda\|_{1,\Omega}^2.$$
As a result we bound

$$ \mathcal{E}_3 \le \frac{c}{4} \|\nabla(u - u_h)\|_{0,\Omega}^2 + Ch^2 \left( \|\lambda\|_{L^\infty(\Omega)}^2 \|u\|_{2,\Omega}^2 + \|\lambda\|_{1,\Omega}^2 \right). \quad (10) $$

We conclude by inserting (8)–(9)–(10) into (7) and by using [15, Lemma 36.1]:

$$ \frac{c}{2} \|u - u_h\|_{h}^{2} \leq Ch^{2} \left( \|f\|_{0,\Omega} \|u\|_{2,\Omega} + \|u\|_{2,\Omega}^{2} + \|\lambda\|_{L^{\infty}(\Omega)}^{2} \|u\|_{2,\Omega}^{2} + \|\lambda\|_{1,\Omega}^{2} \right). $$

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