Convex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations

Abstract

We study convex integration solutions in the context of the modelling of shape-memory alloys. The purpose of the article is twofold, treating both rigidity and flexibility properties: Firstly, we relate the maximal regularity of convex integration solutions to the presence of lower bounds in variational models with surface energy. Hence, variational models with surface energy could be viewed as a selection mechanism allowing for or excluding convex integration solutions. Secondly, we present the first numerical implementations of convex integration schemes for the model problem of the geometrically linearised two-dimensional hexagonal-to-rhombic phase transformation. We discuss and compare the two algorithms from Rüland et al. (J Elast. 2019. https://doi.org/10.1007/s10659-018-09719-3; SIAM J Math Anal 50(4):3791–3841, 2018) and give a numerical estimate of the regularity attained.

Appendices

Fractional Poincaré Inequalities

In this last section, for self-containedness, we recall a possible proof of the fractional Poincaré inequality (for fractional Besov-type spaces), which is used in different forms in Sect. 2. We start with the $L^2$-based version:

Lemma A.1

Let $s\in (0,1)$ and $n\ge 1$. Let $\Omega \subset \mathbb {R}^n$ be an open, bounded $C^{1,1}$ domain. Let $u\in H^{s}(\mathbb {R}^n)$ with $\text {supp}(u) \subset \overline{\Omega }$ and denote $\Omega _{\delta }=\{x\in \Omega : {{\,\mathrm{dist}\,}}(x,\partial \Omega )\ge \delta \}$ for $\delta >0$. Then, there exists $C=C(s,\Omega ,n)>1$ such that

$$\begin{aligned} \Vert u\Vert _{L^2(\Omega _{\delta }{\setminus } \Omega _{2\delta })} \le C \delta ^{s} \Vert u\Vert _{H^{s}(\mathbb {R}^n)}. \end{aligned}$$

(70)

Proof

We use the Caffarelli–Silvestre extension $\overline{u}$ of u (Caffarelli and Silvestre 2007), which is the $H^{1}(\mathbb {R}^{n+1}_+, x_{n+1}^{1-2s})$ solution to the equation

$$\begin{aligned} \nabla \cdot x_{n+1}^{1-2s} \nabla \overline{u}&= 0 \text{ in } \mathbb {R}^{n+1}_+,\\ \overline{u}&= u \text{ on } \mathbb {R}^n \times \{0\}. \end{aligned}$$

For $x_1 \in \Omega _{\delta } {\setminus } \Omega _{2\delta }$, $x_2=p(x_1) \in \mathbb {R}^n {\setminus } \overline{\Omega }$ with $|x_1-x_2|\le 4 \delta$ and $\tilde{\delta } \in [\delta , 2\delta ]$, the fundamental theorem and Hölder’s inequality yield

$$\begin{aligned} |u(x_1)|^2&= |\overline{u}(x_1,0)|^2 \le 2|\overline{u}(x_1,\tilde{\delta })|^2 + 2 \left( \int \limits _{0}^{\tilde{\delta }}|\partial _t \overline{u}(x_1,t)| \mathrm{d}t \right) ^2\\&\le 2|\overline{u}(x_1,\tilde{\delta })|^2 + C \tilde{\delta }^{2s}\int \limits _{0}^{\tilde{\delta }} t^{1-2s}|\partial _t \overline{u}(x_1,t)|^2 \mathrm{d}t\\&\le 2 |\overline{u}(p(x_1),\tilde{\delta })|^2 + C \tilde{\delta }^2 \int \limits _{0}^{1}|\nabla ' \overline{u}(\tau x_1 + (1-\tau )p(x_1), \tilde{\delta })|^2 \mathrm{d}\tau \\&\quad + C \tilde{\delta }^{2s}\int \limits _{0}^{\tilde{\delta }} t^{1-2s}|\partial _t \overline{u}(x_1,t)|^2 \mathrm{d}t\\&\le 2\tilde{\delta }^2 \int \limits _{0}^{1}|\nabla ' \overline{u}(\tau x_1 + (1-\tau )p(x_1), \tilde{\delta })|^2 \mathrm{d}\tau + C \tilde{\delta }^{2s}\int \limits _{0}^{\tilde{\delta }} t^{1-2s}|\partial _t \overline{u}(x_1,t)|^2 \mathrm{d}t\\&\quad + C \tilde{\delta }^{2s}\int \limits _{0}^{\tilde{\delta }} t^{1-2s}|\partial _t \overline{u}(p(x_1),t)|^2 \mathrm{d}t. \end{aligned}$$

Here the constant $C>1$ changes from line to line and depends on s but not on $\delta$. In deriving the above estimate, we applied the fundamental theorem thrice: first in the normal direction (where we then used Hölder’s inequality to insert the weight $t^{1-2s}$), then in the tangential directions and finally once more in the normal direction (where we used the vanishing Dirichlet data for $u(p(x_1))=\overline{u}(p(x_1),0)$). Estimating the integrals involving the normal derivative by using $\delta <\tilde{\delta }\le 2\delta$, we thus infer

$$\begin{aligned} |u(x_1)|^2&\le 2\tilde{\delta }^2 \int \limits _{0}^{1}|\nabla ' \overline{u}(\tau x_1 + (1-\tau )p(x_1), \tilde{\delta })|^2 \mathrm{d}\tau + C \delta ^{2s}\int \limits _{0}^{2\delta } t^{1-2s}|\partial _t \overline{u}(x_1,t)|^2 \mathrm{d}t\\&\quad + C \delta ^{2s}\int \limits _{0}^{2\delta } t^{1-2s}|\partial _t \overline{u}(p(x_1),t)|^2 \mathrm{d}t. \end{aligned}$$

Averaging over $\tilde{\delta } \in [\delta , 2\delta ]$ yields

$$\begin{aligned} |u(x_1)|^2&\le 2\delta ^{-1} \int \limits _{\delta }^{2\delta } \tilde{\delta }^2 \int \limits _{0}^{1}|\nabla ' \overline{u}(\tau x_1 + (1-\tau )p(x_1), \tilde{\delta })|^2 \mathrm{d}\tau \mathrm{d} \tilde{\delta }\\&\quad + C \delta ^{2s}\int \limits _{0}^{2\delta } t^{1-2s}|\partial _t \overline{u}(x_1,t)|^2 \mathrm{d}t + C \delta ^{2s}\int \limits _{0}^{2\delta } t^{1-2s}|\partial _t \overline{u}(p(x_1),t)|^2 \mathrm{d}t\\&= 2\delta ^{-1} \int \limits _{\delta }^{2\delta } \tilde{\delta }^{1+2s} \tilde{\delta }^{1-2s} \int \limits _{0}^{1}|\nabla ' \overline{u}(\tau x_1 + (1-\tau )p(x_1), \tilde{\delta })|^2 \mathrm{d}\tau \mathrm{d} \tilde{\delta }\\&\quad + C \delta ^{2s}\int \limits _{0}^{2\delta } t^{1-2s}|\partial _t \overline{u}(x_1,t)|^2 \mathrm{d}t + C \delta ^{2s}\int \limits _{0}^{2\delta } t^{1-2s}|\partial _t \overline{u}(p(x_1),t)|^2 \mathrm{d}t\\&\le C \delta ^{-1} \delta ^{1+2s} \int \limits _{\delta }^{2\delta } \tilde{\delta }^{1-2s} \int \limits _{0}^{1}|\nabla ' \overline{u}(\tau x_1 + (1-\tau )p(x_1), \tilde{\delta })|^2 \mathrm{d}\tau \mathrm{d} \tilde{\delta }\\&\quad + C \delta ^{2s}\int \limits _{0}^{2\delta } t^{1-2s}|\partial _t \overline{u}(x_1,t)|^2 \mathrm{d}t + C \delta ^{2s}\int \limits _{0}^{2\delta } t^{1-2s}|\partial _t \overline{u}(p(x_1),t)|^2 \mathrm{d}t\\&\le C \delta ^{2s} \int \limits _{0}^{2\delta } \int \limits _{0}^{1} t^{1-2s} |\nabla ' \overline{u}(\tau x_1 + (1-\tau )p(x_1), t)|^2 \mathrm{d}\tau \mathrm{d}t\\&\quad + C \delta ^{2s}\int \limits _{0}^{2\delta } t^{1-2s}|\partial _t \overline{u}(x_1,t)|^2 \mathrm{d}t + C \delta ^{2s}\int \limits _{0}^{2\delta } t^{1-2s}|\partial _t \overline{u}(p(x_1),t)|^2 \mathrm{d}t. \end{aligned}$$

Finally, integrating over $x_1 \in \Omega _{\delta } {\setminus } \Omega _{2\delta }$ leads to

$$\begin{aligned} \Vert u\Vert _{L^2(\Omega _{\delta }{\setminus } \Omega _{2\delta })}^2&\le C \delta ^{2s} \int \limits _{0}^1 \int \limits _{0}^{2\delta } t^{1-2s} \Vert \nabla ' \overline{u}(\tau \cdot + (1-\tau )p(\cdot ),t)\Vert _{L^2(\Omega _{\delta }{\setminus } \Omega _{2\delta })}^2 \mathrm{d}t \mathrm{d}\tau \\&\quad + 2C \delta ^{2s} \Vert t^{\frac{1-2s}{2}} \partial _t \overline{u}\Vert _{L^2((\Omega _{\delta } {\setminus } \Omega _{2\delta }) \times [0,2\delta ])}^2\\&\le C \delta ^{2s} \Vert t^{\frac{1-2s}{2}} \nabla \overline{u}\Vert _{L^2(\Omega \times [0,2\delta ])}^2\\&\le C \delta ^{2s} \Vert t^{\frac{1-2s}{2}} \nabla \overline{u}\Vert _{L^2(\mathbb {R}^{n+1}_+)}^2. \end{aligned}$$

Hence, using that [see Caffarelli and Silvestre (2007)]

$$\begin{aligned} \Vert t^{\frac{1-2s}{2}} \nabla \overline{u}\Vert _{L^2(\mathbb {R}^{n+1}_+)}^2 \le C \Vert u\Vert _{\dot{H}^s(\mathbb {R}^n)}, \end{aligned}$$

we obtain the desired estimate (70). $\square$

Similarly, we have the analogue of this in Sobolev and Besov spaces:

Lemma A.2

Let $s\in (0,1)$, $p\in (1,\infty )$ and $n\ge 1$. Let $\Omega \subset \mathbb {R}^n$ be an open, bounded $C^{1,1}$ domain. Let $u\in W^{s,p}(\mathbb {R}^n)$ with $\text {supp}(u) \subset \overline{\Omega }$ and denote $\Omega _{\delta }=\{x\in \Omega : {{\,\mathrm{dist}\,}}(x,\partial \Omega )\ge \delta \}$ for $\delta >0$. Then, there exists $C=C(s,p,\Omega ,n)>1$ such that

$$\begin{aligned} \Vert u\Vert _{L^p(\Omega _{\delta }{\setminus } \Omega _{2\delta })} \le C \delta ^{s} \Vert u\Vert _{W^{s,p}(\mathbb {R}^n)}. \end{aligned}$$

(71)

Proof

The argument follows essentially as above. However, for the estimate in the normal direction, we use

$$\begin{aligned} |\overline{u}(x_1,0)|^p&\le C |\overline{u}(x_1,\delta )|^p + C\left( \int \limits _{0}^{\delta } \partial _t \overline{u} \mathrm{d}t\right) ^{p}\\&\le C|\overline{u}(x_1,\delta )|^p + C\int \limits _{0}^{\delta } |t^{1-\frac{1}{p}-s }\partial _t \overline{u}|^p \mathrm{d}t \left( \int \limits _{0}^{\delta } t^{-1+\frac{sp}{p-1}}\mathrm{d}t \right) ^{p-1}\\&\le C|\overline{u}(x_1,\delta )|^p + C\delta ^{sp} \int \limits _{0}^{\delta } t^{p-1-s p}|\partial _t \overline{u}|^p \mathrm{d}t . \end{aligned}$$

Combining this with estimates in the tangential directions similarly as in the proof of Lemma A.2, we obtain

$$\begin{aligned} \Vert u\Vert _{L^p(\Omega _{\delta }{\setminus } \Omega _{2\delta })}&\le C \delta ^{s} \Vert t^{1-\frac{1}{p}-s}\nabla \overline{u}\Vert _{L^p(\mathbb {R}^{n+1}_+)} \le C \delta ^{s} \Vert u\Vert _{W^{s,p}(\mathbb {R}^n)}. \end{aligned}$$

Here we used the trace characterisation of $W^{s,p}(\mathbb {R}^n)$, see Lenzmann and Schikorra (2016, Section 10), where the authors rely on the characterisation from Bui and Candy (2015). $\square$

A similar argument using the characterisation of Besov spaces from Lenzmann and Schikorra (2016, Section 10) [where the authors again rely on the characterisation from Bui and Candy (2015)] also yields a similar Poincaré estimate in Besov spaces.

The Pseudocode of Convex Integration Algorithm

In order to allow for a self-contained and more precise understanding of the algorithms used in generating the pictures from above, we include a pseudocode variant of the code which we have used. In the sequel, we make use of the convention that the ith component of a function f is described by f[i].

Algorithm B.1

Let $M \in \mathcal {M}:=\{N \in \mathbb {R}^{2\times 2}: \ e(N)\in {{\,\mathrm{intconv}\,}}(\{e^{(1)}, e^{(2)}, e^{(3)}\})\}$, where $e^{(1)}, e^{(2)}, e^{(3)}$ are as in (68). Let $\Omega \subset \mathbb {R}^2$ be a triangle.

• (1a) Variables. We consider

  • the displacement $u_k: \Omega \rightarrow \mathbb {R}^2$ at step k,

  • a collection of (up to null-sets disjoint) triangles $\widehat{\Omega }_k = \{\Omega _1^k,\dots ,\Omega _{j_k}^k\}$, which cover $\Omega$,

  • and the error in matrix space $\epsilon _k: \widehat{\Omega }_k \rightarrow (0,1)$ at step k, which is constant on each subset of $\widehat{\Omega }_k$.

• (1b) Functions. We consider a “covering function”, which covers a given triangle by “good sets”, on which the deformation $u_k$ will be improved, and a “remainder”. More precisely,

  $$\begin{aligned} {{\,\mathrm{Cover}\,}}_{v}:&\mathcal {T} \times \mathcal {M} \rightarrow \mathcal {R} \times \mathcal {T},\\&(T,M) \mapsto (\{Q_1,\dots ,Q_{j(T,M)}\} , \{T_1,\dots , T_{l(T,M)}\}). \end{aligned}$$

  Here $\mathcal {T}$ is the set of all triangles, $\mathcal {R}$ is a set of certain quadrilaterals [these differ in Rüland et al. 2019, 2018] and $\mathcal {M}:=\{M\in \mathbb {R}^{2\times 2}: \ e(M) \in {{\,\mathrm{intconv}\,}}(\{e^{(1)}, e^{(2)}, e^{(3)}\}) \}$. For each $T \in \mathcal {T}$ and $M \in \mathcal {M}$ we have an (up to null-sets) disjoint covering

  $$\begin{aligned} T = \bigcup \limits _{Q \in {{\,\mathrm{Cover}\,}}_{v}(T,M)[1]} Q \cup \bigcup \limits _{\tilde{T}\in {{\,\mathrm{Cover}\,}}_{v}(T,M)[2]}\tilde{T}, \ \end{aligned}$$

  with

  $$\begin{aligned} \frac{\left| \bigcup \limits _{Q \in {{\,\mathrm{Cover}\,}}_{v}(T,M)[1]} Q \right| }{|T|} \ge v. \end{aligned}$$

  The sets $Q \in {{\,\mathrm{Cover}\,}}_v(T,M)[1]$ are the “good sets”, on which the current displacement gradient $\nabla u_k$ will be modified and pushed towards the energy wells. The sets $\tilde{T} \in {{\,\mathrm{Cover}\,}}_v(T,M)[2]$ are the “remainders” on which the displacement $u_k$ is not changed.

  We further consider a “replacement function”, which improves the current displacement gradient in the sense that the replaced deformation gradient is closer to the wells (or at least closer to the wells on a large portion of the domain). This depends on the current displacement gradient, the underlying domain and the error in matrix space. As an output it yields

  • the level sets (in the form of a finite collection of triangles) of the new improved deformation,

  • a piecewise affine function whose symmetric gradient attains values in ${{\,\mathrm{intconv}\,}}(\{e^{(1)}, e^{(2)}, e^{(3)}\})$,

  • and an updated error in matrix space.

  More precisely,

  $$\begin{aligned} {{\,\mathrm{Replace}\,}}:&\mathcal {R} \times \mathcal {A}_{{{\,\mathrm{intconv}\,}}(\{e^{(1)}, e^{(2)}, e^{(3)}\})} \times (0,1) \\&\quad \rightarrow \mathcal {T} \times \mathcal {A}_{{{\,\mathrm{intconv}\,}}(\{e^{(1)}, e^{(2)}, e^{(3)}\})} \times (0,1),\\&\quad (Q,w_{\text {old}},\epsilon ) \mapsto (\{T_{1},\dots ,T_{j_0}\} , w, \tilde{\epsilon }) \end{aligned}$$

  Here $\mathcal {A}_{{{\,\mathrm{intconv}\,}}(\{e^{(1)}, e^{(2)}, e^{(3)}\})}$ denotes the set of piecewise affine deformations with symmetric gradients in ${{\,\mathrm{intconv}\,}}\{e^{(1)},e^{(2)},e^{(3)}\}$. Furthermore, $w|_{\partial Q}=w_{old}|_{\partial Q}$.

• (2) Initialisation. We begin by setting $u_0(x)=Mx$, $\epsilon _0 = \epsilon _0(M)$, $\widehat{\Omega }_0 = \{\Omega \}$.

• (3) Iteration step. The algorithm proceeds iteratively: Assume that $u_k$, $\epsilon _k$ and $\widehat{\Omega }_k$ with $k\ge 0$ are already given. Then on each $\Omega _{j}^k \in \widehat{\Omega }_k$ for which $e(\nabla u_k)|_{\Omega _j^k} \notin \{e^{(1)}, e^{(2)}, e^{(3)}\}$, apply the function ${{\,\mathrm{Cover}\,}}_v(\Omega _{j}^k, \nabla u_k|_{\Omega _j^k})$. Let ${{\,\mathrm{Cover}\,}}_v(\Omega _{j}^k,\nabla u_k|_{\Omega _j^k})[1]= \{\Omega _{j,1}^k,\dots ,\Omega _{j,m(j,k)}^k\}$.

  For each $\Omega _{j,m}^k \in {{\,\mathrm{Cover}\,}}_v(\Omega _j^k,\nabla u_k|_{\Omega _j^k})[1]$ apply the function ${{\,\mathrm{Replace}\,}}(\Omega _{j,m}^k, u_k |_{\Omega _{j}^k},\epsilon _k)$. This yields

  • (i) an up to null-sets disjoint covering of $\Omega _{j,m}^k$ into triangles

    $$\begin{aligned} \{\Omega _{j,m,1}^k,\dots ,\Omega _{j,m,l(j,m,k)}^k\}:={{\,\mathrm{Replace}\,}}(\Omega _{j,m}^k, u_k|_{\Omega _{j}^k},\epsilon _k)[1]; \end{aligned}$$

  • (ii) a function $v_{j,m,k}:={{\,\mathrm{Replace}\,}}(\Omega _{j,m}^k, u_k|_{\Omega _{j}^k},\epsilon _k)[2]: \Omega _{j,m}^k \rightarrow \mathbb {R}^2$ whose gradient is constant on each of the sets $\Omega _{j,m,l}^k$ with $l\in \{1,\dots , l(j,m,k)\}$ and for which $v_{j,m,k}(x)=u_k(x) \text{ for } \text{ all } x \in \partial \Omega _{j,m}^k$;

  • (iii) a parameter $\tilde{\epsilon }_{j,m,k}:={{\,\mathrm{Replace}\,}}(\Omega _{j,m}^k, u_k |_{\Omega _{j}^k},\epsilon _k)[3]$.

  We then set

  $$\begin{aligned} u_{k+1}|_{\Omega _{j,m}^k}&=v_{j,m,k} \text{ and } u_{k+1}|_{{{\,\mathrm{Cover}\,}}_v(\Omega _j^k,\nabla u_k|_{\Omega _j^k})[2]}=u_k|_{{{\,\mathrm{Cover}\,}}_v(\Omega _j^k,\nabla u_k|_{\Omega _j^k})[2]},\\ \widetilde{\Omega }_{k+1}&= \bigcup \limits _{j,m} {{\,\mathrm{Replace}\,}}(\Omega _{j,m}^k, u|_{\Omega _{j}^k},\epsilon _k)[1] \cup \bigcup \limits _{j} {{\,\mathrm{Cover}\,}}_v(\Omega _j^k,\nabla u_k|_{\Omega _j^k})[2],\\ \epsilon _{k+1}|_{T}&= \left\{ \begin{array}{ll} \tilde{\epsilon }_{j,m,k} &{} \text{ if } T \subset \Omega _{j,m,l}^k \in {{\,\mathrm{Cover}\,}}_v(\Omega _j^k,\nabla u_k|_{\Omega _j^k})[1] \text{ for } \text{ some } j \in \{1,\dots ,j_k\},\\ \epsilon _{k}|_{T} &{} \text{ if } T \in {{\,\mathrm{Cover}\,}}_v(\Omega _j^k,\nabla u_k|_{\Omega _j^k})[2] \text{ for } \text{ some } j \in \{1,\dots ,j_k\}. \end{array} \right. \end{aligned}$$

  If on $\Omega _{j}^k$ we already have $e(\nabla u_k)|_{\Omega _{j}^k} \in \{e^{(1)}, e^{(2)}, e^{(3)}\}$, we set $u_{k+1}|_{\Omega _j^k} = u_{k}|_{\Omega _j^k}$ and

  $$\begin{aligned} \widehat{\Omega }_{k+1}=\widetilde{\Omega }_{k+1}\cup \{\Omega _{j}^{k}\in \widehat{\Omega }_k: \nabla u_k|_{\Omega _{j}^{k}}\in \{e^{(1)}, e^{(2)},e^{(3)}\} \text{ a.e. }\}. \end{aligned}$$

It has been shown in Rüland et al. (2019, 2018) that this procedure is well defined, if the function ${{\,\mathrm{Replace}\,}}$ is chosen appropriately. (There is a slight subtlety in that in those articles we approximate domains by rectangles, but this does not matter for the non-quantitative algorithm which is used here.) As explained above, in spite of their similar overall structure, the algorithms from Rüland et al. (2019, 2018), however, differ substantially in their underlying replacement constructions encoded by the function ${{\,\mathrm{Replace}\,}}$. The function ${{\,\mathrm{Cover}\,}}_v$ is essentially given by a greedy algorithm.