Metodi variazionali nelle scienze dei materiali: l'approccio data-driven

The aim of this master thesis is to provide a summary of the by-now available results in the data-driven approach to elasticity and conductivity problems, with a thorough comparison with its classical counterpart. The present work is mainly based on the paper by S. Conti, S. Müller and M. Ortiz, ``Data-Driven Problems in Elasticity'' and references therein. In general, the two core elements of the prototypical problem in materials sciences are the following: on the one hand, there are constitutive relations, which provide equilibrium and compatibility conditions and which are accepted as invariable. These prescribe, for instance, the balance of forces applied or the local state. On the other, there is the information on the material, obtained as empirical data. The classical and well-established bridge between constitutive relations and the data is the inference of material laws from the latter. Then, such material laws are used to define suitable boundary value problems. However, the possible presence of noisy or incomplete data can lead to error and uncertainty in the formulation of material models, especially in high dimensional phase spaces. For this reason, the data-driven formulation aims to directly relate the boundary value problems with the material data, so avoiding such empirical material modeling. Thus, the modeling process is sidestepped and the cloud of points which arises from the empirical observation is directly used as a set which the solution of the data-driven problem should fit as closely as possible. Rigorously, we consider a phase space $$Z=L^2(\Omega;\R^N)\times L^2(\Omega;\R^N)$$ and then two sets, the first one, $E$, prescribing the universal physical laws, and the second one, $D$, which is described directly through the local empirical measurements $D_{loc}\subseteq \R^{N}$ as $$D:=\{z \in Z \;|\; z(x)\in D_{loc} \text{ a.e. in } \Omega \}.$$ The data-driven solution of the problem, namely $\overline{z}\in Z$, must satisfy the constitutive relations, i.e. $\overline{z}\in E$, and minimize the distance from the data-set $D$. We are therefore led to the analysis of the minimization problem $$\min_{z \in E} \{d(z, D)\},$$ where $d(z, D):=\inf \limits_{y\in D}d(z, y)$ and $d(\cdot,\cdot )$ a suitable metric on $Z$. The thesis also deals with the classical resolution of problems in material sciences. In particular, considerable attention is devoted to the well-known approach based on convex analysis. Indeed, from the classical standpoint, the material problems we are interested in can be described by the minimization of an integral functional of the form $$I(u)=\int_{\Omega} f(x,u(x), \nabla u(x))\, dx$$ over a suitable Sobolev space, i.e. $$\inf_{u\in u_0 + W_0 ^{1,p}} I(u)$$ Such problems are tackled with the Direct Method of the Calculus of Variations, and the main condition for the existence of a minimizer is the weak lower semicontinuity of $I$. This leads naturally to the question of which are the sufficient and necessary conditions on $f$ for $I$ to be weak lower semicontinuous. Whereas, in the scalar case $\Omega \subset \R$ or $N=1$, the convexity of the map $\xi \mapsto f(x,u,\xi)$ is necessary and sufficient, in higher dimension this assumption turns out to be too stringent, being only sufficient and conflicting with modeling considerations. This observation led to the introduction by Charles B. Morrey in 1952 of the notion of quasiconvexity, which is proved to be equivalent to weak lower semicontinuity.