The quasistatic brittle fracture model proposed by G. Francfort and J.-J. Marigo can be Γ-approximated at each time evolution step by the Ambrosio-Tortorelli functional. In this paper, we focus on a modification of this functional which includes additional constraints via penalty terms to enforce the irreversibility of the fracture as well as the applied displacement field. Secondly, we build on this variational model an adapted discretization to numerically compute the time-evolving minimizing solution. We present the derivation of a novel a posteriori error estimator driving the anisotropic adaptive procedure. The main properties of these automatically generated meshes are to be very fine and strongly anisotropic in a very thin neighborhood of the crack, but only far away from the crack tip, while they show a highly isotropic behavior in a neighborhood of the crack tip instead. As a consequence of these properties, the resulting discretizations follow very closely the propagation of the fracture, which is not significantly influenced by the discretization itself, delivering a physically sound prediction of the crack path, with a reasonable computational effort. In fact, we provide numerical tests which assess the balance between accuracy and complexity of the algorithm. We compare our results with isotropic mesh adaptation and we highlight the remarkable improvements both in terms of accuracy and computational cost with respect to simulations in the pertinent most recent literature.

Motivated by variational models in continuum mechanics, we introduce a novel algorithm to perform nonsmooth and nonconvex minimizations with linear constraints in Euclidean spaces. We show how this algorithm is actually a natural generalization of the well-known non-stationary augmented Lagrangian method for convex optimization. The relevant features of this approach are its applicability to a large variety of nonsmooth and nonconvex objective functions, its guaranteed convergence to critical points of the objective energy independently of the choice of the initial value, and its simplicity of implementation. In fact, the algorithm results in a nested double loop iteration. In the inner loop an augmented Lagrangian algorithm performs an adaptive finite number of iterations on a fixed quadratic and strictly convex perturbation of the objective energy, depending on a parameter which is adapted by the external loop. To show the versatility of this new algorithm, we exemplify how it can be used for computing critical points in inverse free-discontinuity variational models, such as the Mumford-Shah functional, and, by doing so, we also derive and analyze new iterative thresholding algorithms.

We present a model for biological network formation originally introduced by Cai and Hu [Adaptation and optimization of biological transport networks, Phys. Rev. Lett. 111 (2013) 138701]. The modeling of fluid transportation (e.g., leaf venation and angiogenesis) and ion transportation networks (e.g., neural networks) is explained in detail and basic analytical features like the gradient flow structure of the fluid transportation network model and the impact of the model parameters on the geometry and topology of network formation are analyzed. We also present a numerical finite-element based discretization scheme and discuss sample cases of network formation simulations.

We introduce a novel constructive approach to define time evolution of critical points of an energy functional. Our procedure, which is different from other more established approaches based on viscosity approximations in infinite dimension, is prone to efficient and consistent numerical implementations, and allows for an existence proof under very general assumptions. We consider in particular rather nonsmooth and nonconvex energy functionals, provided the domain of the energy is finite dimensional. Nevertheless, in the infinite dimensional case study of a cohesive fracture model, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evolution can be indeed recovered as a limit of evolutions of critical points of finite dimensional discretizations of the energy, constructed according to our scheme. To illustrate the results, we provide several numerical experiments both in one and two dimensions. These agree with the crack initiation criterion, which states that a fracture appears only when the stress overcomes a certain threshold, depending on the material.Moreover, we say that a measurable function v : [0, T ] → E is an approximable quasistatic evolution with initial condition v 0 and constraint f , if for every t ∈ [0, T ] there exists a sequence δ k → 0 + and a sequence (v δ k ) k∈N of discrete quasistatic evolutions with time step δ k , initial condition v 0 , and constraint f , such thatWe are now ready to state the main results of the paper (see Theorem 2.15 and Theorem 2.16).Dissipative systems. (0)). Under suitable assumptions on J, ψ, A, and f (see Theorem 2.15) we prove that thereis an approximable quasistatic evolution with initial condition v 0 and constraint f ;(C) Energy inequality: the function s → q(s),ḟ (s) F belongs to L 1 (0, T ) andis defined by (2.9) and ·, · F denotes the duality product in F.Any function v ∈ BV ([0, T ]; E) satisfying (A), (B), and (C) is a rate independent evolution. We also remark that, as a consequence of the local stability (B), the energy inequality becomes an equality in all the nontrivial intervals where the solution v(t) is absolutely continuous (see Theorem 2.18, where also the nondissipative case is treated). In addition, in such intervals the doubly nonlinear inclusion A * q(t) ∈ ∂J(v(t)) + ∂ψ(v(t)) is satisfied. It is however a well-known fact that, due to nonconvexity, our solutions can in general develop time discontinuities, where additional dissipation appears (see [27]). Non dissipative systems. When there is no dissipation we can still prove an existence result, although the evolution obtained is in general expected to be less regular in time. This is due to the fact that the absence of dissipation causes loss of compactness, since simple estimates of the total variation of the approximating solutions are now missing. This is undoubtedly a point of great interest in the analysis of the degenerate case (see also [2] for an abstract approach in the unconstrained setting). To compensate the loss of compactness, we need ...

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