# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

December 2017 , Volume 10 , Issue 6

Issue dedicated to Professor Tomas Roubicek on the occasion of his 60th birthday

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2017, 10(6): ⅰ-ⅲ doi: 10.3934/dcdss.201706i +[Abstract](1285) +[HTML](74) +[PDF](5236.9KB)
Abstract:
2017, 10(6): 1207-1232 doi: 10.3934/dcdss.2017066 +[Abstract](2063) +[HTML](84) +[PDF](590.6KB)
Abstract:

The discretization of a bilaterally constrained total variation minimization problem with conforming low order finite elements is analyzed and three iterative schemes are proposed which differ in the treatment of the non-differentiable terms. Unconditional stability and convergence of the algorithms is addressed, an application to piecewise constant image segmentation is presented and numerical experiments are shown.

2017, 10(6): 1233-1256 doi: 10.3934/dcdss.2017067 +[Abstract](1685) +[HTML](96) +[PDF](524.1KB)
Abstract:

We consider non-dissipative (elastic) rate-type material models that are derived within the Gibbs-potential-based thermodynamic framework. Since the absence of any dissipative mechanism in the model prevents us from establishing even a local-in-time existence result in two spatial dimensions for a spatially periodic problem, we propose two regularisations. For such regularized problems we obtain well-posedness of the planar, spatially periodic problem. In contrast with existing results, we prove ours for a regularizing term present solely in the evolution equation for the stress.

2017, 10(6): 1257-1280 doi: 10.3934/dcdss.2017068 +[Abstract](1317) +[HTML](64) +[PDF](528.3KB)
Abstract:

We address the analysis of the Souza-Auricchio model for shape-memory alloys in the finite-strain setting. The model is formulated in variational terms and the existence of quasistatic evolutions is obtained within the classical frame of energetic solvability. The finite-strain model is proved to converge to its small-strain counterpart for small deformations via a variational convergence argument.

2017, 10(6): 1281-1301 doi: 10.3934/dcdss.2017069 +[Abstract](2007) +[HTML](97) +[PDF](651.4KB)
Abstract:

This paper deals with shape optimization of systems governed by the Stokes flow with threshold slip boundary conditions. The stability of solutions to the state problem with respect to a class of domains is studied. For computational purposes the slip term and impermeability condition are handled by a regularization. To get a finite dimensional optimization problem, the optimized part of the boundary is described by Bézier polynomials. Numerical examples illustrate the computational efficiency.

2017, 10(6): 1303-1327 doi: 10.3934/dcdss.2017070 +[Abstract](1704) +[HTML](78) +[PDF](5095.1KB)
Abstract:

We study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period \begin{document}$\varepsilon$\end{document}. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit \begin{document}$\varepsilon \to 0$\end{document} and show that the effective dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable equi-continuity of the solutions in the limit \begin{document}$\varepsilon \to 0$\end{document}.

2017, 10(6): 1329-1350 doi: 10.3934/dcdss.2017071 +[Abstract](1715) +[HTML](62) +[PDF](502.9KB)
Abstract:

We formulate an inverse optimal design problem as a Mathematical Programming problem with Equilibrium Constraints (MPEC). The equilibrium constraints are in the form of a second-order conic optimization problem. Using the so-called Implicit Programming technique, we reformulate the bilevel optimization problem as a single-level nonsmooth nonconvex problem. The major part of the article is devoted to the computation of a subgradient of the resulting composite objective function. The article is concluded by numerical examples demonstrating, for the first time, that the Implicit Programming technique can be efficiently used in the numerical solution of MPECs with conic constraints on the lower level.

2017, 10(6): 1351-1373 doi: 10.3934/dcdss.2017072 +[Abstract](1642) +[HTML](82) +[PDF](997.0KB)
Abstract:

Recently, a thermodynamically consistent non-linear constitutive equation has been developed to describe the large deformation cyclic response of viscoelastic polyamides (see [17]). In this paper, two boundary value problems within the context of the above model, namely the stress relaxation of a right circular annular cylinder subject to twisting, and the inflation of a sphere are studied. In addition to solving the above problems numerically, investigation of the merits and pitfalls of studying the same boundary value problem for a special class of inhomogeneous body and its homogenized counterpart is undertaken. This study finds that for moderate strains the differences in relaxation time between the actual inhomogeneous body and its homogenized counterparts may be significant.

2017, 10(6): 1375-1391 doi: 10.3934/dcdss.2017073 +[Abstract](1560) +[HTML](93) +[PDF](594.3KB)
Abstract:

An optimal control problem is studied for a quasilinear Maxwell equation of nondegenerate parabolic type. Well-posedness of the quasilinear state equation, existence of an optimal control, and weak Gâteaux-differentiability of the control-to-state mapping are proved. Based on these results, first-order necessary optimality conditions and an associated adjoint calculus are derived.

2017, 10(6): 1393-1411 doi: 10.3934/dcdss.2017074 +[Abstract](1563) +[HTML](67) +[PDF](437.6KB)
Abstract:

The formulation of balance laws in continuum and statistical mechanics is expounded in forms that open the way to revise and review the correspondence instituted, in a manner proposed by Irving and Kirkwood in 1950 and improved by Noll in 1955 and 2010, between the basic balance laws of Cauchy continua and those of standard Hamiltonian systems of particles.

2017, 10(6): 1413-1466 doi: 10.3934/dcdss.2017075 +[Abstract](1290) +[HTML](89) +[PDF](894.8KB)
Abstract:

In this paper we address a model coupling viscoplasticity with damage in thermoviscoelasticity. The associated PDE system consists of the momentum balance with viscosity and inertia for the displacement variable, at small strains, of the plastic and damage flow rules, and of the heat equation. It has a strongly nonlinear character and in particular features quadratic terms on the right-hand side of the heat equation and of the damage flow rule, which have to be handled carefully. We propose two weak solution concepts for the related initial-boundary value problem, namely 'entropic' and 'weak energy' solutions. Accordingly, we prove two existence results by passing to the limit in a carefully devised time discretization scheme. Finally, in the case of a prescribed temperature profile, and under a strongly simplifying condition, we provide a continuous dependence result, yielding uniqueness of weak energy solutions.

2017, 10(6): 1467-1485 doi: 10.3934/dcdss.2017076 +[Abstract](2144) +[HTML](84) +[PDF](602.5KB)
Abstract:

We study the optimal control of a rate-independent system that is driven by a convex quadratic energy. Since the associated solution mapping is non-smooth, the analysis of such control problems is challenging. In order to derive optimality conditions, we study the regularization of the problem via a smoothing of the dissipation potential and via the addition of some viscosity. The resulting regularized optimal control problem is analyzed. By driving the regularization parameter to zero, we obtain a necessary optimality condition for the original, non-smooth problem.

2017, 10(6): 1487-1517 doi: 10.3934/dcdss.2017077 +[Abstract](1984) +[HTML](97) +[PDF](543.1KB)
Abstract:

We study a model for the rate-independent evolution of cohesive zone delamination in a visco-elastic solid, also exposed to dynamics effects. The main feature of this model, inspired by [32], is that the surface energy related to the crack opening depends on the history of the crack separation between the two sides of the crack path, and allows for different responses upon loading and unloading.

Due to the presence of multivalued and unbounded operators featuring non-penetration and the 'memory'-constraint in the strong formulation of the problem, we prove existence of a weaker notion of solution, known as semistable energetic solution, pioneered in [41] and refined in [38].

2017, 10(6): 1519-1537 doi: 10.3934/dcdss.2017078 +[Abstract](1649) +[HTML](69) +[PDF](486.0KB)
Abstract:

We illustrate an alternative derivation of the viscous regularization of a nonlinear forward-backward diffusion equation which was studied in [A. Novick-Cohen and R. L. Pego. Trans. Amer. Math. Soc., 324:331-351]. We propose and discuss a new ''non-smooth'' variant of the viscous regularization and we offer an heuristic argument that indicates that this variant should display interesting hysteretic effects. Finally, we offer a constructive proof of existence of solutions for the viscous regularization based on a suitable approximation scheme.

2017, 10(6): 1539-1561 doi: 10.3934/dcdss.2017079 +[Abstract](2602) +[HTML](90) +[PDF](1482.7KB)
Abstract:

A new quasi-static and energy based formulation of an interface damage model which provides interface traction-relative displacement laws like in traditional trilinear (with bilinear softening) or generally multilinear cohesive zone models frequently used by engineers is presented. This cohesive type response of the interface may represent the behaviour of a thin adhesive layer. The level of interface adhesion or damage is defined by several scalar variables suitably defined to obtain the required traction-relative displacement laws. The weak solution of the problem is sought numerically by a semi-implicit time-stepping procedure which uses recursive double minimization in displacements and damage variables separately. The symmetric Galerkin boundary-element method is applied for the spatial discretization. Sequential quadratic programming is implemented to resolve each partial minimization in the recursive scheme applied to compute the time-space discretized solutions. Sample 2D numerical examples demonstrate applicability of the proposed model.

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