Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis

We derive a thermodynamically consistent general continuum-mechanical model describing mutually coupled martensitic and ferro/paramagnetic phase transformations in electrically-conductive magnetostrictive materials such as NiMnGa. We use small-strain and eddy-current approximations, yet large velocities and electric current injected through the boundary are allowed. Fully nonlinear coupling of magneto-mechanical and thermal effects is considered. The existence of energy-preserving weak solutions is proved by showing convergence of time-discrete approximations constructed by a carefully designed semi-implicit regularized scheme. The research that led to the present paper was partially supported by a grant of the group GNFM of INdA

Coercivity condition for higher moment a priori estimates for nonlinear SPDEs and existence of a solution under local monotonicity

Higher order moment estimates for solutions to nonlinear SPDEs governed by locally-monotone operators are obtained under appropriate coercivity condition. These are then used to extend known existence and uniqueness results for nonlinear SPDEs under local monotonicity conditions to allow derivatives in the operator acting on the solution under the stochastic integral.Comment: 32 page

Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics

This note deals with the analysis of a model for partial damage, where the rate- independent, unidirectional flow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from Roubicek [1, 2] with the methods from Lazzaroni/Rossi/Thomas/Toader [3]. The present analysis encompasses, differently from [2], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [3], a nonconstant heat capacity and a time-dependent Dirichlet loading

A comparison of delamination models: Modeling, properties, and applications

This contribution presents recent results in the modeling and the analysis of delamination problems. It addresses adhesive contact, brittle, and cohesive zone models both in a quasistatic and a viscous, dynamic setting for the bulk part. Also different evolution laws for the delaminating surface are discussed

Three examples concerning the interaction of dry friction and oscillations

We discuss recent work concerning the interaction of dry friction, which is a rate independent effect, and temporal oscillations. First, we consider the temporal averaging of highly oscillatory friction coefficients. Here the effective dry friction is obtained as an infimal convolution. Second, we show that simple models with state-dependent friction may induce a Hopf bifurcation, where constant shear rates give rise to periodic behavior where sticking phases alternate with sliding motion. The essential feature here is the dependence of the friction coefficient on the internal state, which has an internal relaxation time. Finally, we present a simple model for rocking toy animal where walking is made possible by a periodic motion of the body that unloads the legs to be moved

Numerical approach to a model for quasistatic damage with spatial BV-regularization

We address a model for rate-independent, partial, isotropic damage in quasistatic small strain linear elasticity, featuring a damage variable with spatial BV-regularization. Discrete solutions are obtained using an alternate time-discrete scheme and the Variable-ADMM algorithm to solve the constrained nonsmooth optimization problem that determines the damage variable at each time step. We prove convergence of the method and show that discrete solutions approximate a semistable energetic solution of the rate-independent system. Moreover, we present our numerical results for two benchmark problems

Source localization of reaction-diffusion models for brain tumors

We propose a mathematically well-founded approach for locating the source (initial state) of density functions evolved within a nonlinear reaction-diffusion model. The reconstruction of the initial source is an ill-posed inverse problem since the solution is highly unstable with respect to measurement noise. To address this instability problem, we introduce a regularization procedure based on the nonlinear Landweber method for the stable determination of the source location. This amounts to solving a sequence of well-posed forward reaction-diffusion problems. The developed framework is general, and as a special instance we consider the problem of source localization of brain tumors. We show numerically that the source of the initial densities of tumor cells are reconstructed well on both imaging data consisting of simple and complex geometric structures

Existence of Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain in two and three space dimensions. In contrast to previous works, we study a new model recently developed by Abels, Garcke, and Gr\"un for fluids with different densities, which leads to a solenoidal velocity field. The model is given by a non-homogeneous Navier-Stokes system with a modified convective term coupled to a Cahn-Hilliard system. The density of the mixture depends on an order parameter.Comment: 33 page

Convergence of tamed Euler schemes for a class of stochastic evolution equations

We prove stability and convergence of a full discretization for a class of stochastic evolution equations with super-linearly growing operators appearing in the drift term. This is done using the recently developed tamed Euler method, which uses a fully explicit time stepping, coupled with a Galerkin scheme for the spatial discretization