Petr Klouček

On thermodynamic active control of shape memory alloy wires

We propose a model for the control of high-frequency oscillations in shape memory alloy wires. We introduce a notion of generalized solution for a generalized control and in this context prove a local exact controllability result effectively corresponding to an approximate controllability result for the nonconvex pseudoelastic material system.

THE DETACHMENT OF BUBBLES UNDER A POROUS RIGID SURFACE DURING ALUMINUM ELECTROLYSIS

We determine the critical pressure and width of a liquid layer at which the sudden detachment of bubbles at solid–liquid interfaces occurs. We obtain these values by solving numerically a constrained minimization problem corresponding to the conservation of mass density of gas contained in the bubble attached to a rigid surface, and to the conservation of its free energy.

The relaxation of non-quasiconvex variational integrals

Abstract. We show that the Steepest Descent Algorithm in connection with wiggly energies yields minimizing sequences that converge to a global minimum of the associated non-quasiconvex variational integrals. We introduce a multi-level infinite dimensional variant of the Steepest Descent Algorithm designed to compute complex microstructures by forming non-smooth minimizers from the smooth initial guesses. We apply this multi-level method to the minimization of the variational problems associated with martensitic branching.We show that the Steepest Descent Algorithm in connection with wiggly energies yields minimizing sequences that converge to a global minimum of the associated non-quasiconvex variational integrals. We introduce a multi-level infinite dimensional variant of the Steepest Descent Algorithm designed to compute complex microstructures by forming non-smooth minimizers from the smooth initial guesses. We apply this multi-level method to the minimization of the variational problems associated with martensitic branching.

On the asymptotically stochastic computational modeling of microstructures

We consider a class of alloys and ceramics with equilibria described by non-attainable infima of non-quasiconvex variational integrals. Such situations frequently arise when atomic lattice structure plays an important role at the mesoscopic continuum level.We prove that standard variational approaches associated with gradient based relaxation of non-quasiconvex integrals in Banach or Hilbert spaces are not capable of generating relaxing sequences for problems with non-attainable structure.We introduce a variational principle suitable for the computational purposes of approaching non-attainable infima of variational integrals. We demonstrate that this principle is suitable for direct calculations of the Young Measures on a computational example in one dimension.The new variational principle provides the possibility to approximate crystalline microstructures using a Fokker-Planck equation at the meso-scale. We provide an example of such a construction.

Thermal stabilization of shape memory alloy wires

We show that fast, localized heating and cooling of a Shape Memory material can provide a very effective means of damping vibrational energy. We model the thermally induced pseudo-elastic behavior of a NiTi Shape Memory wire using the variant of Landau-Devonshire potential. We assume that the wire consists of martensitic NiTi single crystal. Dynamically, we model the material response using conservation of momentum and a nonlinear heat equation. We use a frame invariant version of the Fourier heat flux which incorporates dependence on the atomic lattice through the stretch. In the settings used in this paper, the computational experiments confirm that circa 80% of the vibrational energy can be eliminated at the moment of the onset of the thermally induced phase transition.

CONVERGENCE OF GIBBS MEASURES ASSOCIATED WITH SIMULATED ANNEALING

We give a sufficient condition for a sequence of Gibbs measures dominated by Lebesgue measure to converge to a singular measure concentrated on a submanifold. The limiting measure is absolutely continuous with respect to Hausdorff (Riemannian) measure on the submanifold, and a formula for its density is given. These results have implications for simulated annealing algorithms on a continuous state space when the set of minimizers of the objective function is more complex than a finite set of points. Under regularity conditions, the limiting measure is concentrated on the highest dimensional submanifold of the set of minimizers, so that lower dimensional components of the minimizing set are essentially lost. A generalization of the main result treats multiple limits within submanifolds, which could be useful for constrained optimization with simulated annealing. An example is given which shows that if the conditions of the theorem do not hold, then unexpected results may occur.

The Computational Modeling of Crystalline Materials Using a Stochastic Variational Principle

We introduce a variational principle suitable for the computational modeling of crystalline materials. We consider a class of materials that are described by non-quasiconvex variational integrals.We are further focused on equlibria of such materials that have non-attainment structure, i.e., Dirichlet boundary conditions prohibit these variational integrals from attaining their infima. Consequently, the equilibrium is described by probablity distributions. The new variational principle provides the possibility to use standard optimization tools to achieve stochastic equilibrium states starting from given initial deterministic states.

A subgrid projection method for relaxation of non-attainable differential inclusions

We propose a subgrid projection method which is suitable for computing microstructures describing non-attainable infima of variational integrals. We document that a descent method in combination with the proposed method yields relaxing sequences which converge in weak-* topology as well as in the sense of approximate Young measures. We show that a sufficient condition for our method to work is to asymptotically form white noise. We present an example which shows that this requires symmetry in the target function.

Tracking Free Boundaries in Fluids Using a Variational Principle

We derive a variational principle suitable for tracking free boundaries in fluids. The variational principle is based on the Lagrangian formulation of the Navier–Stokes equations. The principle is derived from a generalization of the principle of stationary action applied to a Riemannian manifold of volume-preserving flow maps. The dual variational principle for the indicatrices identifying the free boundaries is based on the Wasserstein–Kantorovich metric.

Stochastic Relaxation of Variational Integrals with Non-attainable Infima

We provide an example of a stochastic approach to relaxation of the variational integrals with non-attainable infima in one dimension. We provide an approximation for the coefficients of the Laplace transformation of the Probability Density Function. This approaximation yields the relaxing microstructures.