doc. Dr. rer. nat. Ing. Jan Valdman

Článek

10.1007/s00032-022-00368-9

Katedra aplikované matematiky

We formulate a large-strain model of single-slip crystal elastoplasticity in the framework of energetic solutions. The numerical performance of the model is compared with laboratory experiments on the compression of a stack of papers.

Článek

10.1007/s11228-022-00651-2

Katedra aplikované matematiky

The paper starts with a description of SCD (subspace containing derivative) mappings and the SCD semismooth* Newton method for the solution of general inclusions. This method is then applied to a class of variational inequalities of the second kind. As a result, one obtains an implementable algorithm which exhibits locally superlinear convergence. Thereafter we suggest several globally convergent hybrid algorithms in which one combines the SCD semismooth* Newton method with selected splitting algorithms for the solution of monotone variational inequalities. Finally, we demonstrate the efficiency of one of these methods via a Cournot-Nash equilibrium, modeled as a variational inequality of the second kind, where one admits really large numbers of players (firms) and produced commodities.

Článek

10.3390/math10234412

Katedra aplikované matematiky

The incremental energy minimization principle provides a compact variational formulation for evolutionary boundary problems based on constitutive models of rate-independent dissipative solids. In this work, we develop and implement a versatile computational tool for the resolution of these problems via the finite element method (FEM). The implementation is coded in the MATLAB programming language and benefits from vector operations, allowing all local energy contributions to be evaluated over all degrees of freedom at once. The monolithic solution scheme combined with gradient-based optimization methods is applied to the inherently nonlinear, non-smooth convex minimization problem. An advanced constitutive model for shape memory alloys, which features a strongly coupled rate-independent dissipation function and several constraints on internal variables, is implemented as a benchmark example. Numerical simulations demonstrate the capabilities of the computational tool, which is suited for the rapid development and testing of advanced constitutive laws of rate-independent dissipative solids.