Jan Valdman

Stefan Krömer, Jan Valdman

We present a new penalty term approximating the Ciarlet-Nečas condition (global invertibility of deformations) as a soft constraint for hyperelastic materials. For non-simple materials including a suitable higher order term in the elastic energy, we prove that the penalized functionals converge to the original functional subject to the Ciarlet-Nečas condition. Moreover, the penalization can be chosen in such a way that all low energy deformations, self-interpenetration is completely avoided even for sufficiently small finite values of the penalization parameter. We also present numerical experiments in 2d illustrating our theoretical results.

Farid Bozorgnia, Jan Valdman

This paper is concerned with the two--phase obstacle problem, a type of a variational free boundary problem. We recall the basic estimates of Repin and Valdman (2015) and verify them numerically on two examples in two space dimensions. A solution algorithm is proposed for the construction of the finite element approximation to the two--phase obstacle problem. The algorithm is not based on the primal (convex and nondifferentiable) energy minimization problem but on a dual maximization problem formulated for Lagrange multipliers. The dual problem is equivalent to a quadratic programming problem with box constraints. The quality of approximations is measured by a functional a posteriori error estimate which provides a guaranteed upper bound of the difference of approximated and exact energies of the primal minimization problem. The majorant functional in the upper bound contains auxiliary variables and it is optimized with respect to them to provide a sharp upper bound. A space density of the nonlinear related part of the majorant functional serves as an indicator of the free boundary.

Petr Harasim, Jan Valdman

We verify functional a posteriori error estimate for obstacle problem proposed by Repin. Simplification into 1D allows for the construction of a nonlinear benchmark for which an exact solution of the obstacle problem can be derived. Quality of a numerical approximation obtained by the finite element method is compared with the exact solution and the error of approximation is bounded from above by a majorant error estimate. The sharpness of the majorant error estimate is discussed.

Martin Kružík, Jan Valdman

We study computational behavior of a mesoscopic model describing temperature/external magnetic field-driven evolution of magnetization. Due to nonconvex anisotropy energy describing magnetic properties of a body, magnetization can develop fast spatial oscillations creating complicated microstructures. These microstructures are encoded in Young measures, their first moments then identify macroscopic magnetization. Our model assumes that changes of magnetization can contribute to dissipation and, consequently, to variations of the body temperature affecting the length of magnetization vectors. In the ferromagnetic state, minima of the anisotropic energy density depend on temperature and they tend to zero as we approach the so-called Curie temperature. This brings the specimen to a paramagnetic state. Such a thermo-magnetic model is fully discretized and tested on two-dimensional examples. Computational results qualitatively agree with experimental observations. The own MATLAB code used in our simulations is available for download.

Piotr Skrzypacz, Kaisar Tangirbergen, Jan Valdman

This work investigates a nonlinear two-point boundary value problem arising in diffusion-reaction processes in catalyst slabs with power-law kinetics and fractional reaction order. The model exhibits a free-boundary structure, where an unknown interface separates a dead-core region with vanishing concentration from an active region with positive concentration. We propose a Physics-Informed Neural Network (PINN) framework that incorporates a structured, hard-constrained trial solution embedding the asymptotic behavior near the interface. The dead-core location is treated as a trainable parameter, enabling the simultaneous approximation of the concentration profile and identification of the free boundary without explicit interface tracking. The method is validated against analytical solutions and high-precision numerical shooting. Numerical experiments demonstrate that the approach accurately captures both the solution profile and the free-boundary location while maintaining a computationally manageable training cost.

Miroslav Frost, Martin Kružík, Jan Valdman

A sharp-interface model describing static equilibrium configurations of shape mory alloys by means of interfacial polyconvex energy density introduced by Šilhavý in 2010 and extended to a quasistatic situation by Knüpfer and Kružík in 2016 is computationally tested. Elastic properties of variants of martensite and the austenite are described by polyconvex energy density functions. Volume fractions of particular variants are modeled by a map of bounded variation. Additionally, energy stored in martensite-martensite and austenite-martensite interfaces is measured by an interface-polyconvex function. It is assumed that transformations between material variants are accompanied by energy dissipation which, in our case, is positively and one-homogeneous giving rise to a rate-independent model. Various two-dimensional computational examples are presented and the used computer code is made available for downloads.

Piotr Skrzypacz, Boris Golman, Jan Valdman

The paper deals with dead-core solutions to an isothermal reaction-diffusion problem with power-law kinetics for a single reaction that takes place in a chemical reactor represented by a bounded domain in two dimensions. The model boundary value problem for the stationary non-linear diffusion-reaction equation is solved numerically using an appropriate time-marching method. The spatial discretization is based on the lumped finite element method for piecewise linear functions. The effects of the reaction order and Thiele modulus on the concentration profiles and the size of dead zones are investigated numerically. The paper demonstrates that the formation of multiple dead zones is possible for particular reactor geometries.

Martin Čermák, Stanislav Sysala, Jan Valdman

We propose an effective and flexible way to implement 2D and 3D elastoplastic problems in MATLAB using fully vectorized codes. Our technique is applied to a broad class of the problems including perfect plasticity or plasticity with hardening and several yield criteria. The problems are formulated in terms of displacements, discretized by the implicit Euler method in time and the finite element method in space, and solved by the semismooth Newton method. We discuss in detail selected models with the von Mises and Prager-Drucker yield criteria and four types of finite elements. The related codes are available for download. A particular interest is devoted to the assembling of tangential stiffness matrices. Since these matrices are repeatedly constructed in each Newton iteration and in each time step, we propose another vectorized assembling than current ones known for the elastic stiffness matrices. The main idea is based on a construction of two large and sparse matrices representing the strain-displacement and tangent operators, respectively, where the former matrix remains fixed and the latter one is updated only at some integration points. Comparisons with other available MATLAB codes show that our technique is also efficient for purely elastic problems. In elastoplasticity, the assembly times are linearly proportional to the number of integration points in a plastic phase and additional times due to plasticity never exceed assembly time of the elastic stiffness matrix.

Tomáš Roubíček, Jan Valdman

The quasistatic rate-independent damage combined with linearized plasticity with hardening at small strains is investigated. The fractional-step time discretisation is devised with the purpose to obtain a numerically efficient scheme converging possibly to a physically relevant stress-driven solutions, which however is to be verified a-posteriori by using a suitable integrated variant of the maximum-dissipation principle. Gradient theories both for damage and for plasticity are considered to make the scheme numerically stable with guaranteed convergence within the class of weak solutions. After finite-element approximation, this scheme is computationally implemented and illustrative 2-dimensional simulations are performed.

Immanuel Anjam, Jan Valdman

We propose an effective and flexible way to assemble finite element stiffness and mass matrices in MATLAB. We apply this for problems discretized by edge finite elements. Typical edge finite elements are Raviart-Thomas elements used in discretizations of H(div) spaces and Nedelec elements in discretizations of H(curl) spaces. We explain vectorization ideas and comment on a freely available MATLAB code which is fast and scalable with respect to time.

Tomáš Roubíček, Jan Valdman

The quasistatic, Prandtl-Reuss perfect plasticity at small strains is combined with a gradient, reversible (i.e. admitting healing) damage which influences both the elastic moduli and the yield stress. Existence of weak solutions of the resulted system of variational inequalities is proved by a suitable fractional-step discretisation in time with guaranteed numericalstability and convergence. After finite-element approximation, this scheme is computationally implemented and illustrative 2-dimensional simulations are performed. The model allows e.g. for application in geophysical modelling of re-occurring rupture of lithospheric faults. Resulted incremental problems are solved in MATLAB by quasi-Newton method to resolve elastoplasticity component of the solution while damage component is obtained by solution of a quadratic programming problem.

Leszek Marcinkowski, Talal Rahman, Atle Loneland et al.

A symmetric and a nonsymmetric variant of the additive Schwarz preconditioner are proposed for the solution of a nonsymmetric system of algebraic equations arising from a general finite volume element discretization of symmetric elliptic problems with large jumps in the entries of the coefficient matrices across subdomains. It is shown in the analysis, that the convergence of the preconditioned GMRES iteration with the proposed preconditioners, depends polylogarithmically on the mesh parameters, in other words, the convergence is only weakly dependent on the mesh parameters, and it is robust with respect to the jumps in the coefficients.