arXiv subjects

Jan Valdman

Publications and source records attributed to Jan Valdman.

42 records · Page 3Linked to original sources

arXiv2015

Stress-driven solution to rate-independent elasto-plasticity with damage at small strains and its computer implementation

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.

math.NA↗

arXiv2015

Fast MATLAB assembly of FEM matrices in 2D and 3D: Edge elements

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.

cs.MS↗

arXiv2015

Perfect plasticity with damage and healing at small strains, its modelling, analysis, and computer implementation

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.

math.NA↗

arXiv2014

Additive Schwarz preconditioner for the general finite volume element discretization of symmetric elliptic problems

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.

math.NA↗

arXiv2014

Verification of functional a posteriori error estimates for obstacle problem in 2D

We verify functional a posteriori error estimate proposed by S. Repin for a class of obstacle problems. The obstacle problem is formulated as a quadratic minimization problem with constrains equivalently formulated as a variational inequality. New benchmarks with known analytical solutions in 2D are constructed based on 1D benchmark introduced by P. Harasim and J. Valdman. Numerical approximation of the obstacle problem is obtained by the finite element method using bilinear elements on a rectangular mesh. Error of the approximation is meassured in the energy norm and bounded from above by a functional majorant, whose value is minimized with respect to unknown gradient field discretized by Raviart-Thomas elements and Lagrange multipliers field discretized by piecewise constant functions.

math.NA↗

arXiv2013

Verification of functional a posteriori error estimates for obstacle problem in 1D

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.

math.NA↗