arXiv subjects

Jaroslav Haslinger

Publications and source records attributed to Jaroslav Haslinger.

4 recordsLinked to original sources

arXiv2023

Convex optimization problems inspired by geotechnical stability analysis

This paper is motivated by the limit load, limit analysis and shear strength reduction methods, which are commonly employed in geotechnical stability analysis or similar applications. The aim is to make these methods more approachable by introducing a unified framework based on abstract convex optimization and its parametric studies. We establish suitable assumptions on the abstract problems that capture the selected features of these methods and facilitate rigorous theoretical investigation. Further, we propose continuation techniques tailored to the resulting parametric problem formulations and show that the developed abstract framework could also be useful outside the domain of geotechnical stability analysis. The main results are illustrated with analytical and numerical examples. The numerical example deals with a 3D slope stability problem.

math.NA↗

arXiv2020

An abstract inf-sup problem inspired by limit analysis in perfect plasticity and related applications

This work is concerned with an abstract inf-sup problem generated by a bilinear Lagrangian and convex constraints. We study the conditions that guarantee no gap between the inf-sup and related sup-inf problems. The key assumption introduced in the paper generalizes the well-known Babuska-Brezzi condition. It is based on an inf-sup condition defined for convex cones in function spaces. We also apply a regularization method convenient for solving the inf-sup problem and derive a computable majorant of the critical (inf-sup) value, which can be used in a posteriori error analysis of numerical results. Results obtained for the abstract problem are applied to continuum mechanics. In particular, examples of limit load problems and similar ones arising in classical plasticity, gradient plasticity and delamination are introduced.

math.OC↗

arXiv2015

A new incremental method of computing the limit load in deformation plasticity models

The aim of this paper is to introduce a new incremental procedure that can be used for numerical evaluation of the limit load. Existing incremental type methods are based on parametrization of the energy by the loading parameter $\zeta\in[0,\zeta_{lim})$, where $\zeta_{lim}$ is generally unknown. In the new method, the incremental procedure is operated in terms of an inverse mapping and the respective parameter $\alpha$ is changing in the interval $(0,+\infty)$. Theoretically, in each step of this algorithm, we obtain a guaranteed lower bound of $\zeta_{lim}$. Reduction of the problem to a finite element subspace associated with a mesh $\mathcal T_h$ generates computable bound $\zeta_{lim,h}$. Under certain assumptions, we prove that $\zeta_{lim,h}$ tends to $\zeta_{lim}$ as $h\rightarrow0_+$. Numerical tests confirm practical efficiency of the suggested method.

math.NA↗

arXiv2015

Stokes problem with a solution dependent slip bound: Stability of solutions with respect to domains

We study the Stokes problem in a bounded planar domain $Ω$ with a friction type boundary condition that switches between a slip and no-slip stage. Unlike our previous work [6], in the present paper the threshold value may depend on the velocity field. Besides the usual velocity-pressure formulation, we introduce an alternative formulation with three Lagrange multipliers which allows a more flexible treatment of the impermeability condition as well as optimum design problems with cost functions depending on the shear and/or normal stress. Our main goal is to determine under which conditions concerning smoothness of $Ω$, solutions to the Stokes system depend continuously on variations of $Ω$. Having this result at our disposal, we easily prove the existence of a solution to optimal shape design problems for a large class of cost functionals.

math.AP↗