A nonsmooth optimization approach for hemivariational inequalities with applications to contact mechanics

In this paper we introduce an abstract nonsmooth optimization problem and prove existence and uniqueness of its solution. We present a numerical scheme to approximate this solution. The theory is later applied to a sample static contact problem describing an elastic body in frictional contact with a foundation. This contact is governed by a nonmonotone friction law with dependence on normal and tangential components of displacement. Finally, computational simulations are performed to illustrate obtained results

Comparison of selected clustering algorithms

Algorytm k-średnich (ang. „k-means”) oraz „expectation-maximization” (ang.) oparty na modelu „Gaussian mixture model” (ang.) to jedne z najczęściej stosowanych i najbardziej uznanych algorytmów w dziedzinie klasteryzacji danych. Oba te algorytmy do działania wymagają podania liczby klastrów, na które ma zostać podzielony zbiór danych. Jeśli liczba ta nie jest znana, można zastosować jedno z kryteriów jej doboru. Jednym z najpopularniejszych kryteriów znajdujących zastosowanie przy użyciu algorytmu expectation-maximization jest „Akaike Information Criterion” (ang.). W tej pracy przedstawiony zostanie sposób działania wymienionych algorytmów a także zaprezentowany zostanie autorski algorytm klasteryzacji danych oparty na idei sieci Kohonena – samoorganizującej mapy (ang. „self-organizing map”).K-means and expectation-maximization algorithm based on Gaussian mixture model are one of the most common and best founded algorithms in data clustering. Both algorithms have to be given a number of clusters in order to divide data set. If this number is not known, a criterion that determines it can be used. One of the most popular among them, used with expectation maximization algorithm, is Akaike Information Criterion. In this paper we will present listed algorithms and introduce author's clustering algorithm based on the idea of self-organizing map

Numerical analysis and simulations of contact problem with wear

This paper presents a quasistatic problem of an elastic body in frictional contact with a moving foundation. The model takes into account wear of the contact surface of the body caused by the friction. We recall existence and uniqueness results obtained in Sofonea et al. (2017). The main aim of this paper is to present a fully discrete scheme for numerical approximation together with an error estimation of a solution to this problem. Finally, computational simulations are performed to illustrate the mathematical model

Numerical studies of a hemivariational inequality for a viscoelastic contact problem with damage

This paper is devoted to the study of a hemivariational inequality modeling the quasistatic bilateral frictional contact between a viscoelastic body and a rigid foundation. The damage effect is built into the model through a parabolic differential inclusion for the damage function. A solution existence and uniqueness result is presented. A fully discrete scheme is introduced with the time derivative of the damage function approximated by the backward finite different and the spatial derivatives approximated by finite elements. An optimal order error estimate is derived for the fully discrete scheme when linear elements are used for the velocity and displacement variables, and piecewise constants are used for the damage function. Simulation results on numerical examples are reported illustrating the performance of the fully discrete scheme and the theoretically predicted convergence orders.Comment: 22 pages, 5 figure