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We consider a mathematical model which describes the quasistatic frictionless contact of a viscoelastic body with a rigid-plastic foundation. We describe the mechanical assumptions, list the hypotheses on the data and provide three different variational formulations of the model in which the unknowns are the displacement field, the stress field and the strain field, respectively. These formulations have a different structure. Nevertheless, we prove that they are pairwise dual of each other. Then, we deduce the unique weak solvability of the contact problem as well as the Lipschitz continuity of its weak solution with respect to the data. The proofs are based on recent results on history-dependent variational inequalities and inclusions. Finally, we present numerical simulations in the study of the contact problem, together with the corresponding mechanical interpretations., Comment: 25 pages, 4 figures

PySDM is an open-source Python package for simulating the dynamics of particles undergoing condensational and collisional growth, interacting with a fluid flow and subject to chemical composition changes. It is intended to serve as a building block for process-level as well as computational-fluid-dynamics simulation systems involving representation of a continuous phase (air) and a dispersed phase (aerosol), with PySDM being responsible for representation of the dispersed phase. The PySDM package core is a Pythonic high-performance implementation of the Super-Droplet Method (SDM) Monte-Carlo algorithm for representing collisional growth, hence the name. PySDM has two alternative parallel number-crunching backends available: multi-threaded CPU backend based on Numba and GPU-resident backend built on top of ThrustRTC. The usage examples are built on top of four simple atmospheric cloud modelling frameworks: box, adiabatic parcel, single-column and 2D prescribed flow kinematic models. In addition, the package ships with tutorial code depicting how PySDM can be used from Julia and Matlab.

In this paper we present an abstract nonsmooth optimization problem for which we recall existence and uniqueness results. We show a numerical scheme to approximate its solution. The theory is later applied to a sample static contact problem describing an elastic body in frictional contact with a foundation. This problem leads to a hemivariational inequality which we solve numerically. Finally, we compare three computational methods of solving contact mechanical problems: direct optimization method, augmented Lagrangian method and primal-dual active set strategy., Comment: 21 pages, 5 figures. arXiv admin note: text overlap with arXiv:1903.04241

Super-Droplet Method (SDM) is a probabilistic Monte-Carlo-type model of particle coagulation process, an alternative to the mean-field formulation of Smoluchowski. SDM as an algorithm has linear computational complexity with respect to the state vector length, the state vector length is constant throughout simulation, and most of the algorithm steps are readily parallelizable. This paper discusses the design and implementation of two number-crunching backends for SDM implemented in PySDM, a new free and open-source Python package for simulating the dynamics of atmospheric aerosol, cloud and rain particles. The two backends share their application programming interface (API) but leverage distinct parallelism paradigms, target different hardware, and are built on top of different lower-level routine sets. First offers multi-threaded CPU computations and is based on Numba (using Numpy arrays). Second offers GPU computations and is built on top of ThrustRTC and CURandRTC (and does not use Numpy arrays). In the paper, the API is discussed focusing on: data dependencies across steps, parallelisation opportunities, CPU and GPU implementation nuances, and algorithm workflow. Example simulations suitable for validating implementations of the API are presented., Comment: accepted to ICCS 2021

This work discusses the numerical aspects of representing the diffusional (condensational) growth in particulate systems such as atmospheric clouds. It focuses on the Eulerian modeling approach, in which the evolution of the particle size spectrum is carried out using a fixed-bin discretization associated with inherent numerical diffusion. Focus is on the applications of MPDATA numerical schemes (variants explored include: infinite-gauge, non-oscillatory, third-order-terms and recursive antidiffusive correction). Methodology for handling coordinate transformations associated with both particle size distribution variable choice and numerical grid layout are expounded. Analysis of the performance of the scheme is performed using: (i) an analytically solvable box-model test case, and (ii) the single-column "KiD" test case in which the size-spectral advection due to condensation is solved simultaneously with the spatial advection in the vertical physical coordinate, and in which the supersaturation evolution is coupled with the droplet growth through water mass budget. The single-column problem involves numerical solution of a two-dimensional advection problem (spectral and spatial dimensions). The box-model simulations demonstrate that, for the problem considered, even a tenfold decrease of the spurious numerical spectral broadening can be obtained by a proper choice of the MPDATA variant (maintaining the same spatial and temporal resolution), yet at an increased computational cost. Analyses using the single-column test case reveal that the width of the droplet size spectrum is affected by numerical diffusion pertinent to both spatial and spectral advection. Application of even a single corrective iteration of MPDATA robustly decreases the relative dispersion of the droplet spectrum, roughly by a factor of two at the levels of maximal liquid water content.

In this work, we analyze a non-clamped dynamic viscoelastic contact problem involving thermal effect. The friction law is described by a non-monotone relation between the tangential stress and the tangential velocity. This leads to a system of second-order inclusion for displacement and a parabolic equation for temperature. We provide a fully discrete approximation of the problem and find optimal error estimates without any smallness assumption on the data. The theoretical result is illustrated by numerical simulations.