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A MIXED VIRTUAL ELEMENT METHOD FOR THE boussinesq problem ON POLYGONAL MESHES

Journal of Computational Mathematics 2021年第3期39卷 392-427页

作者： Gabriel N.Gatica Mauricio Munar Filander A.Sequeira CI^2 MA and Departamento de Ingenieria Matemitica Universidad de ConcepcionCasilla 160-CConcepcionChile Escuela de Matemaitica Universidad NacionalCampus Omar DengoHerediaCosta Rica

In this work we introduce and analyze a mixed virtual element method(mixed-VEM)for the two-dimensional stationary boussinesq problem.The continuous formulation is based on the introduction of a pseudostress tensor depending nonlinearly on the velocity,which allows to obtain an equivalent model in which the main unknowns are given by the aforementioned pseudostress tensor,the velocity and the temperature,whereas the pressure is computed via a postprocessing formula.In addition,an augmented approach together with a fixed point strategy is used to analyze the well-posedness of the resulting continuous formulation.Regarding the discrete problem,we follow the approach employed in a previous work dealing with the Navier-Stokes equations,and couple it with a VEM for the convection-diffusion equation modelling the temperature.More precisely,we use a mixed-VEM for the scheme associated with the fluid equations in such a way that the pseudostress and the velocity are approximated on virtual element subspaces of H(div)and H^(1),respectively,whereas a VEM is proposed to approximate the temperature on a virtual element subspace of H^(1).In this way,we make use of the L^(2)-orthogonal projectors onto suitable polynomial spaces,which allows the explicit integration of the terms that appear in the bilinear and trilinear forms involved in the scheme for the fluid equations.On the other hand,in order to manipulate the bilinear form associated to the heat equations,we define a suitable projector onto a space of polynomials to deal with the fact that the diffusion tensor,which represents the thermal conductivity,is variable.Next,the corresponding solvability analysis is performed using again appropriate fixed-point arguments.Further,Strang-type estimates are applied to derive the a priori error estimates for the components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure.The corresponding rates of convergence are also established.Finally,several numerical examples

关键词： boussinesq problem Pseudostress-based formulation Augmented formulation Mixed virtual element method.High-order approximations

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Interior-stress fields produced by a general axisymmetric punch

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Friction 2022年第4期10卷 530-544页

作者： Longxiang YANG Zhanjiang WANG Weiji LIU Guocheng ZHANG Bei PENG School of Automotive Engineering Geely University of ChinaChengdu 641423China Department of Mechanical Engineering Southwest Jiaotong UniversityChengdu 610031China School of Mechatronic Engineering Southwest Petroleum UniversityChengdu 610500China School of Mechanical and Electrical Engineering University of Electronic Science and Technology of ChinaChengdu 611731China Center for Robotics University of Electronic Science and Technology of ChinaChengdu 611731China

This work is a supplement to the work of Sneddon on axisymmetric boussinesq problem in 1965 in which the distributions of interior-stress fields are derived here for a punch with general profile.A novel set of mathematical procedures is introduced to process the basic elastic solutions(obtained by the method of Hankel transform,which was pioneered by Sneddon)and the solution of the dual integral equations.These processes then enable us to not only derive the general relationship of indentation depth D and total load P that acts on the punch but also explicitly obtain the general analytical expressions of the stress fields beneath the surface of an isotropic elastic half-space.The usually known cases of punch profiles are reconsidered according to the general formulas derived in this study,and the deduced results are verified by comparing them with the classical results.Finally,these general formulas are also applied to evaluate the von Mises stresses for several new punch profiles.

关键词： axisymmetric boussinesq problem interior stress Hankel transform dual integral equations

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