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mixed finite element method for Multi-layer Viscoelastic Contact Systems with Time-Dependent Tresca Friction

JOURNAL OF SCIENTIFIC COMPUTING 2025年第2期103卷

作者： Zhang, Zhizhuo Barboteu, Mikael Cao, Jinde Southeast Univ Sch Math Nanjing 211189 Peoples R China Univ Perpignan via Domitia Lab Modelisat Pluridisciplinaire & Simulat LAMPS 52 Ave Paul Alduy F-66860 Perpignan France Minist Transport Key Lab Transport Ind Comprehens Transportat Theor Nanjing Modern Multimodal Transportat Lab Nanjing 211135 Peoples R China

In this study, we designed a framework of the mixed finite element method for solving a class of multi-layer viscoelastic contact systems with time-dependent Tresca friction conditions, which can be used for mechanical modeling of engineering problems such as asphalt pavement. Theoretically, we have proven the convergence properties of the numerical solution of the displacement field under the condition that the dual multipliers do not have uniqueness. Moreover, by combining the midpoint strategy, this numerical method still maintains stability in the numerical simulation of complex contact systems with elastic and viscoelastic layers, thus becoming a unified computational framework for such contact problems. Benefit from this, the numerical method can achieve mechanical simulations of asphalt pavements, thereby offering prospects for reference and application in actual industrial modeling.

关键词： mixed finite element method Lagrangian dual multipliers Multi-layer contact system Viscoelasticity Time-dependent Tresca friction conditions Midpoint strategy Asphalt pavement

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A mixed finite element method for convection-diffusion complementarity problems

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2025年 1-23页

作者： Mezzan, Youness Cadi Ayyad Univ Fac Sci Semlalia Dept Math Marrakech Morocco

This work introduces a numerical scheme designed to solve convection-diffusion complementarity problems. The proposed method integrates a Lagrange-Galerkin approach, which discretizes the material derivative along characteristic paths, with a mixed finite element method. To solve the resulting complementarity saddle point system at the discrete level, an effective primal-dual active set solver is employed. The effectiveness and performance of this approach are validated through rigorous benchmarking against known analytical solutions from the literature.

关键词： Convection-diffusion Characteristics method Complementarity problems mixed finite element method Active set solver

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mixed finite element method for multi-layer elastic contact systems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2025年 457卷

作者： Zhang, Zhizhuo Barboteu, Mikael Nie, Xiaobing Dumont, Serge Abdel-Aty, Mahmoud Cao, Jinde Southeast Univ Sch Math Nanjing 211189 Peoples R China Univ Perpignan Via Domitia Lab Modelisat Pluridisciplinaire & Simulat 52 Ave Paul Alduy F-66860 Perpignan France Sohag Univ Fac Sci Dept Math Sohag 82524 Egypt Ahlia Univ Deanship Grad Studies & Sci Res Manama 10878 Bahrain

With the development of multi-layer elastic systems in the field of engineering mechanics, the corresponding variational inequality theory and algorithm design have received more attention and research. In this study, a class of equivalent saddle point problems with interlayer Tresca friction conditions and the mixed finite element method are proposed and analyzed. Then, the convergence of the numerical solution of the mixed finite element method is theoretically proven, and the corresponding algebraic dual algorithm is provided. Finally, through numerical experiments, the mixed finite element method is not only compared with the layer decomposition method, but also its convergence relationship with respect to the spatial discretization parameter H is verified.

关键词： mixed finite element method Variational inequality Contact problem Tresca's friction law Pavement mechanics

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A mixed finite element method for Vibration Equations of Structurally Damped Beam and Plate

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NUMERICAL methodS FOR PARTIAL DIFFERENTIAL EQUATIONS 2025年第2期41卷

作者： Ye, Yuqian Yin, Zhe Zhu, Ailing Shandong Normal Univ Sch Math & Stat Jinan Peoples R China

This paper employs a mixed finite element method to study the initial-boundary value problems of structurally damped beam-plate vibration. By introducing variables, we transform the fourth-order spatial and second-order temporal differential equations into second-order spatial and first-order temporal coupled equations. We use the elliptic projection operator to establish the mixed finite element semidiscrete and fully discrete formats, prove the existence and uniqueness of solutions, and complete optimal error estimations in the L2$$ {L}<^>2 $$-norm and H1$$ {H}<^>1 $$-norm. When the mixed finite element space is a polynomial of degree k$$ k $$, the convergence order of the numerical solution in the L2$$ {L}<^>2 $$-norm is k+1$$ k+1 $$, and the convergence order in the H1$$ {H}<^>1 $$-norm is k$$ k $$. The effectiveness of the mixed finite element format is verified through numerical examples, and the impact of the damping coefficient on beam and plate vibration is investigated. Simultaneously, the occurrence of strong damping effects is observed.

关键词： beam and plate vibration error estimation mixed finite element method numerical simulation structurally damped

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A combined mixed finite element method and discontinuous Galerkin method for hybrid-dimensional fracture models of two-phase flow

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2025年 459卷

作者： Chen, Shuangshuang Beijing Univ Technol Sch Math Stat & Mech Beijing 100124 Peoples R China

In this paper, a combined numerical method consisting of the mixed finite element method (MFE) for the pressure equation and the discontinuous Galerkin (DG) method for the saturation equation is proposed to solve hybrid-dimensional fracture models of incompressible two-phase flow in porous media. The hybrid-dimensional fracture models treat fractures as ( d - 1)- dimensional interfaces immersed in d-dimensional matrix domains and take fluid exchange between fractures and surrounding matrix into account. Fully implicit approximation schemes combining the MFE-DG method with the backward Euler time discretization for the models with both a single fracture and an intersecting fractures network are all formulated successfully. The stability of the discrete solution is analyzed, and optimal error estimates in H(div)-norm for the velocity and in L2-norm for the pressure are derived, as well as in the discrete H1-norm for the saturation. Numerical experiments with a single fracture and a T-junction intersecting fractures network are conducted to verify the accuracy of our theoretical analysis.

关键词： Two-phase flow Hybrid-dimensional fracture model Intersecting fractures mixed finite element method Discontinuous Galerkin method

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A mixed finite element method ON A STAGGERED MESH FOR NAVIER-STOKES EQUATIONS

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Journal of Computational Mathematics 2008年第6期26卷 816-824页

作者： Houde Han Ming Yan Department of Mathematics University of Science and Technology of China Hefei 230026 China

In this paper, we introduce a mixed finite element method on a staggered mesh for the numerical solution of the steady state Navier-Stokes equations in which the two components of the velocity and the pressure are defined on three different meshes. This method is a conforming quadrilateral Q1 × Q1 - P0 element approximation for the Navier-Stokes equations. First-order error estimates are obtained for both the velocity and the pressure. Numerical examples are presented to illustrate the effectiveness of the proposed method.

关键词： mixed finite element method Staggered mesh Navier-Stokes equations Error estimate.

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mixed finite element method for a Hemivariational Inequality of Stationary Navier-Stokes Equations

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JOURNAL OF SCIENTIFIC COMPUTING 2021年第1期89卷 1-22页

作者： Han, Weimin Czuprynski, Kenneth Jing, Feifei Univ Iowa Dept Math Iowa City IA 52242 USA Univ Iowa Program Appl Math & Computat Sci AMCS Iowa City IA 52242 USA Northwestern Polytech Univ Sch Math & Stat Xian 710129 Shaanxi Peoples R China Northwestern Polytech Univ Xian Key Lab Sci Computat & Appl Stat Xian 710129 Shaanxi Peoples R China

In this paper, we develop and study the mixed finite element method for a hemivariational inequality of the stationary Navier-Stokes equations (NS hemivariational inequality). The NS hemivariational inequality models the motion of a viscous incompressible fluid in a bounded domain, subject to a nonsmooth and nonconvex slip boundary condition. The incompressibility contraint is treated through a mixed formulation. Solution existence and uniqueness are explored. The mixed finite element method is applied to solve the NS hemivariational inequality and error estimates are derived. Numerical results are reported on the use of the P1b/P1 pair, illustrating the optimal convergence order predicted by the error analysis.

关键词： Navier-Stokes equations Hemivariational inequality Existence Uniqueness mixed finite element method Error estimation

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mixed finite-element method FOR MISCIBLE DISPLACEMENT PROBLEMS IN POROUS-MEDIA

SOCIETY OF PETROLEUM ENGINEERS JOURNAL

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SOCIETY OF PETROLEUM ENGINEERS JOURNAL 1984年第4期24卷 391-398页

作者： DARLOW, BL EWING, RE WHEELER, MF RICE UNIV HOUSTONTX 77251 MOBIL RES & DEV CORP PAULSBORONJ 08066

Abstract. Effective numerical simulation of many EOR problems requires very accurate approximation of the Darcy velocities of the respective fluids. In this paper we describe a new method for the accurate determination of the Darcy velocity of the total fluid in the miscible displacement of one incompressible fluid by another in a porous medium. The new mixed finite-element procedure solves for both the pressure and velocity of the total fluid simultaneously as a system of first-order partial differential equations. By solving for u = (− k/μ) ∇p as one term, we minimize the difficulties occurring in standard methods caused by differentiation or differencing of p and multiplication by rough coefficients k/μ.By using mixed finite elements for the pressure equation coupled in a sequential method with a finite element procedure for the concentration of the invading fluid, we are able to treat a variety of problems with variable permeabilities, different mobility ratios, and a fairly general location of injection and production wells. mixed finite-element methods also produce minimal grid-orientation effect. Computational results on a variety of two-dimensional (2D) problems are presented.

关键词： pressure equation mixed finite element method equation reservoir simulation mobility ratio miscible displacement problem fluid dynamics flow in porous media penalty finite element method Flow in porous media Improved and Enhanced Recovery Reservoir Fluid Dynamics Reservoir Simulation

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mixed finite element method and the characteristics-mixed finite element method for a slightly compressible miscible displacement problem in porous media

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MATHEMATICS AND COMPUTERS IN SIMULATION 2015年 107卷 24-45页

作者： Sun, Tongjun Yuan, Yirang Shandong Univ Sch Math Jinan 250100 Peoples R China

Single-phase, two-component slightly compressible miscible flow in porous media, is governed by a system of nonlinear partial differential equations. The density of the mixture is related to the pressure and the concentration. An approximate scheme is introduced for this system, which is constructed by two methods. A standard mixed finite element is used for the pressure equation. A characteristic-mixed finite element method is presented for the concentration equation. Characteristic approximation is applied to handle the convection part of the concentration equation, and a lowest order mixed finite element spatial approximation is adopted to deal with the diffusion part. Thus, the scalar unknown concentration and the diffusive flux can be approximated simultaneously. In order to derive the optimal L-2-norm error estimates, a post-processing step is included in the approximation to the scalar unknown concentration. A numerical experiment is presented finally to validate the theoretical analysis. (C) 2014 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Slightly compressible miscible displacement Characteristic-mixed finite element method mixed finite element method Post-processing step

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mixed finite element method for a beam equation with the p(x)-biharmonic operator

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2023年第1期139卷 57-67页

作者： Almeida, Rui M. P. Duque, Jose C. M. Ferreira, Jorge Panni, Willian S. Univ Beira Interior Math & Applicat Ctr Rua Marques dAvila e Bolama P-6201001 Covilha Portugal Fed Fluminense Univ Dept Exact Sci Ave Trabalhadores 420 BR-27255125 Volta Redonda RJ Brazil

In this paper, we consider a nonlinear beam equation with the p(x)-biharmonic operator, where the exponent p(x) is a given function. We transform the problem into a system of two differential equations and prove the existence, uniqueness and regularity of the weak solution. Then we formulate the discrete problem associated with that system using the finite element method and prove the existence, uniqueness and stability of the discrete solution. We also investigate the order of convergence and prove some error estimates. Next, we apply the Lagrange basis to obtain an algebraic system of equations. Finally, we implement the computational codes in Matlab software considering the one and two dimensional cases and we present examples to illustrate the theory.

关键词： p(x)-biharmonic operator Weak solution Convergence order mixed finite element method Numerical simulations Beam equation

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