The Galerkin-Petrov least squares method is combined with the mixed finite element method to deal with the stationary, incompressible magnetohydrodynamics system of equations with viscosity. A Galerkin-Petrov least sq...The Galerkin-Petrov least squares method is combined with the mixed finite element method to deal with the stationary, incompressible magnetohydrodynamics system of equations with viscosity. A Galerkin-Petrov least squares mixed finite element format for the stationary incompressible magnetohydrodynamics equations is presented. And the existence and error estimates of its solution are derived. Through this method, the combination among the mixed finite element spaces does not demand the discrete Babuska-Brezzi stability conditions so that the mixed finite element spaces could be chosen arbitrartily and the error estimates with optimal order could be obtained.展开更多
A nonlinear Galerkin mixed element (NGME) method and a posteriori error exstimator based on the method are established for the stationary Navier-Stokes equations. The existence and error estimates of the NGME solution...A nonlinear Galerkin mixed element (NGME) method and a posteriori error exstimator based on the method are established for the stationary Navier-Stokes equations. The existence and error estimates of the NGME solution are first discussed, and then a posteriori error estimator based on the NGME method is derived.展开更多
A nonlinear Galerkin mixed element （NGME） method for the stationary incompressible magnetohydrodynamics equations is presented. And the existence and error estimates of the NGME solution are derived.
In this paper,a composite numerical scheme is proposed to solve the threedimensional Darcy-Forchheimer miscible displacement problem with positive semi-definite assumptions.A mixed finite element is used for the fow e...In this paper,a composite numerical scheme is proposed to solve the threedimensional Darcy-Forchheimer miscible displacement problem with positive semi-definite assumptions.A mixed finite element is used for the fow equation.The velocity and pressure are computed simultaneously.The accuracy of velocity is improved one order.The concentration equation is solved by using mixed finite element,multi-step difference and upwind approximation.A multi-step method is used to approximate time derivative for improving the accuracy.The upwind approximation and an expanded mixed finite element are adopted to solve the convection and diffusion,respectively.The composite method could compute the diffusion flux and its gradient.It possibly becomes an eficient tool for solving convection-dominated diffusion problems.Firstly,the conservation of mass holds.Secondly,the multi-step method has high accuracy.Thirdly,the upwind approximation could avoid numerical dispersion.Using numerical analysis of a priori estimates and special techniques of differential equations,we give an error estimates for a positive definite problem.Numerical experiments illustrate its computational efficiency and feasibility of application.展开更多
This paper discusses convergence and complexity of arbitrary,but fixed,order adaptive mixed element methods for the Poisson equation in two and three dimensions.The two main ingredients in the analysis,namely the quas...This paper discusses convergence and complexity of arbitrary,but fixed,order adaptive mixed element methods for the Poisson equation in two and three dimensions.The two main ingredients in the analysis,namely the quasi-orthogonality and the discrete reliability,are achieved by use of a discrete Helmholtz decomposition and a discrete inf-sup condition.The adaptive algorithms are shown to be contractive for the sum of the error of flux in L2-norm and the scaled error estimator after each step of mesh refinement and to be quasi-optimal with respect to the number of elements of underlying partitions.The methods do not require a separate treatment for the data oscillation.展开更多
The non_stationary natural convection problem is studied. A lowest order finite difference scheme based on mixed finite element method for non_stationary natural convection problem, by the spatial variations discreted...The non_stationary natural convection problem is studied. A lowest order finite difference scheme based on mixed finite element method for non_stationary natural convection problem, by the spatial variations discreted with finite element method and time with finite difference scheme was derived, where the numerical solution of velocity, pressure, and temperature can be found together, and a numerical example to simulate the close square cavity is given, which is of practical importance.展开更多
An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same t...An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.展开更多
A splitting positive definite mixed finite element method is proposed for second-order viscoelasticity wave equation. The proposed procedure can be split into three independent symmetric positive definite integro-diff...A splitting positive definite mixed finite element method is proposed for second-order viscoelasticity wave equation. The proposed procedure can be split into three independent symmetric positive definite integro-differential sub-system and does not need to solve a coupled system of equations. Error estimates are derived for both semidiscrete and fully discrete schemes. The existence and uniqueness for semidiscrete scheme are proved. Finally, a numerical example is provided to illustrate the efficiency of the method.展开更多
A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order...A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.展开更多
A numerical method based on the explicit two-step method in time direction and the mixed finite element method in spatial direction is presented for the symmetric regularized long wave(SRLW)equation.The optimal a prio...A numerical method based on the explicit two-step method in time direction and the mixed finite element method in spatial direction is presented for the symmetric regularized long wave(SRLW)equation.The optimal a priori error estimates(O((∆t)^(2)+h^(m+1)+h^(k+1)))for fully discrete explicit two-step mixed scheme are derived.Moreover,a numerical example is provided to confirm our theoretical results.展开更多
This paper proposes a new, simple and efficient method for nonlinear simulation of arch dam cracking from the construction period to the operation period, which takes into account the arch dam construction process and...This paper proposes a new, simple and efficient method for nonlinear simulation of arch dam cracking from the construction period to the operation period, which takes into account the arch dam construction process and temperature loads. In the calculation mesh, the contact surface of pair nodes is located at places on the arch dam where cracking is possible. A new effective iterative method, the mixed finite element method for friction-contact problems, is improved and used for nonlinear simulation of the cracking process. The forces acting on the structure are divided into two parts： external forces and contact forces. The displacement of the structure is chosen as the basic variable and the nodal contact force in the possible contact region of the local coordinate system is chosen as the iterative variable, so that the nonlinear iterative process is only limited within the possible contact surface and is much more economical. This method was used to simulate the cracking process of the Shuanghe Arch Dam in Southwest China. In order to prove the validity and accuracy of this method and to study the effect of thermal stress on arch dam cracking, three schemes were designed for calculation. Numerical results agree with actual measured data, proving that it is feasible to use this method to simulate the entire process of nonlinear arch dam cracking.展开更多
The transient behavior of a semiconductor device consists of a Poisson equation for the electric potential and of two nonlinear parabolic equations for the electron density and hole density. The electric potential equ...The transient behavior of a semiconductor device consists of a Poisson equation for the electric potential and of two nonlinear parabolic equations for the electron density and hole density. The electric potential equation is discretized by a mixed finite element method. The electron and hole density equations are treated by implicit-explicit multistep finite element methods. The schemes are very efficient. The optimal order error estimates both in time and space are derived.展开更多
A least-squares mixed finite element method was formulated for a class of Stokes equations in two dimensional domains. The steady state and the time-dependent Stokes' equations were considered. For the stationary ...A least-squares mixed finite element method was formulated for a class of Stokes equations in two dimensional domains. The steady state and the time-dependent Stokes' equations were considered. For the stationary equation, optimal H-t and L-2-error estimates are derived under the standard regularity assumption on the finite element partition ( the LBB-condition is not required). Far the evolutionary equation, optimal L-2 estimates are derived under the conventional Raviart-Thomas spaces.展开更多
An initial-boundary value problem for shallow equation system consisting of water dynamics equations,silt transport equation, the equation of bottom topography change,and of some boundary and initial conditions is stu...An initial-boundary value problem for shallow equation system consisting of water dynamics equations,silt transport equation, the equation of bottom topography change,and of some boundary and initial conditions is studied, the existence of its generalized solution and semidiscrete mixed finite element(MFE) solution was discussed, and the error estimates of the semidiscrete MFE solution was derived.The error estimates are optimal.展开更多
The vapor deposition chemical reaction processes, which are of extremely extensive applications, can be classified as a mathematical model by the following governing nonlinear partial differential equations containing...The vapor deposition chemical reaction processes, which are of extremely extensive applications, can be classified as a mathematical model by the following governing nonlinear partial differential equations containing velocity vector, temperature field, pressure field, and gas mass field. The mixed finite element （MFE） method is employed to study the system of equations for the vapor deposition chemical reaction processes. The semidiscrete and fully discrete MFE formulations are derived. And the existence and convergence （error estimate） of the semidiscrete and fully discrete MFE solutions are demonstrated. By employing MFE method to treat the system of equations for the vapor deposition chemical reaction processes, the numerical solutions of the velocity vector, the temperature field, the pressure field, and the gas mass field can be found out simultaneously. Thus, these researches are not only of important theoretical means, but also of extremely extensive applied vistas.展开更多
The penalty and hybrid methods are being much used in dealing with the general incompatible element, With the penalty method convergence can always be assured, but comparatively speaking its accuracy is lower, and the...The penalty and hybrid methods are being much used in dealing with the general incompatible element, With the penalty method convergence can always be assured, but comparatively speaking its accuracy is lower, and the condition number and sparsity are not so good. With the hybrid method, convergence can be assured only when the rank condition is satisfied. So the construction of the element is extremely limited. This paper presents the mixed hybrid penalty element method, which combines the two methods together. And it is proved theoretically that this new method is convergent, and it has the same accuracy, condition number and sparsity as the compatible element. That is to say, they are optimal to each other.Finally, a new triangle element for plate bending with nine freedom degrees is constructed with this method (three degreesof freedom are given on each corner -- one displacement and tworotations), the calculating formula of the element stiffness matrix is almost the same as that of the old triangle element for plate bending with nine degrees of freedom But it is converged to true solution with arbitrary irregrlar triangle subdivision. If the true solution u?H3 with this method the linear and quadratic rates of convergence are obtianed for three bending moments and for the displacement and two rotations respectively.展开更多
In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the m...In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.展开更多
<span style="font-family:Verdana;">In this paper, for the initial and boundary value problem of beams with</span> <span style="font-family:Verdana;">structural damping, by introdu...<span style="font-family:Verdana;">In this paper, for the initial and boundary value problem of beams with</span> <span style="font-family:Verdana;">structural damping, by introducing intermediate variables, the original </span><span style="font-family:Verdana;">fourth-order problem is transformed into second-order partial differential equations, and the mixed finite volume element scheme is constructed, and the existence, uniqueness and convergence of the scheme are analyzed</span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">.</span></span></span><span><span><span style="font-family:Verdana;"> Numerical examples are provided to confirm the theoretical results. In the end, we test the value of <em>δ</em></span><span style="font-family:Verdana;"> to observe its influence on the model.</span></span></span>展开更多
Beam is one of the common structures in engineering,with the development of technology,homogeneous beams no longer meet the needs of engineering structural design,for this reason,people have researched the non-homogen...Beam is one of the common structures in engineering,with the development of technology,homogeneous beams no longer meet the needs of engineering structural design,for this reason,people have researched the non-homogeneous beams.In this paper,we study the mixed finite element method for the vibration problem of non-homogeneous damped beams.The fourth-order differential equations are transformed into a system of low-order partial differential equations by introducing intermediate variables,constructing a semidiscrete extended mixed finite element format,proving the existence and uniqueness of the solution of the format,and utilizing the elliptic projection operator for the error estimation.The time derivative term is discretized by the central difference,and the fully discrete mixed element format is given to prove the stability and convergence of the format.The feasibility and effectiveness of the mixed method are verified by numerical examples,and the effects of different damping coefficientsμon beam vibration are investigated.展开更多
This study enhances the application of cross-sectional warping considered mixed finite element(WMFE)formulation to accurately determine natural vibration,static displacement response,and shear and normal stress evalua...This study enhances the application of cross-sectional warping considered mixed finite element(WMFE)formulation to accurately determine natural vibration,static displacement response,and shear and normal stress evaluation with very close to the precision of solid finite elements(FEs)in two-phase/multi-phase functionally graded(FG)laminated composite beams strength using carbon nanotubes(CNTs).The principles of three dimensional(3D)elasticity theory are used to derive constitutive equations.The mixed finite element(MFE)method is improved by accounting for warping effects by displacement-based FEs within the cross-sectional domain.The MFE with two nodes has a total of 24 degrees of freedom.The two-phase material consists of a polymer matrix reinforced with aligned CNTs that are FG throughout the beam thickness.The multi-phase FG beam is modeled as a three-component composite material,consisting of CNTs,a polymer matrix,and fibers.The polymer matrix is reinforced by longitudinally aligned fibers and randomly dispersed CNT particles.The fiber volume fractions are considered to change gradually through the thickness of the beam following a power-law variation.The W-MFE achieves satisfactory results with fewer degrees of freedom than 3D solid FEs.Benchmark examples examine the effects of ply orientation,configuration,and fiber gradation on FG beam behavior.展开更多
基金Project supported by the National Natural Science Foundation of China (Nos.10471100 and 40437017)the Science and Technology Foundation of Beijing Jiaotong University
文摘The Galerkin-Petrov least squares method is combined with the mixed finite element method to deal with the stationary, incompressible magnetohydrodynamics system of equations with viscosity. A Galerkin-Petrov least squares mixed finite element format for the stationary incompressible magnetohydrodynamics equations is presented. And the existence and error estimates of its solution are derived. Through this method, the combination among the mixed finite element spaces does not demand the discrete Babuska-Brezzi stability conditions so that the mixed finite element spaces could be chosen arbitrartily and the error estimates with optimal order could be obtained.
文摘A nonlinear Galerkin mixed element (NGME) method and a posteriori error exstimator based on the method are established for the stationary Navier-Stokes equations. The existence and error estimates of the NGME solution are first discussed, and then a posteriori error estimator based on the NGME method is derived.
基金Project supported by the National Natural Science Foundation of China (Nos.10471100 and 40437017)
文摘A nonlinear Galerkin mixed element （NGME） method for the stationary incompressible magnetohydrodynamics equations is presented. And the existence and error estimates of the NGME solution are derived.
基金supported by the Natural Science Foundation of Shandong Province(ZR2021MA019)the National Natural Science Foundation of China(11871312)。
文摘In this paper,a composite numerical scheme is proposed to solve the threedimensional Darcy-Forchheimer miscible displacement problem with positive semi-definite assumptions.A mixed finite element is used for the fow equation.The velocity and pressure are computed simultaneously.The accuracy of velocity is improved one order.The concentration equation is solved by using mixed finite element,multi-step difference and upwind approximation.A multi-step method is used to approximate time derivative for improving the accuracy.The upwind approximation and an expanded mixed finite element are adopted to solve the convection and diffusion,respectively.The composite method could compute the diffusion flux and its gradient.It possibly becomes an eficient tool for solving convection-dominated diffusion problems.Firstly,the conservation of mass holds.Secondly,the multi-step method has high accuracy.Thirdly,the upwind approximation could avoid numerical dispersion.Using numerical analysis of a priori estimates and special techniques of differential equations,we give an error estimates for a positive definite problem.Numerical experiments illustrate its computational efficiency and feasibility of application.
基金supported by National Natural Science Foundation of China (Grant Nos. 11171219,11161130004)E-Institutes of Shanghai Municipal Education Commission(Grant No. E03004)+1 种基金supported by Shanghai Leading Discipline Project(Grant No. N.S30405)Shanghai Normal University Research Program (Grant No. SK201202)
文摘This paper discusses convergence and complexity of arbitrary,but fixed,order adaptive mixed element methods for the Poisson equation in two and three dimensions.The two main ingredients in the analysis,namely the quasi-orthogonality and the discrete reliability,are achieved by use of a discrete Helmholtz decomposition and a discrete inf-sup condition.The adaptive algorithms are shown to be contractive for the sum of the error of flux in L2-norm and the scaled error estimator after each step of mesh refinement and to be quasi-optimal with respect to the number of elements of underlying partitions.The methods do not require a separate treatment for the data oscillation.
文摘The non_stationary natural convection problem is studied. A lowest order finite difference scheme based on mixed finite element method for non_stationary natural convection problem, by the spatial variations discreted with finite element method and time with finite difference scheme was derived, where the numerical solution of velocity, pressure, and temperature can be found together, and a numerical example to simulate the close square cavity is given, which is of practical importance.
基金Supported by the National Natural Science Foundation of China (10601022)Natural Science Foundation of Inner Mongolia Autonomous Region (200607010106)Youth Science Foundation of Inner Mongolia University(ND0702)
文摘An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.
基金This work was supported in part by the National Natural Science Foundation of China （Grant No, 11061021）, the Program of Higher-level Talents of Inner Mongolia University （No. Z200901004）, and the Scientific Research Projection of Higher Schools of Inner Mongolia （Nos. N J10006, N J10016, NJZZ12011）.
文摘A splitting positive definite mixed finite element method is proposed for second-order viscoelasticity wave equation. The proposed procedure can be split into three independent symmetric positive definite integro-differential sub-system and does not need to solve a coupled system of equations. Error estimates are derived for both semidiscrete and fully discrete schemes. The existence and uniqueness for semidiscrete scheme are proved. Finally, a numerical example is provided to illustrate the efficiency of the method.
基金supported by the National Natural Science Foundation of China (No. 10601022)NSF ofInner Mongolia Autonomous Region of China (No. 200607010106)513 and Science Fund of InnerMongolia University for Distinguished Young Scholars (No. ND0702)
文摘A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.
基金supported by the National Natural Science Fund of China(11061021,11301258 and 11361035)the Scientific Research Projection of Higher Schools of Inner Mongolia(NJZZ12011 and NJZY13199)+1 种基金the Natural Science Fund of Inner Mongolia Province(2012MS0106 and 2012MS0108)the Program of Higher-level talents of Inner Mongolia University(125119).
文摘A numerical method based on the explicit two-step method in time direction and the mixed finite element method in spatial direction is presented for the symmetric regularized long wave(SRLW)equation.The optimal a priori error estimates(O((∆t)^(2)+h^(m+1)+h^(k+1)))for fully discrete explicit two-step mixed scheme are derived.Moreover,a numerical example is provided to confirm our theoretical results.
基金supported by the National Nature Science Foundation of China (Grant No 90510017)
文摘This paper proposes a new, simple and efficient method for nonlinear simulation of arch dam cracking from the construction period to the operation period, which takes into account the arch dam construction process and temperature loads. In the calculation mesh, the contact surface of pair nodes is located at places on the arch dam where cracking is possible. A new effective iterative method, the mixed finite element method for friction-contact problems, is improved and used for nonlinear simulation of the cracking process. The forces acting on the structure are divided into two parts： external forces and contact forces. The displacement of the structure is chosen as the basic variable and the nodal contact force in the possible contact region of the local coordinate system is chosen as the iterative variable, so that the nonlinear iterative process is only limited within the possible contact surface and is much more economical. This method was used to simulate the cracking process of the Shuanghe Arch Dam in Southwest China. In order to prove the validity and accuracy of this method and to study the effect of thermal stress on arch dam cracking, three schemes were designed for calculation. Numerical results agree with actual measured data, proving that it is feasible to use this method to simulate the entire process of nonlinear arch dam cracking.
文摘The transient behavior of a semiconductor device consists of a Poisson equation for the electric potential and of two nonlinear parabolic equations for the electron density and hole density. The electric potential equation is discretized by a mixed finite element method. The electron and hole density equations are treated by implicit-explicit multistep finite element methods. The schemes are very efficient. The optimal order error estimates both in time and space are derived.
文摘A least-squares mixed finite element method was formulated for a class of Stokes equations in two dimensional domains. The steady state and the time-dependent Stokes' equations were considered. For the stationary equation, optimal H-t and L-2-error estimates are derived under the standard regularity assumption on the finite element partition ( the LBB-condition is not required). Far the evolutionary equation, optimal L-2 estimates are derived under the conventional Raviart-Thomas spaces.
文摘An initial-boundary value problem for shallow equation system consisting of water dynamics equations,silt transport equation, the equation of bottom topography change,and of some boundary and initial conditions is studied, the existence of its generalized solution and semidiscrete mixed finite element(MFE) solution was discussed, and the error estimates of the semidiscrete MFE solution was derived.The error estimates are optimal.
基金Project supported by the National Natural Science Foundation of China (Nos.10471100 and 40437017)the Science and Technology Foundation of Beijing Jiaotong University
文摘The vapor deposition chemical reaction processes, which are of extremely extensive applications, can be classified as a mathematical model by the following governing nonlinear partial differential equations containing velocity vector, temperature field, pressure field, and gas mass field. The mixed finite element （MFE） method is employed to study the system of equations for the vapor deposition chemical reaction processes. The semidiscrete and fully discrete MFE formulations are derived. And the existence and convergence （error estimate） of the semidiscrete and fully discrete MFE solutions are demonstrated. By employing MFE method to treat the system of equations for the vapor deposition chemical reaction processes, the numerical solutions of the velocity vector, the temperature field, the pressure field, and the gas mass field can be found out simultaneously. Thus, these researches are not only of important theoretical means, but also of extremely extensive applied vistas.
文摘The penalty and hybrid methods are being much used in dealing with the general incompatible element, With the penalty method convergence can always be assured, but comparatively speaking its accuracy is lower, and the condition number and sparsity are not so good. With the hybrid method, convergence can be assured only when the rank condition is satisfied. So the construction of the element is extremely limited. This paper presents the mixed hybrid penalty element method, which combines the two methods together. And it is proved theoretically that this new method is convergent, and it has the same accuracy, condition number and sparsity as the compatible element. That is to say, they are optimal to each other.Finally, a new triangle element for plate bending with nine freedom degrees is constructed with this method (three degreesof freedom are given on each corner -- one displacement and tworotations), the calculating formula of the element stiffness matrix is almost the same as that of the old triangle element for plate bending with nine degrees of freedom But it is converged to true solution with arbitrary irregrlar triangle subdivision. If the true solution u?H3 with this method the linear and quadratic rates of convergence are obtianed for three bending moments and for the displacement and two rotations respectively.
文摘In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.
文摘<span style="font-family:Verdana;">In this paper, for the initial and boundary value problem of beams with</span> <span style="font-family:Verdana;">structural damping, by introducing intermediate variables, the original </span><span style="font-family:Verdana;">fourth-order problem is transformed into second-order partial differential equations, and the mixed finite volume element scheme is constructed, and the existence, uniqueness and convergence of the scheme are analyzed</span><span style="font-family:Verdana;"><span style="font-family:Verdana;"><span style="font-family:Verdana;">.</span></span></span><span><span><span style="font-family:Verdana;"> Numerical examples are provided to confirm the theoretical results. In the end, we test the value of <em>δ</em></span><span style="font-family:Verdana;"> to observe its influence on the model.</span></span></span>
文摘Beam is one of the common structures in engineering,with the development of technology,homogeneous beams no longer meet the needs of engineering structural design,for this reason,people have researched the non-homogeneous beams.In this paper,we study the mixed finite element method for the vibration problem of non-homogeneous damped beams.The fourth-order differential equations are transformed into a system of low-order partial differential equations by introducing intermediate variables,constructing a semidiscrete extended mixed finite element format,proving the existence and uniqueness of the solution of the format,and utilizing the elliptic projection operator for the error estimation.The time derivative term is discretized by the central difference,and the fully discrete mixed element format is given to prove the stability and convergence of the format.The feasibility and effectiveness of the mixed method are verified by numerical examples,and the effects of different damping coefficientsμon beam vibration are investigated.
基金funding provided by the Scientific and Technological Research Council of Türkiye(TÜBİTAK).
文摘This study enhances the application of cross-sectional warping considered mixed finite element(WMFE)formulation to accurately determine natural vibration,static displacement response,and shear and normal stress evaluation with very close to the precision of solid finite elements(FEs)in two-phase/multi-phase functionally graded(FG)laminated composite beams strength using carbon nanotubes(CNTs).The principles of three dimensional(3D)elasticity theory are used to derive constitutive equations.The mixed finite element(MFE)method is improved by accounting for warping effects by displacement-based FEs within the cross-sectional domain.The MFE with two nodes has a total of 24 degrees of freedom.The two-phase material consists of a polymer matrix reinforced with aligned CNTs that are FG throughout the beam thickness.The multi-phase FG beam is modeled as a three-component composite material,consisting of CNTs,a polymer matrix,and fibers.The polymer matrix is reinforced by longitudinally aligned fibers and randomly dispersed CNT particles.The fiber volume fractions are considered to change gradually through the thickness of the beam following a power-law variation.The W-MFE achieves satisfactory results with fewer degrees of freedom than 3D solid FEs.Benchmark examples examine the effects of ply orientation,configuration,and fiber gradation on FG beam behavior.
