A mixed differential quadrature and finite element free vibration and buckling analysis of thick beams on two-parameter elastic foundations

As a first endeavor, a mixed differential quadrature (DQ) and finite element (FE) method for boundary value structural problems in the context of free vibration and buckling analysis of thick beams supported on two-parameter elastic foundations is presented. The formulations are based on the two-dimensional theory of elasticity. The problem domain along axial direction is discretized using finite elements. The resulting system of equations and the related boundary conditions are discretized in the thickness direction and in strong-form using DQM. The method benefits from low computational efforts of the DQ in conjunction with the effectiveness of the FE method in general geometry and systematic boundary treatment resulting in highly accurate and fast convergence behavior solution. The boundary conditions at the top and bottom surface of the beams are implemented accurately. The presented formulations provide an effective analysis tool for beams free of shear locking. Comparisons are made with results from elasticity solutions as well as higher-order beam theory.     

在动载荷作用下框架结构大变形分析的微分代数方法

采用弧坐标首先建立了在动载荷作用下,具有不连续性条件和初始位移的框架结构大变形分析的非线性数学模型．其次, 在空间区域内, 采用微分求积单元法（DQEM）来离散非线性数学模型, 并提出了在使用DQEM来求解结构大变形分析中,多个变量具有间断性条件的有效方法,得到了一组非线性DQEM的离散化方程,它是时间域内的一组具有奇异性的非线性微分-代数方程．同时也给出了求解非线性微分-代数方程组的一个解法A·D2作为应用,求解了受集中力和分布力作用的框架和组合框架的大变形静动力学问题,并与现有结果进行了比较．数值算例表明,处理多个变量具有间断性条件的方法和求解代数-微分系统的方法是一个有效的和一般的方法,它具有较少的节点、 较小的计算工作量、 较高的精度、良好的收敛性、 操作简单以及应用广泛等优点．     

The differential quadrature element method irregular element torsion analysis model

The differential quadrature element method (DQEM) has been proposed. The element weighting coefficient matrices are generated by the differential quadrature (DQ) or generic differential quadrature (GDQ). By using the DQ or GDQ technique and the mapping procedure the governing differential or partial differential equations, the transition conditions of two adjacent elements and the boundary conditions can be discretized. A global algebraic equation system can be obtained by assembling all of the discretized equations. This method can convert a generic engineering or scientific problem having an arbitrary domain configuration into a computer algorithm. The DQEM irregular element torsion analysis model is developed.     

Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method

In this article, a combination of the finite element (FE) and differential quadrature (DQ) methods is used to solve the eigenvalue (buckling and free vibration) equations of rectangular thick plates resting on elastic foundations. The elastic foundation is described by the Pasternak (two-parameter) model. The three dimensional, linear and small strain theory of elasticity and energy principle are employed to derive the governing equations. The in-plane domain is discretized using two dimensional finite elements. The spatial derivatives of equations in the thickness direction are discretized in strong-form using DQM. Buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with Pasternak elastic foundation are investigated using the proposed FE-DQ method. The results obtained by the mixed method have been verified by the few analytical solutions in the literature. It is concluded that the mixed FE-DQ method has good convergancy behavior; and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom.     

Numerical analysis of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D incompressible flow with large data

In this article, we develop a branch of nonsingular solutions of a Picard multilevel stabilization of mixed finite volume method for the 2D/3D stationary Navier‐Stokes equations without relying on the unique solution condition. The method presented consists of capturing almost all information of initial problem (the nonlinear problems) on the coarsest mesh and then performs one Picard defect correction (the linear problems) on each subsequent mesh based on previous information thus only solving one large linear systems. What is more, the method presented can results in a better coefficient matrix in the model presented with small viscosity. Theoretical results show that the method presented is derived with the convergence rate of the same order as the corresponding finite volume method/finite element method solving the stationary Navier‐Stokes equations on a fine mesh. Therefore, the method presented is definitely more efficient than the standard finite volume method/finite element method. Finally, numerical experiments clearly show the efficiency of the method presented for solving the stationary Navier‐Stokes equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 30–50, 2018     

A differential quadrature analysis of unsteady open channel flow

A rapid, convergent and accurate differential quadrature method (DQM) is employed for numerical simulation of unsteady open channel flow. To the best of authors’ knowledge, this is the first attempt to use the DQM in open channel hydraulics. The Saint-Venant equations and the related nonhomogenous, time dependent boundary conditions are discretized in spatial and temporal domain by DQ rules. The unknowns in the entire domain are computed by satisfying governing equations, boundary and initial conditions simultaneously. By employing DQM, accurate results can be obtained using dramatically less grid points in spatial and time domain. The stability of DQM solution is not sensitive to choosing time step or Courant number unlike other methods. Although numerical problems such as instability, oscillation and underestimation near critical depth can be seen by using other methods but DQM solution is smooth and accurate in this case. The results are sensitive to grid distribution in time domain. In light of this, Chebyshev–Gauss–Lobatto distribution performance is excellent. To validate the DQM solutions, the obtained results are compared with those of the characteristic method. In conclusion, DQM is a potential powerful method with minimum computational effort for unsteady flow simulation.     

ERROR ESTIMATES FOR TWO-SCALE COMPOSITE FINITE ELEMENT APPROXIMATIONS OF NONLINEAR PARABOLIC EQUATIONS

We study spatially semidiscrete and fully discrete two-scale composite finite element method for approximations of the nonlinear parabolic equations with homogeneous Dirich-let boundary conditions in a convex polygonal domain in the plane.This new class of finite elements,which is called composite finite elements,was first introduced by Hackbusch and Sauter[Numer.Math.,75(1997),pp.447-472]for the approximation of partial differential equations on domains with complicated geometry.The aim of this paper is to introduce an efficient numerical method which gives a lower dimensional approach for solving par-tial differential equations by domain discretization method.The composite finite element method introduces two-scale grid for discretization of the domain,the coarse-scale and the fine-scale grid with the degrees of freedom lies on the coarse-scale grid only.While the fine-scale grid is used to resolve the Dirichlet boundary condition,the dimension of the finite element space depends only on the coarse-scale grid.As a consequence,the resulting linear system will have a fewer number of unknowns.A continuous,piecewise linear composite finite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods.We have derived the error estimates in the L∞(L2)-norm for both semidiscrete and fully discrete schemes.Moreover,numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.     

Analysis of a domain decomposition method for the nearly elastic wave equations based on mixed finite element methods

This paper concerns the numerical solution of the nearly elasticwave equations with the first-order absorbing boundary condition;these equations describe the motion of a nearly elastic solidin the frequency domain. Two mixed finite elements, the Johnson-Mercierelement and the Arnold-Douglas-Gupta element, are adapted andanalyzed for the problem. The resulting mixed finite elementequations are complex-valued and are neither Hermitian nor definite.As a result, most standard iterative methods fail to convergefor the systems. To solve the mixed finite element equations,a parallelizable domain decomposition iterative method is proposed.The convergence of the method is demonstrated and a rate ofconvergence of the form 1 - Ch is derived. These results arevalid for the case when the original domain is decomposed intosubdomains which consist of an individual element associatedwith the above two mixed finite elements.     

A computationally efficient modification of mixed finite element methods for flow problems with full transmissivity tensors

The mixed finite element method for approximately solving flow equations in porous media has received a good deal of attention in the literature. The main idea is to solve for the head/pressure and fluid velocity (Darcy velocity) simultaneously to obtain a higher order approximation of the fluid velocity. In the case of a diagonal transmissivity tensor the algebraic equations resulting from the discretization can be reduced to a system of algebraic equations for the head/pressure variable alone. This reduction results in a smaller number of unknows to be solved for in an iterative method such as preconditioned conjugate gradient method. The fluid velocity is then obtained from an algebraic relationship. In the case of full transmissivity tensor, the algebraic reduction is more difficult. This paper investigates some algorithms resulting from the modification of the mixed finite element that take advantage of the mixed finite element method for the diagonal tensor case. The resulting schemes are more efficient implementations that maintain the same order of accuracy as the original schemes. © 1993 John Wiley & Sons, Inc.     

A finite element method for granular flow through a frictional boundary

A finite element method for the flow of dry granular solids through a domain involving a frictional contact boundary is formulated. The granular material is assumed as a compressible viscous-elastic–plastic continuum. Based on the principles of continuum mechanics, a complete set of equations is developed. The resulting boundary value problem is solved by the finite element method in space and by the finite difference method in time. The derivation of the finite element equations and the mathematical framework of the numerical technique are presented, together with two illustrative examples to demonstrate the validity of the technique.     

区域分裂法求非线性抛物方程的扩张混合元解

针对非线性抛物方程,给出了全离散的扩张混合元格式,利用一个建立在非重叠型区域分裂技巧上的并行迭代法求解了最后的非线性代数方程组,证明了迭代法的收敛性并给出了最优阶的误差估计.     

The primal mixed finite element method and the LBB condition

Spurious or kinematic modes have posed a major obstacle to the implementation of the mixed finite element method. This research shows that spurious modes resulting from the approximation spaces not satisfying the LBB condition do not prevent a well posed problem. When the LBB condition is not satisfied, the resulting matrix equations are singular. A direct solution method is presented for the efficient solution of the possibly singular equations. Orthogonal flux basis functions are introduced to simplify the problem. Then the solution procedure is based on nested domain decomposition. This solution procedure is shown to be competitive with direct solution methods for the displacement finite element method. Examples are included to demonstrate various aspects of the LBB condition and the solution procedure.     

A Positivity-preserving Finite Element Method for Quantum Drift-diffusion Model

In this paper, we propose a positivity-preserving finite element method for solving the three-dimensional quantum drift-diffusion model. The model consists of five nonlinear elliptic equations, and two of them describe quantum corrections for quasi-Fermi levels. We propose an interpolated-exponential finite element (IEFE) method for solving the two quantum-correction equations. The IEFE method always yields positive carrier densities and preserves the positivity of second-order differential operators in the Newton linearization of quantum-correction equations. Moreover, we solve the two continuity equations with the edge-averaged finite element (EAFE) method to reduce numerical oscillations of quasi-Fermi levels. The Poisson equation of electrical potential is solved with standard Lagrangian finite elements. We prove the existence of solution to the nonlinear discrete problem by using a fixed-point iteration and solving the minimum problem of a new discrete functional. A Newton method is proposed to solve the nonlinear discrete problem. Numerical experiments for a three-dimensional nano-scale FinFET device show that the Newton method is robust for source-to-gate bias voltages up to 9V and source-to-drain bias voltages up to 10V.     

Numerical analysis of a PML model for time-dependent Maxwell’s equations

In this paper, we study a perfectly matched layer model for the three-dimensional time-dependent Maxwell’s equations. We develop both semi- and fully-discrete finite element methods for solving the truncated PML problem by Nedelec edge elements. Optimal convergence rates are proved for both semi- and fully-discrete schemes. To our knowledge, this is the first error analysis obtained for time domain finite element method for PML models.     

The coupling of boundary integral and finite element methods for the bidimensional exterior steady stokes problem

In this paper, we represent a new numerical method for solving the steady-state Stokes equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior one. The solution in the exterior domain is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocitypressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. This is studied by means of an abstract framework, well adapted to the model problem, in which convergence results and optimal error estimates are derived. Computer results will be discussed in a forthcoming paper.     

A power penalty approach to a mixed quasilinear elliptic complementarity problem

In this paper, a power penalty approximation method is proposed for solving a mixed quasilinear elliptic complementarity problem. The mixed complementarity problem is first reformulated as a double obstacle quasilinear elliptic variational inequality problem. A nonlinear elliptic partial differential equation is then defined to approximate the resulting variational inequality by using a power penalty approach. The existence and uniqueness of the solution to the partial differential penalty equation are proved. It is shown that, under some mild assumptions, the sequence of solutions to the penalty equations converges to the unique solution of the variational inequality problem as the penalty parameter tends to infinity. The error estimates of the convergence of this penalty approach are also derived. At last, numerical experimental results are presented to show that the power penalty approximation method is efficient and robust.

     

椭圆型方程的并行迭代区域分裂法——两个子区域情形

§1.问题的分析 设Ω?R~2是一有界开区域,是定义在Ω上的椭圆算子,其中对X∈Ω,[a_(i·j)(X)]_i,j=1,2对称且一致正定;a_(ij)(X)分片连续且上,下有界,a(X)≥0.我们求解如下问题: Lu=f,在Ω中, u=0,在?Ω上, (1.1)其中f∈H~(-1)(Ω),u∈H_0~1(Ω).这里取齐次Dirichlet边界条件,仅仅是为了叙述问题的方便.(1.1)的变分形式是     

解重调和问题混合有限元方程的直接方法

§1.引言 考虑如下重调和方程的齐次边值问题: △~2w=f,在Ω中, w=?w/?v=0,在Ω上.(1.1)其中Ω是平面凸多边形区域,?Ω是Ω的边界,?/?v表示?Ω上的外法向导数.     

A Time Finite Element Method Based on the Generalized Variational Calculus

In this contribution, a new finite element method in the temporal domain is presented, in which the time step size is introduced as an additional variable. Thus, the variation of the time integral of the Lagrangean resulting from Hamilton's principle has to be carried out with respect to the rules of the generalized variational calculus. Apart from the usual time integral of the Euler‐Lagrange differential equations, the so‐called transversality condition is obtained as an additional result representing a time‐boundary term, which is used to obtain an optimal step size in the time domain.     

Immersed finite element methods for 4th order differential equations

We propose three new finite element methods for solving boundary value problems of 4th order differential equations with discontinuous coefficients. Typical differential equations modeling the small transverse displacement of a beam and a thin plate formed by multiple uniform materials are considered. One important feature of these finite element methods is that their meshes can be independent of the interface between different materials. Finite element spaces based on both the conforming and mixed formulations are presented. Numerical examples are given to illustrate capabilities of these methods.