Three-dimensional simulation of flat rolling through a combined finite element–boundary element approach

This paper presents an innovative approach for analysing three-dimensional flat rolling. The proposed approach is based on a solution resulting from the combination of the finite element method with the boundary element method. The finite element method is used to perform the rigid–plastic numerical modelling of the workpiece allowing the estimation of the roll separating force, rolling torque and contact pressure along the surface of the rolls. The boundary element method is applied for computing the elastic deformation of the rolls. The combination of the two numerical methods is made using the finite element solution of the contact pressure along the surface of the rolls to define the boundary conditions to be applied on the elastic analysis of the rolls. The validity of the proposed approach is discussed by comparing the theoretical predictions with experimental data found in the literature.     

A finite element method for interface problems in domains with smooth boundaries and interfaces

This paper is concerned with the analysis of a finite element method for nonhomogeneous second order elliptic interface problems on smooth domains. The method consists in approximating the domains by polygonal domains, transferring the boundary data in a natural way, and then applying a finite element method to the perturbed problem on the approximate polygonal domains. It is shown that the error in the finite element approximation is of optimal order for linear elements on a quasiuniform triangulation. As such the method is robust in the regularity of the data in the original problem.     

节点应力连续的四边形单元

节点应力连续的四边形单元Q4-CNS是一种基于单位分解理论的混合的有限元无网格法.Q4-CNS可以视作FE-LSPIM QUAD4的发展.Q4-CNS形函数的导数在节点处是连续的,因此可以自然的得到节点应力,而不需要使用节点应力磨平算法.数值实验表明,与传统四边形单元(QUAD4)相比,Q4-CNS具有更好的计算精度和更高的收敛速度.在扭曲网格下,Q4-CNS也能取得满意的数值精度.然而,QUAD4的数值精度则会随着网格的扭曲明显的变差.基于Kirchhoff-Love假设的非协调板单元计算中,不仅要求形函数在单元的交界面上要保持C0连续性,而且要求形函数在节点处具有C1连续性,所以在任意的四边形单元上构造满足插值条件的非协调板单元形函数较为困难.Q4-CNS形函数的导数在节点处是连续的,所以Q4-CNS在求解基于Kirchhoff-Love假设的板单元问题中具有潜在的应用价值.     

A Characteristic Finite Volume Element Method for the Air Pollution Model

In this article, a characteristic finite volume element method is presented for solving air pollution models. The convection term is discretized using the characteristic method and diffusion term is approximated by finite volume element method. Compared with standard finite volume element method, our proposed method is more accurate and efficient, especially suitable to solve convection-dominated problems. The proposed numerical schemes are analyzed for convergence in L ₂ norm. Some numerical results are presented to demonstrate the efficiency and accuracy of the method.     

A wavelet-based stochastic finite element method of thin plate bending

A wavelet-based stochastic finite element method is presented for the bending analysis of thin plates. The wavelet scaling functions of spline wavelets are selected to construct the displacement interpolation functions of a rectangular thin plate element and the displacement shape functions are expressed by the spline wavelets. A new wavelet-based finite element formulation of thin plate bending is developed by using the virtual work principle. A wavelet-based stochastic finite element method that combines the proposed wavelet-based finite element method with Monte Carlo method is further formulated. With the aid of the wavelet-based stochastic finite element method, the present paper can deal with the problem of thin plate response variability resulting from the spatial variability of the material properties when it is subjected to static loads of uncertain nature. Numerical examples of thin plate bending have demonstrated that the proposed wavelet-based stochastic finite element method can achieve a high numerical accuracy and converges fast.     

FLIM - A Variant of FEM based on Line Integration

A variant of the finite element method (FEM) for modelling and solving partial differential equations based on triangular and tetrahedral meshes is proposed. While FEM is based on integration over finite elements, the new approach - briefly denoted as FLIM hereafter - uses integration along edges (finite lines). The stiffness matrix, which - for linear triangles and tetrahedra - is identical with the one obtained with FEM, as well as the load vector can solely be obtained by summing up the edge contributions. This new variant requires much lower storage than FEM, especially for three-dimensional problems, but yields the same approximation error and convergence rate as the finite element method. It is shown that its performance, when applied to linear problems, is in close agreement with the performance of the finite element method. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

ERROR ESTIMATES FOR SPARSE OPTIMAL CONTROL PROBLEMS BY PIECEWISE LINEAR FINITE ELEMENT APPROXIMATION

Optimization problems with L¹-control cost functional subject to an elliptic partial differential equation(PDE)are considered.However,different from the finite dimensiona l¹-regularization optimization,the resulting discretized L¹norm does not have a decoupled form when the standard piecewise linear finite element is employed to discretize the continuous problem.A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the L¹-norm.In this paper,a new discretized scheme for the L¹-norm is presented.Compared to the new discretized scheme for L¹-norm with the nodal quadrature formula,the advantages of our new discretized scheme can be demonstrated in terms of the order of approximation.Moreover,finite element error estimates results for the primal problem with the new discretized scheme for the L¹-norm are provided,which confirms that this approximation scheme will not change the order of error estimates.To solve the new discretized problem,a symmetric Gauss-Seidel based majorized accelerated block coordinate descent(sGS-mABCD)method is introduced to solve it via its dual.The proposed sGS-mABCD algorithm is illustrated at two numerical examples.Numerical results not only confirm the finite element error estimates,but also show that our proposed algorithm is efficient.     

Weighted extended b-splines for one-dimensional electromagnetic problems

This paper considers the weighted extended b-splines as basis function for finite element method in electromagnetics and compares with the standard finite element method applied to the two-point boundary value problems with different boundary conditions. This new approach, which provides more accurate results than standard finite element method, is presented to compare other numerical techniques and applied to one-dimensional electromagnetic problems. Computed results are compared with other numerical results in literature.     

A TWO-LEVEL FINITE ELEMENT GALERKIN METHOD FOR THE NONSTATIONARY NAVIER-STOKES EQUATIONS II： TIME DISCRETIZATION

In this article we consider the fully discrete two-level finite element Galerkin method for the two-dimensional nonstationary incompressible Navier-Stokes equations. This method consists in dealing with the fully discrete nonlinear Navier-Stokes problem on a coarse mesh with width $H$ and the fully discrete linear generalized Stokes problem on a fine mesh with width $h << H$. Our results show that if we choose $H=O(h^{1/2}$) this method is as the same stability and convergence as the fully discrete standard finite element Galerkin method which needs dealing with the fully discrete nonlinear Navier-Stokes problem on a fine mesh with width $h$. However, our method is cheaper than the standard fully discrete finite element Galerkin method.     

Geometrically nonlinear analysis with a 4-node membrane element formulated by the quadrilateral area coordinate method

Recently, a 4-node quadrilateral membrane element AGQ6-I, has been successfully developed for analysis of linear plane problems. Since this model is formulated by the quadrilateral area coordinate method (QACM), a new natural coordinate system for developing quadrilateral finite element models, it is much less sensitive to mesh distortion than other 4-node isoparametric elements and free of various locking problems that arise from irregular mesh geometries. In order to extend these advantages of QACM to nonlinear applications, the total Lagrangian (TL) formulations of element AGQ6-I was established in this paper, which is also the first time that a plane QACM element being applied in the implicit geometrically nonlinear analysis. Numerical examples of geometrically nonlinear analysis show that the presented formulations can prevent loss of accuracy in severely distorted meshes, and therefore, are superior to those of other 4-node isoparametric elements. The efficiency of QACM for developing simple, effective and reliable serendipity plane membrane elements in geometrically nonlinear analysis is demonstrated clearly.     

Adaptive finite element analysis of structures under transient dynamic loading using modal superposition

In this paper, the error estimation and adaptive strategy developed for the linear elastodynamic problem under transient dynamic loading based on the Z–Z criterion is utilized for 2D and plate bending problems. An automatic mesh generator based on “growth meshing” is utilized effectively for adaptive mesh refinement. Optimal meshes are obtained iteratively corresponding to the prescribed domain discretization error limit and for a chosen number of basis modes satisfying modal truncation errors. Numerous examples show the effectiveness of the integrated approach in achieving the target accuracy in finite element transient dynamic analysis.     

Limit analysis decomposition and finite element mixed method

This paper proposes an original decomposition approach to the upper bound method of limit analysis. It is based on a mixed finite element approach and on a convex interior point solver, using linear or quadratic discontinuous velocity fields. Presented in plane strain, this method appears to be rapidly convergent, as verified in the Tresca compressed bar problem in the linear velocity case. Then, using discontinuous quadratic velocity fields, the method is applied to the celebrated problem of the stability factor of a Tresca vertical slope: the upper bound is lowered to 3.7776-value to be compared to the best published lower bound 3.7752-by succeeding in solving a nonlinear optimization problem with millions of variables and constraints.     

Two-grid methods for characteristic finite volume element solution of semilinear convection-diffusion equations

Two-grid methods for characteristic finite volume element solutions are presented for a kind of semilinear convection-dominated diffusion equations. The methods are based on the method of characteristics, two-grid method and the finite volume element method. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H). And the fine-grid solution (with grid size h) can be obtained by a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. The two-grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h^1/3).     

On the semi-discrete stabilized finite volume method for the transient Navier–Stokes equations

A stabilized finite volume method for solving the transient Navier–Stokes equations is developed and studied in this paper. This method maintains conservation property associated with the Navier–Stokes equations. An error analysis based on the variational formulation of the corresponding finite volume method is first introduced to obtain optimal error estimates for velocity and pressure. This error analysis shows that the present stabilized finite volume method provides an approximate solution with the same convergence rate as that provided by the stabilized linear finite element method for the Navier–Stokes equations under the same regularity assumption on the exact solution and a slightly additional regularity on the source term. The stability and convergence results of the proposed method are also demonstrated by the numerical experiments presented.     

Navier-Stokes方程的最佳有限元非线性Galerkin算法

1．引言对于Navier－Stokes方程有限元数值求解方面的研究已有很多的文章和专著,多数是采用有限元Galerkin算法,例见文献[1-4]．然而,由于Navier-Stokes方程在大雷诺数时有其强的非线性性和对时间土的长期依赖性,用计算机求解Navier－Stokes方程在速度和容量方面是难以承受的．为了克服这些困难,最近人们提出了有限元非线性Galerkin算法,见文献卜8］,然而这种算法只是在某一有限时刻之后具有好的收敛速度,在初始时刻的某一区间不能达到好的收敛速度．本文应用Taylor展开技术导出了数值求解二维非定常Navier－Stokes方程的最佳…     

对流扩散方程的有限体积-有限元方法的误差估计

本文结合有限体积方法和有限元方法处理非线性对流扩散问题,非线性对流项利用有限体积方法处理,扩散项利用有限元方法离散,并给近似解的误差估计。     

and Norms Error Estimates in Finite Element Method for Linear Parabolic Interface Problems

We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L ²(Ω) and H ¹(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose.     

Unstructured finite volume method for matrix free explicit solution of stress-strain fields in two dimensional problems with curved boundaries in equilibrium condition

In this paper, a plane stress structural solver which uses a matrix free unstructured finite volume method based on Galerkin approach is introduced for solution of weak form of two dimensional Cauchy equations on linear triangular element meshes. The developed shape function free Galerkin finite volume structural solver explicitly computes stresses and displacements in cartesian coordinate directions for the two dimensional solid mechanic problems in equilibrium condition. The accuracy of the introduced algorithm is assessed by comparison of computed results of two plane-stress cases with curved boundaries under uniformly distributed loads with available analytical solutions. The results of the introduced method are presented in terms of stress and strain contours and its effective parameters on convergence behaviour to equilibrium condition are assessed.     

Radially symmetric weighted extended b-spline model

In this paper, the weighted extended basis splines approach in the finite element method is applied to the electrostatic, electromagnetic wave and bioheat problems for inhomogeneous boundary conditions and radially symmetric structures. This new method, which does not need mesh generation, overcomes some of the drawbacks of using meshes and piecewise-uniform or linear trial functions. Two-dimensional radially symmetric electrostatic and electromagnetic wave equations are evaluated. We also attempt to propose a three-dimensional radially symmetric unexposed human eye model for simulating changes in corneal temperature using these new finite elements in conjunction with linear, quadratic and cubic b-splines. Our findings indicate that weighted extended basis spline solutions improve the standard finite element method. The simulation results which are verified using the values reported in the literature, point out to better efficiency in terms of the accuracy level.     

ANALYSIS OF A MIXED FINITE ELEMENT METHOD FOR A PHASE FIELD BENDING ELASTICITY MODEL OF VESICLE MEMBRANE DEFORMATION

In this paper, we study numerical approximations of a recently proposed phase fieldmodel for the vesicle membrane deformation governed by the variation of the elastic bend-ing energy. To overcome the challenges of high order nonlinear differential systems and thenonlinear constraints associated with the problem, we present the phase field bending elas-ticity model in a nested saddle point formulation. A mixed finite element method is thenemployed to compute the equilibrium configuration of a vesicle membrane with prescribedvolume and surface area. Coupling the approximation results for a related linearized prob-lem and the general theory of Brezzi-Rappaz-Raviart, optimal order error estimates for thefinite element approximations of the phase field model are obtained. Numerical results areprovided to substantiate the derived estimates.