对流扩散方程的混合时间间断时空有限元方法

构造并分析二阶对流扩散方程的混合时间间断时空有限元格式．利用混合有限元方法将二阶方程降阶,利用空间连续而时间允许间断的时空有限元方法离散低阶方程．证明数值解的稳定性、存在唯一性和收敛性．最后通过数值结果验证该算法的有效性和可行性．     

Sobolev方程的时间间断Galerkin有限元方法

引入Sobolev方程的等价积分方程,构造Sobolev方程的新的时间间断Galerkin有限元格式.该格式不仅保持有限元解在时间剖分点处的间断特性,而且避免了传统时空有限元格式中跳跃项的出现,从而降低了格式理论分析和数值模拟的复杂性.证明了Sobolev方程的时间间断而空间连续的时空有限元解的稳定性、存在唯一性、L2...     

输运方程谱间断流线扩散耦合方法

本文讨论了谱有限元方法,构造了求解Ｂｏｌｔｚｍ ａｎｎ 方程球谐函数谱展开和间断流线扩散有限元耦合格式．建立了这种耦合方法的稳定性及最优阶收敛性误差估计．得到了比标准有限元更高的精度．     

Stokes方程的稳定化间断有限元法

本文对定常的Stokes方程提出了一种新的间断有限元法,通过对通常的间断Galerkin有限元法应用稳定化思想,建立了一个相容的稳定间断有限元格式,对速度和压力的任意分片多项式空间Pl(K),Pm(K)的间断有限元逼近证明了解的存在唯一性,给出了关于速度和压力的L2 范数的最优误差估计．     

板的低阶间断与连续有限元法统一分析

基于Reissner-Mindlin板问题的间断Galerkin有限元逼近, 建立了一个对挠度空间和角位移空间取连续或间断元都适用的低阶有限元离散格式. 取剪切力空间为分片常数元, 挠度空间和角位移空间无论取间断元还是连续元, 格式都是一致稳定的, 并给出了H¹范数估计及L²范数估计. 作为应用,对几类低阶有限元空间讨论. 结果表明, 该格式对常见的低阶有限元空间都适用, 并且若至少有一个元连续时, 该格式需要的空间比[1,2]中的都要简单.       

四阶线性抛物型积分-微分方程的混合间断时空有限元法

构造四阶抛物型积分-微分方程的混合间断时空有限元格式,利用混合有限元方法将高阶方程降阶,利用空间连续而时间允许间断的时空有限元方法离散方程,证明离散解的稳定性,存在唯一性和收敛性.     

一类四阶抛物型积分-微分方程的混合间断时空有限元法

构造四阶抛物型积分-微分方程的混合间断时空有限元格式,利用混合有限元方法将高阶方程降阶,利用空间连续而时间允许间断的时空有限元方法离散方程,证明离散解的稳定性,存在唯一性和收敛性.     

电报方程的时间间断时空有限元方法

为同时高精度逼近速度和位移,利用时间间断的时空有限元与降阶的思想,对一类电报方程的初边值问题建立一种时间间断时空有限元格式.利用有限差分方法与有限元方法相结合的技巧,证明了格式的稳定性和收敛性,得到了速度的L∞(L2)模和位移的L∞(H1)模最优误差估计.最后用数值算例验证了理论分析结果和所提算法的有效性.     

对流扩散方程的时空间断有限元方法

研究对流扩散方程的时空间断Galerkin有限元方法,该方法采用时,空两个变量都允许间断的基函数,更适用于移动网格,自适应算法以及并行计算.本文利用拉格朗日欧拉方法,采用F.Brezzi数值流通量,给出对流扩散方程的间断时空有限元离散格式,并证明格式的相容性,强制性,稳定性,解的存在唯一性,以及总体误差估计.     

Stokes方程的压力梯度局部投影间断有限元法

本文对定常的Stokes方程提出了一种新的间断有限元法,通过将通常的间断Galerkin有限元法与压力梯度局部投影相结合,建立了一个稳定的间断有限元格式,对速度和压力的任意分片多项式空间P_l(K),P_m(K)的间断有限元逼近证明了解的存在唯一性,给出了关于速度和压力的L~2范数的最优误差估计.     

Numerical simulation based on POD for two-dimensional solute transport problems

A proper orthogonal decomposition (POD) method is applied to a usual finite element scheme for two-dimensional solute transport problems such that it is reduced into a reduced finite element formulation with lower dimensions and high enough accuracy. Numerical examples show that the results of numerical computations are consistent with accurate solutions. Moreover, this validates the feasibility and efficiency of POD method.     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

最佳等参元

等参元及其参数变换的插值方法。是有限元分析的有力工具之一,在工程计算中,得到广泛的应用. 在有限元分析中,当采用等参元时,一旦单元的等参坐标变换的Jacobi矩阵发生奇异,就要中止计算,下机修改原有的单元剖分,直到所有单元的Jacobi矩阵均非奇异. [1]突破原等参元的规定,给出了八节点Serendipity等参元的修改公式;[2]也给出了类似的修改公式.上述均以数值例子说明新公式的优点.而[3—6]完整、系统地给出     

Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes

A new weak Galerkin (WG) finite element method is introduced and analyzed in this article for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter‐free. Optimal order error estimates in a discrete H² norm is established for the corresponding WG finite element solutions. Error estimates in the usual L² norm are also derived, yielding a suboptimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1003–1029, 2014     

A stabilized mixed formulation for finite strain deformation for low-order tetrahedral solid elements

A stabilized mixed finite element formulation for four-noded tetrahedral elements is introduced for robustly solving small and large deformation problems. The uniqueness of the formulation lies within the fact that it is general in that it can be applied to any type of material model without requiring specialized geometric or material parameters. To overcome the problem of volumetric locking, a mixed element formulation that utilizes linear displacement and pressure fields was implemented. The stabilization is provided by enhancing the rate of deformation tensor with a term derived using a bubble function approach. The element was implemented through a user-programmable element of the commercial finite element software ANSYS. Using the ANSYS platform, the performance of the element was evaluated by comparing the predicted results with those obtained using mixed quadratic tetrahedral elements and hexahedral elements with a B-bar formulation. Based on the quality of the results, the new element formulation shows significant potential for use in simulating complex engineering processes.     

A coupled finite element-differential quadrature element method and its accuracy for moving load problem

In this paper a mixed method, which combines the finite element method and the differential quadrature element method (DQEM), is presented for solving the time dependent problems. In this study, the finite element method is first used to discretize the spatial domain. The DQEM is then employed as a step-by-step DQM in time domain to solve the resulting initial value problem. The resulting algebraic equations can be solved by either direct or iterative methods. Two general formulations using the DQM are also presented for solving a system of linear second-order ordinary differential equations in time. The application of the formulation is then shown by solving a sample moving load problem. Numerical results show that the present mixed method is very efficient and reliable.     

伪双曲方程的新混合有限元方法

构造分析一类二阶伪双曲方程的H1-Galerkin扩展混合有限元方法,该方法采用了扩展混合有限元方法和H1-Galerkin混合有限元方法相结合的技巧.新的格式同时保持了扩展混合有限元方法和H1-Galerkin混合有限元方法的优点.该混合格式与标准的混合格式相比能同时逼近三个变量:未知函数、梯度和流量(系数乘以梯度),并且不必满足LBB相容性条件.     

Formulation of hybrid Trefftz finite element method for elastoplasticity

The present investigation provides a hybrid Trefftz finite element approach for analysing elastoplastic problems. A dual variational functional is constructed and used to derive hybrid Trefftz finite element formulation for elastoplasticity of bulky solids. The formulation is applicable to either strain hardening or elastic-perfectly plastic materials. A solution algorithm based on initial stress formulation is introduced into the new element model. The performance of the proposed element model is assessed by three examples and comparison is made with results obtained by other approaches. The hybrid Trefftz finite element approach is demonstrated to be particularly suited for nonlinear analysis of two-dimensional elastoplastic problems.     

Discontinuous Stable Elements for the Incompressible Flow

In this paper, we derive a discontinuous Galerkin finite element formulation for the Stokes equations and a group of stable elements associated with the formulation. We prove that these elements satisfy the new inf–sup condition and can be used to solve incompressible flow problems. Associated with these stable elements, optimal error estimates for the approximation of both velocity and pressure in L ² norm are obtained for the Stokes problems, as well as an optimal error estimate for the approximation of velocity in a mesh dependent norm.