A MODIFIED WEAK GALERKIN FINITE ELEMENTMETHOD FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION-REACTION PROBLEMS

In this work,a modified weak Galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion equations.The key feature of the proposed method is to replace the classical gradient and divergence operators by the modified weak gradient and modified divergence operators,respectively.We apply the backward finite difference method in time and the modified weak Galerkin finite element method in space on uniform mesh.The stability analyses are presented for both semi-discrete and fully-discrete modified weak Galerkin finite element methods.Optimal order of convergences are obtained in suitable norms.We have achieved the same accuracy with the weak Galerkin method while the degrees of freedom are reduced in our method.Various numerical examples are presented to support the theoretical results.It is theoretically and numerically shown that the method is quite stable.

Modified Layer-Removal Method for Measurement of Residual Stress in Pre-stretched Aluminium Alloy Plate

Residual stress is one of the factors affecting the machining deformation of monolithic structure parts in the aviation industry. Thus, the studies on machining deformation rules induced by residual stresses largely depend on correctly and efficiently measuring the residual stresses of workpieccs. A modified layer-removal method is proposed to measure residual stress by analysing the characteristics of a traditional, layer-removal method. The coefficients of strain release are then deduced according to the simulation results using the finite element method （FEM）. Moreover, the residual stress in a 7075T651 aluminium alloy plate is measured using the proposed method, and the results are then analyzed and compared with the data obtained by the traditional methods. The analysis indicates that the modified layer-removal method is effective and practical for measuring the residual stress distribution in pre-stretched aluminium alloy plates.

AN EFFICIENT GPU ACCELERATION FORMAT FOR FINITE ELEMENT ANALYSIS

This paper proposes a new Graphics Processing Unit(GPU)-accelerated storage format to speed up Sparse Matrix Vector Products(SMVPs) for Finite Element Method(FEM) analysis of electromagnetic problems.A new format called Modified Compile Time Optimization(MCTO) format is used to reduce much execution time and design for hastening the iterative solution of FEM equations especially when rows have uneven lengths.The MCTO-applied FEM is about 10 times faster than conventional FEM on a CPU,and faster than other row-major ordering formats on a GPU.Numerical results show that the proposed GPU-accelerated storage format turns out to be an excellent accelerator.

The Characteristics-mixed Finite Element Method for Enhanced Oil Recovery Simulation and Optimal Order L^2 Error Estimate

This article discusses the enhanced oil recovery numerical simulation of the chemical flooding(such as surfactants, alcohol, polymers) composed of two-dimensional multicomponent, ultiphase and incompressible mixed fluids. After the oil field is waterflooded, there is still a large amount of crude oil left in the oil deposit. By adding certain chemical substances to the fluid injected, its driving capacity can be greatly increased. The mathematical model of two-dimensional enhanced oil recovery simulation can be described

Determination of processing power and optimum billet radius of modified backward extrusion by upper bound approach

A recently developed backward extrusion method entitled “modified backward extrusion” was presented using an upper bound analysis. For this purpose deformation area was divided into four distinct zones and a kinematically admissible velocity field for each of them was suggested. Total dissipated power was calculated for the deformation zones and the extrusion power wascomputed. The correlations of important geometrical parameters with extrusion force and dissipated powers were shown. Finding the initial billet size, a challenging area in the modified backward extrusion method, was discussed and the optimum billet radius was obtained, considering the minimum relative extrusion pressure. Finite element analyses were conducted and the results werecompared with the upper bound analysis. Finally, experiments were executed on commercially pure aluminium and a good agreement between upper bound and finite element analyses with experimental values was observed.

Buckling Analysis of Tapered Continuous Columns by Using Modified Buckling Mode Shapes

Elastic critical buckling load of a column depends on various parameters,such as boundary conditions,material,and crosssection geometry.The main purpose of this work is to present a new method for investigating the buckling load of tapered columns subjected to axial force.The proposed method is based on modified buckling mode shape of tapered structure and perturbation theory.The mode shape of the damaged structure can be expressed as a linear combination of mode shapes of the intact structure.Variations in length in piecewise form can be positive or negative.The method can be used for single-span and continuous columns.Comparison of results with those of finite element and Timoshenko methods shows the high accuracy and efficiency of the proposed method for detecting buckling load.

A SECOND ORDER MODIFIED CHARACTERISTICS VARIATIONAL MULTISCALE FINITE ELEMENT METHOD FOR TIME-DEPENDENT NAVIER-STOKES PROBLEMS

In this paper, by combining the second order characteristics time discretization with the variational multiscale finite element method in space we get a second order modified characteristics variational multiscale finite element method for the time dependent Navier- Stokes problem. The theoretical analysis shows that the proposed method has a good convergence property. To show the efficiency of the proposed finite element method, we first present some numerical results for analytical solution problems. We then give some numerical results for the lid-driven cavity flow with Re = 5000, 7500 and 10000. We present the numerical results as the time are sufficient long, so that the steady state numerical solutions can be obtained.

A Robust Modified Weak Galerkin Finite Element Method for Reaction-Diffusion Equations

In this paper,a robust modified weak Galerkin(MWG)finite element method for reaction-diffusion equations is proposed and investigated.An advantage of this method is that it can deal with the singularly perturbed reaction-diffusion equations.Another advantage of this method is that it produces fewer degrees of freedom than the traditional WG method by eliminating the element boundaries freedom.It is worth pointing out that,in our method,the test functions space is the same as the finite element space,which is helpful for the error analysis.Optimalorder error estimates are established for the corresponding numerical approximation in various norms.Some numerical results are reported to confirm the theory.

A Modified Polynomial Preserving Recovery and Its Applications to A Posteriori Error Estimates

A modified polynomial preserving gradient recovery technique is proposed. Unlike the polynomial preserving gradient recovery technique,the gradient recovered with the modified polynomial preserving recovery(MPPR) is constructed element-wise, and it is discontinuous across the interior edges.One advantage of the MPPR technique is that the implementation is easier when adaptive meshes are involved.Superconvergence results of the gradient recovered with MPPR are proved for finite element methods for elliptic boundary problems and eigenvalue problems under adaptive meshes. The MPPR is applied to adaptive finite element methods to construct asymptotic exact a posteriori error estimates.Numerical tests are provided to examine the theoretical results and the effectiveness of the adaptive finite element algorithms.

Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation

We present the finite difference/element method for a two-dimensional modified fractional diffusion equation.The analysis is carried out first for the time semi-discrete scheme,and then for the full discrete scheme.The time discretization is based on the L1-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term.We use finite element method for the spatial approximation in full discrete scheme.We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent.Moreover,the optimal convergence rate is obtained.Finally,some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.

A FINITE ELEMENT/BOUNDARY ELEMENT——MODIFIED MODAL DECOMPOSITION METHOD FOR VIBRATION AND SOUND RADIATION FROM SUBMERGED SHELL OF REVOLUTION

A finite element / boundary element-modified modal decomposition method (FBMMD) is presented for predicting the vibration and sound radiation from submerged shell of revolution. Improvement has been made to accelerate the convergence to FBMD method by means of introducing the residual modes which take into accaunt the quasi -state contributiort of all neglected modes. As an example, the vibration and sound radiation of a submerged spherical shell excited by axisymmetric force are studied in cases of ka=l,2,3 and 4. From the calculated results we see that the FBMMD method shows a significant improvement to the accuracy of surface sound pressure, normal displacement and directivity patterns of radiating sound, especially to the directivity patterns.

Derivative of a Determinant with Respect to an Eigenvalue in the Modified Cholesky Decomposition of a Symmetric Matrix, with Applications to Nonlinear Analysis

In this paper, we obtain a formula for the derivative of a determinant with respect to an eigenvalue in the modified Cholesky decomposition of a symmetric matrix, a characteristic example of a direct solution method in computational linear algebra. We apply our proposed formula to a technique used in nonlinear finite-element methods and discuss methods for determining singular points, such as bifurcation points and limit points. In our proposed method, the increment in arc length (or other relevant quantities) may be determined automatically, allowing a reduction in the number of basic parameters. The method is particularly effective for banded matrices, which allow a significant reduction in memory requirements as compared to dense matrices. We discuss the theoretical foundations of our proposed method, present algorithms and programs that implement it, and conduct numerical experiments to investigate its effectiveness.

考虑化学反应的高聚物压密劈裂注浆仿真研究

考虑浆液化学反应原理,综合运用扩展有限元理论、修正剑桥模型和高聚物浆液膨胀力计算模型,建立了模拟高聚物在土体中压密劈裂注浆动态过程的二维仿真分析方法。通过模型试验验证了所提出方法的适用性,进而分析了浆液膨胀力、浆脉形态、土体孔隙比随时间变化规律及注浆孔埋深、土体断裂韧度对浆脉扩展过程的影响。结果表明:受高聚物化学反应进程驱动,在浆液与土体耦合作用下,浆液膨胀力初期随时间近似呈线性增长,达到峰值后快速下降并趋于稳定;浆脉长度和宽度发展速度不同步,裂缝启裂至二次扩展阶段,浆脉长度基本保持不变,宽度线性增大,裂缝二次扩展后,浆脉长度近似线性增大,而宽度增长速率趋缓;受浆液挤密作用影响,裂缝两侧一定范围内土体孔隙比显著降低,沿垂直于裂缝面方向,距注浆孔中心越近,孔隙比越小;孔隙比随时间整体呈下降趋势,随着裂缝的扩展和后期膨胀压力的下降,紧邻裂缝面两侧土体孔隙比有一定恢复,随后趋于稳定;随着注浆孔埋深和土体断裂韧度的增加,浆脉长度逐渐减小,宽度不断增大,两者变化速率基本保持不变;浆脉扩展稳定时间随埋深的增大而提前,随断裂韧度的增大而延后。

局部环向变厚度方钢管混凝土柱偏压力学性能研究

针对海洋环境中飞溅区造成的钢管混凝土柱局部环向腐蚀现象,采用机械削减方式模拟钢管壁厚局部环向腐蚀,进行了7个方钢管混凝土柱偏压试验,揭示了削弱率和偏心率对其偏压承载力的影响,并通过数值模拟对削弱率进行拓展分析.提出了基于削弱率的偏压承载力降低系数,代入中、美规范进行偏压承载力计算.结果表明:钢管的局部环向变厚度严重削弱了方钢管混凝土柱的偏压承载力和侧向挠曲能力;较大的加载偏心率也降低了其偏压承载力,但增大了其侧向挠曲能力.引入偏压承载力降低系数后,根据中、美规范计算得到的偏压承载力与试验值均吻合较好;相比于我国规范,美国规范在计算偏压承载力时相对更加保守.

Highly accurate symplectic element based on two variational principles

For the stability requirement of numerical resultants, the mathematical theory of classical mixed methods are relatively complex. However, generalized mixed methods are automatically stable, and their building process is simple and straightforward. In this paper, based on the seminal idea of the generalized mixed methods, a simple, stable, and highly accurate 8-node noncompatible symplectic element(NCSE8) was developed by the combination of the modified Hellinger-Reissner mixed variational principle and the minimum energy principle. To ensure the accuracy of in-plane stress results, a simultaneous equation approach was also suggested. Numerical experimentation shows that the accuracy of stress results of NCSE8 are nearly the same as that of displacement methods, and they are in good agreement with the exact solutions when the mesh is relatively fine. NCSE8 has advantages of the clearing concept, easy calculation by a finite element computer program, higher accuracy and wide applicability for various linear elasticity compressible and nearly incompressible material problems. It is possible that NCSE8 becomes even more advantageous for the fracture problems due to its better accuracy of stresses.

XFEM和修正剑桥模型模拟高聚物劈裂注浆方法研究

针对自膨胀高聚物材料在土体中劈裂扩散仿真手段尚不完备的现状,初步建立了一种模拟高聚物对土体劈裂压密作用的仿真方法。该方法采用扩展有限元离散求解区域土体介质,用修正剑桥模型描述土体力学特性,利用高聚物密度-围压关系迭代计算作用于裂缝面的浆液膨胀压力,实现了对土体劈裂缝扩展过程的数值模拟,通过与试验结果的对比验证了该方法的适用性。算例分析表明,应用该方法能够模拟膨胀力作用下土体裂缝启裂过程、扩展走向和浆脉厚度变化,获得裂缝周围土体变形模量、孔隙比、应力场和密度场分布特征,为深入研究膨胀性高聚物注浆材料对土体劈裂压密机制奠定了基础。

热传导问题杂交基本解有限元法虚拟源点的探究

针对热传导问题,提出了杂交基本解有限元法.首先,假设两个独立场:一个为利用基本解线性组合近似的单元域内温度场,另一个为使用与传统有限元法相同形式的辅助网线温度场.然后,利用修正变分泛函将上述两个独立场关联起来,并导出有限元列式.然而,该方法的准确性很大程度上取决于源点的分布和数量,通常将源点布置在单元外部两种虚拟边界上:与单元相似的边界和圆形边界.此外,还提出了双重虚拟边界,并与上述两种源点布局方式进行对比.通过两个典型数值算例,验证了该文方法在不同源点布局下的有效性和对网格畸变的不敏感性.

A numerical technique based on collocation method for solving modified Kawahara equation

In this article,a numerical solution of the modified Kawahara equation is presented by septic B-spline collocation method.Applying the von-Neumann stability analysis,the present method is shown to be unconditionally stable.L 2 and L∞error norms and conserved quantities are given at selected times.The accuracy of the proposed method is checked by test problems including motion of the single solitary wave,interaction of solitary waves and evolution of solitons.

基于扩展有限元法数值模拟双材料界面裂纹问题

扩展有限元法通过在间断区域引入富集函数,在处理强弱不连续问题上较有限元法有极大的优势。本研究给出了基于扩展有限元法的双材料界面裂纹位移逼近方程及相互作用积分的数值离散方法和单元刚度矩阵的积分策略,材料界面弱不连续性用改进扩展有限元模拟,裂纹贯穿部分用强不连续的Heaviside函数模拟,裂纹尖端分别用2种不同渐近裂纹尖端富集函数模拟,用Matlab编制相应的扩展有限元程序。算例表明,数值模拟结果和参考文献的结果拟合的较好。

Graphical method based on modified maximum force criterion to indicate forming limit curves of 22MnB5 boron steel sheets at elevated temperatures

A new approach for predicting forming limit curves(FLCs)at elevated temperatures was proposed herein.FLCs are often used to predict failure and determine the optimal forming parameters of automotive parts.First,a graphical method based on a modified maximum force criterion was applied to estimate the FLCs of 22MnB5 boron steel sheets at room temperature using various hardening laws.Subsequently,the predicted FLC data at room temperature were compared with corresponding data obtained from Nakazima's tests to obtain the best prediction.To estimate the FLC at elevated temperatures,tensile tests were conducted at various temperatures to determine the ratios of equivalent fracture strains between the corresponding elevated temperatures and room temperature.FLCs at elevated temperatures could be established based on obtained ratios.However,the predicted FLCs at elevated temperatures did not agree well with the corresponding FLC experimental data of Zhou et al.A new method was proposed herein to improve the prediction of FLCs at elevated temperatures.An FLC calculated at room tem-perature was utilized to predict the failure of Nakazima's samples via finite element simulation.Based on the simulation results at room temperature,the mathematical relationships between the equivalent ductile fracture strain versus stress triaxiality and strain ratio were established and then combined with ratios between elevated and room temperatures to calculate the FLCs at different temperatures.The predicted FLCs at elevated temperatures agree well with the corresponding experimental FLC data.