固体中短波传播单位分解有限元法的解析积分

针对固体中短波传播数值模拟的单位分解有限元法中单元矩阵积分的被积函数的强烈振荡特性,应用直角坐标系下标准有限元形函数和单元内的波动方向知识提出了一种单元矩阵的解析积分方案。它对于平面三,六,四,八和九节点的直边单位分解有限单元是完全解析的,对于与这些单元相应的曲边单元则是半解析的。数值结果显示所提出的积分方案在计算效率上比高斯-勒让德积分有大幅度提高。     

Boundary-type finite element method for wave propagation analysis

The boundary-type finite element method has been investigated and applied to the Helmholz and mild-slope equations. Four types of interpolation function are examined based on trigonometric function series. Three-node triangular, four-node quadrilateral, six-node triangular and eight-node quadrilateral elements are tested; these are all non-conforming elements. Three types of numerical example show that the three-node triangular and four-node quadrilateral elements are useful for practical analysis.     

Finite element analysis of wave propagation in fluid-saturated porous media

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A triangular quasi-conforming finite element for transient dynamic analysis

A type of 3 node triangular element is constructed by the Quasi-conforming method, which may be used to solve the equation of a type of inverse problem of wave propagation after Laplace transformation Δu−A ² u=0. The strains in the element are approximated by an exponential function and the string-net function between neighbouring elements is approximated by one dimensional general solution of the equation. Furthermore the strain field satisfies the equation, and therefore in the derivation of the element formulation, no shape function is needed. In this sense, it is a kind of hybrid element. Compared with the ordinary linear triangular element, the new one features higher precision with coarse meshes. Some numerical tests are presented. The project is supported by the National Natural Science Foundation of China.     

A depth‐integrated non‐hydrostatic finite element model for wave propagation

This paper presents a two‐dimensional finite element model for simulating dynamic propagation of weakly dispersive waves. Shallow water equations including extra non‐hydrostatic pressure terms and a depth‐integrated vertical momentum equation are solved with linear distributions assumed in the vertical direction for the non‐hydrostatic pressure and the vertical velocity. The model is developed based on the platform of a finite element model, CCHE2D. A physically bounded upwind scheme for the advection term discretization is developed, and the quasi second‐order differential operators of this scheme result in no oscillation and little numerical diffusion. The depth‐integrated non‐hydrostatic wave model is solved semi‐implicitly: the provisional flow velocity is first implicitly solved using the shallow water equations; the non‐hydrostatic pressure, which is implicitly obtained by ensuring a divergence‐free velocity field, is used to correct the provisional velocity, and finally the depth‐integrated continuity equation is explicitly solved to satisfy global mass conservation. The developed wave model is verified by an analytical solution and validated by laboratory experiments, and the computed results show that the wave model can properly handle linear and nonlinear dispersive waves, wave shoaling, diffraction, refraction and focusing. Copyright © 2013 John Wiley & Sons, Ltd.     

A novel virtual node method for polygonal elements

A novel polygonal finite element method （PFEM） based on partition of unity is proposed, termed the virtual node method （VNM）. To test the performance of the present method, numerical examples are given for solid mechanics problems. With a polynomial form, the VNM achieves better results than those of traditional PFEMs, including the Wachspress method and the mean value method in standard patch tests. Compared with the standard triangular FEM, the VNM can achieve better accuracy. With the ability to construct shape functions on polygonal elements, the VNM provides greater flexibility in mesh generation. Therefore, several fracture problems are studied to demonstrate the potential implementation. With the advantage of the VNM, the convenient refinement and remeshing strategy are applied.     

固体非傅立叶温度场的时域间断Galerkin有限元法

运用时域间断Galerkin有限元法[1],对高频非傅立叶热波动问题[2-3]进行分析。其主要特点是:取温度及温度的时间导数为基本未知量,对其分别进行3次Hermite插值和线性插值。在保证节点温度自动保持连续的基础上,温度的时间导数在离散时域存在间断。数值结果表明所提出的方法能够滤掉虚假的数值震荡,能够良好地模拟固体中的非傅立叶热波动行为。     

裂纹扩展过程中线性内聚力模型计算的半解析有限元法

提出了求解基于线性内聚力模型的平面裂纹扩展问题的半解析有限元法,利用弹性平面扇形域哈密顿体系的方程,通过分离变量法及共轭辛本征函数向量展开法,推导了一个环形和一个圆形奇异超级解析单元列式,组装这两个超级单元能准确地描述裂纹表面作用有双线性内聚力的平面裂纹尖端场。将该解析元与有限元相结合,构成半解析的有限元法,可求解任意几何形状和载荷的基于线性内聚力模型的平面裂纹扩展问题。典型算例的计算结果表明本文方法简单有效,具有令人满意的精度。     

The partition‐of‐unity method for linear diffusion and convection problems: accuracy,stabilization and multiscale interpretation

We investigate the effectiveness of the partition‐of‐unity method (PUM) for convection–diffusion problems. We show that for the linear diffusion equation, an exponential enrichment function based on an approximation of the analytic solution leads to improved accuracy compared to the standard finite‐element method. It is illustrated that this approach can be more efficient than using polynomial enrichment to increase the order of the scheme. We argue that the PUM enrichment, can be interpreted as a subgrid‐scale model in a multiscale framework, and that the choice of enrichment function has consequences for the stabilization properties of the method. The exponential enrichment is shown to function as a near optimal subgrid‐scale model for linear convection. Copyright © 2003 John Wiley & Sons, Ltd.     

Two-grid partition of unity method for second order elliptic problems

A two-grid partition of unity method for second order elliptic problems is proposed and analyzed. The standard two-grid method is a local and parallel method usually leading to a discontinuous solution in the entire computational domain. Partition of unity method is employed to glue all the local solutions together to get the global continuous one, which is optimal in HI-norm. Furthermore, it is shown that the L^2 error can be improved by using the coarse grid correction. Numerical experiments are reported to support the theoretical results.     

直接计算应力强度因子的扩展有限元法

系统地给出了直接计算应力强度因子的扩展有限元法。该方法以常规有限元法为基础,利用单位分解法思想,通过在近似位移表达式中增加能够反映裂纹面的不连续函数及反映裂尖局部特性的裂尖渐进位移场函数,间接体现裂纹面的存在,从而无需使裂纹面与有限元网格一致,无需在裂尖布置高密度网格,也不需要后处理就可以直接计算出应力强度因子,并且大大简化了前后处理工作。最后通过两个简单算例验证了该方法的精度,分析了影响计算结果的因素,并与采用J积分计算的应力强度因子作了对比,得出了两种方法计算精度相当的结论。     

Non‐hydrostatic 3D free surface layer‐structured finite volume model for short wave propagation

In this paper a layer‐structured finite volume model for non‐hydrostatic 3D environmental free surface flow is presented and applied to several test cases, which involve the computation of gravity waves. The 3D unsteady momentum and mass conservation equations are solved in a collocated grid made of polyhedrons, which are built from a 2D horizontal unstructured mesh, by just adding several horizontal layers. The mesh built in such a way is unstructured in the horizontal plane, but structured in the vertical direction. This procedure simplifies the mesh generation and at the same time it produces a well‐oriented mesh for stratified flows, which are common in environmental problems. The model reduces to a 2D depth‐averaged shallow water model when one single layer is defined in the mesh. Pressure–velocity coupling is achieved by the Semi‐Implicit Method for Pressure‐Linked Equations algorithm, using Rhie–Chow interpolation to stabilize the pressure field. An attractive property of the model proposed is the ability to compute the propagation of short waves with a rather coarse vertical discretization. Several test cases are solved in order to show the capabilities and numerical stability of the model, including a rectangular free oscillating basin, a radially symmetric wave, short wave propagation over a 1D bar, solitary wave runup on a vertical wall, and short wave refraction over a 2D shoal. In all the cases the numerical results are compared either with analytical or with experimental data. Copyright © 2008 John Wiley & Sons, Ltd.     

Gauge finite element method for incompressible flows

A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E, Liu J‐G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and differs from the standard gauge variable that arises from decomposing a compressible velocity field. It has the advantage that an additional boundary condition can be assigned to the gauge variable, thus eliminating the issue of a pressure boundary condition associated with the original primitive variable formulation. The computational task is then reduced to solving standard heat and Poisson equations, which are approximated by straightforward, piecewise linear (or higher‐order) finite elements. This method can achieve high‐order accuracy at a cost comparable with that of solving standard heat and Poisson equations. It is naturally adapted to complex geometry and it is much simpler than traditional finite element methods for incompressible flows. Several numerical examples on both structured and unstructured grids are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

In-plane wave motion in finite element model

The analysis method of lattice dynamics in classical physics is extended to study the properties of in-plane wave motion in the hybrid-mass finite element model in this paper. The dispersion equations of P and SV waves in the discrete model are first obtained by means of separating the characteristic equation of the motion equation, and then used to analyse the properties of P-and SV-homogeneous, inhomogeneous waves and other types of motion in the model. The dispersion characters, cut-off frequencies of P and SV waves, the polarization drift and appendent anisotropic property of wave motion caused by the discretization are finally discussed. The project sponsored by the Earthquake Science Foundation under Contract No. 90141     

Elasto-optic technique for measurement of elastic wave propagation in solids

The modulation of the optical path of the beam of a laser vibrometer in a specimen under acoustic excitation is measured at two planes, separated by a precisely known distance. The phase shift and the decrease in magnitude are used to calculate the phase velocity and attenuation, respectively. The method is demonstrated for a homogeneous specimen, and the results compare favorably with those obtained by a conventional ultrasonic technique. The method is then applied to measure specular and first diffraction-order reflection from a coplanar periodic array of particles in an elastic matrix and phase velocity spectra in a tetragonal periodic particulate composite. As expected, in a periodic composite the establishment of dispersive Floquet-type waves is observed throughout the entire periodic particulate composite.     

随机结构有限元分析的递推求解方法

提出了一种新的谱随机有限元分析方法——递推求解方法。该方法将随机结构的随机响应表示成非正交多项式展式,建立了和摄动法类似的一系列确定的递推方程,并通过确定性有限元方法对这些递推方程进行静力问题求解。算例表明,当随机量出现较大涨落时,计算结果相对于传统摄动法有不小的改进。     

Nonlinear quasi-conforming finite element method

The nonlinear quasi-conforming FEM is presented based on the basic concept of the quasiconforming finite element. First, the incremental principle of stationary potential energy is discussed. Then, the formulation process of the nonlinear quasi-conforming FEM is given. Lastly, two computational examples of shells are given.     

A deforming finite element mesh for use in moving one-dimensional boundary wave problems

The finite element method is developed to solve the problem of wave run-up on a mild, plane slope. A novel approach to implementing a deforming mesh of one-dimensional, three-node, isoparametric elements is described and demonstrated. The discrete time interval (DTI), arbitrary Lagrangian–Eulerian (ALE) and space–time element (STE) methods are used to solve the unsteady one-dimensional shallow water wave equations. The boundary condition required is simply the seaward water surface elevation, and although the method has only been tested for monochromatic waves, it should be equally valid for any sea state which can be described as a water surface elevation as a function of time. All three solution methods are shown to given good results. Time histories of the terms of the governing equations are calculated and used to demonstrate how the ALE and STE methods account for mesh deformation. The model could be extended to two dimensions, which would have practical application to the run-up of obliquely incident waves. © 1997 John Wiley & Sons, Ltd.     

Galerkin meshless methods based on partition of unity quadrature

Introduction Meshlessmethodsarenewmethodsofnumericalcomputationwhichhavebeendeveloped rapidlyinrecentyears.Inthesemethods,onlynodesareneeded,meshinformationistotally unnecessary.Thiscanavoidorpartlyavoidthedifficultyofmeshgeneration.Duetohigh accuracyandstability,Galerkinmeshlessmethodsareappliedbroadly,butitisunavoidable tocomputetheintegrationoverthewholephysicaldomaininGalerkinweakform,whichisa greatchallengeforGalerkinmeshlessmethodsbecauseoftheabsenceofmesh.TocarryouttheintegrationinGal…     

Synthesis of highly convergent 2D and 3D finite elements for short acoustic wave simulation

New higher-order finite elements of enhanced convergence properties for acoustic wave simulation are presented in the paper. The element matrices are obtained by combining modal synthesis and optimization techniques in order to achieve minimum errors of higher modes of the computational domain. As a result, simulation models of propagating wave pulses require a smaller number of finite element divisions per wavelength compared to the conventional element model thus significantly reducing computational costs. Though finite element matrices are obtained in optimization, the resulting patterns of the matrices are versatile and further can be used in any wave propagation model. The mass matrices of the elements are diagonal, so explicit time integration schemes are applicable. The usage of new elements is especially efficient in situations where wavelengths of the simulated signal are much shorter than the dimensions of the computational domain. This is referred to as short wave propagation analysis. The results of wave propagation simulation for ultrasonic measurements are presented as application examples. The B-scans and computed dispersion curves are provided for visual interpretation of the results.