二维多流体域耦合声场的光滑有限元解法

针对声学有限元分析中四节点等参单元计算精度低,对网格质量敏感的问题,将光滑有限元法引入到多流体域耦合声场的数值分析中,提出了二维多流体域耦合声场的光滑有限元解法。该方法在Helmholtz控制方程与多流体域耦合界面的声压/质点法向速度连续条件的基础上,得到二维多流体耦合声场的离散控制方程,并采用光滑有限元的分区光滑技术将声学梯度矩阵形函数导数的域内积分转换形函数的域边界积分,避免了雅克比矩阵的计算。以管道二维多流体域耦合内声场为数值分析算例,研究结果表明,与标准有限元相比,对单元尺寸较大或扭曲严重的四边形网格模型,光滑有限元的计算精度更高。因此光滑有限元能很好地应用于大尺寸单元或扭曲严重的网格模型下二维多流体域耦合声场的预测,具有良好的工程应用前景。      

A priori mesh quality metric error analysis applied to a high-order finite element method

Characterization of computational mesh’s quality prior to performing a numerical simulation is an important step in insuring that the result is valid. A highly distorted mesh can result in significant errors. It is therefore desirable to predict solution accuracy on a given mesh. The HiFi/SEL high-order finite element code is used to study the effects of various mesh distortions on solution quality of known analytic problems for spatial discretizations with different order of finite elements. The measured global error norms are compared to several mesh quality metrics by independently varying both the degree of the distortions and the order of the finite elements. It is found that the spatial spectral convergence rates are preserved for all considered distortion types, while the total error increases with the degree of distortion. For each distortion type, correlations between the measured solution error and the different mesh metrics are quantified, identifying the most appropriate overall mesh metric. The results show promise for future a priori computational mesh quality determination and improvement.     

Free Mesh Method: fundamental conception, algorithms and accuracy study

The finite element method (FEM) has been commonly employed in a variety of fields as a computer simulation method to solve such problems as solid, fluid, electro-magnetic phenomena and so on. However, creation of a quality mesh for the problem domain is a prerequisite when using FEM, which becomes a major part of the cost of a simulation. It is natural that the concept of meshless method has evolved. The free mesh method (FMM) is among the typical meshless methods intended for particle-like finite element analysis of problems that are difficult to handle using global mesh generation, especially on parallel processors. FMM is an efficient node-based finite element method that employs a local mesh generation technique and a node-by-node algorithm for the finite element calculations. In this paper, FMM and its variation are reviewed focusing on their fundamental conception, algorithms and accuracy.     

二维Helmholtz边界超奇异积分方程解析研究

基于常规边界元法及超奇异边界积分方程复线性耦合的Burton-Miller方法应用于无限域声学问题的最大难点在于处理超奇异积分（二维问题）.目前,此类超奇异积分主要使用各种弱奇异/正则化方法求解,而这些弱奇异/正则化方法具有时间消耗大等弱点.基于围道积分定理,本文给出一种使用常值单元的二维Helmholtz边界超奇异积分的解析表达式.在有限部分积分意义下,所有的奇异和超奇异积分可以解析表达.数值算例表明该解析表达式是有效的.     

A Pseudo-Stokes Mesh Motion Algorithm

This work presents a moving mesh methodology based on the solution of a pseudo flow problem. The mesh motion is modeled as a pseudo Stokes problem solved by an explicit finite element projection method. The mesh quality requirements are satisfied by employing a null divergent velocity condition. This methodology is applied to triangular unstructured meshes and compared to well known approaches such as the ones based on diffusion and pseudo structural problems. One of the test cases is an airfoil with a fully meshed domain. A specific rotation velocity is imposed as the airfoil boundary condition. The other test is a set of two cylinders that move toward each other. A mesh quality criterion is employed to identify critically distorted elements and to evaluate the performance of each mesh motion approach. The results obtained for each test case show that the pseudo-flow methodology produces satisfactory meshes during the moving process.     

三维静电场线性插值边界元中的解析积分方法

提出求解三维静电场的三角形线性插值边界元解析积分方法.针对含1/R和1/R²的积分项,将单元形状函数分解为常数项、含x的线性项和含y的线性项,从而将边界单元积分简化为6个基本积分组合,并导出其解析计算公式,避免了因形状函数改变而导致的重复计算.该方法不仅可以准确计算远离奇异情况下的边界元积分,而且可以准确计算一阶和二阶接近奇异积分以及一阶奇异积分.计算结果表明,在接近奇异积分和奇异积分比较突出的问题中,当数值积分方法不能给出正确结果时,用同样的边界元网格,解析积分方法可以给出正确的结果,提高了三维静电场线性插值边界元法的计算精度.     

A finite-element method using dispersion reduced spline elements for room acoustics simulation

This paper presents a finite element method (FEM) using hexahedral 27-node spline acoustic elements (Spl27) with low numerical dispersion for room acoustics simulation in both the frequency and time domains, especially at higher frequencies. Dispersion error analysis in one dimension is performed to increase the accuracy of FEM using Spl27 by modifying the numerical integration points of element stiffness and mass matrices. The basic accuracy and efficiency of the FEM using the improved Spl27, which uses modified integration points, are presented through numerical experiments using benchmark problems in both the frequency and time domains, revealing that FEM using the improved Spl27 in both domains provides more accurate results than the conventional method does, and with fewer degrees of freedom. Moreover, the effectiveness of FEM using the improved Spl27 over that using hexahedral 27-node Lagrange elements is shown for time domain analysis of the sound field in a practical sized room.     

Isogeometric finite element analysis of interior acoustic problems

Isogeometric Analysis (IGA) can bridge the gap between geometrical and numerical modeling. To this end, the same basis functions used in Computer Aided Design are applied to represent geometry and approximate physical field in analysis. In this paper, the IGA is firstly introduced to finite element method (FEM) for interior acoustic problems. The domain is parameterized by Non-Uniform Rational B-Spline (NURBS) in the algorithm, which simplifies the mesh generation greatly and furthermore supplies an exact representation of curved boundaries. In addition, the IGA FEM possesses a distinct feature of flexible order-elevation technique without modifying the geometry. Several numerical examples are presented to validate the accuracy and demonstrate the merits of the IGA FEM in the analysis of interior acoustic problems.     

An approximation for the boundary optimal control problem of a heat equation defined in a variable domain

In this paper, we consider a numerical approximation for the boundary optimal control problem with the control constraint governed by a heat equation defined in a variable domain. For this variable domain problem, the boundary of the domain is moving and the shape of theboundary is defined by a known time-dependent function. By making use of the Galerkin finite element method, we first project the original optimal control problem into a semi-discrete optimal control problem governed by a system of ordinary differential equations. Then, based on the aforementioned semi-discrete problem, we apply the control parameterization method to obtain an optimal parameter selection problem governed by a lumped parameter system, which can be solved as a nonlinear optimization problem by a Sequential Quadratic Programming （SQP） algorithm. The numerical simulation is given to illustrate the effectiveness of our numerical approximation for the variable domain problem with the finite element method and the control parameterization method.     

A coupled edge-/face-based smoothed finite element method for structural-acoustic problems

The edge-based smoothed finite element method (ES-FEM) and the face-based smoothed finite element method (FS-FEM) developed recently have shown great efficiency in solving solid mechanics problems with triangular and tetrahedral meshes. In this paper, a coupled ES-/FS-FEM model is extended to solve the structural-acoustic problems consisting of a plate structure interacting with the fluid medium. Three-node triangular elements and four-node tetrahedral elements are used to discretize the two-dimensional (2D) plate and three-dimensional (3D) fluid, respectively, as they can be generated easily and even automatically for complicated geometries. The field variable in each element is approximated using the linear shape functions, which is exactly the same as that in the standard FEM. The gradient field of the problem is obtained particularly using the gradient smoothing operation over the edge-based and face-based smoothing domains in 2D and 3D, respectively. The gradient smoothing technique can provide a proper softening effect to the model, effectively solve the problems caused by the well-known “overly-stiff” phenomenon existing in the standard FEM, and hence significantly improve the accuracy of the solution for the coupled systems. Intensive numerical studies have been conducted to verify the effectiveness of the coupled ES-/FS-FEM for structural-acoustic problems.     

有限周期电磁结构的区域分解快速算法

针对有限周期电磁结构,提出一种高效率的有限元分裂与互连算法.把原求解区域划分成若干个子区域,显著地降低了问题的复杂度.根据广义变分原理,采用拉格朗日乘子在子区域之间交换信息,并建立其相应的粗问题.研究子区域系数矩阵的可逆性.通过引入基本子区域,实现可扩展并行计算,且尤其适合于分析光子晶体等有限周期结构.     

局部时间步长间断有限元方法求解三维欧拉方程

使用间断有限元方法求解三维流体力学方程.空间剖分采用非结构四面体网格,为了克服显格式在单元网格尺寸差别较大时计算效率低下的问题,在格式中采用局部时间步长技术(LTS),即控制方程在空间、时间上积分得到一种单步格式,既可以局部计算每个单元又避免了Runge-Kutta高精度格式处理三维问题时存储量过大的问题.为了提高流体力学方程计算精度,在计算单元边界的数值流通量时使用任意高阶精度方法(ADER).数值算例表明格式稳定有效.     

变系数瞬态热传导问题边界元格式的特征正交分解降阶方法

提出了一种基于边界元法求解变系数瞬态热传导问题的特征正交分解(POD)降阶方法,重组并推导出变系数瞬态热传导问题适合降阶的边界元离散积分方程,建立了变系数瞬态热传导问题边界元格式的POD降阶模型,并用常数边界条件下建立的瞬态热传导问题的POD降阶模态,对光滑时变边界条件瞬态热传导问题进行降阶分析.首先,对一个变系数瞬态热传导问题,建立其边界域积分方程,并将域积分转换成边界积分;其次,离散并重组积分方程,获得可用于降阶分析的矩阵形式的时间微分方程组;最后,用POD模态矩阵对该时间微分方程组进行降阶处理,建立降阶模型并对其求解.数值算例验证了本文方法的正确性和有效性.研究表明:1)常数边界条件下建立的低阶POD模态矩阵,能够用来准确预测复杂光滑时变边界条件下的温度场结果;2)低阶模型的建立,解决了边界元法中采用时间差分推进技术求解大型时间微分方程组时求解速度慢、算法稳定性差的问题.     

流动数值模拟中一种并行自适应有限元算法

给出了一种流动数值模拟中的基于误差估算的并行网格自适应有限元算法.首先,以初网格上获得的当地事后误差估算值为权,应用递归谱对剖分方法划分初网格,使各子域上总体误差近似相等,以解决负载平衡问题.然后以误差值为判据对各子域内网格进行独立的自适应处理.最后应用基于粘接元的区域分裂法在非匹配的网格上求解N-S方程.区域分裂情形下N-S方程有限元解的误差估算则是广义Stokes问题误差估算方法的推广.为验证方法的可靠性,给出了不可压流经典算例的数值结果.     

Two-Level Stabilized Finite Volume Methods for the Stationary Navier-Stokes Equations

In this work, two-level stabilized finite volume formulations for the 2D steady Navier-Stokes equations are considered. These methods are based on the local Gauss integration technique and the lowest equal-order finite element pair. Moreover, the two-level stabilized finite volume methods involve solving one small Navier-Stokes problem on a coarse mesh with mesh size $H$, a large general Stokes problem for the Simple and Oseen two-level stabilized finite volume methods on the fine mesh with mesh size $h$=$\mathcal{O}(H^2)$ or a large general Stokes equations for the Newton two-level stabilized finite volume method on a fine mesh with mesh size $h$=$\mathcal{O}(|\log h|^{1/2}H^3)$. These methods we studied provide an approximate solution $(\widetilde{u}_h^v,\widetilde{p}_h^v)$ with the convergence rate of same order as the standard stabilized finite volume method, which involve solving one large nonlinear problem on a fine mesh with mesh size $h$. Hence, our methods can save a large amount of computational time.     

Method of Lines Based on Finite Element Discrete Technique to Analyze Eigenvalues of Waveguides

In this paper, a novel numerical technique to solve electromagnetic boundary value problem is described. This finite element based method of lines is developed by combining finite element method and the method of lines, so that it not only has high flexibility to treat geometrically and compositionally complex problems but also maintains high accuracy of semi-analytical technique. The geometry discretization, element mapping, element trial function and standard ordinary differential equation construction are discussed in details. Our numerical result demonstrates that this method can successfully solve the complex problems with fewer mesh lines when compared with conventional method of lines.     

含冷却水管大体积混凝土温度场计算的一种新方法

水管冷却是混凝土坝施工期的主要温度控制措施,提出一种新的水管冷却数值模拟理论和计算方法.该方法将水管置于常规混凝土单元内部,在单元中把混凝土与水管的接触面作为散热面纳入控制方程的边界条件,把混凝土通过冷却水管壁面耗散的能量叠加到常规泛函中,根据此复合泛函由变分原理建立含冷却水管混凝土的有限元支配方程.编制相应的三维计算程序,在程序中水管网格的拓扑信息由程序自动完成,冷却水管可以从混凝土单元任意位置穿过,因此不增加网格剖分的难度,算例表明该方法能准确模拟混凝土的冷却效应.     

An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors.     

Numerical simulation of laser-generated ultrasound in non-metallic material by the finite element method

The use of a pulsed laser for the generation of the elastic waves in non-metallic materials in the thermoelastic regime is investigated by using finite element method (FEM), taking into account not only thermal diffusion and the finite spatial and temporal shape of the laser pulse, but also optical penetration and the temperature dependence of material properties. The optimum finite element model is established based on analysis of two important parameters, meshing size and time step, and the stability of solution. Temperature distributions and temperature gradient fields in non-metallic material for different time steps are obtained, this temperature field is equivalent to a bulk force source to generate ultrasonic wave. The laser-generated ultrasound waveforms at the epicenter and surface acoustic waveforms (SAWs) are obtained and the influence of optical penetration into the material on the temperature field and the ultrasound waveforms are analyzed. The numerical results indicate that the heat penetration into non-metallic material is caused mainly by the optical penetration, and the ultrasound waveforms, especially the shape of the precursor, are strongly dependent on the optical penetration depth into non-metallic material.     

导热几何形状反演的变分原理及边界条件的确立

本文采用变域变分原理,建立了导热几何形状反演问题的变分原理,同时获得了该问题所需满足的边界条件和附加条件.该变分原理能将未知形状的几何变量及控制方程结合在一个变分泛函中,使得数学描述简洁、紧凑,且几何变量及控制方程的求解能耦合地进行.介绍了运用该变分原理并结合有限元方法进行数值计算的方法. 关键词： 几何形状反演 变分原理 有限元 导热