A new meshless method for steady-state heat conduction problems in anisotropic and inhomogeneous media

Summary A new meshless method is developed to analyze steady-state heat conduction problems with arbitrarily spatially varying thermal conductivity in isotropic and anisotropic materials. The analog equation is used to construct equivalent equations to the original differential equation so that a simpler fundamental solution of the Laplacian operator can be employed to take the place of the fundamental solutions related to the original governing equation. Next, the particular solution is approximated by using radial basis functions, and the corresponding homogeneous solution is solved by means of the virtual boundary collocation method. As a result, a new method fully independent of mesh is developed. Finally, several numerical examples are implemented to demonstrate the efficiency and accuracy of the proposed method. The numerical results show good agreement with the actual results.This work was supported by the National Natural Science Foundation of China (No. 10472082) and Australian Research Council.     

Singular boundary method for solving plane strain elastostatic problems

This study documents the first attempt to apply the singular boundary method (SBM), a novel boundary only collocation method, to two-dimensional (2D) elasticity problems. Unlike the method of fundamental solutions (MFS), the source points coincide with the collocation points on the physical boundary by using an inverse interpolation technique to regularize the singularity of the fundamental solution of the equation governing the problems of interest. Three benchmark elasticity problems are tested to demonstrate the feasibility and accuracy of the proposed method through detailed comparisons with the MFS, boundary element method (BEM), and finite element method (FEM).     

Singular boundary method for 2D dynamic poroelastic problems

This paper extends a strong-form meshless boundary collocation method, named the singular boundary method (SBM), for the solution of dynamic poroelastic problems in the frequency domain, which is governed by Biot equations in the form of mixed displacement–pressure formulation. The solutions to problems are represented by using the fundamental solutions of the governing equations in the SBM formulations. To isolate the singularities of the fundamental solutions, the SBM uses the concept of the origin intensity factors to allow the source points to be placed on the physical boundary coinciding with collocation points, which avoids the auxiliary boundary issue of the method of fundamental solutions (MFS). Combining with the origin intensity factors of Laplace and plane strain elastostatic problems, this study derives the SBM formulations for poroelastic problems. Five examples for 2D poroelastic problems are examined to demonstrate the efficiency and accuracy of the present method. In particular, we test the SBM to the multiply connected domain problem, the multilayer problem and the poroelastic problem with corner stress singularities, which are all under varied ranges of frequencies.     

A boundary-type finite element model for water surface wave problems

A new combinative method of boundary-type finite elements and boundary solutions is presented to study wave diffraction-refraction and harbour oscillation problems. The numerical model is based on the mild-slope equation. The key feature of this method is that the discretized matrix equation can be formulated only by the calculation of a line integral, since the interpolation equation which satisfies the governing equation in each element is used. The numerical solutions are compared with existing analytical, experimental, observed and other numerical results. The present method is shown to be an effective and accurate method for water surface wave problems.     

基本解方法与边界节点法求解Helmholtz方程的比较研究

基本解方法和边界节点法是基于径向基函数的两种重要无网格边界离散数值技术。针对Helmholtz方程,本文比较研究这两种数值方法在不同计算区域问题上的计算精度、插值矩阵对称性、病态性及计算成本。数值试验结果表明,两种方法都可以有效求解边界数据准确的Helmholtz问题。在数值离散过程中,两种方法都可以通过调整配置点的位...     

三维势流场的比例边界有限元求解方法

比例边界有限元法(SBFEM)是线性偏微分方程的一种新的数值求解方法。该方法只对计算域边界利用Galerkin方法进行数值离散,相对于有限元方法(FEM)减少了一个空间坐标的维数,而在减少的空间坐标方向利用解析方法进行求解;相对于边界元法(BEM),比例边界有限元方法不需要基本解,避免了奇异积分的计算,所以它结合了有限元和边界元方法的优点。本文建立了利用比例边界有限元法求解三维Laplace方程的数值模型并用于计算三维物体周围的水流场,将计算结果与解析解和边界元方法进行了对比,结果表明此方法可以很好地模拟水流场,且具有较高的计算精度。     

A new microstructure-dependent Saint–Venant torsion model based on a modified couple stress theory

In this paper a new modified couple stress model is developed for the Saint–Venant torsion problem of micro-bars of arbitrary cross-section. The proposed model is derived from a modified couple stress theory and has only one material length scale parameter. Using a variational procedure the governing differential equation and the associated boundary conditions are derived in terms of the warping function. This is a fourth order partial differential equation representing the analog of a Kirchhoff plate having the shape of the cross-section and subjected to a uniform tensile membrane force with mixed Neumann boundary conditions. Since the fundamental solution of the equation is known, the problem could be solved using the direct Boundary Element Method (BEM). In this investigation, however, the Analog Equation Method (AEM) solution is applied and the results are cross checked using the Method of Fundamental Solutions (MFS). Several micro-bars of various cross-sections are analyzed to illustrate the applicability of the developed model and to reveal the differences between the current model and an existing one which, however, contains two additional constants related to the microstructure. Moreover, useful conclusions are drawn from the micron-scale torsional response of micro-bars, giving thus a better insight in the gradient elasticity approach of the deformable bodies.     

Asymptotic approximation method for elliptic variational inequality of first kind简

An asymptotic approximation method is proposed to solve a particular elliptic variational inequality of first kind associated with unilateral obstacle problems. In this method, the free boundary is first captured, and then the method of the fundamental solution (MFS) is used to find the solution of the Dirichlet problem for Laplace’s equation in the non-coincidence set. Numerical examples are given to show the efficiency of the method.     

A refined theory of elastic thick plates for extensional deformation

Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein.     

A Quasilinear Based Procedure for Saturated/Unsaturated Water Movement in Soils

The quasilinear form of Richards equation for one-dimensional unsaturated flow in soils can be readily solved for a wide variety of conditions. However, it cannot explain saturated/unsaturated flow and the constant diffusivity assumption, used to linearise the transient quasilinear equation, can introduce significant error. This paper presents a quasi-analytical solution to transient saturated/unsaturated flow based on the quasilinear equation, with saturated flow explained by a transformed Darcy's equation. The procedure presented is based on the modified finite analytic method. With this approach, the problem domain is divided into elements, with the element equations being solutions to a constant coefficient form of the governing partial differential equation. While the element equations are based on a constant diffusivity assumption, transient diffusivity behaviour is incorporated by time stepping. Profile heterogeneity can be incorporated into the procedure by allowing flow properties to vary from element to element. Two procedures are presented for the temporal solution; a Laplace transform procedure and a finite difference scheme. An advantage of the Laplace transform procedure is the ability to incorporate transient boundary condition behaviour directly into the analytical solutions. The scheme is shown to work well for two different flow problems, for three soil types. The technique presented can yield results of high accuracy if the spatial discretisation is sufficient, or alternatively can produce approximate solutions with low computational overheads by using large sized elements. Error was shown to be stable, linearly related to element size.     

奇异边界法模拟二维弹性力学问题

奇异边界法是一种新的边界型无网格数值离散方法。该方法使用基本解作为插值基函数,在继承传统边界型方法优点的同时,不需要费时费力的网格划分和奇异积分,数学简单,编程容易,是一个真正的无网格方法。为避免配置点与插值源点重合时带来的基本解源点奇异性,该方法提出了源点强度因子的概念,从而将边界型强格式方法的核心归结为求解源点强度因子。本文首次将该方法应用于求解平面弹性力学问题。数值算例表明,本文算法稳定,效率高,并可达到很高的计算精度。     

双调和方程奇异边界法的改进及在Kirchhoff板弯曲中的应用

奇异边界法(SBM)是一种基于边界离散的无网格数值方法,在很多科学计算和工程领域中得到广泛的应用．该方法在处理复杂几何区域或者多连通区域时比基本解方法(MFS)数值计算更为稳定,具有易于实施、精度高等优点．SBM数值计算的关键之处在于源强度因子的计算,特别是相对于Laplace方程更为复杂的双调和方程的边界条件下源强度因子的计算．在高阶导数边界条件下,采用反插或者“加减项”原理计算源强度因子相对繁琐．本文对双调和方程的SBM进行了改进,将其中一个插值基函数改进为非奇异基函数形式,避免计算该基函数的源强度因子,极大简化了SBM的数值计算．本文改进对MFS同样有效,可以作为对传统MFS数值算法的补充．数值算例结果表明,本文提出的改进均能得到误差很小的数值解,且算法稳定,计算效率较高．     

Meshless methods for the inverse problem related to the determination of elastoplastic properties from the torsional experiment

The problem of determining the elastoplastic properties of a prismatic bar from the given experimental relation between the torsional moment M and the angle of twist per unit length of the rod’s length θ is investigated as an inverse problem. The proposed method to solve the inverse problem is based on the solution of some sequences of the direct problem by applying the Levenberg-Marquardt iteration method. In the direct problem, these properties are known, and the torsional moment is calculated as a function of the angle of twist from the solution of a non-linear boundary value problem. This non-linear problem results from the Saint-Venant displacement assumption, the Ramberg–Osgood constitutive equation, and the deformation theory of plasticity for the stress–strain relation. To solve the direct problem in each iteration step, the Kansa method is used for the circular cross section of the rod, or the method of fundamental solutions (MFS) and the method of particular solutions (MPS) are used for the prismatic cross section of the rod. The non-linear torsion problem in the plastic region is solved using the Picard iteration.     

弹性力学中一种新的边界轮廓法

利用基本解的特性,将面力积分方程化成仅含有Ｃａｕｃｈｙ主值积分的形式,基于这种边界积分方程,提出了一种新的边界轮廓法,对于三维问题,该方法只须计算沿边界单元界线的线积分,对二维问题,则只需计算边界单元两点的热函数之差,无须进行数值积分计算,实例计算说明该方法是有效的。     

TRIAL FUNCTION METHOD FOR THE LONGITUDINAL VIBRATION OF RODS WITH VARIABLE CROSS-SECTIONS

给出了一种试探函数法,并研究了变截面杆的纵振动问题. 先给出振动控制方程的特殊函数形式的试探解,然后要求此解满足控制方程,反过来确定了控制方程各种可能的系数函数（即截面变化函数）并得到了控制方程的精确解. 作为例子,给出了一种变截面杆在3 种边界条件下的频率方程,计算出了固有频率. 研究表明,试探函数法简单、直接,适合于研究变截面杆的纵振动问题. 对于杆扭转振动、薄膜振动以及管中波传播等问题,该方法同样有推广应用价值.     

Helmholtz边界积分方程中奇异积分间接求解方法

提出了间接求解传统Helmholtz边界积分方程CBIE的强奇异积分和自由项系数,以及Burton-Miller边界积分方程BMBIE中的超强奇异积分的特解法。对于声场的内域问题,给出了满足Helmholtz控制方程的特解,间接求出了CBIE中的强奇异积分和自由项系数。对于声场外域对应的BMBIE中的超强奇异积分,按Guiggiani方法计算其柯西主值积分需要进行泰勒级数展开的高阶近似,公式繁复,实施困难。本文给出了满足Helmholtz控制方程和Sommerfeld散射条件的特解,提出了间接求出超强奇异积分的方法。推导了轴对称结构外场问题的强奇异积分中的柯西主值积分表达式,并通过轴对称问题算例证明了本文方法的高效性。数值结果表明,对于内域问题,采用本文特解法的计算结果优于直接求解强奇异积分和自由项系数的结果,且本文的特解法可避免针对具体几何信息计算自由项系数,因而具有更好的适用性。对于外域问题,两者精度相当,但本文的特解法可避免对核函数进行高阶泰勒级数展开,更易于数值实施。     

Numerical method for unsteady compressible boundary layers driven by compression or expansion wave

A numerical analysis is presented for the unsteady compressible laminar boundary layer driven by a compression or expansion wave. Approximate or series expansion methods have been used for the problems because of the characteristics of the governing equations, such as non-linearity, coupling with the thermal boundary layer equation and initial conditions. Here a transformation of the governing equations and the numerical linearization technique are introduced to deal with the difficulties. First, the governing equations are transformed for the initial conditions by Howarth and semisimilarity variables. These transformations reduce the number of independent variables from three to two and the governing equations from partial to ordinary differential equations at the initial point. Next, the numerical linearization technique is introduced for the non-linearity and the coupling with the thermal boundary layer equation. Because the non-linear terms are linearized without sacrifice of numerical accuracy, the solutions can be obtained without numerical iterations. Therefore the exact numerical solution, not approximate or series expansion, can be obtained. Compared with the approximate or series expansion method, this method is much improved. Results are compared with the series expansion solutions.     

A NEW STATE SPACE SOLUTION FOR RECTANGULAR THICK LAMINATES WITH CLAMPED EDGES Hu Wenfeng Liu Yihua

通过把固支边上的边界位移函数作为状态变量引入状态方程,得到了含固支边矩形单层与叠层厚板的精确解. 在求解过程中,将非齐次状态方程的求解变为齐次状态方程的求解,省略了求解待定常数的中间过程,使求解过程变得简单. 所得到的解能够严格满足固支边界条件,在同一材料层内不需作分层处理,因而更加精确. 此外,对固支边的应力提出了新的计算方法,能够得到更精确的边界应力. 算例表明,本文解比现有精确解收敛快,与有限元解吻合的更好,尤其是在固支边处体现得更加明显.     

A generalised finite difference scheme based on compact integrated radial basis function for flow in heterogeneous soils

In the present paper, we develop a generalised finite difference approach based on compact integrated radial basis function (CIRBF) stencils for solving highly nonlinear Richards equation governing fluid movement in heterogeneous soils. The proposed CIRBF scheme enjoys a high level of accuracy and a fast convergence rate with grid refinement owing to the combination of the integrated RBF approximation and compact approximation where the spatial derivatives are discretised in terms of the information of neighbouring nodes in a stencil. The CIRBF method is first verified through the solution of ordinary differential equations, 2–D Poisson equations and a Taylor‐Green vortex. Numerical comparisons show that the CIRBF method outperforms some other methods in the literature. The CIRBF method in conjunction with a rational function transformation method and an adaptive time‐stepping scheme is then applied to simulate 1–D and 2–D soil infiltrations effectively. The proposed solutions are more accurate and converge faster than those of the finite different method used with a second‐order central difference scheme. Additionally, the present scheme also takes less time to achieve target accuracy in comparison with the 1D‐IRBF and higher order compact schemes.     

The MFS for the Cauchy problem in two-dimensional steady-state linear thermoelasticity

We study the reconstruction of the missing thermal and mechanical fields on an inaccessible part of the boundary for two-dimensional linear isotropic thermoelastic materials from over-prescribed noisy (Cauchy) data on the remaining accessible boundary. This problem is solved with the method of fundamental solutions (MFS) together with the method of particular solutions (MPS) via the MFS-based particular solution for two-dimensional problems in uncoupled thermoelasticity developed in Marin and Karageorghis, 2012a, Marin and Karageorghis, 2013. The stabilisation/regularization of this inverse problem is achieved by using the Tikhonov regularization method (Tikhonov and Arsenin, 1986), whilst the optimal value of the regularization parameter is selected by employing Hansen’s L-curve method (Hansen, 1998).