Calculating transport of water from a conduit to the porous matrix by boundary distributed source method

This study presents the development of a suitable numerical method for porous media flow with free and moving boundary (Stefan) problems arising in systems with wetted and unwetted regions of porous media. A non-singular version of the method of fundamental solutions (MFS), termed the boundary distributed source method (BDS), is applied. Darcy flow and homogenous isotropic porous media is assumed. The solution is represented in terms of the fundamental solution of the Laplace equation in two-dimensional Cartesian coordinates. The desingularisation is achieved through boundary distributed sources of the fundamental solution and indirect calculation of the derivatives of the fundamental solution. Respectively, the artificial boundary, characteristic for the classical, singular MFS is not present. The novel BDS is compared with the MFS and the analytical solutions for several numerical examples with excellent agreement. A sensitivity study of the solution, regarding the discretization and the free parameters is performed. The main contributions of the study are the application of the BDS to free and moving boundary problems and the comparison of BDS with MFS for these types of problems. The developed model can be applied to various geohydrological problems.     

A scaled boundary finite element method applied to electrostatic problems

The scaled boundary finite element method (SBFEM) is a novel semi-analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. In this paper, the SBFEM is firstly extended to solve electrostatic problems. Two new SBFE coordination systems are introduced. Based on Laplace equation of electrostatic field, the derivations (based on a new variational principle formulation) and solutions of SBFEM equations for both bounded domain and unbounded domain problems are expressed in details, the solution for the inclusion of prescribed potential along the side-faces of bounded domain is also presented in details, then the total charges on the side-faces can be semi-analytically solved, and a particular solution for the potential field in unbounded domain satisfying the constant external field is solved. The accuracy and efficiency of the method are illustrated by numerical examples with complicated field domains, potential singularities, inhomogeneous media and open boundaries. In comparison with analytic solution method and other numerical methods, the results show that the present method has strong ability to resolve singularity problems analytically by choosing the scaling centre at the singular point, has the inherent advantage of solving the open boundary problems without truncation boundary condition, has efficient application to the problems with inhomogeneous media by placing the scaling centre in the bi-material interfaces, and produces more accurate solution than conventional numerical methods with far less number of degrees of freedom. The method in electromagnetic field calculation can have broad application prospects.     

On the propagation of waves from spherical and cylindrical cavities in anisotropic materials

The propagation of waves from a spherical or cylindrical cavity in an inhomogeneous anisotropic elastic solid is considered. In the first instance, integral transforms are used to provide solutions to specific boundary value problems involving elastic media exhibiting certain inhomogeneities. It is then noted that the Bergman integral operator method provides a more general analysis. Finally, an asymptotic approach having a wide range of application is discussed and employed to construct wavefront and high-frequency expansions for the solution field in general media.     

数值流形方法用于岩石爆破数值计算的可行性浅析

数值流形方法是一种新的可综合用于求解连续与非连续介质力学的数值计算方法,通过对该方法的基本理论与求解思想的分析表明,该方法应用在岩石爆破数值模拟中是一个可行的数值分析方法,并对其在实际应用中应注意和需要解决的问题进行了分析,对其应用于实际具有参考作用.     

Three-dimensional boundary-element computation of potential flow in fractured rock

The use of the boundary-element method (BEM) for three-dimensional potential flow problems is described herein. The specific application for which the program is designed is the flow in fractured porous media. There are several enhancements to the usual boundary-element calculations. The method uses linear elements to approximate better the boundary conditions and the solution on the surfaces as compared to constant elements; accurate integration over high-aspect-ratio triangles is achieved by computing all the integrals in closed form, as opposed to numerical cubature or mixed exact-numerical integration schemes; and the zoning for inhomogeneous regions uses an efficient method which reduces the size of the problem and avoids difficulties with singularities.     

Axial symmetric stationary heat conduction analysis of non-homogeneous materials by triple-reciprocity boundary element method

The heat conduction problems in homogeneous media can be easily solved by the boundary element method. The spatial variations of heat sources as well as material coefficients gives rise to domain integrals in integral formulations for solution of boundary value problems in functionally gradient materials (FGM), since the fundamental solutions are not available for partial differential equations with variable coefficients, in general. In this paper, we present the development of the triple reciprocity method for solution of axial symmetric stationary heat conduction problems in continuously non-homogeneous media with eliminating the domain integrals. In this method, the spatial variations of domain “sources” are approximated by introducing new potential fields and using higher order fundamental solutions of the Laplace operator.     

弹性介质模糊有限元控制方程的快速解法

线弹性模糊有限元方法是分析弹性介质体模糊特性对结构响应产生不确定性影响的有效方法。即使对弹性介质体而言,模糊有限元控制方程的求解时间问题也是困扰其推广应用的主要障碍。为获得可靠可行的模糊有限元控制方程的快速求解方法,在深入研究弹性介质体的模糊源特点基础上,提出当引起结构模糊特性的力学参数为单源模糊数时,可以利用单源模糊数的运算特点来求解模糊有限元的控制方程,进而利用合成运算求解结构的模糊位移和模糊应力的分布。推导了基于单源模糊数运算的弹性介质模糊应力和模糊位移的计算表达式。应用模糊有限元求解的区间解法和快速解法对算例进行比较分析,结果表明了快速解法的正确性。     

Vortex motions of solid media in dynamic problems of the elasticity theory

Within the framework of a problem of constructing a mathematical theory for vortex motions of solid media, we investigated a number of problems of the dynamic elasticity theory with application of numerical and analytical approaches. The solution for a self-similar problem in the finite form is presented, which we were able to divide uniquely into potential and vortex components.     

Hybrid numerical scheme for time-evolving wave fields

Many problems in geophysics, acoustics, elasticity theory, cancer treatment, food process control and electrodynamics involve study of wave field synthesis (WFS) in some form or another. In the present work, modelling of wave propagation phenomena is studied as a static problem, using finite element method and treating time as an additional spatial dimension. In particular, WFS problems are analysed using discrete methods. It is shown that a fully finite element-based scheme is very natural and effective method for the solution of such problems. Distributed WFS in the context of two-dimensional problems is outlined and incorporation of any geometric or material non-linearities is shown to be straightforward. This has significant implications for problems in geophysics or biological media, where material inhomogeneities are quite prevalent. Numerical results are presented for several problems referring to media with material inhomogeneities and predefined absorption profiles. The method can be extended to three-dimensional problems involving anisotropic media properties in a relatively straightforward manner. Copyright © 2006 John Wiley & Sons, Ltd.     

A dual-reciprocity boundary element method for a class of elliptic boundary value problems for non-homogeneous anisotropic media

A dual-reciprocity boundary element method is proposed for the numerical solution of a two-dimensional boundary value problem (BVP) governed by an elliptic partial differential equation with variable coefficients. The BVP under consideration has applications in a wide range of engineering problems of practical interest, such as in the calculation of antiplane stresses or temperature in non-homogeneous anisotropic media. The proposed numerical method is applied to solve specific test problems.     

RVE-based multilevel method for periodic heterogeneous media with strong scale mixing

A Representative Volume Element based multilevel (multigrid) solution method for wave propagation problems in periodic heterogeneous media is developed. The intergrid transfer operators are constructed from the solution of the Representative Volume Element (RVE) problem. It is shown that the convergence of the RVE-based multilevel method improves with increasing material heterogeneity and decreasing time integration step. Numerical results confirm theoretical estimates.     

Finite-volume micromechanics of periodic materials: Past,present and future

The finite-volume method is now a well-established tool in the numerical engineering community for simulation of a wide range of problems in fluid and solid mechanics. Its acceptance by the mechanics of heterogeneous media community, however, continues to be slow, often characterized by confusion with the finite-element method or so-called higher-order theories. Herein, we provide a brief historical perspective on the evolution of this important technique in the fluid mechanics community, its transition to the solution of solid mechanics boundary-value problems initiated in Europe in 1988, and the recent developments aimed at the solution of unit cell problems of periodic heterogeneous media. The differences and similarities with the finite-element method are highlighted, and the resulting tangible advantages of the finite-volume technique discussed and illustrated. Finally, our most recent results in this area are presented which demonstrate the method’s capability of solving unit cell problems with complex architectures in a variety of settings and applications, while revealing undocumented effects of interest in the development of new material microstructures with targeted response. Recent attempts to develop alternative versions of this technique are also discussed, together with our ongoing work to generalize the finite-volume micromechanics approach in order to further enhance its predictive capabilities and efficiency.     

Radiative heat transfer in participating media — A review

This paper presents an overview of various exact analytic and approximate numerical methods for the solution of radiative heat transfer problems in participating media. Review of each method is followed by its strengths and limitations. Importance of radiative heat transfer analysis and difficulties in the solution of radiative transfer problems have been emphasized.     

Advanced Kantorovich method for biharmonic problems

In this paper a method for solving biharmonic problems involving a mixed numerical-analytical approach is described. The algorithm of this method is given and the efficiency of its application for the solution of biharmonic problems is discussed. The recommendations about an application of this method for solving stationary three-dimensional problems in the theory of elasticity are given.     

Boundary elements and nonclassical differences in the problems of nonlinear theory of elasticity

We propose a method of simultaneous application of the fundamental solution of the Lame equations, boundary elements, nonclassical finite-difference relations, and the procedure of continuation of solutions in a parameter to the solutions two-dimensional problems for physically nonlinear inhomogeneous media. Carpathian Division of the Institute of Geophysics, Ukrainian Academy of Sciences. L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 36, No. 1, pp. 32–38, January–February, 2000.     

On the use of space-time finite elements in the solution of elasto-dynamic fracture problems

The use of a discontinuous Galerkin (DG) formulation for the solution of dynamic fracture problems in linear elastic media with and without cohesive zones is explored. The results are compared with closed-form as well as numerical solutions available from the literature. The effectiveness of the space-time finite element method in the study of dynamic fracture problems is demonstrated, especially in those cases in which dynamic fracture occurs along with time discontinuous loading.     

Reduced-order model technique for the analysis of anisotropic inhomogeneous media: application to liquid-crystal displays

Electromagnetic wave propagation in anisotropic inhomogeneous media is computed by a novel reduced-order model technique, which is based on the restriction of the Marcuvitz-Schwinger equations on Krylov subspaces and on the application of the singular-value decomposition. The model is derived from the standard coupled-wave method and includes both wide-angle diffraction and light scattering at dielectric interfaces. The method, currently implemented for two-dimensional problems, was applied to the analysis of different liquid-crystal test cells. Numerical results are compared with those obtained through the application of the coupled-wave method and the Jones method and with experimental microscopic measurements.     

Light transport in three-dimensional semi-infinite scattering media

The three-dimensional radiative transfer equation is solved for modeling the light propagation in anisotropically scattering semi-infinite media such as biological tissue, considering the effect of internal reflection at the interfaces. The two-dimensional Fourier transform and the modified spherical harmonics method are applied to derive the general solution to the associated homogeneous problem in terms of analytical functions. The obtained solution is used for solving boundary-value problems, which are important for applications in the biomedical optics field. The derived equations are successfully verified by comparisons with Monte Carlo simulations.     

Thermal stresses in a viscoelastic trimaterial

A general solution for a three dissimilar sandwiched medium subjected to a point heat source and a three-phase composite cylinder subjected to a remote uniform heat flow is presented in this work. The solutions to the heat conduction and thermoelastic problems are derived based on the method of analytic continuation associated with the alternation technique. A rapidly convergent series solution for both the temperature and the stress field, which is expressed in terms of an explicit general term of the corresponding homogeneous potential, is obtained in an elegant form. The hereditary integral in conjunction with the Kelvin–Maxwell model is applied to simulate the thermoviscoelastic properties while a thermorheologically simple material is considered. Based on the correspondence principle, the Laplace transformed thermoviscoelastic solution is directly determined from the corresponding thermoelastic one. The real-time solution can then be solved numerically by taking inverse Laplace transform. Two typical examples concerning interfacial stresses for plane-layered media and circularly cylindrical-layered media are discussed in detail.     

Hybrid finite element-Monte Carlo method for coupled conduction and gas radiation enclosure heat transport

A hybrid two-dimensional finite element-Monte Carlo numerical solution method has been developed for solving complex transient non-linear gas radiation enclosure problems. Solid conducting media are coupled to conducting gaseous or particulate participating media and both may be internally energy generating. Bilinear isoparametric elements are used, allowing geometrically complex enclosures with internal objects to be present. Radiative energy transport within the gaseous enclosures is accounted for using a Monte Carlo method which permits a broad range of complexities. Included here are: diffuse-grey and specularly reflecting walls with temperature and spectrally dependent, and non-homogeneous emissivities or absorptivities; absorbing/emitting gases with temperature and spectrally dependent, and non-homogeneous emissivities or absorptivities; and isotropic or anisotropic scattering gases or particles. This method was verified by solving several standard steady state problems and comparing the solutions to those found using the conventional finite element, the P–N and the exchange factor methods. Solution times are comparable to those of the finite element method. A more complex transient problem solution is then given to demonstrate the versatility and power of the method.