An Iterative FEM for Solving Wave Problems of a Halfspace

An iterative finite element method for solving wave problems of a halfspace is presented in this paper. The halfspace is first truncated by introducing a fictive finite boundary on which some fictive boundary conditions must be imposed. A finite computational domain is in each iteration subjected to actual boundary conditions on real boundary and to fictive Dirichlet or Neumann boundary conditions on the fictive boundary. The radiation condition is satisfied by using DtN operator. The DtN operator is not introduce in the finite element formulation on the fictive boundary so any finite elements can be used. The method is simple and specially useful for computing higher harmonics.     

Laplace方程边值问题的边界积分方程法

§ 1.　Introduction　　Inengineeringandtechnology ,theproblemofstaticelectricfieldscanbeattributedtotheboundaryproblemofLaplaceequationofstaticeletricpotentialfunction .Themethodsofclassi calmathematicalphysicscanbeonlyusedtosolveboundaryproblemofverysimpledomainandspecialboundarycondition .Althoughthemethodsoflimitedelementscanbeusedtosolvetheproblemsonarbitrarydomain ,butitneedstopartitionthewholedomainandtocalculateverycomplex .Theapproachofboundaryintegralequationistosolverelatedproblemsb…     

A coherent analysis of Stokes flows under boundary conditions of friction type

This paper is concerned with the stationary and nonstationary flow of viscous incompressible fluid under boundary conditions of friction type, which are certain nonlinear boundary conditions similar to the so-called Signorini boundary condition in elasticity. We assume that the flow is governed by the linear Stokes equation, while the boundary condition is nonlinear. From the methodological viewpoint, the analysis is carried out in a coherent way, starting from study of the related boundary value problems for the stationary flow by means of the theory of variational inequalities, and getting to wellposedness of the initial boundary value problems for the nonstationary flow by means of the nonlinear semigroup theory. From the viewpoint of applications, we mention original motivations and include some new generalizations like the cases of anisotropic friction and inhomogeneous boundary value.     

AN ARTIFICIAL BOUNDARY CONDITION FOR THE INCOMPRESSIBLE VISCOUS FLOWS IN A NO-SLIP CHANNEL

1.IntroductionWhencomputingthenumericals0luti0nsofviscousfluidfl0wproblemsinallun-boundedd0main,0neoftenintroducesartificialboundaries,andsetsupanartificialbopundarycondition0nthem;thenthe0riginalproblemisreducedtoaproblemonab0undedc0mputationald0main.InordertoIimitthecomputatio11alcost,theseboundariesmustnotbet00farfromthedomainofinterest.Theref0re,theartificialboundaryc0nditi0nsmustbegoodapprotimationt0the"exact"boundaryconditions(sothatthes0lutionoftheproblemintheboundeddonlainisequaltothes…     

无限域拉普拉斯方程问题的高阶边界条件及其应用

对无限域Laplace方程问题,推导出了高阶边界条件.在采用数值方法的有限域的外边界上应用高阶边界条件,可以在保证计算精度的前提下缩小数值求解域,从而减小计算工作量和少占用计算机内存.数值算例表明,一阶边界条件近似于精确边界条件,它明显地优于经典边界条件和二阶边界条件.     

An analytic series method for Laplacian problems with mixed boundary conditions

Mixed boundary value problems are characterised by a combination of Dirichlet and Neumann conditions along at least one boundary. Historically, only a very small subset of these problems could be solved using analytic series methods (“analytic” is taken here to mean a series whose terms are analytic in the complex plane). In the past, series solutions were obtained by using an appropriate choice of axes, or a co-ordinate transformation to suitable axes where the boundaries are parallel to the abscissa and the boundary conditions are separated into pure Dirichlet or Neumann form. In this paper, I will consider the more general problem where the mixed boundary conditions cannot be resolved by a co-ordinate transformation. That is, a Dirichlet condition applies on part of the boundary and a Neumann condition applies along the remaining section. I will present a general method for obtaining analytic series solutions for the classic problem where the boundary is parallel to the abscissa. In addition, I will extend this technique to the general mixed boundary value problem, defined on an arbitrary boundary, where the boundary is not parallel to the abscissa. I will demonstrate the efficacy of the method on a well known seepage problem.     

A principle of images for absorbing boundary conditions

When one uses high-order finite difference schemes for the wave equation, for instance fourth order schemes, the treatment of boundary conditions poses a real difficulty since one needs several additional equations (for the nodes close to the boundary), while one single scalar boundary condition is available. In the case of perfectly reflecting boundary conditions, namely the homogeneous Neumann or Dirichlet conditions, this difficulty can be overcomed by the use of the well-known image principle, which permits the extension of the equation outside of the domain of calculation by an appropriate symmetrization of the data. We propose in this article a generalization of this principle to the absorbing boundary conditions. Through a symmetrization process, we are led to introduce a damped wave equation with a damping term supported by the boundary. The treatment of the boundary condition is then replaced by the approximation of this new damped wave equation in the whole space. The theoretical justification of our approach is based on new energy estimates for the wave equation (when high-order absorbing boundary conditions are used), and constitutes an alternative to the use of the well-known Kreiss criterion to prove the stability of the associated initial boundary value problems. © 1994 John Wiley & Sons, Inc.     

一个扩散问题的自然边界元法与有限元法组合

本文讨论由Helmholtz方程描述的扩散问题的自然边界元法与有限元法的组合．取一个圆作为公共边界，用Fourier展开建立边界积分方程，将无界区域上的问题化为有界区域上的非局部边值问题．在变分方程中公共边界上的未知量只包含函数本身而不包含其法向导数，从而减少了未知数的数目，并且边界元剐度矩阵只有极少量不同的元素，有利于数值计算．这种组台方法优越于建立在直接边界元法基础上的组合方法．文中证明了变分解的唯一性，数值解的收敛性和误差估计．最后讨论了数值技术并给出一个算倒．     

Optimización de forma de modelos paramétricos bidimensionales basada en ponderación geométrica y reglas evolutivas

This paper presents a method for shape optimization of two-dimensional models subjected to simple or multiple load cases. The optimization is performed iteratively using evolutionary rules, based on the stress level. These rules determine regions on the boundary where the material is underused or overused; the objective is that the model evolves to a minimum weight model with a high and homogeneous stress level. The evolution is performed by modifying the boundary of the model slowly. Since the boundary is defined by parametric B-spline curves, generated changes result in a smooth boundary. The main proposal of the method is that boundary modifications are given by a set of displacements that, its magnitude and direction, take into account the geometrical information of the neighborhood, unlike other methods that simply generate displacement perpendicular to the boundary. Finally, the paper presents the design of a frame bike in order to show the good performance of the method.     

Travel Time Tomography

We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary by measuring for every point and direction of entrance of a geodesic the point of exit and direction of exit and its length. The linearization of these two problems is tensor tomography. The question is whether one can determine a symmetric two-tensor from its integrals along geodesics. We emphasize recent results on boundary and lens rigidity and in tensor tomography in the partial data case, with further applications.     

Sets of finite perimeter and the Hausdorff-Gauss measure on the Wiener space

In Euclidean space, the integration by parts formula for a set of finite perimeter is expressed by the integration with respect to a type of surface measure. According to geometric measure theory, this surface measure is realized by the one-codimensional Hausdorff measure restricted on the reduced boundary and/or the measure-theoretic boundary, which may be strictly smaller than the topological boundary. In this paper, we discuss the counterpart of this measure in the abstract Wiener space, which is a typical infinite-dimensional space. We introduce the concept of the measure-theoretic boundary in the Wiener space and provide the integration by parts formula for sets of finite perimeter. The formula is presented in terms of the integration with respect to the one-codimensional Hausdorff-Gauss measure restricted on the measure-theoretic boundary.     

Two perturbation formulations of the nonlinear dynamics of a cable excited by a boundary motion

A nonlinear cable excited by an inclined boundary motion, termed as cable's moving boundary problem, is attacked by two different perturbation approaches, i.e., the boundary modulation formulation and the quasi-static drift formulation. The former transforms the boundary motion into a weak modulation on cable's high-order dynamics, while the latter introduces a hybrid mode expansion using an empirical drift shape function. In both formulations, the inclined boundary motion induces three different excitation effects, i.e., longitudinal direct, vertical boundary kinematic, and high-order parametric, all of which being characterized by the parametric modulation factors. Detailed comparative studies indicate that the modulation factors in the two formulations are exactly equivalent to each other only if a new drift shape function, well defined in the boundary modulation formulation, is used for the quasi-static drift formulation. In contrast, the empirical shape functions lead only to an approximate equivalence for intermediate/large boundary motion inclinations. Moreover, for small inclinations, the two formulations induce possible quantitative and qualitative differences. The approximate analytical framework is validated and shown to be computationally efficient, by comparison with the finite difference method.     

Differentiability and its asymptotic analysis for nonlinear singularly perturbed boundary value problem

In this paper, we show differentiability of solutions with respect to the given boundary value data for nonlinear singularly perturbed boundary value problems and its corresponding asymptotic expansion of small parameter. This result fills the gap caused by the solvability condition in Esipova’s result so as to lay a rigorous foundation for the theory of boundary function method on which a guideline is provided as to how to apply this theory to the other forms of singularly perturbed nonlinear boundary value problems and enlarge considerably the scope of applicability and validity of the boundary function method. A third-order singularly perturbed boundary value problem arising in the theory of thin film flows is revisited to illustrate the theory of this paper. Compared to the original result, the imposed potential condition is completely removed by the boundary function method to obtain a better result. Moreover, an improper assumption on the reduced problem has been corrected.     

Weak Galerkin finite element method for valuation of American options

We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.     

Micro/nano sliding plate problem with Navier boundary condition

For Newtonian flow through micro or nano sized channels, the no-slip boundary condition does not apply and must be replaced by a condition which more properly reflects surface roughness. Here we adopt the so-called Navier boundary condition for the sliding plate problem, which is one of the fundamental problems of fluid mechanics. When the no-slip boundary condition is used in the study of the motion of a viscous Newtonian fluid near the intersection of fixed and moving rigid plane boundaries, singular pressure and stress profiles are obtained, leading to a non-integrable force on each boundary. Here we examine the effects of replacing the no-slip boundary condition by a boundary condition which attempts to account for boundary slip due to the tangential shear at the boundary. The Navier boundary condition, possesses a single parameter to account for the slip, the slip length ℓ, and two solutions are obtained; one integral transform solution and a similarity solution which is valid away from the corner. For the former the tangential stress on each boundary is obtained as a solution of a set of coupled integral equations. The particular case solved is right-angled corner flow and equal slip lengths on each boundary. It is found that when the slip length is non-zero the force on each boundary is finite. It is also found that for a suffciently large distance from the corner the tangential stress on each boundary is equal to that of the classical solution. The similarity solution involves two restrictions, either a right-angled corner flow or a dependence on the two slip lengths for each boundary. When the tangential stress on each boundary is calculated from the similarity solution, it is found that the similarity solution makes no additional contribution to the tangential stress of that of the classical solution, thus in agreement with the findings of the integral transform solution. Values of the radial component of velocity along the line θ = π /4 for increasing distance from the corner for the similarity and integral transform solutions are compared, confirming their agreement for sufficiently large distances from the corner. (Received: November 9, 2005)     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.     

Biharmonic equation in a highly oscillating domain and homogenization of an associated control problem

We consider a Dirichlet boundary control problem posed in an oscillating boundary domain governed by a biharmonic equation. Homogenization of a PDE with a non-homogeneous Dirichlet boundary condition on the oscillating boundary is one of the hardest problems. Here, we study the homogenization of the problem by converting it into an equivalent interior control problem. The convergence of the optimal solution is studied using periodic unfolding operator.     

Absorbing boundary conditions for electromagnetic wave propagation

In this paper, the theoretical perfectly absorbing boundary condition on the boundary of a half-space domain is developed for the Maxwell system by considering the system as a whole instead of considering each component of the electromagnetic fields individually. This boundary condition allows any wave motion generated within the domain to pass through the boundary of the domain without generating any reflections back into the interior. By approximating this theoretical boundary condition a class of local absorbing boundary conditions for the Maxwell system can be constructed. Well-posedness in the sense of Kreiss of the Maxwell system with each of these local absorbing boundary conditions is established, and the reflection coefficients are computed as a plane wave strikes the artificial boundary. Numerical experiments are also provided to show the performance of these local absorbing boundary conditions

     

Artificial Multilevel Boundary Element Preconditioners

A hierarchical multilevel preconditioner is constructed for an efficient solution of a first kind boundary integral equation with the single layer potential operator discretized by a boundary element method. This technique is based on a hierarchical clustering of all boundary elements as used in fast boundary element methods. This hierarchy is applied to define a sequence of nested boundary element spaces of piecewise constant basis functions as used in the definition of the preconditioning multilevel operator.