Anisotropic mesh adaptation for numerical solution of boundary value problems

We present an efficient mesh adaptation algorithm that can be successfully applied to numerical solutions of a wide range of 2D problems of physics and engineering described by partial differential equations. We are interested in the numerical solution of a general boundary value problem discretized on triangular grids. We formulate a necessary condition for properties of the triangulation on which the discretization error is below the prescribed tolerance and control this necessary condition by the interpolation error. For a sufficiently smooth function, we recall the strategy how to construct the mesh on which the interpolation error is below the prescribed tolerance. Solving the boundary value problem we apply this strategy to the smoothed approximate solution. The novelty of the method lies in the smoothing procedure that, followed by the anisotropic mesh adaptation (AMA) algorithm, leads to the significant improvement of numerical results. We apply AMA to the numerical solution of an elliptic equation where the exact solution is known and demonstrate practical aspects of the adaptation procedure: how to control the ratio between the longest and the shortest edge of the triangulation and how to control the transition of the coarsest part of the mesh to the finest one if the two length scales of all the triangles are clearly different. An example of the use of AMA for the physically relevant numerical simulation of a geometrically challenging industrial problem (inviscid transonic flow around NACA0012 profile) is presented. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

A Collocation Code for Singular Boundary Value Problems in Ordinary Differential Equations

We present a MATLAB package for boundary value problems in ordinary differential equations. Our aim is the efficient numerical solution of systems of ODEs with a singularity of the first kind, but the solver can also be used for regular problems. The basic solution is computed using collocation methods and a new, efficient estimate of the global error is used for adaptive mesh selection. Here, we analyze some of the numerical aspects relevant for the implementation, describe measures to increase the efficiency of the code and compare its performance with the performance of established standard codes for boundary value problems.     

An experimental comparison between the aqueous pH scale and the electron donicity scale

We have made an experimental comparison of the aqueous pH scale with the Lewis base donicity scale. This requires an indicator that can function in both kinds of system. We find that the zeta-potential zero charge point for certain oxide materials provides a suitable indicator. The materials are: SiO₂, Al₂O₃, TiO₂, and MgO. For each material, we measure the zeta potential in aqueous media as a function of pH. In each case, the zeta potential varies systematically with pH, and changes sign at some value of pH, that we refer to as the zero charge point. We then do similar measurements in a series of organic liquids having different donicities. In this case, the zeta potential varies systematically with the donicity and may change sign. In such cases, we determine the donicity that corresponds to the zero charge point. For a given oxide material, we can then relate a zero charge point on the pH scale to one on the donicity scale. This gives us a consistent experimental alignment of the pH and donicity scales, relative to one another. We compare our results with thermodynamic data, relating to the pH scale, and the hydrogen electrode, and find good agreement.     

氟电极测定铝的新方法——Apple Ⅱ-ISE智能离子计的应用

本文提出一种氟电极测铝的新方法,即二次回归校正曲线法及分段抛物插值法。由于汇集了1.该法依据的数学模型与实验数据相符;2.AppleⅡ-ISE智能离子计具有较强的数据处理功能;3.氟电极性能良好等三个特点,故本法准确可靠,快速方便,具有一定理论意义和较大实用价值。     

COST-BENEFIT ANALYSIS APPLIED TO WILDERNESS PRESERVATION—OPTION VALUE UNCERTAINTY AND DITONICITY

It has been recognized for some time that when cost-benefit analysis is applied to irreversible environmental decisions, such as that of developing or preserving wilderness land, there can be an option value associated with the preservation decision, which arises when there is future uncertainty with respect to the benefits of development or preservation. In this paper the provenance of option value is examined and it is shown that an important cause is a special kind of uncertainty, viz. the possibility of reversals in direction of the relative valuations of wilderness land and developed land, a property we refer to as ditonicity. It is shown that the more ditonic the relative valuation process the greater the deviance between the certainty-equivalence development policy and the stochastically optimal one, and thus by implication the greater the option value. In the two cases with zero ditonicity, when relative wilderness valuations always increase or always decrease (even though in a stochastic fashion), there is zero option value. The model used assumes that service flows from wilderness and developed land are size-dependent, with future relative values known only in terms of a stochastic process, which can take jumps up or down of the same proportional size, at random times. Development can be partial or total and can occur in impulses at any time over an infinite time horizon.     

The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation

In this article we describe a numerical method to solve a nonhomogeneous diffusion equation with arbitrary geometry by combining the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunction expansion method (EEM). This forms a meshless numerical scheme of the MFS‐MPS‐EEM model to solve nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for any time and any shape. Nonhomogeneous diffusion equation with complex domain can be separated into a Poisson equation and a homogeneous diffusion equation using this model. The Poisson equation is solved by the MFS‐MPS model, in which the compactly supported radial basis functions are adopted for the MPS. On the other hand, utilizing the EEM the diffusion equation is first translated to a Helmholtz equation, which is then solved by the MFS together with the technique of the singular value decomposition (SVD). Since the present meshless method does not need mesh generation, nodal connectivity, or numerical integration, the computational effort and memory storage required are minimal as compared with other numerical schemes. Test results for two 2D diffusion problems show good comparability with the analytical solutions. The proposed algorithm is then extended to solve a problem with irregular domain and the results compare very well with solutions of a finite element scheme. Therefore, the present scheme has been proved to be very promising as a meshfree numerical method to solve nonhomogeneous diffusion equations with time‐independent source terms of any time frame, and for any arbitrary geometry. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Models for Stationary Max-Stable Random Fields

Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.     

Integration of an Equilibrium System in an Enhanced Theory of Bending of Elastic Plates

The complete integral of the system of partial differential equations governing the equilibrium bending of elastic plates with transverse shear deformation and transverse normal strain is constructed by means of complex variable methods. The process helps to elucidate the physical meaning of certain analytic constraints imposed on the asymptotic behavior of the solutions and shows that in the case of an infinite plate, any analytic solution has finite energy if and only if the bending and twisting moments, the transverse shear force, the displacements in vertical planes, and two other characteristic quantities vanish at infinity. An example is discussed to illustrate the theory.     

双周期圆截面纤维复合材料平面问题的解析法

结合双准周期Riemann边值问题理论与Eshelby等效夹杂原理，为双周期圆截面纤维复合材 料平面问题发展了一个实用有效的解析方法，获得了问题的全场级数解并与有限元结果进行 了比较. 该方法为非均匀材料的力学性质分析和复合材料等新材料的微结构设计提供了 一个有效的计算工具，也可用来评估有限元等数值与近似方法的精度.     

On the Solution of Mixed Problems in Linear Anti-plane Piezoelectricity

In the present paper we consider interior and exterior mixed boundary value problems of anti-plane shear in the static theory of linear piezoelectricity. Using the boundary integral equation method we reduce the problems to systems of singular integral equations with discontinuous coefficients to which the classical Nöether’s theorems on existence of the solution can be applied. This allows us to establish well-posedness results and to obtain integral solutions of the corresponding mixed boundary value problems for a rather general class of piezoelectric materials. Mathematics Subject Classifications (2000) 45E05, 45F15, 74F15.