Husain J. Al-Gahtani

Iterative Coupling of BE and FE Methods in Elastostatics

Iterative domain decomposition coupling is one of the recent approaches for combining the boundary element method (BEM) and the finite element method (FEM). The domain of the original problem is subdivided into two sub-domains, which are separately modeled by the FEM and BEM. Successive renewal of the variables on the interface of the two sub-domains is performed through an iterative procedure to reach the final convergence. In this paper, we investigate the iterative method. We also establish the convergence conditions. A simple numerical example is given to elaborate on the effect of different factors such as initial guess, boundary conditions, and geometrical and material properties of the sub-domains on solution convergence.

An overlapping domain decomposition approach for coupling the finite and boundary element methods

Abstract An overlapping iterative domain decomposition approach for the coupling of the finite element method (FEM) and the boundary element method (BEM) is presented in this paper. In this proposed method, the domain of the original problem is subdivided into a FEM sub-domain and a BEM sub-domain, such that the two sub-domains partially overlap over a common region. The common region is modeled by both methods. A brief discussion on the existing iterative coupling methods and their limitations are given in the first part of this paper. In the second part, the proposed overlapping method is described and the convergence conditions are presented. Two numerical examples are given to demonstrate the capability of the proposed method for handling cases where the Neumann boundary conditions are specified on the entire external boundary of the FEM or BEM sub-domains.

Convergence of the domain decomposition finite element-boundary element coupling methods

In this work, we analyze three available domain decomposition methods. We also establish the convergence conditions. The theoretical analysis provides an interval in which a relaxation parameter has to be chosen in order to achieve convergence. Moreover, it allows the selection of the relaxation parameter so that convergence is achieved with the minimum number of iterations. Several example problems are given for elaboration.

Application of the boundary element method to rubber-like elasticity

A boundary element method solution for the problem of finite deformation of rubber-like materials is presented. The solution is obtained by solving the nonlinear boundary element equations using an incremental-iterative procedure. The procedure is then implemented in a computer code that is tested through three numerical examples. In the first example, the plane strain problem of a thick-walled cylindrical pressure vessel is considered. In the second and third examples, which represent the plane stress case, a square rubber sheet with and without a central hole is subjected to finite plane deformation.

Chloride-Induced Corrosion of Steel in Concrete: An Overview on Chloride Diffusion and Prediction of Corrosion Initiation Time

Initiation of corrosion of steel in reinforced concrete (RC) structures subjected to chloride exposures mainly depends on coefficient of chloride diffusion, , of concrete. Therefore, is one of the key parameters needed for prediction of initiation of reinforcement corrosion. Fick’s second law of diffusion has been used for long time to derive the models for chloride diffusion in concrete. However, such models do not include the effects of various significant factors such as chloride binding by the cement, multidirectional ingress of chloride, and variation of with time due to change in the microstructure of concrete during early period of cement hydration. In this paper, a review is presented on the development of chloride diffusion models by incorporating the effects of the key factors into basic Fick’s second law of diffusion. Determination of corrosion initiation time using chloride diffusion models is also explained. The information presented in this paper would be useful for accurate prediction of corrosion initiation time of RC structures subjected to chloride exposure, considering the effects of chloride binding, effect of time and space on , and interaction effect of multidirectional chloride ingress.

RBF-based meshless method for the free vibration of beams on elastic foundations

In this paper, solution is obtained for the free vibration differential equations of motion of an axially loaded beam on elastic foundation using a meshless method. Use is made of the multiquadrics radial basis function (RBF) in obtaining the numerical solution for four different cases: (1) one end clamped, the other end simply supported; (2) both ends clamped; (3) both ends simply supported; and (4) a simple beam on elastic foundation with end rotational springs. The approach is easier to implement and program as compared to grid/mesh-based methods such as the finite difference method (FDM) and the finite element method (FEM). Accuracy of the results obtained using the proposed method was verified using the analytical results available in the literature for the first three cases considered. Numerical results of the fourth case were aimed at justifying the use of the numerical scheme for a problem whose analytical solution is not readily available and to show the high accuracy of the RBF method. The results prove that the method require much less number of nodes to converge to the correct solution as compared to FDM.

Integral-based solution for a class of second order boundary value problems

An integral formulation for the solution of a class of second order boundary value problems which are described by the equation d^2yy/dx^2 + P(x,y, dy/dx, d^2yy/dx^2)) = 0, x @e (0,a), is presented. The resulting integral equations are then solved by expressing the dependent variable y as a power series which made the computation of various integrals possible. The proposed method is tested through some examples to show the applicability of the method to solve a wide range of second order differential equations including the nonlinear ones.

Series-Based Solution for Analysis of Simply Supported Rectangular Thin Plate with Internal Rigid Supports

In this study, Navier’s solution for the analysis of simply supported rectangular plates is extended to consider rigid internal supports. The proposed method offers a more accurate solution for the bending moment at the critical section and therefore serves as a better analytical solution for design purposes. To model the plate-support interaction, the patched areas representing the contact between the plate and supports are divided into groups of cells. The unknown internal reactions at the centers of the divided cells are obtained by satisfying the compatibility conditions at the centers of the cells. Three numerical examples are presented to demonstrate the accuracy of the proposed analytical solution. The given examples reveal good agreements with those obtained by the finite element analysis. In addition, they show the advantage of the new solution as compared to the existing analytical solution which inaccurately estimates the location and magnitude of the maximum bending moment.

RBF-Based Meshless Method for Large Deflection of Elastic Thin Rectangular Plates with Boundary Conditions Involving Free Edges

An RBF-based meshless method is presented for the analysis of thin plates undergoing large deflection. The method is based on collocation with the multiquadric radial basis function (MQ-RBF). In the proposed method, the resulting coupled nonlinear equations are solved using an incremental-iterative procedure. The accuracy and efficiency of the method are verified through several numerical examples. The inclusion of the free edge boundary condition proves that this method is accurate and efficient in handling such complex boundary value problems.

Simplified Formulation of Stress Concentration Factors for Spherical Pressure Vessel–Cylindrical Nozzle Juncture

Parametric study of the thin shell solution of internally pressurized spherical vessel–cylindrical nozzle juncture is used to develop simplified closed-form formulas of stress concentration factor (SCF) as functions of the key vessel–nozzle geometric parameters known to influence the solution. The SCF values are not based on the vessel stresses alone; nozzle stresses are also analyzed and the corresponding SCF determined. Therefore, for a given vessel–nozzle juncture, the designer will be left with adequate information upon which to decide the controlling SCF. Predictions by the proposed equations are validated using finite element method (FEM). Consequently, design charts are presented based on both the vessels and nozzles SCF as predicted by the proposed expressions.