Engineering problems involving unbounded domains are encountered in many areas, such as soil-structure interaction, consolidation of soils, and underground structures. The unbounded problems can be effectively and accurately analyzed by using appropriate infinite elements. Because the concept of infinite elements is the same as that of finite elements except for the infinity of the element domain, the infinite elements can be readily implemented in an usual finite element method. Thus, in the infinite element method, the near-field region is modeled by conventional finite elements, while the far-field region is represented by infinite elements.
There exist many geotechnical problems where the unknown variables decay with respect to $1/r^n$(n=0.5,1,2,…). Each type of decay characteristics was previously modeled by using each infinite element in which only one type of decay characteristics could be represented. Such an infinite element could be formulated only if analytical solutions or the decay characteristics of variables were known. And accurate results could be obtained only if the selected infinite elements for the analysis were appropriate for the decay characteristics of the problem. Therefore, it has been necessary to develop an infinite element to represent various types of decay characteristics in unbounded media.
This dissertation presents p-version infinite elements for analyzing $1/r^n$ type of decay problems in an unbounded soil. In order to model exterior unbounded regions, two kinds of infinite elements were developed. They are horizontal and radial infinite elements. Orthogonal polynomial series of Laguerre polynomials multiplied by an exponential decay function were proposed for representing $1/r^n$ type decay functions. The proposed polynomials were validated by checking the orthogonality of the polynomials, the regular condition and the accuracy of approximation. The shape functions of the p-version infinite elements were derived by using the orthogonal polynomial series.
To verify the proposed p-version infinite elements, some numerical example analyses were carried out : a rigid strip footing and a flexible disk footing on a homogeneous half-space soil or on an unbounded layered soil which is horizontally infinite ; a circular tunnel and a square tunnel excavated in a homogeneous infinite soil or in an unbounded layered soil with initial earth stress fields. Compared with the analytical solutions and the numerical solutions obtained by a finite element method, it has been found that the proposed p-version infinite element method offers accurate solutions. Infinite element analysis by increasing the polynomial order of a p-version infinite element, gives very similar results to those obtained from the finite element analysis by increasing the domain size. And the results of the infinite element analysis converged to exact solutions.
Therefore it seems that the proposed infinite element could efficiently represent the decay characteristics of the displacement in unbounded soils. Making use of the proposed p-version infinite elements, the total number of elements and the degrees of freedom for analysis could be significantly reduced. Unsteady state seepage analyses were also carried out for a semi-infinite soil with various boundary conditions. Comparison of the results with those of analytical solutions and other methods showed that the present method also offers good solutions in this type of problems. Using the proposed infinite elements, the unbounded domain problems with various decay characteristics can be efficiently analyzed.
