Vedrana Kozulić

An improved collocation method for solving the Henry problem

The original Henry problem is characterized with severe (albeit unphysical) sea boundary condition difficult to handle with numerical methods. In this paper, we present an improved numerical solution of the Henry problem. Presented numerical model solves the steady-state dimensionless equations by the collocation method named Fup Fragment Collocation Method (FFCM) and uses R(bf) basis functions of Fup(2) (x, y) type. This method enables application of the classical formulation and high approximation accuracy as proven in comparison with published solutions. Particular difficulty in solving the original Henry problem is in the accurate representation of the seepage face due to the fixed sea concentration at the sea boundary. The results of the original Henry problem formulation and problem with modified boundary conditions indicate the accuracy and robustness of the FFCM in describing the discharge area of the considered problem.

Computational Modeling of Structural Problems Using Atomic Basis Functions

This paper presents the application of the Fup n (x) basis functions in numerical modeling of different engineering problems. Fup n (x) basis functions belong to a class of atomic functions which are infinitely-differentiable functions with compact support. The collocation method has been applied in the development of numerical models. A system of algebraic equations is formed in which the differential equation of the problem is satisfied in collocation points of a closed domain while boundary conditions are satisfied exactly at the domain boundary. Using this way, the required accuracy of approximate solution is obtained simply by an increase in the number of basis functions. So, this concept represents a fully mesh free method. The properties of the atomic basis functions enable a hierarchic expansion of an approximate solution base either in the entire domain or in its segments. Presented numerical models are illustrated by examples of the torsion of prismatic bars, elasto-plastic analyses of beam bending and thin plate bending problems. The results of the analyses are compared with the existing exact and relevant numerical solutions. It can be concluded that the possibility of hierarchically expanding the number of basis functions in the domain significantly accelerates the convergence of a numerical procedure in a simple way. Values of the main solution function, e.g. displacements, and all the values derived from the main solution of the problem such as stresses, bending moments and transversal forces, are calculated in the same points and with the same degree of accuracy since numerical integration is avoided.

Application of the Solution Structure Method in Numerically Solving Poisson’s Equation on the Basis of Atomic Functions

This paper summarizes the main principles of the solution structure method and presents it in combination with atomic basis functions and a collocation technique. The solution of a boundary value p...

Numerical Solution of Poisson’s Equation in an Arbitrary Domain by Using Meshless R-Function Method

This paper describes a numerical procedure that uses solution structure method, atomic basis functions and a collocation technique. Solution structure method is based on the theory of R-functions. The solution of a boundary value problem is expressed in the form of formulae called solution structure which depends on three components: the first component describes the geometry of the domain exactly in analytical form, the second describes all boundary conditions exactly, while the third component is called differential component because it contains information about governing equation. Unknown differential component of the solution structure is represented by a linear combination of basis functions. Here, we propose to use atomic basis functions because of their good approximation properties. To determine the coefficients of linear combination in the solution structure, a collocation technique is used. Combination of atomic basis functions and solution structure method gives the meshfree method that can be applied for solving boundary value problems in domains of arbitrarily complex geometry with complex boundary conditions. This paper summarizes the main principles of the proposed method and presents its application to solution of the torsion problem.