SOME NUMERICAL PROPERTIES OF THE BOUNDARY ELEMENT METHOD USING FUNDAMENTAL SOLUTIONS IN A HALF-PLANE IN A TWO-DIMENSIONAL FORMULATION

In the numerical analysis of elasticity problems by the boundary element method, the boundary integral equation, which is widely used at present, is formulated using the Kelvin fundamental solution in infinite space. However, for many practical problems where the objects are of semi-infinite or finite size, the half-plane fundamental solution can also be used instead of the Kelvin fundamental solution. In the half-plane fundamental solution model, we see that both the loading point and the observation point are defined relative to the free surface of the half-plane, so that the half-plane fundamental solution is determined not only by the distance between the two points, but also by their relative position with respect to the surface. Thus, it is possible to investigate the different numerical properties of the boundary element method based on the half-plane fundamental solution compared to solutions based on the Kelvin fundamental solution. In this paper, some numerical properties of the half-plane fundamental solution in the analysis of two-dimensional objects of finite size are studied. From the numerical analysis and examples presented in the paper, we see that the fundamental half-plane solution can be used satisfactorily even with a smaller number of elements for the numerical analysis of two-dimensional finite-size problems. In addition, the following interesting numerical properties are noted in comparison with the fundamental Kelvin solution: in contrast to the two-dimensional Kelvin solution, the fundamental half-plane solution gives a "directional" effect of the calculated deformation. In this case, attention should be paid to the location of the symmetry axis of the deformed state of the objects (if any), parallel to the symmetry axis of the half-plane of the model, the error of the numerical results increases with the increase in the distance of the object from the surface of the half-plane model, which also makes it impossible to use the fundamental solution. Therefore, it is better to place the object as close as possible to the surface of the half-plane model. Finally, it should also be noted that the above numerical properties of the solutions using the fundamental half-plane solution turned out to be correct in the case of using any higher-order elements, although the data presented here were obtained only using constant elements.