Finite and infinite element method applied to water wave diffraction problems

Abstract

In this work, three types of infinite elements are developed to solve the problem of linear water wave diffraction by objects in a 2D unbounded domain. The infinite elements, which model the far field wave potential stretching to infinity, are coupled to conventional finite elements, which model the near field wave potential. This coupling greatly economises the finite element analysis. The original mapped infinite element, due to Zienkiewicz et al [102], is improved to model objects of large aspect ratio more economically. This infinite element (Type 1) can now be used on the exterior of an ellipse (or other shapes) rather than a circle circumscribing the object. The element is validated by solving the problem of diffraction of water waves by an ellipse with different angles of wave incidence. The results are compared with their equivalent analytical solutions and the errors are very small being less than 1.0%.The wave envelope approach, due to Astley et al [9], is employed to develop a simple mapped wave envelope infinite element. The element mass, stiffness and damping matrices are derived from first principles using the weighted residual approach. This element (Type 2) can also be used on the exterior of any shapes. The element is validated by solving the problem of wave diffraction by circular and elliptical vertical cylinders for different angles of wave incidence. The results are compared with their equivalent analytical solutions and again the errors are very small. The problem of wave diffraction by multiple objects is also considered. A new wave envelope mapped infinite element (Type 3) is developed to tackle such a problem. Examples involving diffraction of water waves by arrays of circular and elliptical cylinders are solved. For circular cylinders, the results are compared with their equivalent analytical solutions which show excellent agreement. A comparison is made between the three types of infinite elements by solving the diffraction of waves by circular and elliptical vertical cylinders. The results show that all three types of infinite elements can give accurate results in the near field for a given single diffracting object. Type 1 would be a preferable choice in situations where computing resource is the main concern and reliable solutions are required only in the near field. Type 2 or 3 would give very accurate solutions both in the near and far fields. Type 3 is the most suitable choice for modelling any number, shape and configuration of bodies.