one of the simplest numerical integration method which provides a large saving in computational efforts, is the well known one-point Gauss quadrature which is widely used for 4 nodes quadrilateral elements. On the other hand, the biggest disadvantage to one-point integration is the need to control the zero energy modes, called hourglassing modes, which arise. The efficiency of four different anti-hourglassing approaches, Flanagan (elastic approach), DYNA3D, Hansbo and Liu have been investigated. The first two approaches have been used in 2 and 3-D explicit codes and the latter have been employed in 2-D implicit codes. For 2-D explicit codes, the computational time was reduced by 55% and 60% for elastic and DYNA3D, respectively. However, for 3-D codes the reduction was dependent on the number of elements and was obtained between 50% and 70%. Also, the error due to the application of elastic methods was less than that for DYNA3D when the results were compared with those obtained from 2-points Gauss quadrature. Nevertheless, the convergence occurred more rapidly and the oscillations were damped out more quickly for DYNA3D approach. For implicit codes, the anti-hourglassing methods had no effect on the computations and therefore a 2-points Gauss quadrature is recommended for implicit codes as it provide the results more accurately

Using underwater shock waves produced by underwater explosion of high explosives (HE) has been studied, as a manufacturing process of metals, for many years. Although early studies on this subject had mainly military motivation, many investigations have recently been conducted in other industrial fields. The main aim of these research are to understand the details of the motion of spherical bubble produced by HE explosion as well as the subsequent flow field generated by the motion of a strong shock wave in the water. Due to the limitation of computing facilities, the early numerical simulations of underwater explosion phenomenon were performed by crude numerical methods (e.g., Stenberg et al. 1971). In those works the linear artificial viscosity of von Neumann was used to prevent the numerical instability close to the shock (the so-called q-method). Flores and Holt (1981) applied Glimms method to underwater explosion of an uniform spherical bubble. Their study showed the capability of new methods to study the problem. Molyneaux et al. (1996) used the finite element program DYNA3D and observed that numerical tools are able to predict well both the magnitude and form of the pressure transient field. Their simulation was performed for a cylindrical charge. In the present study, a third order Godunov method has been utilized in the framework of a Lagrangian approach to investigate the entire flow field generated by underwater explosion of a spherical bubble. The developed code is based on the Piecewise Parabolic Method (PPM) of Collela (Colella et. al.1984). The original PPM which was developed for ideal gases, here is extended to treat the real gas effect. The water and the explosion products were treated using Mie-Gruneisen and JWL equation of states, respectively. The processes of shock and expansion waves reflection from the origin and interaction with the gas/water interface are fully resolved with the present method.