SPH in a Total Lagrangian formalism

To correct some of the main shortcomings of conventional SPH, a version of this method based on the Total Lagrangian formalism, T. Rabczuk, T. Belytschko and S. Xiao (2004), is developed. The resulting scheme removes the spatial discretisation instability inherent in conventional SPH, J. Monaghan (1992). The Total Lagrangian framework is combined with the mixed correction which ensures linear completeness and compliance with the patch test, R. Vignjevic, J. Campbell, L. Libersky (2000). The mixed correction utilizes Shepard Functions in combination with a correction to derivative approximations. Incompleteness of the kernel support combined with the lack of consistency of the kernel interpolation in conventional SPH results in fuzzy boundaries. In corrected SPH, the domain boundaries and field variables at boundaries are approximated with the default accuracy of the method. Additionally, these corrections are introduced into the Total Lagrangian SPH and compared to the conventional SPH and to a number of selected corrected variants, G. Johnson, R. Stryk, S. Beissel. (1996), J. Bonet, S. Kulasegaram (2002), and P. Randles, L. Libersky (1996). The resulting Total Lagrangian SPH scheme not only ensures fist order consistency but also alleviates the particle deficiency (kernel support incompleteness) problem. Furthermore a number of improvements to the kernel derivative approximation are proposed. To illustrate the performance of the Total Lagrangian SPH and the mixed correction, four numerical examples ranging from simple I D dynamic elasticity to 3D real engineering problems are also provided.