Numerical Methods

Abstract

This paper provides a comprehensive survey of the numerical methods related to geotechnical problems, most of which were reported in papers appearing in Soils and Foundations. The reason why most of the reviewed papers are concentrated in Soils and Foundations is that if we were to include papers appearing in other journals in theˆeld of geotechnical engineering, closely related to numerical methods, e.g., Computers and Geotechnics, Int. J. Numer. Anal. Meth. Geomech., etc., we would have to deal with almost all the papers in those journals. Firstly, we present a description of the current status of the numerical methods, and then give a brief review of the literature covering several topics in geotechnical applications. The scope of the review is limited, and thus, the authors do not profess to cover the entire range of literature.

Arbitrary mesh‐moving velocity‐based space‐time finite element method for large deformation analysis of solids

“…The analytical solutions to the present problem for the hypoelastic material are given by $$$$ {\sigma}_{xx}=\frac{E}{1-{\nu}^2}\log \left(1+\frac{u_{x,\mathrm{right}}}{L_0}\right),\kern0.3em {\sigma}_{yy}={\sigma}_{xy}=0, $$$$ where $$$ {u}_{x,\mathrm{right}} $$$ is the horizontal displacement at the right edge. A similar form of the exact solution under plane stress conditions can be found in Reference 59. On the other hand, the analytical solutions for the Neo–Hookean hyperelastic material are given by $$$$ {\sigma}_{xx}=-\mu {b}_{yy}\exp \left[-\frac{\mu }{\lambda}\left(1-{b}_{yy}\right)\right]+\frac{\mu }{b_{yy}}\exp \left[\frac{\mu }{\lambda}\left(1-{b}_{yy}\right)\right], $$$$ $$$$ {b}_{xx}=\frac{1}{b_{yy}}\exp \left[\frac{2\mu }{\lambda}\left(1-{b}_{yy}\right)\right],\kern0.3em {b}_{xy}=0,\kern7.0em $$$$ …”