Preliminary Assessment of Higher-Order u-p Elements for Poromechanics Applications

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The requirement of C 0 continuity for the pressure interpolation in coupled u-p formulations necessitates the presence of pressure degrees of freedom along element edges (for two-dimensional elements) and faces (for three-dimensional elements) in continuous pressure elements. This translates into a larger number of pressure degrees of freedom per element as compared to mixed elements with discontinuous pressure interpolations. Not surprisingly, the number of potential mixed elements that exhibit stable response, particularly under incompressible or nearly incompressible conditions, is smaller than in the case of elements with discontinuous pressure interpolations. Traditionally, u-p finite element analyses of geomaterials employed two- and three-dimensional mixed finite elements with linear pressure interpolations and a quadratic displacement interpolation. Since such elements are known to generally yield stable results, they have been used rather extensively in the past and form a basis against which to compare the performance of newer mixed elements with continuous pressure approximations. To a lesser degree, two- and three-dimensional mixed finite elements with continuous pressure interpolations that employ a so-called displacement “bubble” function have also been used in u-p finite element analyses of geomaterials. “Bubble” functions are additional displacement degrees of freedom that are included at an interior node in the element. In both “Taylor-Hood” and “bubble” function elements the pressure interpolation is linear. Higher-order pressure interpolations are relatively rare, being confined to quartic triangular u-p elements (in displacement) with a quadratic pressure interpolation. Not surprisingly, systematic assessments of such elements have not appeared in the literature. This paper describes the preliminary phases of just such an assessment for two-dimensional elements. INTRODUCTORY REMARKS The requirement of C 0 continuity for the pressure interpolation in coupled up formulations means that by their nature, continuous pressure elements must have