On the static interaction of fluid and gas loaded multi-chamber systems in large deformation finite element analysis

Completing previous studies [D. Berry, H. Yang, Formulation and experimental verification of a pneumatic finite element, Int. J. Numer. Methods Engrg. 39 (1996) 1097–1114; J. Bonet, R.D. Wood, J. Mahaney, P. Heywood, Finite element analysis of air supported membrane structures, Comput. Methods Appl. Mech. Engrg. 190 (2000) 579–595; T. Rumpel, Effiziente Diskretisierung von statischen Fluid-Struktur-Problemen bei großen Deformationen. Ph.D. thesis, Institut für Mechanik der Universität Karlsruhe (TH), 2003; T. Rumpel, K. Schweizerhof, Volume-dependent pressure loading and its influence on the stability of structures, Int. J. Numer. Methods Engrg. 56 (2003) 211–238; T. Rumpel, K. Schweizerhof, Hydrostatic fluid loading in non-linear finite element analysis, Int. J. Numer. Methods Engrg. 59 (2004) 849–870.] on the deformation dependence of gas or fluid loadings with the assumption of finite gas and fluid volumes, the current contribution focuses on the influence of modifications of the size and shape of a finite structure filled with combinations of gas and incompressible or compressible heavy fluid. In contrast to the previous contributions (Berry and Yang, 1996; Bonet et al., 2000; Rumpel, 2003; Rumpel and Schweizerhof, 2003; Rumpel and Schweizerhof, 2004), where different finite element formulations for some specific load cases were derived, this derivation provides a general formulation of a multi chamber system filled with gas and additional compressible heavy fluid. By introducing two stiffness parameters it is possible to reduce the formulation to the pure pneumatic or hydrostatic load cases described in earlier works. Further on it will be shown how to manage the modification of the load cases during the loading process. The linearization of the corresponding virtual work expression, necessary for a Newton-type solution algorithm, leads to additional terms for the volume dependence (Bonet et al., 2000; Rumpel, 2003; Rumpel and Schweizerhof, 2003) and the expected fully symmetric stiffness matrix. The discretization of the solid structures with finite elements leads to standard stiffness matrix forms for the structures plus the so-called load stiffness matrices [K. Schweizerhof, E. Ramm, Displacement dependent pressure loads in non-linear finite element analyses, Comput. Struct. 18 (1984) 1099–1114.] and several dyadic rank updates – depending on the type of filling – for each filled structure part, assuming that loaded surface segments are identical with the element surfaces. A further focus is on a specific solution scheme exploiting the dyadic rank update structure. Some numerical examples show the effectiveness of the approach and the necessity to take the corresponding terms in the variational expression and in the linearization into account.