Hydrostatic fluid loading in non‐linear finite element analysis

Abstract

SUMMARYDeformation dependent pressure loading on solid structures is created by the interaction of fluids with the deformable surface of a structure. Such fairly simple load models are valid for static and quasistatic analyses and they are a very efficient tool to represent the influence of a fluid on the behaviour of structures. Completing previous studies on the deformation dependence of the loading with the assumption of finite gas volumes, the current contribution focuses on the influence of modifications of the size and shape of a finite structure containing an incompressible fluid with a free surface. The linearization of the corresponding virtual work expression necessary for a Newton-type solution leads to additional terms for the volume dependence. Investigating these terms the conservativeness of the problem can be proven by the symmetry of the linearized form. Similarly to gas loading, the discretization of the structure with finite elements leads to standard stiffness matrix forms plus the so-called load stiffness matrices and a rank-one update for each filled structure part, if the loaded surface segments are identical with element surfaces. Some numerical examples show the effectiveness of the approach and the necessity to take the corresponding terms in the variational expression and the following linearization into account.

Simulation of Hydroforming of Metal Sheets with an Efficient FE‐Formulation Based on an Analytical Meshfree Description of a Compressible Fluid

“…After linearizing the variational form of the fluid potential (2) and separate transformations a symmetric system of equations can be derived [1], [2], [4]. Using standard finite element isoparametric mapping with N denoting the matrix of shape functions of the wetted surface elements and d the vector of the corresponding discrete nodal displacements the so-called load stiffness matrix K f elem is obtained, which consists of the symmetric terms from the change of normal vector ∆n and a change of the position dependent pressure ∆p…”

On the Stability Analysis of Thin Walled Shell Structures Containing Gas or Fluid

“…, [7], [8] and [9]) is taken, which yields a special structure of the nonlinear equations representing the change of the gas or fluid volume or alternatively the change of the wetted part of the shell surface. Finally this procedure leads first to the so-called load-stiffness matrix [11], to which several rank-one updates depending on the volume containing either gas or fluid or both are added.…”

“…For this purpose we refer to the former contributions of the authors group [7], [8] and [9] and the works of Bonet et al [1], which dealt with gas and fluid supported shell structures and considered both the shape and volume dependence of the hydrostatic pressure distribution. With these derivations it is possible to come up with an analytical formulation of the fluid/gas only described by surface integrals over the surrounding wetted structure.…”

“…The normal change parts for the gas and fluid domains can be adapted from [7] and [8]. For the fluid domain the additional part due to the non-constant pressure distribution p x t must be considered.…”

“…we obtain from the normal change parts the symmetric load stiffness matrices for each structural element in contact with gas and/or fluid (see also [11], [5], [6], [7], [8]):…”