Klaus-Jürgen Bathe

A finite element formulation for nonlinear incompressible elastic and inelastic analysis

We introduce a displacement-pressure (up) finite element formulation for the geometrically and materially nonlinear analysis of compressible and almost incompressible solids. The (up) formulation features the a priori replacement of the pressure computed from the displacement field by a separately interpolated pressure; this replacement is performed without reference to any specific material description. Considerations for incremental nonlinear analysis (including contact boundary conditions) are discussed and various (up) elements are studied. Numerical examples show the performance of the (up) formulation for two- and three-dimensional problems involving isotropic, orthotropic, rubber-like and elasto-plastic materials.

A solution method for static and dynamic analysis of three-dimensional contact problems with friction

Abstract A solution method is presented for the analysis of contact between two (or more) three-dimensional bodies. The surfaces of the contacting bodies are discretized using quadrilateral surface segments. A Lagrange multiplier technique is employed to impose that, in the contact area, the surface displacements of the contacting bodies are compatible with each other. Distributed contact tractions over the surface segments are calculated from the externally applied forces, inertia forces and internal element stresses. Using the segment tractions, Coulombs law of friction is enforced in a global sense over each surface segment. The time integration of dynamic response is performed using the Newmark method with parameters δ = 1 2 and α = 1 2 . Using these parameters the energy and momentum balance criteria for the contacting bodies are satisfied accurately when a reasonably small time step is used. The applicability of the algorithm is illustrated by selected sample numerical solutions to static and dynamic contact problems.

Some practical procedures for the solution of nonlinear finite element equations

Abstract Procedures for the solution of incremental finite element equations in practical nonlinear analysis are described and evaluated. The methods discussed are employed in static analysis and in dynamic analysis using implicit time integration. The solution procedures are implemented, and practical guidelines for their use are given.

The inf-sup test

We briefly review the inf-sup condition for the finite element solution of problems in incompressible elasticity, and then propose a numerical test on whether the inf-sup condition is passed. The evaluation of elements with this test is simple, and various results are presented. This inf-sup test will prove useful for many discretizations of constrained variational problems.

Finite element analysis of fluid flows fully coupled with structural interactions

Abstract Some advances in capabilities for analysis of fluid flows fully coupled with structural interactions are presented. Incompressible Navier–Stokes and compressible Navier–Stokes or Euler fluids and the full interaction with structures undergoing large deformations, nonlinear material response and contact conditions can be considered. The analysis capabilities are available in the ADINA System, and are integrated within computer-aided design using the available ADINA modeler and CAD interfaces. Various analysis cases are presented to illustrate the solution capabilities.

The inf–sup condition and its evaluation for mixed finite element methods

Abstract The objective of this paper is to review the general inf–sup condition for mixed finite element methods and summarize numerical procedures (inf–sup tests) for the evaluation of the inf–sup expressions specific to various problem areas. The inf–sup testing of a given mixed finite element discretization is most important in order to assess its reliability and solution effectiveness. The problem areas considered are (almost) incompressible analysis of solids and fluids, acoustic fluids, the analysis of plates and shells, and the solution of convection-dominated flows.

A geometric and material nonlinear plate and shell element

Abstract A displacement-based versatile and effective finite element is presented for linear and geometric and material nonlinear analysis of plates and shells. The element is formulated by interpolating the element geometry using the mid-surface nodal point coordinates and mid-surface nodal point normals. A total and an updated Lagrangian formulation are presented, that allow very large displacements and rotations. In linear analysis of plates, the element reduces to well-established plate bending elements based on classical plate theory, whereas in linear analysis of shells and geometrically nonlinear analysis of plates and shells by use of the element, in essence, a very general shell theory is employed. The element has been implemented as a variable-number-nodes element and can also be employed as a fully compatible transition element to model shell intersections and shell-solid regions. In the paper various sample solutions are presented that illustrate the effectiveness of the element in practical analysis.

Fundamental considerations for the finite element analysis of shell structures

Abstract The objective in this paper is to present fundamental considerations regarding the finite element analysis of shell structures. First, we review some well-known results regarding the asymptotic behaviour of a shell mathematical model. When the thickness becomes small, the shell behaviour falls into one of two dramatically different categories; namely, the membrane-dominated and bending-dotninated cases. The shell geometry and boundary conditions decide into which category the shell structure falls, and a seemingly small change in these conditions can result into a change of category and hence into a dramatically different shell behaviour. An effective finite element scheme should be applicable to both categories of shell behaviour and the rate of convergence in either case should be optimal and independent of the shell thickness. Such a finite element scheme is difficult to achieve but it is important that existing procedures be analysed and measured with due regard to these considerations. To this end, we present theoretical considerations and we propose appropriate shell analysis test cases for numerical evaluations.

Analysis of fluid-structure interactions. a direct symmetric coupled formulation based on the fluid velocity potential

Abstract We present a symmetric finite element method for solving fluid-structure interaction problems. The formulation uses velocity potentials and a hydrostatic pressure as unknowns in each fluid region, and displacements as unknowns in the solid. The hydrostatic pressure is an unknown variable at only one node per fluid region. A C matrix (multiplied by time derivatives of the nodal variables, but not a damping matrix) enforces the coupling between the variables. The resulting matrix equations are banded and symmetric, making them easy to incorporate in standard displacement-based finite element codes. Several test cases indicate that this approach works well for static, transient, and frequency analyses.

An evaluation of the MITC shell elements

Abstract Based on fundamental considerations for the finite element analysis of shells, we evaluate in the present paper the performance of the MITC general shell elements. We give the results obtained in the analysis of judiciously selected test problems and conclude that the elements are effective for general engineering applications.