Klaus-Jürgen Bathe
2
1 INTRODUCTION
Finite element procedures are now widely used in engineering and the sciences and we can
expect a continued growth in the use of these methods. The simulations of dynamic events are
of much interest [1-4].
Considering the analysis and design of civil and mechanical engineering structures,
frequently, physical tests can only be performed to a limited extent. Hence, the results of
simulations of such structures cannot be compared with test data. It is then very important to
use reliable finite element methods in order to have the highest possible confidence in the
computed results.
The objective in this paper is to briefly survey our recent developments of finite element
procedures for nonlinear dynamic analysis. In our research we have continuously focused on
the reliability of methods. Of course, any simulation starts with the selection of a
mathematical model, and this model must be chosen judiciously. However, once an
appropriate mathematical model has been selected, for the questions asked, the finite element
solution of that model needs to be obtained reliably, effectively, and ideally to a controlled
accuracy.
In the following sections we briefly present our recent developments regarding the finite
element analysis of shells, wave propagation problems, highly nonlinear dynamic long-
duration events, normal modes of proteins, and fluid-structure interactions.
2 ON RELIABILITY OF FINITE ELEMENT METHODS
Once a mathematical model has been chosen, it is important that well-founded, reliable and
of course efficient numerical methods be used for solution. By reliability of a finite element
procedure we mean that in the solution of a well-posed mathematical model, the procedure
will always, for a reasonable finite element mesh, give a reasonable solution — and if the
mesh is reasonably fine, an accurate solution of the chosen mathematical model is obtained
[3].
By reliability of a finite element procedure we also mean that if some analysis
conditions are changed, and seemingly only slightly, in the mathematical model, then for a
given finite element mesh, time integration scheme, and so on, the accuracy of the finite
element solution does not drastically decrease, unless there are distinct physical reasons.
These conditions on analysis methods are very difficult to achieve and require theoretical
depth in the understanding of the methods, and thorough and extensive testing based on
theoretical insights. These conditions also rule out the use of methods that require the setting
and problem-solution-dependent adjusting of numerical parameters to achieve stability of a
procedure.
3 SHELL ELEMENTS
The fundamental requirements in the development of shell elements are that the
discretization should satisfy the consistency condition, the ellipticity condition, and ideally
the inf-sup condition [4-7]
() ()
,,
sup sup
hh
hh h
hh
VV
h
VV
bb
cE
∈∈
≥∀∈
vv
η v η v
η
vv
(1)
where V is the complete (continuous) displacement space, V
h
is the finite element
displacement space,
h
E
is the finite element strain space, b(.,.) is the applicable bilinear form,
and
c is a constant independent of the shell thickness t and the element size h. To show