Non-linear Finite Element Structural Analysis
Introduction
In most professional structural design practice,
prediction of the response of structures during an earthquake is carried
out by the analysis of simple linear elastic finite element models and
quite a bit of "hand waving" to predict the non-linear response from linear
elastic results. Linear elastic models are used because they are
computationally inexpensive and most importantly, they are fast.
However, exact non-linear finite element structural analysis methods have
been around for years, but have long been considered merely an academic
interest because they are notoriously slow and computationally intensive.
With the advent of parallel computing and the ever increasing speed of
processors, exact non-linear analyses will no longer be considered only
an academic interest.
A Terse Treatment of Non-linear Structural Analysis
At a given load step in the linear elastic analysis
of a structure, the state determination for the structural elements is
trivial and does not require iteration to satisfy any form of equilibrium.
For a load step in a non-linear analysis, there are three levels of iteration
required. Iteration is required to ensure global, or overall structural
equilibrium, i.e. the balance between applied loads and internal resistance.
This internal resistance is provided by the elements, where a second level
of iteration is required to ensure that the loads transferred to the elements
are equilibrated by the elements' resisting forces. These element
resisiting forces are provided by the elements' constitutive modeling,
for example, steel or reinforced concrete. The constitutive model
provides a third level iteration where the internal element deformations
are required to satisfy a prescribed stress-strain relationship.
Why Parallelize?
In forming the global equations of equilibrium for
a structure, looping over all elements in the finite element model is required.
This looping is done so that elements can be queried for their tangent
stiffness and their resisting forces. In a sequential computing environment,
elements are queried one by one for their tangent stiffness, which is then
assembled into the global system of equilibrium equations. After
the equations have been solved, the elements are asked to provide, one
by one, their resisting forces. This process of forming tangent stiffness,
solving equations, and forming resisting forces is repeated until global
equilibrium is satisfied for the current load step. In an earthquake
analysis, there will be thousands of load steps, so in a sequential computing
environment, this analysis will take quite some time, even with current
computer processing capabilities. By parallelizing the element operations
of forming tangent stiffness and resisting forces, run-time will be significantly
reduced.
Other Areas of Interest
Dynamic load balancing of a non-linear finite element
analysis is required. As elements begin to experience non-linear
response, the amount of computation required to form tangent stiffness
and resisting forces increase as more iterations will be required to satisfy
equilibrium. During an earthquake analysis, it is not known before
which elements will exhibit non-linear behavior, therefore, an efficient
form of dynamic load balancing is required to maintain equity among processors.
In addition, substructure domain decomposition techniques may improve data
locality and take advantage of architecture specific memory features.
Current Work
This subject has been studied rather intensively
by many researchers. Significant work in parallelizing the finite
element method for structural analysis is ongoing here at the University
of California under the direction of Dr.
Frank McKenna.
References
Cook, Robert D., David S. Malkus and Michael E. Plesha,
Concepts
and Applications of Finite Element Analysis, Third Edition, John Wiley
& Sons, 1989.
McKenna, Francis, Object-Oriented Finite Element
Programming: Frameworks for Analysis, Algorthms, and Parallel Computing,
Ph.D. Thesis, University of California, 1997.
Michael H. Scott
mhscott@ce.berkeley.edu
January 26, 2000