Nonlinear finite element analyses confront users with many choices. An understanding of the fundamental concepts of
nonlinear finite element analysis is necessary if you do not want to use the finite element program as a black box.
The purpose of this manual is to describe the numerical methods included in Radioss.
In a Cartesian coordinates system, the coordinates of a material point in a reference or initial configuration are denoted
by . The coordinates of the same point in the deformed or final configuration are denoted by .
Kinematic constraints are boundary conditions that are placed on nodal velocities. They are mutually exclusive for each degree
of freedom (DOF), and there can only be one constraint per DOF.
The stability of solution concerns the evolution of a process subjected to small perturbations. A process is considered
to be stable if small perturbations of initial data result in small changes in the solution. The theory of stability
can be applied to a variety of computational problems.
A large variety of materials is used in the structural components and must be modeled in stress analysis problems.
For any kind of these materials a range of constitutive laws is available to describe by a mathematical approach the
behavior of the material.
Explicit scheme is generally used for time integration in Radioss, in which velocities and displacements are obtained by direct integration of nodal accelerations.
The performance criterion in the computation was always an essential point in the architectural conception of Radioss. At first, the program has been largely optimized for the vectored super-calculators like CRAY. Then, a first parallel
version SMP made possible the exploration of shared memory on processors.
Finite element discretizations with Lagrangian meshes are commonly classified as either an
updated Lagrangian formulation or a total Lagrangian formulation. Both formulations use a
Lagrangian description. That means that the dependent variables are functions of the material
(Lagrangian) coordinates and time. In the geometrically nonlinear structural analysis the
configuration of the structure must be tracked in time. This tracking process necessary involves
a kinematic description with respect to a reference state. Three choices called "kinematic
descriptions" have been extensively used:
Total Lagrangian description (TL)
The FEM equations are formulated with respect to a fixed reference configuration which is
not changed throughout the analysis. The initial configuration is often used; but in special
cases the reference could be an artificial base configuration.
Updated Lagrangian description (UL)
The reference is the last known (accepted) solution. It is kept fixed over a step and
updated at the end of each step.
Corotational description (CR)
The FEM equations of each element are referred to two systems. A fixed or base
configuration is used as in TL to compute the rigid body motion of the element. Then the
deformed current state is referred to the corotated configuration obtained by the rigid body
motion of the initial reference.
The updated Lagrangian and corotational formulations are the approaches used in Radioss. These two approaches are schematically presented in Figure 1.
By default, Radioss uses a large strain, large displacement
formulation with explicit time integration. The large displacement formulation is obtained by
computing the derivative of the shape functions at each cycle. The large strain formulation is
derived from the incremental strain computation. Hence, stress and strains are true stresses and
true strains.
In the updated Lagrangian formulation, the Lagrangian coordinates are considered
instantaneously coincident with the Eulerian spatial coordinates. This leads to the following
simplifications:(1)
(2)
The derivatives are with respect to the spatial (Eulerian) coordinates. The weak form involves
integrals over the deformed or current configuration. In the total Lagrangian formulation, the
weak form involves integrals over the initial (reference) configuration and derivatives are
taken with respect to the material coordinates.
The corotational kinematic description is the most recent of the formulations in geometrically
nonlinear structural analysis. It decouples small strain material nonlinearities from geometric
nonlinearities and handles naturally the question of frame indifference of anisotropic behavior
due to fabrication or material nonlinearities. Several important works outline the various
versions of CR formulation. 12345
Some new generation of Radioss elements are based on this
approach. Refer to Element Library for more details.
Note: A similar
approach to CR description using convected-coordinates is used in some branches of fluid
mechanics and theology. However, the CR description maintains orthogonality of the moving
frames. This will allow achieving an exact decomposition of rigid body motion and deformational
modes. On the other hand, convected coordinates form a curvilinear system that fits the change
of metric as the body deforms. The difference tends to disappear as the mesh becomes finer.
However, in general case the CR approach is more convenient in structural mechanics.
1Belytschko T. and Hsieh B.H., “Nonlinear Transient Finite Element
Analysis with convected coordinates”, Int. Journal Num. Methods in
Engineering, 7 255-271, 1973.
2Argyris J.H., “An excursion into the large rotations”, Computer
Methods in Applied Mechanics and Engineering, 32, 85-155, 1982.
3Crisfield M.A., “A consistent corotational formulation for nonlinear
three-dimensional beam element”, Computer Methods in Applied Mechanics
and Engineering, 81, 131-150, 1990.
4Simo J.C., “A finite strain beam formulation, Part I: The
three-dimensional dynamic problem”, Computer Methods in Applied Mechanics
and Engineering, 49, 55-70, 1985.
5Wempner G.A., “F.E., finite rotations and small strains of flexible
shells”, IJSS, 5, 117-153, 1969.