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.
Element degeneration is the collapsing of an element by one or more edges. For example: making an eight node element into
a seven node element by giving nodes 7 and 8 the same node number.
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.
Hourglass modes are element distortions that have zero strain energy. Thus, no stresses are
created within the element. There are 12 hourglass modes for a brick element, 4 modes for
each of the 3 coordinate directions. represents the hourglass mode vector, as defined by
Flanagan-Belytschko. 1 They produce linear strain modes, which cannot be
accounted for using a standard one point integration scheme.
To correct this phenomenon, it is necessary to introduce anti-hourglass forces and moments.
Two possible formulations are presented hereafter.
Kosloff and Frasier
Formulation
The Kosloff-Frasier hourglass formulation 2 uses a simplified hourglass vector. The hourglass
velocity rates are defined as:(1)
Where,
Non-orthogonal hourglass mode shape vector
Node velocity vector
Direction index, running from 1 to 3
Node index, from 1 to 8
Hourglass mode index, from 1 to 4
This vector is not perfectly orthogonal to the rigid body and deformation modes.
All hourglass formulations except the physical stabilization formulation for solid elements
in Radioss use a viscous damping technique. This allows the
hourglass resisting forces to be given by:(2)
Where,
Material density
Sound speed
Dimensional scaling coefficient defined in the input
Volume
Flanagan-Belytschko
Formulation
In the Kosloff-Frasier formulation seen in Kosloff and Frasier Formulation, the hourglass base vector is not perfectly orthogonal to the rigid body and deformation
modes that are taken into account by the one point integration scheme. The mean
stress/strain formulation of a one point integration scheme only considers a fully linear
velocity field, so that the physical element modes generally contribute to the hourglass
energy. To avoid this, the idea in the Flanagan-Belytschko formulation is to build an
hourglass velocity field which always remains orthogonal to the physical element modes. This
can be written as:(3)
The linear portion of the velocity field can be expanded to give:(4)
Decomposition on the hourglass vectors base gives 1:(5)
Where,
Hourglass modal velocities
Hourglass vectors base
Remembering that and factorizing Equation 5
gives:(6)
(7)
is the hourglass shape vector used in place of in Equation 2.
Physical Hourglass
Formulation
You also try to decompose the internal force vector as:(8)
In elastic case, you have:(9)
The constant part is evaluated at the quadrature point just like other one-point
integration formulations mentioned before, and the non-constant part (Hourglass) will be
calculated as:
Taking the simplification of (that is the Jacobian matrix of Strain Rate, Equation 1 is diagonal), you
have:(10)
with 12 generalized hourglass stress rates calculated by:(11)
and(12)
Where, , , are permuted between 1 to 3 and has the same definition than in Equation 6.
Extension to nonlinear materials has been done simply by replacing shear modulus
by its effective tangent values which is evaluated at the quadrature
point. For the usual elastoplastic materials, use a more sophistic procedure which is
described in Advanced Elasto-plastic Hourglass Control.
Advanced
Elasto-plastic Hourglass Control
With one-point integration formulation, if the
non-constant part follows exactly the state of constant part for the case of elasto-plastic
calculation, the plasticity will be under-estimated due to the fact that the constant
equivalent stress is often the smallest one in the element and element will be stiffer.
Therefore, defining a yield criterion for the non-constant part seems to be a good idea to
overcome this drawback.
Plastic yield criterion
The von Mises type of criterion is written by:(13)
for any point in the solid element, where is evaluated at the quadrature point.
As only one criterion is used for the non-constant part, two choices are possible:
taking the mean value, that is,
taking the value by some representative points, for example: eight Gauss
points
The second choice has been used in this element.
Elastro-plastic hourglass stress calculation
The incremental hourglass stress is computed by:
Elastic increment
Check the yield criterion
If , the hourglass stress correction will be done by un
radial return
1Flanagan D. and Belytschko T., “A Uniform Strain Hexahedron and
Quadrilateral with Orthogonal Hourglass Control”, Int. Journal Num.
Methods in Engineering, 17 679-706, 1981.
2Kosloff D. and Frazier G., “Treatment of hourglass pattern in low
order finite element code”, International Journal for Numerical and
Analytical Methods in Geomechanics, 1978.