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.

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.

The Radioss solid tetrahedron element is a 4 node element with one
integration point and a linear shape function.

This element has no hourglass. But the drawbacks are the low convergence and the shear locking.

10-Node Solid Tetrahedron

The Radioss solid tetrahedron element is a 10 nodes element
with 4 integration points and a quadratic shape function as shown in Figure 1.

Introducing volume coordinates in an isoparametric frame:

${L}_{1}=r$

${L}_{2}=s$

${L}_{3}=t$

${L}_{4}=1-{L}_{1}-{L}_{2}-{L}_{3}$

The shape functions are expressed by:(1)${\Phi}_{1}=\left(2{L}_{1}-1\right){L}_{1}$(2)${\Phi}_{2}=\left(2{L}_{2}-1\right){L}_{2}$(3)${\Phi}_{3}=\left(2{L}_{3}-1\right){L}_{3}$(4)${\Phi}_{4}=\left(2{L}_{4}-1\right){L}_{4}$(5)${\Phi}_{5}=4{L}_{1}{L}_{2}$(6)${\Phi}_{6}=4{L}_{2}{L}_{3}$(7)${\Phi}_{7}=4{L}_{3}{L}_{1}$(8)${\Phi}_{8}=4{L}_{1}{L}_{4}$(9)${\Phi}_{9}=4{L}_{2}{L}_{4}$(10)${\Phi}_{10}=4{L}_{3}{L}_{4}$

Location of the 4 integration points is expressed by ^{1}.

$${L}_{1}$$

$${L}_{2}$$

$${L}_{3}$$

$${L}_{4}$$

a

α

$\beta $

$\beta $

$\beta $

b

$\beta $

α

$\beta $

$\beta $

c

$\beta $

$\beta $

α

$\beta $

d

$\beta $

$\beta $

$\beta $

α

With,

$\alpha =0\cdot 58541020$ and $\beta =0\cdot 13819660$.

a, b, c, and d are the 4 integration points.

Advantages and Limitations

This element has various advantages:

No hourglass

Compatible with powerful mesh generators

Fast convergence

No shear locking.

But there are some drawbacks too:

Low time step

Not compatible with ALE formulation

Time Step

The time step for a regular tetrahedron is computed as:(11)$dt=\frac{{L}_{c}}{c}$

Where, ${L}_{c}$ is the characteristic length of element depending on tetra
type. The different types are:
(12)${L}_{c}=a\sqrt{\frac{2}{3}};\text{\hspace{0.17em}}{L}_{c}=0.816a$(13)${L}_{c}=a\sqrt{\frac{5/2}{6}};\text{\hspace{0.17em}}{L}_{c}=0.264a$

For another regular tetra obtained by the assemblage of four hexa as shown in Figure 4, the characteristic length is:
(14)${L}_{c}=a\frac{\sqrt{2/3}}{4};\text{\hspace{0.17em}}{L}_{c}=0.204a$

Below is a comparison of the 3 types of elements (8-nodes brick, 10-nodes tetra and
20-nodes brick). The results are shown in Figure 6 for a plastic strain contour.

^{1}Hammet P.C., Marlowe O.P. and Stroud A.H., “Numerical integration
over simplexes and cones”, Math. Tables Aids Comp, 10, 130-7,
1956.