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Theory Manual
This manual provides detailed information about the theory used in the Altair Radioss Solver.
Large Displacement Finite Element Analysis Theory Manual
Element Library
Radioss element library contains elements for one, two or three dimensional problems.
Solid Tetrahedron Elements

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What's New
View new features for Radioss 2024.1.
Overview
Radioss^® is a leading explicit finite element solver for crash and impact simulation.
Tutorials
Discover Radioss functionality with interactive tutorials.
User Guide
This manual provides details on the features, functionality, and simulation methods available in Altair Radioss.
Reference Guide
This manual provides a detailed list of all the input keywords and options available in Radioss.
Example Guide
This manual presents examples solved using Radioss with regard to common problem types.
Verification Problems
This manual presents solved verification models.
Frequently Asked Questions
This section provides quick responses to typical and frequently asked questions regarding Radioss.
Theory Manual
This manual provides detailed information about the theory used in the Altair Radioss Solver.
- Large Displacement Finite Element Analysis Theory Manual
  - Introduction
    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.
  - Basic Equations
  - Finite Element Formulation
  - Dynamic Analysis
  - Element Library
    Radioss element library contains elements for one, two or three dimensional problems.
    - Solid Hexahedron Elements
    - Solid Tetrahedron Elements
    - Shell Elements
    - Solid-Shell Elements
    - Truss Elements (TYPE2)
    - Beam Elements (TYPE3)
    - One Degree of Freedom Spring Elements (TYPE4)
    - General Spring Elements (TYPE8)
    - Pulley Type Spring Elements (TYPE12)
    - Beam Type Spring Elements (TYPE13)
    - Integrated Beam Elements (TYPE 18)
      Beam type /PROP/TYPE18 uses a shear beam theory or Timoshenko formulation like /PROP/TYPE3, but the section inputs (area, inertia) can be default values and can also be discretized by sub-sections; numerical integrations are used to calculate internal forces.
    - Multistrand Elements (TYPE28)
      Multistrand elements are n-node springs where matter is assumed to slide through the nodes.
    - Spring Type Pretensioners (TYPE32)
  - Kinematic Constraints
    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.
  - Linear Stability
    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.
  - Interfaces
    Interfaces solve the contact and impact conditions between two parts of a model.
  - Material Laws
    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.
  - Monitored Volume
    An airbag is defined as a monitored volume. A monitored volume is defined as having one or more 3 or 4 node shell property sets.
  - Static
    Explicit scheme is generally used for time integration in Radioss, in which velocities and displacements are obtained by direct integration of nodal accelerations.
  - Radioss Parallelization
    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.
- ALE, CFD and SPH Theory Manual
- Appendix A: Basic Relations of Elasticity
User Subroutines
This manual describes the interface between Altair Radioss and user subroutines.

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Solid Tetrahedron Elements

4-Node Solid Tetrahedron

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.

Figure 1. (a) Isoparametric 10 Node Tetrahedron; (b) Nodal Mass Distribution

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:

Φ_{1} = (2 L_{1} - 1) L_{1}

Φ_{2} = (2 L_{2} - 1) L_{2}

Φ_{3} = (2 L_{3} - 1) L_{3}

Φ_{4} = (2 L_{4} - 1) L_{4}

Φ_{5} = 4 L_{1} L_{2}

Φ_{6} = 4 L_{2} L_{3}

Φ_{7} = 4 L_{3} L_{1}

Φ_{8} = 4 L_{1} L_{4}

Φ_{9} = 4 L_{2} L_{4}

Φ_{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	$α$	$β$	$β$	$β$
b	$β$	$α$	$β$	$β$
c	$β$	$β$	$α$	$β$
d	$β$	$β$	$β$	$α$

With,

$α = 0 \cdot 58541020$ and $β = 0 \cdot 13819660$ .

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:

d t = \frac{L_{c}}{c}

Where,

L_{c}

is the characteristic length of element depending on tetra type. The different types are:

Figure 2. Regular 4 Nodes Tetra

L_{c} = a \sqrt{\frac{2}{3}}; L_{c} = 0.816 a

Figure 3. Regular 10 Nodes Tetra

L_{c} = a \sqrt{\frac{5 / 2}{6}}; L_{c} = 0.264 a

For another regular tetra obtained by the assemblage of four hexa as shown in Figure 4, the characteristic length is:

Figure 4. Other Regular Tetra

L_{c} = a \frac{\sqrt{2 / 3}}{4}; L_{c} = 0.204 a

CPU Cost and Time/Element/Cycle

The CPU cost is shown in Figure 5:

Figure 5. CPU Cost in TEC

Example: Comparison

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.

Figure 6. Comparison (plastic strain max = 60%)

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.