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Theory Manual
This manual provides detailed information about the theory used in the Altair Radioss Solver.
ALE, CFD and SPH Theory Manual
ALE Formulation
ALE or Arbitrary Lagrangian Eulerian formulation is used to model the interaction between fluids and solids; in particular, the fluid loading on structures. It can also be used to model fluid-like behavior, as seen in plastic deformation of materials.
Stability
The Courant condition (neglecting viscosity effects) is used to determine the stability of an ALE process.

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What's New
View new features for Radioss 2025.
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
- ALE, CFD and SPH Theory Manual
  - ALE Formulation
    ALE or Arbitrary Lagrangian Eulerian formulation is used to model the interaction between fluids and solids; in particular, the fluid loading on structures. It can also be used to model fluid-like behavior, as seen in plastic deformation of materials.
    - Referential Domain
    - Conservation of Momentum
    - Conservation of Mass
    - Conservation of Internal Energy
      Conservation of internal energy is used to model temperature dependent material behavior. It also allows an energy balance evaluation.
    - Rezoned Quantities
    - ALE Materials
    - Numerical Integration
    - Improved Integration Method
    - Momentum Transport Force
    - Stability
      The Courant condition (neglecting viscosity effects) is used to determine the stability of an ALE process.
    - ALE Kinematic Conditions
    - Automatic Grid Computation
    - Fluid-Structure Interaction (TYPE 1 Interface)
    - Example: High Velocity Impacts
      A typical application of the ALE method is using high velocity impacts.
  - Computational Aero-Acoustic
  - Smooth Particle Hydrodynamics
    Smooth Particle Hydrodynamics (SPH) is a meshless numerical method based on interpolation theory. It allows any function to be expressed in terms of its values at a set of disordered point's so-called particles.
- Appendix A: Basic Relations of Elasticity
User Subroutines
This manual describes the interface between Altair Radioss and user subroutines.

Stability

The Courant condition (neglecting viscosity effects) is used to determine the stability of an ALE process.

The maximum time step is calculated by:

Δ t \leq k \frac{Δ l}{c + ν - w}

Where,

$k$: Coefficient
$Δ l$: Smallest characteristic length of an element
$c$: Material speed of sound
$ν$: Material velocity
$w$: Grid velocity

The speed of sound is determined by:

c = \sqrt{\frac{1}{ρ} \frac{\partial p}{\partial ρ} + \frac{4}{3} \frac{μ}{ρ}}

Where,

$ρ$: Density
$μ$: Dynamic viscosity
$p$: Pressure

The relative velocity between the material and grid motion ( $ν$ - $w$ ) is computed by:

ν - w = \sqrt{\frac{1}{N} {\sum_{i = 1}^{3} \sum_{I = 1}^{N} (ν_{i}^{I} - w_{i}^{I})}^{2}}

Where,

$N$: Number of nodes of the considered element (usually $N$ =8)