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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
    - Direct Integration Method: Explicit Scheme
      - Newarks Method
      - Central Difference Algorithm
      - Numerical Starting Procedure
      - Algorithm Flow Chart
      - Body Drop Example
      - Numerical Stability
        The definition of numerically stability is similar to the stability of mechanical systems. A numerical procedure is stable if small perturbations of initial data result in small changes in the numerical solution.
      - Explicit Scheme Stability
      - Courant Condition Stability
        Radioss uses elements with a lumped mass approach. This reduces computational time considerably as no matrix inversion is necessary to compute accelerations.
      - Time Step Control
      - Time Step Control Limitations
    - Large Scale Eigen Value Computation
    - Combine Modal Reduction
  - Element Library
    Radioss element library contains elements for one, two or three dimensional problems.
  - 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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Newarks Method

Newmark's method is a one step integration method. The state of the system at a given time $t_{n + 1} = t_{n} + h$ is computed using Taylor's formula:

f (t_{n} + h) = f (t_{n}) + h f^{'} (t_{n}) + \frac{h^{2}}{2} f^{(2)} (t_{n}) + ... + \frac{h^{s}}{s!} f^{(s)} (t_{n}) + R_{s}

R_{s} = \frac{1}{s!} \int_{t_{n}}^{t_{n} + h} f^{(s + 1)} (τ) {[t_{n} + h - τ]}^{s} d τ

The preceding formula allows the computation of displacements and velocities of the system at time $t_{n + 1}$ :

{\dot{u}}_{n + 1} = {\dot{u}}_{n} + \int_{t_{n}}^{t_{n + 1}} \ddot{u} (τ) d τ

u_{n + 1} = u_{n} + h {\dot{u}}_{n} + \int_{t_{n}}^{t_{n + 1}} (t_{n + 1} - τ) \ddot{u} (τ) d τ

The approximation consists in computing the integrals for acceleration in Equation 3 and in Equation 4 by numerical quadrature:

\int_{t_{n}}^{t_{n + 1}} \ddot{u} (τ) d τ = (1 - γ) h {\ddot{u}}_{n} + γ h {\ddot{u}}_{n + 1} + r_{n}

\int_{t_{n}}^{t_{n + 1}} (t_{n + 1} - τ) \ddot{u} (τ) d τ = (\frac{1}{2} - β) h^{2} {\ddot{u}}_{n} + β h^{2} {\ddot{u}}_{n + 1} + {r^{'}}_{n}

By replacing Equation 3 and Equation 4, you have:

{\dot{u}}_{n + 1} = {\dot{u}}_{n} + (1 - γ) h {\ddot{u}}_{n} + γ h {\ddot{u}}_{n + 1}

u_{n + 1} = u_{n} + h {\dot{u}}_{n} + (\frac{1}{2} - β) h^{2} {\ddot{u}}_{n} + β h^{2} {\ddot{u}}_{n + 1} + {r^{'}}_{n}

According to the values of

γ

and

β

, different algorithms can be derived:

$γ = 0, β = 0$ : pure explicit algorithm. It can be shown that it is always unstable. An integration scheme is stable if a critical time step exists so that, for a value of the time step lower or equal to this critical value, a finite perturbation at a given time does not lead to a growing modification at future time steps.
$γ = 1 / 2, β = 0$ : central difference algorithm. It can be shown that it is conditionally stable.
$γ = 1 / 2, β = 1 / 2$ : Fox & Goodwin algorithm.
$γ = 1 / 2, β = 1 / 6$ : linear acceleration.
$γ = 1 / 2, β = 1 / 4$ : mean acceleration. This integration scheme is the unconditionally stable algorithm of maximum accuracy.