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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
Basic Equations
Equilibrium Equations

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Radioss^® is a leading explicit finite element solver for crash and impact simulation.
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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
  - 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
    - Material and Spatial Coordinates
      In a Cartesian coordinates system, the coordinates of a material point in a reference or initial configuration are denoted by $X$ . The coordinates of the same point in the deformed or final configuration are denoted by $x$ .
    - Mesh Definition
    - Vicinity Transformation
    - Kinematic Description
    - Kinetic Description
    - Stress Rates
    - Stresses in Solids
    - Lagrangian and Corotational Formulations
    - Equilibrium Equations
    - Virtual Power Principle
    - Virtual Power Term Names
    - Small Strain Formulation
  - Finite Element Formulation
  - Dynamic Analysis
  - 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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Equilibrium Equations

Let $Ω$ be a volume occupied by a part of the body in the current configuration, and $Γ$ the boundary of the body. In the Lagrangian formulation, $Ω$ is the volume of space occupied by the material at the current time, which is different from the Eulerian approach where a volume of space through which the material passes is examined. $τ$ is the traction surface on $Γ$ and $b$ are the body forces.

Force equilibrium for the volume is then:

\int_{Γ}^{} τ_{i} d Γ + \int_{Ω} ρ b_{i} d Ω = \int_{Ω} ρ \frac{\partial v_{i}}{\partial t} d Ω

Where, $ρ$ is the material density.

The Cauchy true stress matrix at a point of $Γ$ is defined by:

τ_{i} = n_{j} σ_{j i}

Where, $n$ is the outward normal to $Γ$ at that point. Using this definition, Equation 1 is written:

\int_{Γ} n_{j} σ_{j i} d Γ + \int_{Ω} ρ b_{i} d Ω = \int_{Ω} ρ \frac{\partial v_{i}}{\partial t} d Ω

Gauss' theorem allows the rewrite of the surface integral as a volume integral so that:

\int_{Γ} n_{j} σ_{j i} d Γ = \int_{Ω} \frac{\partial σ_{i j}}{\partial x_{j}} d Ω

As the volume is arbitrary, the expression can be applied at any point in the body providing the differential equation of translation equilibrium:

\frac{\partial σ_{i j}}{\partial x_{j}} + ρ b_{i} = ρ \frac{\partial v_{i}}{\partial t}

Use of Gauss' theorem with this equation leads to the result that the true Cauchy stress matrix must be symmetric:

σ = σ^{T}

So that at each point there are only six independent components of stress. As a result, moment equilibrium equations are automatically satisfied, thus only the translational equations of equilibrium need to be considered.