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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
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.
Conservative Smoothing of Velocities

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User Guide
This manual provides details on the features, functionality, and simulation methods available in Altair Radioss.
Reference Guide
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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
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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
- 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.
  - 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.
    - SPH Approximation of a Function
    - Corrected SPH Approximation of a Function
    - SPH Integration of Continuum Equations
    - Artificial Viscosity
    - Time Step Control Stability
      The stability conditions of explicit scheme in SPH formulation can be written over cells or on nodes.
    - Conservative Smoothing of Velocities
    - SPH Cell Distribution
- Appendix A: Basic Relations of Elasticity
User Subroutines
This manual describes the interface between Altair Radioss and user subroutines.

Conservative Smoothing of Velocities

It can be shown that the SPH method is unstable in tension. The instability is shown to result from an effective stress with a negative modulus (imaginary sound speed) being produced by the interaction between the constitutive relation and the kernel function, and is not caused by the numerical time integration algorithm. ^¹ D.S Balsara ^² states, use special filtering of velocities (so called conservative smoothing, because momentum quantities are not modified):

V_{i} (s m o o t h e d) = v_{i} + α_{c s} \sum_{j} \frac{2 m_{j}}{ρ_{i} + ρ_{j}} (v_{j} - v_{i}) \frac{W_{i} (j) + W_{j} (i)}{2}

¹ Swegle J.W., Hicks D.L., and Attaway S.W., Smoothed Particle Hydrodynamics: Stability Analysis, Journal of Computational Physics, Vol. 116, pp. 123-134, 1995.

² Balsara D.S., Von Neumann Stability Analysis of Smoothed Particle Hydrodynamics Suggestions for Optimal Algorithms, Journal of Computational Physics, Vol. 121, pp. 357-372, 1995.