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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.
Artificial Viscosity

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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
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Artificial Viscosity

As usual in SPH ^¹ implementations, viscosity is rather an inter-particles pressure than a bulk pressure. It was shown that the use of Equation 1 and Equation 2 generates a substantial amount of entropy in regions of strong shear even if there is no compression.

π_{i j} = \frac{- q_{b} \frac{c_{i} + c_{j}}{2} μ_{i j} + q_{α} μ_{i j}^{2}}{\frac{(ρ_{i} + ρ_{j})}{2}}

with

μ_{i j} = \frac{d_{i j} (v_{i} - v_{j}) • (X_{i} - X_{j})}{{‖ X_{i} - X_{j} ‖}^{2} + ε d_{i j}^{2}}

Where, $X_{i}$ (resp. $X_{j}$ ) indicates the position of particle I (resp. $j$ ) and $c_{i}$ (resp $c_{j}$ ) is the sound speed at location $i$ (resp. $j$ ), and $q_{a}$ and $q_{b}$ are constants. This leads us to introduce Equation 3 and Equation 4. ^² The artificial viscosity is decreased in regions where vorticity is high with respect to velocity divergence.

π_{i j} = \frac{- q_{b} \frac{c_{i} + c_{j}}{2} μ_{i j} + q_{α} μ_{i j}^{2}}{\frac{(ρ_{i} + ρ_{j})}{2}}

with

μ_{i j} = \frac{d_{i j} (v_{i} - v_{j}) • (X_{i} - X_{j})}{{‖ X_{i} - X_{j} ‖}^{2} + ε d_{i j}^{2}} \frac{(f_{i} + f_{j})}{2}, f_{k} = \frac{‖ {\nabla \cdot v |}_{k} ‖}{‖ {\nabla \cdot v |}_{k} ‖ + ‖ {\nabla \times v |}_{k} ‖ + ε^{'} \frac{c_{k}}{d_{k}}}

Default values for $q_{a}$ and $q_{b}$ are respectively set to 2 and 1.

¹ Monaghan J.J., Smoothed Particle Hydrodynamics, Annu.Rev.Astron.Astro-phys; Vol. 30; pp. 543-574, 1992.

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.