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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.
SPH Approximation of a Function

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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.

SPH Approximation of a Function

Let

\prod f (x)

the integral approximation of a scalar function

f

in space:

図 1.

\prod f (x) = \int_{Ω} f (y) W (x - y, h) d y

with

h

the so-called smoothing length and

W

a kernel approximation such that:

図 2.

\forall x, \int_{Ω} W (x - y, h) d y = 1

and in a suitable sense

図 3.

\forall x, \lim_{h \to 0} W (x - y, h) = δ (x - y)

$δ$ denotes the Dirac function.

Let a set of particles

i

=1, n at positions

x_{i}

(

i

=1,n) with mass

m_{i}

and density

ρ_{i}

. The smoothed approximation of the function

f

is (summation over neighboring particles and the particle

i

itself):

図 4.

\prod_{s} f (x) = \sum_{i = 1, n} \frac{m_{i}}{ρ_{i}} f (x_{i}) W (x - y, h)

The derivatives of the smoothed approximation are obtained by ordinary differentiation.

図 5.

\nabla f (x) = \sum_{i = 1, n} \frac{m_{i}}{ρ_{i}} f (x_{i}) \nabla W (x - y, h)

The following kernel ^¹ which is an approximation of Gaussian kernel by cubic splines was chosen (図 1):

図 6.

r \leq h \Rightarrow W (r, h) = \frac{3}{2 π h^{3}} [\frac{2}{3} - {(\frac{r}{h})}^{2} + \frac{1}{2} {(\frac{r}{h})}^{3}]

図 7.

h \leq r \leq 2 h \Rightarrow W (r, h) = \frac{1}{4 π h^{3}} {(2 - \frac{r}{h})}^{3}

and

図 8.

2 h \leq r \Rightarrow W (r, h) = 0

図 9. Kernel Based on Spline Functions

This kernel has compact support, so that for each particle $i$ , only the closest particles contribute to approximations at $i$ (this feature is computationally efficient). The accuracy of approximating 式 1 by 式 4 depends on the order of the particles.

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