DefinitionRepresentativeDensity

Computation of the representative density

The representative density is computed from the representative masses and the appropriate volumina of the MESHFREE points (given by %ind_Vi% or (0.3*Y%ind_h%^3), refer to the parameter whichVi in RepresentativeMass_iData). Version 1: \begin{align} \overset{\scriptscriptstyle\frown}{\rho }_{i}=\frac{\sum\limits_{j}{W_{ij}\cdot \overset{\scriptscriptstyle\frown}{m}_{j}}}{\sum\limits_{j}{W_{ij}\cdot \overset{\scriptscriptstyle\frown}{V}_{j}}}\end{align} where the weight kernel is defined as \begin{align} W_{ij}=\exp \left( -\alpha _{W}\cdot r_{ij}^{2} \right)/\overset{\scriptscriptstyle\frown}{V}_{j}^{{{\beta }_{W}}}\text{ with }r_{ij}=\frac{\left\| \mathbf{x}_{j}-\mathbf{x}_{i} \right\|}{\frac{1}{2}\left( h_{j}+h_{i} \right)}\end{align} And \( \alpha _{W} > 0 \text{, }\beta_W =0,1\) to be given by the user. Remark:

• \( \beta =0\): average mass divided by average volume \( \overset{\scriptscriptstyle\frown}{\rho }_{i}=\frac{{\sum\limits_{j}{\exp \left( -\alpha _{W}\cdot r_{ij}^{2} \right)\cdot \overset{\scriptscriptstyle\frown}{m}_{j}}}/{\sum\limits_{j}{\exp \left( -\alpha _{W}\cdot r_{ij}^{2} \right)}}\;}{{\sum\limits_{j}{\exp \left( -\alpha _{W}\cdot r_{ij}^{2} \right)\cdot \overset{\scriptscriptstyle\frown}{V}_{j}}}/{\sum\limits_{j}{\exp \left( -\alpha _{W}\cdot r_{ij}^{2} \right)}}\;}\) (the default!!!)
• \( \beta =1\): average local representative density, i.e. \( \overset{\scriptscriptstyle\frown}{\rho }_{i}=\frac{\sum\limits_{j}{\exp \left( -\alpha _{W}\cdot r_{ij}^{2} \right)\cdot \frac{\overset{\scriptscriptstyle\frown}{m}_{j}}{\overset{\scriptscriptstyle\frown}{V}_{j}}}}{\sum\limits_{j}{\exp \left( -\alpha _{W}\cdot r_{ij}^{2} \right)}}\)

Version 2: Determine the representative density clusterwise by \begin{align} \overset{\scriptscriptstyle\frown}{\rho }_{i}=\frac{\sum\limits_{j\in {{J}_{cluster}}}{\overset{\scriptscriptstyle\frown}{m}_{j}}}{\sum\limits_{j\in {{J}_{cluster}}}{\overset{\scriptscriptstyle\frown}{V}_{j}}}\end{align} which leads to less local fluctuations of the representative density.

Smoothing of the representative density

In order to avoid fluctuations in version 1 due to heavy refinement or coarsening of the pointcloud (in fact heavy redistribution of the local representative massses), the representative density can be smoothed. See DEBUG_GeneralParameter(78) and DEBUG_GeneralParameter(79) . This will solve all surface buckling phenomena due to VOLUME_correction_local using representative masses.