projY()

The projection of a MESHFREE-entity is done by a smooth, least-squares approximation of Shepard-type. Depending on the given parameters, the projection is done from a different chamber, only for specific types of points, or with a specific kernel. The MESHFREE-used least squares approximation naturaly fall back to the Shepard apprximation if order 1 is chosen. The explicit formulation is \begin{align} \tilde{u}(\mathbf{x}) = \frac{ \sum \limits_{j}^{N(\mathbf{x})} W(\mathbf{x}_j,\mathbf{x}) \cdot u(\mathbf{x}_j)}{ \sum \limits_{j}^{N(\mathbf{x})} W(\mathbf{x}_j,\mathbf{x}) }\end{align} with \( W(\mathbf{x}_j,\mathbf{x}) = \exp\left( -\alpha \frac{\|\mathbf{x}_j-\mathbf{x}\|^2}{h(\mathbf{x}_j)^2} \right)\), where \( \mathbf{x}_j\) are the neighbors and \( h(\mathbf{x}_j)\) the smoothing length. The basic projection of the MESHFREE-entity %ind_Entity% (see __Indices__) is invoked as: The values of an entity from a different chamber with chamber index iChamber can be projected by: WhatPointsShouldBeUsed:

• %EQN_Proj_INT% (force the projection using only interior points)
• %EQN_Proj_BND% (force the projection using only boundary points)
• default: %EQN_Proj_ALL% (force the projection using all types of points, i.e. interior and boundary points)

alphaKernel: This option controls the weight function by setting the parameter \( \alpha\) (see above). Note: Given a projection task in iChamber at the location \( \bf{x}\), then MESHFREE will search for the closest neighbor point at location \( \bf{x}_i\) in iChamber. The neighbors for the projection task around \( \bf{x}\) are determined by the neighbor list of \( \bf{x}_i\). Thus, the choice of the parameter NEIGHBOR_FilterMethod will have a big influence on the results of the projection. Please remember that NEIGHBOR_FilterMethod > 1 prevents the neighbor search from "looking through" thin walls.