There is no recipe for providing `support sizes' that fits all problems when it comes to EFF; at best, only `guidelines' can be provided. The nodal distribution, support size, and the underlying PDE that is being solved are linked, and hence the notion of best/optimum can not be stated in general terms. Of course, the only fact that must be honored is any point in 2D must lie within the cover of at least three nodal supports if a linear basis is being used. This ensures that the approximation can be computed at any point in the domain.
The concept of (a,p) regular particle distribution given in (Han, W. and Meng, X., 21001, “Error analysis of the reproducing kernel particle method”, Computer Methods in Applied Mechanics and Engineering Vol. 190, pp. 6157-6181) may be referred.
In principle there is no restriction on the shape of the support of the nodal weight function, but there should be compelling reasons to warrant to make it nonconvex. It would be difficult to create weights on nonconvex support that are smooth (R-functions might be one route), and if not, then a nonconvex polygonal support would lead to non-smooth weights at the vertices. If departure from isotropic (circular in two dimensions) supports is the impetus (e.g., if one has nonuniform nodal discretizations) , one can use weight that have anisotropic (ellipsoidal) supports that are still convex but take the spatial nodal density into account.
Ya Sukumar I understand your point. But if derivative of shape functions is constructed directly based on reproduction of derivative of monomial space then one can bypass the issue of differentiability requirement of window function. In such cases local integrability is the only criteria that the window function should satisfy. Therefore construction of weights (sufficiently differentiable) on nonconvex support will not be a difficult job. However I doubt whether in such cases error estimate of RKPM given in (Han, W. and Meng, X., Error analysis of reproducing kernel particle method, CMAME, Vol 190, pp. 6157-6184) is equally valid or not. Do you have any suggestion on this?
If one has nonuniform nodal discretizations ,How do we decied on anisotropic weights.
Suppose if we are deciding a domain of influence region using Natural neighbours, should the natural neighbour nodes have same weights, irrespective of amount of random ness in distribution?
Nodal supports
There is no recipe for providing `support sizes' that fits all problems when it comes to EFF; at best, only `guidelines' can be provided. The nodal distribution, support size, and the underlying PDE that is being solved are linked, and hence the notion of best/optimum can not be stated in general terms. Of course, the only fact that must be honored is any point in 2D must lie within the cover of at least three nodal supports if a linear basis is being used. This ensures that the approximation can be computed at any point in the domain.
The concept of (a,p) regular particle distribution
The concept of (a,p) regular particle distribution given in (Han, W. and Meng, X., 21001, “Error analysis of the reproducing kernel particle method”, Computer Methods in Applied Mechanics and Engineering Vol. 190, pp. 6157-6181) may be referred.
Thank you sirs for the
Thank you sirs for the comments and references
convexity of the support of kernel function in meshfree method
Is it always required to have convex support of the kernel function in RKPM/MLS or any other meshfree approximation scheme based on moving window?
Re: convex supports
In principle there is no restriction on the shape of the support of the nodal weight function, but there should be compelling reasons to warrant to make it nonconvex. It would be difficult to create weights on nonconvex support that are smooth (R-functions might be one route), and if not, then a nonconvex polygonal support would lead to non-smooth weights at the vertices. If departure from isotropic (circular in two dimensions) supports is the impetus (e.g., if one has nonuniform nodal discretizations) , one can use weight that have anisotropic (ellipsoidal) supports that are still convex but take the spatial nodal density into account.
In reply to Re: convex supports by N. Sukumar
Re:Re:convex supports
Ya Sukumar I understand your point. But if derivative of shape functions is constructed directly based on reproduction of derivative of monomial space then one can bypass the issue of differentiability requirement of window function. In such cases local integrability is the only criteria that the window function should satisfy. Therefore construction of weights (sufficiently differentiable) on nonconvex support will not be a difficult job. However I doubt whether in such cases error estimate of RKPM given in (Han, W. and Meng, X., Error analysis of reproducing kernel particle method, CMAME, Vol 190, pp. 6157-6184) is equally valid or not. Do you have any suggestion on this?
regarding iregular nodal distributions
Dear sirs,
If one has nonuniform nodal discretizations ,How do we decied on anisotropic weights.
Suppose if we are deciding a domain of influence region using Natural neighbours, should the natural neighbour nodes have same weights, irrespective of amount of random ness in distribution?
Raj