My answer is not directly related to the main one, but to the point raised by Victoria Rosca. In my opinion, old papers regarding meshfree Galerkin methods are now misleading. There have been many papers "studying" the relation between the support size (over regular/irregular grid) and the quadrature used, but none of them give sound mathematical reasons to explain why they get accuracy and convergence problems. More than a decade after the paper cited above, the integration issue in meshfree Galerkin methods is now close to be solved. If you have a consistent and stable integration scheme, the effect of the support size and the irregularity of the grid on the stiffness matrix shouldn't affect more than it affects the approximation itself.

You might be interested in Reading these papers:

I. Babuˇska, U. Banerjee, J. E. Osborn, and Q. L. Li. Quadrature for meshless methods. International Journal for Numerical Methods in Engineering, 76(9):143–1470, 2008.

I. Babuˇska, U. Banerjee, J. E. Osborn, and Q. Zhang. Effect of numerical integration on meshless methods. Computer Methods in Applied Mechanics and Engineering, 198(37–40):2886–2897, 2009.

Q. Duan, X. Gao, B. Wang, , X. Li, H. Zhang, T. Belytschko, and Y. Shao. Consistent element-free Galerkin method. International Journal for Numerical Methods in Engineering, 99(2):79–101, 2014.

A. Ortiz-Bernardin, M. A. Puso, and N. Sukumar. Improved robustness for nearly-incompressible large deformation meshfree simulations on Delaunay tessellations. Computer Methods in Applied Mechanics and Engineering, 2015. http://dx.doi.org/10.1016/j.cma.2015.05.009.

Still waiting for response please!

I try to answer this question in my opinion.

the distribution of  discrete nodes has great influence to numerical result. how do you generate irregular nodes? In this book :Meshfree Approximation with MATLAB , matlab function: CreatPoints.m can do it in some different manners.

function [points, N] = CreatePoints(N,s,gridtype)

% Computes a set of N points in [0,1]^s

% Note: could add variable interval later

% Inputs:

% N: number of interpolation points

% s: space dimension

% gridtype: 'c'=Chebyshev, 'f'=fence(rank-1 lattice),

% 'h'=Halton, 'l'=latin hypercube, 'n'=random normal,

% 'r'=random uniform, 's'=Sobol, 'u'=uniform

% Outputs:

% points: an Nxs matrix (each row contains one s-D point)

% N: might be slightly less than original N for

% Chebyshev and gridded uniform points

% Calls on: chebsamp, lattice, haltonseq, lhsamp, i4_sobol,

% gridsamp

% Also needs: fdnodes,gaussj,i4_bit_hi1,i4_bit_lo0,i4_xor

switch gridtype

case 'c'

ppd = zeros(1,s);

for j=1:s

ppd(j) = floor(nthroot(N,s+1-j));

N = N/ppd(j);

end

gam = 0.5*ones(1,s); % density for point distribution, 0.5=Chebyshev

points = chebsamp([zeros(1,s); ones(1,s)], ppd, gam);

N = prod(ppd);

case 'f'

points = lattice(N,s); % N should be(?) power of 2

case 'h'

points = haltonseq(N,s);

case 'l'

points = lhsamp(N,s);

case 'n'

randn('state',47);

points = 0.5 + 0.1*randn(N,s);

case 'r'

rand('state',47);

points = rand(N,s);

case 's'

sp = max(2,s);

points = zeros(N,sp);

seed = 0; for i=1:N; [points(i,:) seed] = i4_sobol(sp,seed); end

points = points(:,1:s);

points = N*points/(N-1);

case 'u'

ppd = zeros(1,s);

for j=1:s

ppd(j) = floor(nthroot(N,s+1-j));

N = N/ppd(j);

end

points = gridsamp([zeros(1,s); ones(1,s)], ppd);

N = prod(ppd);

otherwise

error('Please use c, f, h, n, r, s or u data types')

end

Not all irregular nodes can obtain a accurate result with the MLS collocation method in my experience. the size of influence domain of each evaluated pointed can be determined as this : $ d =\alpha *(p+1)* \delta h$, where p is the order of polynomial basis, and α is the dimensionless size parameter, \delta h  is is the average nodal spacing.

Thank you all for your kind attention.

I have solved the above challenge using the following approach.

1. Firstly, I used a nearest neighbor search algorithm to located the closest node to each of the field nodes;

2. The approach is more convenient for circular support domain - though it can be used for rectangular domain as well.

3. I then obtain the distance between the field node and its nearest neighbor; this distance was then taken/used as the radius of the support domain.

4. I subsequently obtain the size of the influence domain by using the formula: dm(i) = dmax*delta (i) where i is the field node and delta is the radius of the support domain.

Note that instead of using a for loop over all the nodes, I simply perform the step 1 above for all the field nodes hence size of delta is (numnod * 1) and dm is also (numnod*1) since dmax is a scalar which ranges from 2.0 - 4.0 for static analysis.

Thanks!

N.B: Please feel free to comment on my approach.

The influence of the grid irregularity (thus the size of the support domain) is studied in the paper 

Analysis of Thin Plates by the Element-FreeGalerkin Method

Petr Krysl and Ted Belytschko

October 7, 1999

https://www.academia.edu/8869421/Analysis_of_thin_plates_by_the_element-free_Galerkin_method

My answer is not directly related to the main one, but to the point raised by Victoria Rosca. In my opinion, old papers regarding meshfree Galerkin methods are now misleading. There have been many papers "studying" the relation between the support size (over regular/irregular grid) and the quadrature used, but none of them give sound mathematical reasons to explain why they get accuracy and convergence problems. More than a decade after the paper cited above, the integration issue in meshfree Galerkin methods is now close to be solved. If you have a consistent and stable integration scheme, the effect of the support size and the irregularity of the grid on the stiffness matrix shouldn't affect more than it affects the approximation itself.

You might be interested in Reading these papers:

I. Babuˇska, U. Banerjee, J. E. Osborn, and Q. L. Li. Quadrature for meshless methods. International Journal for Numerical Methods in Engineering, 76(9):143–1470, 2008.

I. Babuˇska, U. Banerjee, J. E. Osborn, and Q. Zhang. Effect of numerical integration on meshless methods. Computer Methods in Applied Mechanics and Engineering, 198(37–40):2886–2897, 2009.

Q. Duan, X. Gao, B. Wang, , X. Li, H. Zhang, T. Belytschko, and Y. Shao. Consistent element-free Galerkin method. International Journal for Numerical Methods in Engineering, 99(2):79–101, 2014.

A. Ortiz-Bernardin, M. A. Puso, and N. Sukumar. Improved robustness for nearly-incompressible large deformation meshfree simulations on Delaunay tessellations. Computer Methods in Applied Mechanics and Engineering, 2015. http://dx.doi.org/10.1016/j.cma.2015.05.009.

You can try to use : the concept of natural neighbor to detrmine the influence/support domain, you can refeer to the two papers of L.DINIS : http://www.sciencedirect.com/science/article/pii/S0045782506003446

http://www.sciencedirect.com/science/article/pii/S0955799707001415

I hope this will help you

Best ragards

Nice reply, Alejandro. I will read your papers soon.

Best regards,

Phu