In his book G. R. Liu says that strong form meshfree methods ("collocation methods") are generally less robust than the methods based on weak forms (like EFG, MLPG, PIM, ...). However, they are simpler and faster. Are there studies that compare the numerical characteristics of these variations of meshfree methods? For instance (i) presenting computational time x numerical precision ; (ii) perturbing the nodes distribution and analyzing their influence on the solution precision; (iii) etc ...?
Thank you, Reza.
I am aware of these studies, but I think they do not address a fundamental question: even if the strong forms are less robust and precise, their computational time is much lower than the equivalent weak forms. Wouldn´t this create a "compensation" that could generate results that for the same computational time are more precise than the equivalent weak form results?
Best regards,
Renato.
Dear Renato,
I am totaly agree with you in the computational cost point of view. However, as I stated before, the major concerns about the restraints on the smoothness of the approximation related to the strong form and much more importantly the stability of the algorithm may justify the consequences of the computational costs. Moreover, all weak forms are not inefficient. Weak forms discretized by finite element method (FEM) can be solved efficiently; however, they may not be instrumental to cope with problems , for example, the large deformations and crack propagation ones. I think the heavy computational you have mentioned would be a major concern when one exploits the common meshfree methods to deal with the problem. However, using a parallel implementation can be a remedy to the computational cost. I think some stability and convergence behaviour issues associated with the collocation approaches reduce the motivation to employ them; even though they can be faster and less extensive than the weak forms.
Best wishes,
Reza.
Hello Reza,
Yes, my affirmation on the inefficiency of weak forms only applies to the meshfree context, not to FEM. I agree that parallel implementations would remedy the computational costs but this applies to all meshfree formulations, not only to those based on weak forms.
I work with weak form meshfree formulations, but in my application area (Electromagnetism) people (especially those with a finite difference background) are using more and more meshfree strong form formulations. I wonder why this is happening and I think that a study that compares all the aspects of strong and weak meshfree formulations is something that must be done to clarify the issue.
Best regards,
Renato.
Hello Renato,
I think that the answers for the questions you detailed lack in the Literature. Also, I agree with you and Reza about the weak and strong formulations. We have successfully worked with weak form for the EFGM applied to boundary value problems of Poisson (electromagnetism) and Schroedinger (Quantum well and dot) in 2-D with material interfaces. The approximations provided by the EFGM are more accurate than the ones provided by the FEM, however, the computational cost is higher for the EFGM, as expected. Thus, the application requirements guide the choice by accuracy or fastness.
Best regards,
Gleber.
I think the main reasons that hinder the application of strong forms in solving engineering problems may be (a) the difficulty in dealing with derivative boundary conditions (or Neumann boundary condition); (b) the results are quite related to the node distribution; (c) the results are also strongly related to the shape parameters used in RBF collocation method; (d) the convergency maybe not monotonous; (e) the global matrix maybe ill-conditioned in some circumstances; (f)and many others.
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