Constrained Moving Least Squares (CMLS) Method

Dear Alejandro

Thank you very much for creating the comment to my notes on the imposing displacement BC's in meshfree method.

My M.Sc thesis is mixed formulation and adaptive refinement in the new meshless method named Discrete Least Square Meshless (DLSM) that introduced by my thesis advisor Prof. M.H. Afshar. In this approach we use MLS approximation to construct shape functions. In mixed formulation displacements and stresses obtained simultaneously, therefore for imposing Dirichlet and Newmann BC’s only one penalty coefficient is used. Now my questions are,

1)  How determine amount of penalty coefficient?

2) Can I use maximum-entropy approximation instead of MLS for imposing BC’s without penalty coefficient in mixed formulation for any meshfree method?

Thank you

J. Amani

Dear Alejandro1)In  solving the differential equations governing the planar elasticity problems,  order of approximation of stresses is 2, therefore second derivative of MLS shape functions should be used. In paper with entitle: “Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations“ by Rahmani & Afshar was shown that second derivative of MLS shape functions have more discontinuity than first derivatives. Since in mixed formulation the resulting governing equations are of the first order, both the displacements and stress boundary conditions are of the Dirichlet type.  Thus in mixed formulation order of approximation decrease to 1, therefore higher accuracy is obtained. only in the paper by Prof. Atluri “Meshless Local Petrov-Galerkin (MLPG) mixed collocation method for elasticity problems” mixed formulations in meshless methods was used. 2) No, penalty coefficient method is differing from Lagrange multipliers. In Liu’s book “mesh free methods - moving beyond the finite element method” have been explained that in using Lagrange multipliers stiffness matrix may have nonsymmetrical, while in using penalty coefficient method in my approach stiffness matrix is symmetrical. Also the computational time maybe increase when using Lagrange multipliers specially in using sampling points with nodal points.    3)No, mixed method variously used in finite element method for analysis of plane elasticity problems. For stability the finite element spaces are required to satisfy the Ladyzhenskaya-Babuska-Brezzi (LBB) condition . But in mixed formulation only formulation was changed compared with previous formulation (Irreducible formulation). Results of mixed formulation are better compared with irreducible formulation when amount of penalty coefficients have been determined with some efforts. I want to Solve the penalty problem with:1)Change MLS approximation to max-ent approximation to avoid penalty coefficient.2) Change MLS approximation to CMLS approximation to avoid penalty coefficient. The formulation of CMLS from Liu’s book is attached.Which of the above can solve imposing BC’s in mixed formulation?Kindest regardJafar