I'm not sure precisely what you mean by this question, but I can guess. I suspect that what you mean is that, given the same number of nodes, a meshless method will yield a discrete structural response that is less stiff than the same produced with finite elements.
This might seem counter-intuitive when one thinks about the overlapping domains of shape functions employed in meshless methods. Even though meshless methods are usually constructed to only exhibit linear precision (i.e. they can exactly reproduce a linear field), they typically exhibit a little more than that. This is not the case, however, with finite element shape functions. So, with a given number of nodes, meshless shape functions have more ability to represent the exact solution. As a result, the structure appears less stiff.
I guess your question is on the global stiffness matrix K and integration techinques. In using the numerical quadrature scheme, a background mesh of cells is required for the integration of computing the stiffness matrix. This background mesh is similar to the mesh of elements in FEM, which is no overlap or gap is permitted, and independent on support domains ( or influence domains), which are used for constructing shape functions and interpolating field variables. So, the stiffness matrix of the structure does not increase.
In another way, if the subdomains used for integration are overlap ( and no gap ), you can use a set of contribution functions, whose sum is unity. Then, the global stiffness matrix also does not increase. However, the contribution function for each nodal stiffness matrix Kij (integration on a subdomain) must be chosen reasonablely.
The main objective of my research is to directly design shapes (e.g. an airfoil) with meshless methods. (direct design is a new class of design procedures in aerodynamics that nodal coordinates are derived as dependent variable). at the first stage I tried to solve some sample problem relating to the main problem e.g. ideal flow around a cylinder or an airfoil with MLPG. after that I devised a formulation for direct design with MLPG but it did not work. In fact, it worked in a very limited situation. the main problem was accuracy of the meshfree method. the parameters affecting accuracy in this method(MLPG) are more that methods like FEM. the other important problem is that the accuracy depends strongly on these parameters and has not a predictable behavior with changes in parameters and this problem dependent accuracy is less than methods like FEM. I want to know that do I have a misunderstanding about the method or this behavior is natural?
re: meshless vs. FEM
Thi,
I'm not sure precisely what you mean by this question, but I can guess. I suspect that what you mean is that, given the same number of nodes, a meshless method will yield a discrete structural response that is less stiff than the same produced with finite elements.
This might seem counter-intuitive when one thinks about the overlapping domains of shape functions employed in meshless methods. Even though meshless methods are usually constructed to only exhibit linear precision (i.e. they can exactly reproduce a linear field), they typically exhibit a little more than that. This is not the case, however, with finite element shape functions. So, with a given number of nodes, meshless shape functions have more ability to represent the exact solution. As a result, the structure appears less stiff.
Stiffness matrix in meshless methods
Thi Dang,
I guess your question is on the global stiffness matrix K and integration techinques. In using the numerical quadrature scheme, a background mesh of cells is required for the integration of computing the stiffness matrix. This background mesh is similar to the mesh of elements in FEM, which is no overlap or gap is permitted, and independent on support domains ( or influence domains), which are used for constructing shape functions and interpolating field variables. So, the stiffness matrix of the structure does not increase.
In another way, if the subdomains used for integration are overlap ( and no gap ), you can use a set of contribution functions, whose sum is unity. Then, the global stiffness matrix also does not increase. However, the contribution function for each nodal stiffness matrix Kij (integration on a subdomain) must be chosen reasonablely.
Quoc-Duan
accuracy problem in meshfree methods
The main objective of my research is to directly design shapes (e.g. an airfoil) with meshless methods. (direct design is a new class of design procedures in aerodynamics that nodal coordinates are derived as dependent variable). at the first stage I tried to solve some sample problem relating to the main problem e.g. ideal flow around a cylinder or an airfoil with MLPG. after that I devised a formulation for direct design with MLPG but it did not work. In fact, it worked in a very limited situation. the main problem was accuracy of the meshfree method. the parameters affecting accuracy in this method(MLPG) are more that methods like FEM. the other important problem is that the accuracy depends strongly on these parameters and has not a predictable behavior with changes in parameters and this problem dependent accuracy is less than methods like FEM. I want to know that do I have a misunderstanding about the method or this behavior is natural?
Thanks,
Hossein Bazmara