A New Stabilized Nodal Integration Approach Page: 4 of 13
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2 Michael Anthony Puso, Edward Zywicz, and J.S. Chen
to implement essential boundary conditions and the nodal adjacency only
includes near neighbors. The NN/NEM approach has been implemented in the
context of a meshless method by exploiting the concept of alpha shapes [4, 6] to
determine/treat the free surface of cloud of points. In this work, the integration
method in [10, 5] is modified by applying an additional stabilization term. In
what follows: Section 2 introduces the global weak form, the NN interpolation
scheme and the SCNI approach with added stabilization, Section 3 presents
results demonstrating the necessity of the added stabilization and effectiveness
of the proposed approach.
The formulation will be introduced in the context of linear elasticity and the
straightforward extensions to the nonlinear regime will be given at the end of
2.1 Galerkin Method
Considering a body occupying the domain 2, the governing equations of mo-
tion are given
pu= zVa+b (1)
where u is the displacement field, b is the body force and a a(s) is the
Cauchy stress in terms of strain
(u) 1/2(Vu + VuT) (2)
Employing test function v, the weak form of (1) can then be written
J pvu dQ+ fe ) :a (e(v))d2 =z v -bd + f v - td (3)
where applied tractions t are specified on the boundary Ft.
2.2 Discrete Form
Following standard procedure, the discrete displacement field is defined
uh -- OJ(X) XI X E 2h (4)
in terms of shape functions 0, and nodal displacements ui over all nodes
I 1, N on the discretized domain Qh. Unlike finite elements, the definition
of the domain f2h in meshless methods is not straightforward and is defined
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Puso, M; Zywicz, E & Chen, J S. A New Stabilized Nodal Integration Approach, article, February 8, 2006; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc896093/m1/4/: accessed April 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.