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Strain smoothing for stabilization and regularization of Galerkin meshfree methods. (English) Zbl 1114.65131
Griebel, Michael (ed.) et al., Meshfree methods for partial differential equations III. Selected papers based on the presentations at the 3rd international workshop, Bonn, Germany, September 12–15, 2005. Berlin: Springer (ISBN 3-540-46214-7/pbk). Lecture Notes in Computational Science and Engineering 57, 57-75 (2007).
Summary: We introduce various forms of strain smoothing for stabilization and regularization of two types of instability: (1) numerical instability resulting from nodal domain integration of weak form, and (2) material instability due to material strain softening and localization behavior. For numerical spatial instability, we show that the conforming strain smoothing in stabilized conforming nodal integration only suppresses zero energy modes resulting from nodal domain integration. When the spurious nonzero energy modes are excited, additional stabilization is proposed.
For problems involving strain softening and localization, regularization of the ill-posed problem is needed. We show that the gradient type regularization method for strain softening and localization can be formulated implicitly by introducing a gradient reproducing kernel strain smoothing. It is also demonstrated that the gradient reproducing kernel strain smoothing also provides a stabilization to the nodally integrated stiffness matrix. For application to modeling of fragment penetration processes, a nonconforming strain smoothing as a simplification of conforming strain smoothing is also introduced.
For the entire collection see [Zbl 1104.65003].
Reviewer: Reviewer (Berlin)

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation