Exact imposition of essential boundary condition and material interface continuity in Galerkin‐based meshless methods

Abstract

Summary Represented by the element free Galerkin method, the meshless methods based on the Galerkin variational procedure have made great progress in both research and application. Nevertheless, their shape functions free of the Kronecker delta property present great troubles in enforcing the essential boundary condition and the material continuity condition. The procedures based on the relaxed variational formulations, such as the Lagrange multiplier‐based methods and the penalty method, strongly depend on the problem in study, the interpolation scheme, or the artificial parameters. Some techniques for this issue developed for a particular method are hard to extend to other meshless methods. Under the framework of partition of unity and strict Galerkin variational formulation, this study, taking Poisson's boundary value problem for instance, proposes a unified way to treat exactly both the material interface and the nonhomogeneous essential boundary as in the finite element analysis, which is fit for any partition of unity‐based meshless methods. The solution of several typical examples suggests that compared with the Lagrange multiplier method and the penalty method, the proposed method can be always used safely to yield satisfactory results. Copyright © 2016 John Wiley & Sons, Ltd.

Smoothed numerical manifold method with physical patch‐based smoothing domains for linear elasticity

“…The standard Lagrange multiplier method caused the boundary locking and oscillation, and some techniques are suggested to overcome it, for example, the enriched formulation 59 . By contrast, the second is to adjust the approximation of the primary variable to let it satisfy the Dirichlet boundary condition exactly, for example, the one proposed by Zheng et al 60 for the element‐free Galerkin method and another approach proposed by van den Boom et al 61 for the discontinuity‐enriched FEM 62,63 . The Nitsche method 58 has received substantial attention recently and has been applied to the meshfree method 64,65 and the partition‐of‐unity methods 66‐71 .…”

Acoustic scattering in nonhomogeneous media and the problem of discontinuous gradients: Analysis and inf‐sup stability in the method of finite spheres

“…The second way to treat discontinuous gradients is based on Lagrange multipliers 60 . The third way is the use of some penalty method, 61 which requires the specification of numerical parameters. Comparisons between jump functions and Lagrange multipliers, 60 and between Lagrange multipliers and penalty methods 61 have been made.…”

“…The third way is the use of some penalty method, 61 which requires the specification of numerical parameters. Comparisons between jump functions and Lagrange multipliers, 60 and between Lagrange multipliers and penalty methods 61 have been made. The use of Lagrange multipliers yields slightly better results, 60 using also less quadrature points in the numerical integration.…”

Effects of time integration schemes on discontinuous deformation simulations using the numerical manifold method

“…It has been recognized that the NMM has great potential to be further developed in simulating problems with massive discontinuities, for it adopts a dual cover system, ie, mathematical cover (MC) overlapping the domain of interest, physical cover (PC) formulating the physical problem, and the OCIs for the simulations of complicated dynamic problems. In the past 2 decades, various developments have been conducted to improve performance of the NMM, which can be referred in some studies . Enjoying the benefits of the FEM and DDA, the NMM is also suffering from the high computational costs arising from the inherent implicit time iteration scheme and the OCIs for contacts.…”

“…In the past 2 decades, various developments have been conducted to improve performance of the NMM, which can be referred in some studies. [38][39][40][41][42][43][44][45][46][47] Enjoying the benefits of the FEM and DDA, the NMM is also suffering from the high computational costs arising from the inherent implicit time iteration scheme and the OCIs for contacts. When nonlinear analyses are encountered in such discontinuous contact problems, the OCIs require no tension and no penetration at all contacts, which additionally highs up the computational cost to achieve a convergence state within each time step.…”