Imposing essential boundary conditions in mesh-free methods

By S. Fernández‐Méndez, Antonio Huerta. 2004

494 citations recorded by the source corpus.

Imposing essential boundary conditions is a key issue in mesh-free methods. The mesh-free interpolation does not verify the Kronecker delta property and, therefore, the imposition of prescribed values is not as straightforward as for the finite element method. The aim of this paper is to present a general overview on the existing techniques to enforce essential boundary conditions in Galerkin based mesh-free methods. Special attention is paid to the mesh-free coupling with finite elements for the imposition of prescribed values and to methods based on a modification of the Galerkin weak form. Particular examples are used to analyze and compare their performance in different situations.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/9832eb2acb5ee7d25c16a66366445c827ef49065

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In Computational Mechanics

In Electrical and Electronic Engineering

In Mechanics of Materials

Citations in the corpus

The extended/generalized finite element method: An overview of the method and its applications (2010, unknown): “Fernández-Méndez [74] and Griebel and Schweitzer [94].”
Small and large deformation analysis with the p- and B-spline versions of the Finite Cell Method (2012, methodology): “(17) and (18) by omitting the so-called consistency terms that involve the computation of the normal vector n [59,63].”
Meshfree Methods: Progress Made after 20 Years (2017, unknown): “It has been shown that for two-dimensional problems, if the penalty parameter is of order h−2 optimal convergence can be achieved (Fernández-Méndez and Huerta 2004).”
A Physics-Constrained Data-Driven Approach Based on Locally Convex Reconstruction for Noisy Database (2019, methodology): “Thus, certain techniques are needed to impose kinematically admissible approximation on the essential boundary, such as Lagrange multipliers method [77], Nitsche’s method [78,79], boudary singular kernel method [80], and transformation methods [58,80].”
On methods for stabilizing constraints over enriched interfaces in elasticity (2009, methodology): “The method has witnessed a resurgence in recent years, with applications ranging from boundary conditions in meshfree methods [15, 16] to constraints between overlapping finite element meshes [17].”
Aspects of the use of orthogonal basis functions in the element‐free Galerkin method (2009, methodology): “Once shape functions have been determined, the procedure to turn a governing pde into a system of discrete algebraic equations, via a weighted residual approach for instance, follows much the same path as with finite element (FE) methods, although there are serious issues with imposition of essentia”
Subdivision-stabilised immersed b-spline finite elements for moving boundary flows (2012, methodology): “In recent years, the Nitsche method has been introduced to an increasing number of application areas, including interface problems [2], boundary enforcement in immersed and element-free methods [35, 36, 37], and fluid-structure interaction [12].”
Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement (2005, methodology): “For a state-of-the-art summary of meshless methods, see [23–40].”
An Isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces (2012, methodology): “Since they are not interpolatory at the ends, the hierarchical basis requires weak enforcement of Dirichlet constraints, for which a simple penalty method [76, 77] with penalty parameter β = 10(6) is applied here.”
A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method (2009, methodology): “Note that the choice of this parameter was discussed in the mesh-free framework in (Fernández-Méndez and Huerta 2004).”
Imposing Dirichlet boundary conditions with Nitsche's method and spline‐based finite elements (2010, methodology): “Nitsche’s method has recently emerged as an attractive alternative for weakly imposing Dirichlet boundary conditions in meshfree [7] and embedded finite-element methods [8, 9].”
A discontinuous‐Galerkin‐based immersed boundary method (2008, methodology): “The quite sophisticated construction proposed by Möes et al [48] of a Λh that avoids locking is another example showing that the Lagrange-multiplier approach, which is also the method of choice in many extended finite element (XFEM) and meshless methods[22], is not that straightforward.”
A bubble‐stabilized finite element method for Dirichlet constraints on embedded interfaces (2007, methodology): “Fernandez-Mendez and Huerta [8] proposed solving a local eigenvalue problem to identify the stability constant, but this is not an efficient strategy if required at every time step.”
The Finite Cell Method: A Review in the Context of Higher-Order Structural Analysis of CAD and Image-Based Geometric Models (2015, methodology): “Possible technologies are the penalty method [9,55,199], the Lagrange multiplier method [29,55,56,58,59,61,201] or Nitsche’s method [17,30,43,51,53,62,64,66].”