Exact imposition of inhomogeneous Dirichlet boundary conditions based on weighted finite cell method and level-set function

Zhang, Weihong; Zhao, Linying

doi:10.1016/j.cma.2016.04.036

Cited by 10 publications

(5 citation statements)

References 29 publications

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“…where b i ðxÞ denotes the ith weighting coefficient associated with A −1 g i in which basic properties of b i ðxÞ are as follows [23]. b i = δ ij , i, j = 1, 2 ⋯ N, with δ ij denoting the Kronecker delta function.…”

Section: Ab Boundary Conditions With Resolution Adaptedmentioning

confidence: 99%

WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition

Koubaiti¹,

Elkhalfi²,

El-Mekkaoui

2020

Abstract and Applied Analysis

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The objective of this article is to discuss the existence and the uniqueness of a weighted extended B-spline- (WEB-spline-) based discrete solution for the 2D Navier-Lamé equation of linear elasticity with a different type of mixed boundary condition called CA,B boundary condition. Along with the usual weak mixed formulation, we give existence and uniqueness results for weak solution. Then, we illustrate the performance of Ritz–Galerkin schemes for a model problem and applications in linear elasticity. Finally, we discuss several implementation aspects. The numerical tests confirm that, due to the new integration routines, the weighted B-spline solvers have become considerably more efficient.

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Section: Ab Boundary Conditions With Resolution Adaptedmentioning

confidence: 99%

WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition

Koubaiti¹,

Elkhalfi²,

El-Mekkaoui

2020

Abstract and Applied Analysis

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“…Owing to this feature, a smoothly constructed u a that satisfies the inhomogeneous Dirichlet boundary conditions without being approximated on the grid leads to a more accurate solution. For example, the transfinite interpolation technique [16] has been used to construct the required boundary value function in the finite cell method.…”

Section: Implicit Boundary Methodsmentioning

confidence: 99%

“…They showed results with several different choices of interpolation spaces in the Lagrange multiplier method and several different penalty values in penalty and Nitsche's methods. Zhang et al [16] summarised energy functionals, interpolation forms and linear systems for different kinds of implicit boundary methods in a brief survey, besides the three methods mentioned above. Ramos et al [6] surveyed different variants of the original Lagrange multiplier method which aim to overcome the stabilisation problem to be discussed later.…”

Section: Weak Imposition Of Dirichlet Boundary Conditions and Problemmentioning

confidence: 99%

Weak impositions of Dirichlet boundary conditions in solid mechanics: A critique of current approaches and extension to partially prescribed boundaries

Augarde

Coombs

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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“…The main contribution of this work is equally applicable to standard FEs and the MPM. The IB approach is conceptually similar to the earlier work of Höllig and coworkers in that they both directly modify interpolation functions to impose the essential boundary conditions . The weighted extended B‐spline method of Höllig uses an R‐function to weight the interpolation functions to impose homogeneous Dirichlet boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

“…However, a clear distinction between the work of Höllig and coworkers and the IB method is that the latter allows inhomogeneous Dirichlet boundary conditions to be imposed independent of the discretisation. The very recent work of Zhang and Zhao provides an extension to the IB method through generalising the boundary value function. They combine these advances with the finite cell method, and the level set method to represent the boundaries, to provide a general method for the imposing of inhomogeneous Dirichlet boundary conditions in two dimensions for thermo‐elasticity.…”

Section: Introductionmentioning

confidence: 99%

Imposition of essential boundary conditions in the material point method

Cortis

Coombs

Augarde

et al. 2017

Numerical Meth Engineering

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SummaryThere is increasing interest in the material point method (MPM) as a means of modelling solid mechanics problems in which very large deformations occur, e.g. in the study of landslides and metal forming; however, some aspects vital to wider use of the method have to date been ignored, in particular methods for imposing essential boundary conditions in the case where the problem domain boundary does not coincide with the background grid element edges. In this paper, we develop a simple procedure originally devised for standard finite elements for the imposition of essential boundary conditions, for the MPM, expanding its capabilities to model boundaries of any inclination. To the authors' knowledge, this is the first time that a method has been proposed that allows arbitrary Dirichlet boundary conditions (zero and nonzero values at any inclination) to be imposed in the MPM. The method presented in this paper is different from other MPM boundary approximation approaches, in that (1) the boundaries are independent of the background mesh, (2) artificially stiff regions of material points are avoided, and (3) the method does not rely on mirroring of the problem domain to impose symmetry. The main contribution of this work is equally applicable to standard finite elements and the MPM.

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Exact imposition of inhomogeneous Dirichlet boundary conditions based on weighted finite cell method and level-set function

Cited by 10 publications

References 29 publications

WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition

WEB-Spline Finite Elements for the Approximation of Navier-Lamé System with CA,B Boundary Condition

Weak impositions of Dirichlet boundary conditions in solid mechanics: A critique of current approaches and extension to partially prescribed boundaries

Imposition of essential boundary conditions in the material point method

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