Stabilized low-order finite elements for frictional contact with the extended finite element method

Liu, Fushen; Borja, Ronaldo I.

doi:10.1016/j.cma.2010.03.030

Cited by 98 publications

(80 citation statements)

References 46 publications

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“…More recently, Liu and Borja (2010b) proposed a stabilized formulation for loworder elements, with the option of either Lagrange multipliers or penalty regularization to enforce the interfacial constraints. Unfortunately, the method still retains a free stabilization parameter as well as a free penalty parameter for the regularized case.…”

Section: Introductionmentioning

confidence: 99%

An efficient finite element method for embedded interface problems

Dolbow

Harari

2008

Numerical Meth Engineering

217

223

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We focus on developing a computationally efficient finite element method for interface problems. Finite element methods are severely constrained in their ability to resolve interfaces. Many of these limitations stem from their inability in independently representing interface geometry from the underlying discretization. We propose an approach that facilitates such an independent representation by embedding interfaces in the underlying finite element mesh. This embedding, however, raises stability concerns for existing algorithms used to enforce interfacial kinematic constraints. To address these stability concerns, we develop robust methods to enforce interfacial kinematics over embedded interafces.We begin by examining embedded Dirichlet problems -a simpler class of embedded constraints. We develop both stable methods, based on Lagrange multipliers, and stabilized methods, based on Nitsche's approach, for enforcing Dirichlet constraints over three-dimensional embedded surfaces and compare and contrast their performance. We then extend these methods to enforce perfectly-tied kinematics for elastodynamics with explicit time integration. In particular, we examine the coupled aspects of spatial and temporal stability for Nitsche's approach. We address the incompatibility of Nitsche's method for explicit time integration by (a) proposing a modified weighted stress variational form, and (b) proposing a novel mass-lumping procedure.We revisit Nitsche's method and inspect the effect of this modified variational iv form on the interfacial quantities of interest. We establish that the performance of this method, with respect to recovery of interfacial quantities, is governed significantly by the choice for the various method parameters viz. stabilization and weighting. We establish a relationship between these parameters and propose an optimal choice for the weighting. We further extend this approach to handle non-linear, frictional sliding constraints at the interface. The naturally non-symmetric nature of these problems motivates us to omit the symmetry term arising in Nitsche's method.We contrast the performance of the proposed approach with the more commonly used penalty method. Through several numerical examples, we show that with the proposed choice of weighting and stabilization parameters, Nitsche's method achieves the right balance between accurate constraint enforcement and flux recovery -a balance hard to achieve with existing methods. Finally, we extend the proposed approach to intersecting interfaces and conduct numerical studies on problems with junctions and complex topologies.v To Amma (mom) and Nannagaru (dad), vi

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Section: Introductionmentioning

confidence: 99%

An efficient finite element method for embedded interface problems

Dolbow

Harari

2008

Numerical Meth Engineering

217

223

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“…Other techniques use different stabilized formulations. For example [34] uses a polynomial stabilization valid for linear elements or [11] penalizes the jump in the multiplier linear elements. In other works the idea of extending the solution of internal elements to the intersected elements was explored [14,25].…”

Section: Introductionmentioning

confidence: 99%

A modified perturbed Lagrangian formulation for contact problems

et al. 2015

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The aim of this work is to propose a formulation to solve both small and large deformation contact problems using the finite element method. We consider both standard finite elements and the so-called immersed boundary elements. The method is derived from a stabilized Nitsche formulation. After introduction of a suitable Lagrange multiplier discretization the method can be simplified to obtain a modified perturbed Lagrangian formulation. The stabilizing term is iteratively computed using a smooth stress field. The method is simple to implement and the numerical results show that it is robust. The optimal convergence rate of the finite element solution can be achieved for linear elements.

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“…Another alternative used in [23,26] would be to define the multiplier interpolation in the trace space of the displacements u h | Γ on the boundary.…”

Section: Interpolation Of the Lagrange Multipliersmentioning

confidence: 99%

“…Another alternative, widely used in fluid dynamics [22,23], is to use a procedure to stabilize the solution of the problem. Some methods to implement Dirichlet boundary conditions with stabilized solution in Cartesian meshes, can also be found in [24,25,26,28,27]. In this paper we propose a stabilization method suitable for h-refinement based on the use of hierarchical Cartesian grids where stabilization term does not depend on the solution of the current mesh.…”

Section: Introductionmentioning

confidence: 99%

Imposing Dirichlet boundary conditions in hierarchical Cartesian meshes by means of stabilized Lagrange multipliers

Tur

Albelda

Nadal

et al. 2014

Numerical Meth Engineering

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This is the pre-peer reviewed version of the following article: Tur, M., Albelda, J., Nadal, E. and Ródenas, J. J. (2014) AbstractThe use of Cartesian meshes independent of the geometry has some advantages over the traditional meshes used in the finite element method. The main advantage is that their use together with an appropriate hierarchical data structure reduces the computational cost of the Finite Element analysis. This improvement is based on the substitution of the traditional mesh generation process by an optimized procedure for intersecting the Cartesian mesh with the boundary of the domain and the use efficient solvers based on the hierarchical data structure. One major difficulty associated to the use of Cartesian grids is the fact that the mesh nodes do not, in general, lie over the boundary of the domain, increasing the difficulty to impose Dirichlet boundary conditions. In this paper, Dirichlet boundary conditions are imposed by means of the Lagrange multipliers technique. A new functional has been added to the initial formulation of the problem that has the effect of stabilizing the problem. The technique here presented allows for a simple definition of the Lagrange multipliers field, that even allow us to directly condense the degrees of freedom of the Lagrange multipliers at element level.

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Stabilized low-order finite elements for frictional contact with the extended finite element method

Cited by 98 publications

References 46 publications

An efficient finite element method for embedded interface problems

An efficient finite element method for embedded interface problems

A modified perturbed Lagrangian formulation for contact problems

Imposing Dirichlet boundary conditions in hierarchical Cartesian meshes by means of stabilized Lagrange multipliers

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