Convergence rates for linear elasticity systems on perforated domains

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Li Wang ◽  
Qiang Xu ◽  
Peihao Zhao
Keyword(s):  
Linear Elasticity ◽  
Convergence Rates ◽  
Perforated Domains

scholarly journals A Stabilized Hybrid Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity Problem

2018 ◽  
Vol 18 (3) ◽  
pp. 467 ◽  
Author(s):  
Antônio José Boness Santos ◽  
C O Faria ◽  
Abimael F D Loula
Keyword(s):  
Discontinuous Galerkin ◽  
Linear Elasticity ◽  
Convergence Rates ◽  
Rates Of Convergence ◽  
Processing Technique ◽  
Elasticity Problem ◽  
Hybrid Finite Element Method ◽  
Hybrid Finite Element ◽  
Numerical Studies ◽  
Optimal Convergence Rates

In this work, a primal hybrid finite element method for nearly incom pressible linear elasticity problem on triangular meshes is shown. This method consists of coupling local discontinuous Galerkin problems to the primal variable with a global problem for the Lagrange multiplier, which is identified as the trace of the displacement field. Also, a local post-processing technique is used to recover stress approximations with improved rates of convergence in H(div) norm. Numerical studies show that the method is locking free even using equal or different orders for displacement and stress fields and optimal convergence rates are obtained.

scholarly journals A Reduced Local Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Xuehai Huang
Keyword(s):  
Discontinuous Galerkin ◽  
Linear Elasticity ◽  
Convergence Rates ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Local Discontinuous Galerkin Method ◽  
Element Space ◽  
Local Discontinuous Galerkin ◽  
Ldg Method ◽  
Nearly Incompressible Linear Elasticity

A reduced local discontinuous Galerkin (RLDG) method for nearly incompressible linear elasticity is proposed in this paper, which is locking-free. RLDG method can be formally regarded as a special case of LDG method withC11=0. However, RLDG method is actually not covered by LDG method, whereC11must be chosen to be positive to ensure the stability of LDG method. RLDG method can also be considered as the localization of some symmetric nonconforming mixed finite element method. The implementation of RLDG method is discussed. By introducing a lifting operator as LDG method, RLDG method can be rewritten as primal formulation with unknown displacement only. Next, we obtain that the convergence rates of the approximation to stress tensor in energy norm and displacement inL2-norm areO(hk)andO(hk+1), respectively, which are both uniform with respect toλ. Moreover, we obtain aH(div)-conforming displacement by projecting the displacement and corresponding numerical trace of RLDG method into the Raviart-Thomas element space. And then we analyze the error estimates of this postprocessed displacement inH(div)-seminorm andL2-norm, which are also uniform with respect toλ. Finally, some numerical results are shown to demonstrate the theoretical results.

Isogeometric BDDC deluxe preconditioners for linear elasticity

2018 ◽  
Vol 28 (07) ◽  
pp. 1337-1370 ◽  
Author(s):  
L. F. Pavarino ◽  
S. Scacchi ◽  
O. B. Widlund ◽  
S. Zampini
Keyword(s):  
Linear Elasticity ◽  
Convergence Rates ◽  
Gradient Methods ◽  
Three Dimensions ◽  
Preconditioned Conjugate Gradient ◽  
Algebraic Equations ◽  
Good Convergence ◽  
Finite Element Approximations ◽  
Preconditioned Conjugate Gradient Methods ◽  
Primal Space

Balancing Domain Decomposition by Constraints (BDDC) preconditioners have been shown to provide rapidly convergent preconditioned conjugate gradient methods for solving many of the very ill-conditioned systems of algebraic equations which often arise in finite element approximations of a large variety of problems in continuum mechanics. These algorithms have also been developed successfully for problems arising in isogeometric analysis. In particular, the BDDC deluxe version has proven very successful for problems approximated by Non-Uniform Rational B-Splines (NURBS), even those of high order and regularity. One main purpose of this paper is to extend the theory, previously fully developed only for scalar elliptic problems in the plane, to problems of linear elasticity in three dimensions. Numerical experiments supporting the theory are also reported. Some of these experiments highlight the fact that the development of the theory can help to decrease substantially the dimension of the primal space of the BDDC algorithm, which provides the necessary global component of these preconditioners, while maintaining scalability and good convergence rates.

A NEW NONCONFORMING MIXED FINITE ELEMENT METHOD FOR LINEAR ELASTICITY

2006 ◽  
Vol 16 (07) ◽  
pp. 979-999 ◽  
Author(s):  
SON-YOUNG YI
Keyword(s):  
Finite Element ◽  
Linear Elasticity ◽  
Convergence Rates ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Displacement Boundary Condition ◽  
Displacement Boundary ◽  
Optimal Convergence Rates ◽  
Optimal Order Convergence

We have developed new nonconforming mixed finite element methods for linear elasticity with a pure traction (displacement) boundary condition based on the Hellinger–Reissner variational principle using rectangular elements. Convergence analysis yields an optimal (suboptimal) convergence rate of [Formula: see text] for the L2-error of the stress and [Formula: see text] for the displacement in the pure traction (displacement) boundary problem. However, numerical experiments have yielded optimal-order convergence rates for both stress and displacement in both problems and have shown superconvergence for the displacement at the midpoint of each element. Moreover, we observed that the optimal convergence rates are still valid for large λ.

Computing cell tractions from particle displacements on silicone rubber substrata

1994 ◽  
Vol 52 ◽  
pp. 162-163
Author(s):  
Tim Oliver ◽  
Akira Ishihara ◽  
Ken Jacobsen ◽  
Micah Dembo
Keyword(s):  
Plane Stress ◽  
Linear Elasticity ◽  
Silicone Rubber ◽  
Mathematical Analysis ◽  
Elastic Recovery ◽  
Video Images ◽  
Cell Traction Forces ◽  
Latex Beads ◽  
Cell Traction ◽  
Better Than

In order to better understand the distribution of cell traction forces generated by rapidly locomoting cells, we have applied a mathematical analysis to our modified silicone rubber traction assay, based on the plane stress Green’s function of linear elasticity. To achieve this, we made crosslinked silicone rubber films into which we incorporated many more latex beads than previously possible (Figs. 1 and 6), using a modified airbrush. These films could be deformed by fish keratocytes, were virtually drift-free, and showed better than a 90% elastic recovery to micromanipulation (data not shown). Video images of cells locomoting on these films were recorded. From a pair of images representing the undisturbed and stressed states of the film, we recorded the cell’s outline and the associated displacements of bead centroids using Image-1 (Fig. 1). Next, using our own software, a mesh of quadrilaterals was plotted (Fig. 2) to represent the cell outline and to superimpose on the outline a traction density distribution. The net displacement of each bead in the film was calculated from centroid data and displayed with the mesh outline (Fig. 3).

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