A residual-based a posteriori error estimator for the plane linear elasticity problem with pure traction boundary conditions

Domínguez, Carolina; Gatica, Gabriel N.; Márquez, Antonio

doi:10.1016/j.cam.2015.07.020

Cited by 4 publications

(2 citation statements)

References 33 publications

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“…However, recent applications of medical image quantification have highlighted the importance of guaranteeing certain accuracy and convergence when estimating stress tensor fields [38] and rotation tensor fields [37], due to their important connection to medical conditions. Further developments to improve the accuracy of the numerical solution, that are naturally developed within the finite-element framework adopted in this work, are the introduction of a-posteriori mesh refinement methods, where recent results in linear elasticity for mixed formulations with Neumann boundary conditions [16] can be extended to the case of DIR problems addressed here. Table 3: Errors and convergence rates for the augmented scheme.…”

Section: Discussionmentioning

confidence: 99%

Primal and Mixed Finite Element Methods for Deformable Image Registration Problems

A.¹,

Gatica²,

Hurtado³

2018

SIAM J. Imaging Sci.

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Deformable image registration (DIR) represent a powerful computational method for image analysis, with promising applications in the diagnosis of human disease. Despite being widely used in the medical imaging community, the mathematical and numerical analysis of DIR methods remain understudied. Further, recent applications of DIR include the quantification of mechanical quantities apart from the aligning transformation, which justifies the development of novel DIR formulations where the accuracy and convergence of fields other than the aligning transformation can be studied. In this work we propose and analyze a primal, mixed and augmented formulations for the DIR problem, together with their finite-element discretization schemes for their numerical solution. The DIR variational problem is equivalent to the linear elasticity problem with a nonlinear source term that depends on the unknown field. Fixed point arguments and small data assumptions are employed to derive the well-posedness of both the continuous and discrete schemes for the usual primal and mixed variational formulations, as well as for an augmented version of the later. In particular, continuous piecewise linear elements for the displacement in the case of the primal method, and Brezzi-Douglas-Marini of order 1 (resp. Raviart-Thomas of order 0) for the stress together with piecewise constants (resp. continuous piecewise linear) for the displacement when using the mixed approach (resp. its augmented version), constitute feasible choices that guarantee the stability of the associated Galerkin systems. A-priori error estimates derived by using Strang-type Lemmas, and their associated rates of convergence depending on the corresponding approximation properties are also provided. Numerical convergence tests and DIR examples are included to demonstrate the applicability of the method.

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Section: Discussionmentioning

confidence: 99%

Primal and Mixed Finite Element Methods for Deformable Image Registration Problems

A.¹,

Gatica²,

Hurtado³

2018

SIAM J. Imaging Sci.

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“…The mixed finite element method is an important approach for the linear elasticity problem with the pure traction boundary condition (see [6][7][8][9][10][11][12][13] and references therein). However, it is difficult to construct a stable mixed finite element method for the linear elasticity problem.…”

Section: Introductionmentioning

confidence: 99%

A stabilized nonconforming Crouzeix–Raviart finite element method for the elasticity eigenvalue problem with the pure traction boundary condition

Zhang

Han

Yang

2023

Math Methods in App Sciences

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The linear elasticity problem has important applications in structural analysis and engineering design. It is shown that the classical Crouzeix–Raviart finite element method is unstable to solve the linear elasticity eigenvalue problem with the pure traction boundary condition. In this paper, first we propose a new scheme with the stabilization terms. Then we prove the a priori error estimates for approximate eigenpairs. Finally, we carry out numerical experiments using the stabilized scheme. Theoretical analysis and numerical results show that this scheme is locking free and efficient.

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A posteriori error analysis of mixed finite element methods for stress-assisted diffusion problems

Gatica¹,

Gómez-Vargas

Ruiz–Baier

2022

Journal of Computational and Applied Mathematics

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A residual-based a posteriori error estimator for the plane linear elasticity problem with pure traction boundary conditions

Cited by 4 publications

References 33 publications

Primal and Mixed Finite Element Methods for Deformable Image Registration Problems

Primal and Mixed Finite Element Methods for Deformable Image Registration Problems

A stabilized nonconforming Crouzeix–Raviart finite element method for the elasticity eigenvalue problem with the pure traction boundary condition

A posteriori error analysis of mixed finite element methods for stress-assisted diffusion problems

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