Equilibrium Finite Element Formulations

Summary We revisit compatible finite element formulations for Kirchhoff plates and propose a new general degree hybridized approach that strictly imposes C1 continuity. These new elements are triangular and based on nodal polynomial approximation functions that only use displacement and rotation degrees of freedom for assembly, and thereby “nearly” impose C1 continuity. This condition is then strictly enforced by adding appropriately chosen hybrid constraints and the corresponding Lagrange multipliers. Unlike all other existing approaches, this formulation allows for the definition of elements of arbitrary degree considering a single polynomial basis for each element, without using degrees of freedom associated with second‐order derivatives. The convergence is compared with that of alternative approaches in terms of numbers of elements and degrees of freedom.

Section: Numerical Examples Efficiency and Convergence Studiesmentioning

confidence: 63%

Section: Discussionmentioning

confidence: 99%

Section: A Semihybrid Formulation For Kirchhoff Platesmentioning

confidence: 99%

Section: A Semihybrid Formulation For Kirchhoff Platesmentioning

confidence: 99%

Section: A Semihybrid Formulation For Kirchhoff Platesmentioning

confidence: 99%

See 3 more Smart Citations

A general degree semihybrid triangular compatible finite element formulation for Kirchhoff plates

Almeida

Maunder

Tiago

2019

Towards simplified and optimized a posteriori error estimation using PGD reduced models

Allier

Chamoin

2017

Summary The paper deals with the use of model order reduction within a posteriori error estimation procedures in the context of the finite element method. More specifically, it focuses on the constitutive relation error concept, which has been widely used over the last 40 years for FEM verification of computational mechanics models. A technical key‐point when using constitutive relation error is the construction of admissible fields, and we propose here to use the proper generalized decomposition to facilitate this task. In addition to making the implementation into commercial FE software easier, it is shown that the use of proper generalized decomposition enables to optimize the verification procedure and to get both accurate and reasonably expensive upper bounds on the discretization error. Numerical illustrations are presented to assess the performance of the proposed approach.

Recovery of strong equilibrium from displacement‐based finite element models of Reissner‐Mindlin plates

Maunder

Almeida

2018

Summary In this paper, we extend to Reissner‐Mindlin plate bending problems a technique, originally proposed in the context of two‐dimensional and three‐dimensional continua, for recovering fully equilibrated stresses from the solution of a compatible finite element model. The technique involves partition of unity functions and the analyses of overlapping star patches modelled with hybrid equilibrium plate elements. The patches are subjected to balanced systems of loads composed of partitioned and fictitious loads, where the latter are derived from the stresses of the compatible solution. The special case of assumed linear displacement fields of both deflection and rotation for the compatible model is included. This case requires additional fields of stress resultants to correct possible rotational imbalances of star patches, and these are derived elementwise. Other cases of nonconforming elements are briefly considered. Numerical examples are presented to illustrate the effectiveness of these techniques in terms of the deviation of the recovery, which compares the complementary strain energy of a recovered solution with that obtained by a global equilibrated analysis based on the same stress approximations.