A general degree hybrid equilibrium finite element for Kirchhoff plates

Summary In this paper, we propose a novel approach to combine two complementary numerical models of the dynamical behaviour of mechanical systems: primal (compatible) and dual (equilibrated). This combined model provides an improved estimate of the eigenfrequencies of the system by feeding each model with information from the other. Numerical solutions obtained from both the fundamental models and from the proposed approach are presented and studied. This approach will be applicable to any eigenvalue problem associated with a PDE that can be expressed by complementary numerical models.

Section: Notation Compatibility Equilibrium and Constitutive Relatmentioning

confidence: 99%

Improved eigenfrequencies of mechanical systems by combining complementary models

Almeida

Maunder

2018

“…Different numerical methods may be used for the computation of solutions that are equilibrated in x . In Section 7, a direct finite element solution of the complementary problem, equivalent to those proposed, for example, in [15,[29][30][31][32], is used to find the optimal solution, in the energy sense, amongst all equilibrated solutions considered for problems in 2D and 3D elasticity and plates. This approach requires the complete solution of a new problem and the application of non-standard finite element techniques, which are not always well understood.…”

Section: Conclusion and Future Developmentsmentioning

confidence: 99%

A basis for bounding the errors of proper generalised decomposition solutions in solid mechanics

Almeida

2013

This paper presents an approach that extends the classical error bounding techniques to parametric models. It departs from appropriate pairs of complementary solutions of a linear elastic problem, obtained using a Proper Generalised Decomposition methodology, to determine approximations of selected local outputs and strict bounds of the error of these approximations. The paper starts by presenting the procedures used to obtain the complementary solutions. The properties, the convergence characteristics of the global error and the determination of an indicator of the error distribution are illustrated for a very simple example. The demonstration of the procedure used for determining local outputs and their bounds, also accompanied by illustrative examples, completes the paper. This integral, when computed using a numerical quadrature formula, is in general approximate, because the solutions in space are not constant in j . Nevertheless, preliminary tests indicate that influence of the number of Gauss points on the quality of the PGD approximations is not significant.There is minimum number of points that must be used, those that are necessary for the integration of the explicit polynomial of degree 2 p j , that is, p j C 1 Gauss points. Using a higher quadrature rule increases the quality of the solution, but the change is not significant. Using fewer integration points results in singular Galerkin forms of the equations used to determine the parameters in E 1 and in E 2 . , to whom I extend my gratitude. Preparing these presentations was crucial to have the concepts fresh and organised, ready to respond to a new challenge.

“…13.911144 kNm) as a Kirchhoff model based on quadratic triangular equilibrium elements having the mesh shown in Figure (b) defined by the primitive elements in the macro‐element model. Such Kirchhoff models have been recently developed by Moitinho de Almeida and Maunder .…”

Section: Numerical Examplesmentioning

confidence: 99%

A hybrid equilibrium element for folded plate and shell structures

Maunder

Izzuddin

2013

SUMMARYThis paper extends hybrid equilibrium formulation concepts, previously used with success for planar problems, to the analysis of folded plates and curved shells. A 2D hybrid equilibrium flat shell quadrilateral element is formulated for linear analysis, where detailed consideration is given to the implication of slope discontinuity when the element is used for non‐planar domains. Benchmark plate bending, folded plate and curved shell problems are modelled using equilibrium and conforming elements for comparison. In models of the latter two problems, torsional moments may be released along lines of slope discontinuity, and the effects of this assumption for the folded plate are studied by analysing a third type of model composed of 3D solid brick elements. The comparisons demonstrate an excellent performance from the new hybrid equilibrium analysis method for folded plates and curved shells. Copyright © 2013 John Wiley & Sons, Ltd.