BUCKLING ANALYSIS UNDER COMBINED LOADING OF THIN-WALLED PLATE ASSEMBLIES USING BUBBLE FUNCTIONS

Bubble functions are finite element modes that are zero on the boundary of the element but nonzero at the other point. The present paper adds bubble functions to the ordinary Complex Finite Strip Method(CFSM) to calculate the elastic local buckling stress of plates and plate assemblies. The results indicate that the use of bubble functions greatly improves the convergence of the Finite Strip Method(FSM) in terms of strip subdivision, and leads to much smaller storage required for the structure stiffness and stability matrices. Numerical examples are given, including plates and plate structures subjected to a combination of longitudinal and transverse compression, bending and shear. This study illustrates the power of bubble functions in solving stability problems of plates and plate structures.

Mixed Finite Element Formats of any Order Based on Bubble Functions for Stationary Stokes Problem

Mixed element formats of any order based on bubble functions for the stationary Stokes problem are derived in triangular and tetrahedral meshes and the convergence of these formats are proved.

Multiscale Finite Element Modelling of Flow Through Porous Media with Curved and Contracting Boundaries to Evaluate Different Types of Bubble Functions

The Brinkman equation is used to model the isothermal flow of the Newtonian fluids through highly permeable porous media.Due to the multiscale behaviour of this flow regime the standard Galerkin finite element schemes for the Brinkman equation require excessive mesh refinement at least in the vicinity of domain walls to yield stable and accurate results.To avoid this,a multiscale finite element method is developed using bubble functions.It is shown that by using bubble enriched shape functions the standard Galerkin method can generate stable solutions without excessive near wall mesh refinements.In this paper the performances of different types of bubble functions are evaluated.These functions are used in conjunction with bilinear Lagrangian elements to solve the Brinkman equation via a penalty finite element scheme.

Bubble-Enriched Smoothed Finite Element Methods for Nearly-Incompressible Solids

This work presents a locking-free smoothed finite element method(S-FEM)for the simulation of soft matter modelled by the equations of quasi-incompressible hyperelasticity.The proposed method overcomes well-known issues of standard finite element methods(FEM)in the incompressible limit:the over-estimation of stiffness and sensitivity to severely distorted meshes.The concepts of cell-based,edge-based and node-based S-FEMs are extended in this paper to three-dimensions.Additionally,a cubic bubble function is utilized to improve accuracy and stability.For the bubble function,an additional displacement degree of freedom is added at the centroid of the element.Several numerical studies are performed demonstrating the stability and validity of the proposed approach.The obtained results are compared with standard FEM and with analytical solutions to show the effectiveness of the method.

Bubble-enhanced quadrilateral finite element formulation for nonlinear analysis of geotechnical problems

We investigate a new numerical procedure based on a bubble-enriched finite element formulation in combination with the implicit backward Euler scheme for nonlinear analysis of strip footings and stability of slopes.The soil body is modeled as a perfect plastic Mohr-Coulomb material.The displacement field is approximated by a 4-node quadrilateral element discretization enhanced with bubble modes.Collapse loads and failure mechanisms in cohesive frictional soil are determined by solving a few Newton-Raphson iterations.Numerical results of the present approach are verified by both analytical solutions and other numerical solutions available in the literature.

FINITE ELEMENT SOLUTION OF MULTI-SCALE TRANSPORT PROBLEMS USING THE LEAST SQUARES-BASED BUBBLE FUNCTION ENRICHMENT

This paper presents a technique for deriving least-squares-based polynomial bubble functions to enrich the standard linear finite elements,employed in the formulation of Galerkin weighted-residual statements.The element-level linear shape functions are enhanced using supplementary polynomial bubble functions with undetermined coefficients.The enhanced shape functions are inserted into the model equation and the residual functional is constructed and minimized by using the method of the least squares,resulting in an algebraic system of equations which can be solved to determine the unknown polynomial coefficients in terms of element-level nodal values.The stiffness matrices are subsequently formed with the standard finite elements assembly procedures followed by using these enriched elements which require no additional nodes to be introduced and no extra degree of freedom incurred.Furthermore,the proposed technique is tested on a number of benchmark linear transport equations where the quadratic and cubic bubble functions are derived and the numerical results are compared against the exact and standard linear element solutions.It is demonstrated that low order bubble enriched elements provide more accurate approximations for the exact analytical solutions than the standard linear elements at no extra computational cost in spite of using relatively crude meshes.On the other hand,it is observed that a satisfactory solution of the strongly convection-dominated transport problems may require element enrichment by using significantly higher order polynomial bubble functions in addition to the use of extremely fine computational meshes.

On the Linear Scheme for the Reissner-Mindlin Plate Problem

In this paper we give the optimal selection of the bubble function in the linear scheme proposed by recent paper [1]for the Reissner-Mindlin plate problem，

MIXED FINITE ELEMENT METHODS BASED ON RIESZ-REPRESENTING OPERATORS FOR THE SHELL PROBLEM

To solve the shell problem, we propose a mixed finite element method with bubble-stabili -zation term and discrete Riesz-representation operators. It is shown that this new method is coercive, implytng the well-known X-ellipticity and the Inf-Sup condition being circumvented, and the resulting linear system is symmetrically positively definite, with a condition number being at most O(h-2). Further, an optimal error bound is attained.