Terenzio Scapolla

Hierarchic finite elements for thin Naghdi shell model

Abstract Two approaches have traditionally been used when general shell structures have been analysed. The first approach has been devised by Kirchhoff and Love and later the model has been improved by Koiter. A second class of models is based on the notion of surface introduced by Cosserat. Naghdi has developed this model, where the Reissner-Mindlin-type assumptions are taken into account. In this paper we consider the shell model arising from the Naghdi formulation. It is known that finite element schemes for this model suffer from both shear and membrane locking. Several solutions to avoid the numerical locking have been proposed. Here a displacement finite element scheme is developed using C0 finite elements of hierarchic type with degrees ranging from one to four. Two severe test problems are solved. The results show that good performances are achieved by using high-order finite elements to solve the shell problem in its displacement formulation. The numerical results indicate that high-order elements perform very well in both test problems and match all the available benchmark results.

A wavelet-based method for numerical solution of nonlinear evolution equations

Abstract We describe an adaptive algorithm for solving one-dimensional system of nonlinear partial differential equations. Different strategies are considered for the discretization in time while a multiscale collocation method is applied for the discretization in space. In particular we look at the so called Rothe method which is based on first time then space discretization. Numerical experiments are presented for a set of nonlinear problems from the literature.

Hierarchic finite elements for moderately thick to very thin plates

We consider the numerical solution of Reissner-Mindlin plates. The model is widely used for thin to moderately thick plates. Generally low order finite elements (with degree one or two) are used for the approximation. Unfortunately the solution degenerates very rapidly for small thickness (locking phenomenon). Non standard formulations of the problem are usually combined with low order finite elements to weaken or possibly eliminate the locking of the numerical solution. We introduce a family of hierarchic finite elements and we present a set of numerical results for the plate problem in its plain formulation. We show that reliable solutions are achieved for all thicknesses of practical interest by using high order finite elements. Moreover, the hierarchic structure allows a low cost assessment of the quality of the solution.

A wavelet-like Galerkin method for numerical solution of variational inequalities arising in elastoplasticity

The well-known problem of elasticity that may be written as a variational equation [4] has been recently extended to non-linear elastoplastic behaviours [13] giving rise to a class of variational inequalities of second kind [9]. This paper presents a wavelet Galerkin method for the numerical solution of such a class of problems, extensively studied and solved in the past by means of more traditional approaches. The novelty of the scheme presented herein is represented by the capability of wavelets to locate regions where plasticity is growing without the need of any further indicator. This is useful for adaptive remeshing as well as for gaining insight into internal variable models with applications to dynamic analysis and damage identification. Copyright

Hierarchic finite elements with selective and uniform reduced integration for Reissner-Mindlin plates

It is well known that the numerical solution of Reissner-Mindlin plates degenerates very rapidly for small thickness (locking phenomenon) when standard finite elements are used for the approximation. We have introduced a family of hierarchic high order finite elements in order to assess reliability and robustness with respect to the locking behavior. In a previous note we have given numerical results obtained with exact numerical integration. In this paper we present the results obtained with selective and uniform reduced integration. The results show that, compared with exact integration, selective reduced techniques improve the quality of the numerical performance and are preferable since computational cost is made smaller.

A hierarchic family of C 1 finite elements for 4 th order elleptic problems

This paper describes a new family of high order finite elements for the approximation of plate bending problems. The elements are conforming and the C1-continuity condition is satisfied by using standard polynomials; the family is hierarchic, that is when the degree of the polynomials is increased new shape functions are added to the previous ones. Explicit expressions of the shape functions are given on general triangles. The hierarchic extension of the approximation spaces, also for a further increase of the degree of the elements, is described in details.

On the robustness of hierarchic finite elements for Reissner-Mindlin plates

The discretization of the Reissner-Mindlin model for plates requires finite elements of class only C0. This is perhaps the main advantage respect to the Kirchhoff formulation, where C1 finite elements need to be used for a conforming approximation. However, despite its simple approach, the numerical approximation of the Reissner-Mindlin plate is not straightforward. The inclusion of transverse shear strain effect in standard finite element models introduces undesirable numerical effects. Standard low order finite elements are not able to meet the Kirchhoff constraint enforced when the thickness becomes smaller and therefore are subject to the locking phenomenon. The approximate solution is very sensitive to the plate thickness and unsatisfactory results are obtained for small thickness. We have solved the Reissner-Mindlin plate problem in its plain formulation with the use of high order finite elements. We have developed a hierarchic family of finite elements and have performed several numerical tests to analyze the behavior with respect to the thickness. In this paper we are particularly interested in the investigation of the robustness properties of the family of finite elements. In this direction we analyse the performance of the elements when very small values of the thickness of the plate are considered. The numerical results indicate a large range of robustness for the higher order elements.

Solving cylindrical shell problems with a non-standard finite element

The subject of this work is the numerical analysis of a special finite element for the numerical solution of Naghdi cylindrical shell problems. Its discretization with standard finite elements suffers from the locking phenomenon, i.e., when small thickness shells are solved, the numerical results are affected by a large error. Several solutions to avoid the numerical locking have been proposed: mixed formulations, reduced and/or selective integration. In this paper we present a finite element based on a non-standard formulation of the discrete approximation of the shell problem. The main feature of the element proposed is the introduction of a linear operator that reduces the influence on the numerical solution of both shear and membrane energy terms. The performance of the new element is tested solving a benchmark problem involving very thin shells and severely distorted decompositions. The results show both properties of convergence and robustness.

Combining hierarchic high order and mixed-interpolated finite elements for Reissner-Mindlin plate problems

Abstract The Reissner—Mindlin plate bending model describes the deformation of a plate subject to a transverse loading when transverse shear deformation is taken into account. Despite its simple approach, the discretization of the Reissner—Mindlin model is not straightforward. The inclusion of the transverse shear strain effect in standard finite element models introduces undesirable numerical effects. The approximate solution is very sensitive to the plate thickness and, for small thickness, it is very far from the true solution. The phenomenon is known as locking of the numerical solution. The most common way to avoiding the locking problem is to use non-standard finite elements and/or modify the variational formulation. Recently, numerical experiences with high order finite elements applied to the plain Reissner—Mindlin formulation have shown a consistent improvement in the quality of the results. Meanwhile some mixed-interpolated finite elements have been suggested and shown to be locking free. In this paper we propose the combination of the two classes of elements introducing a family of hierarchic high order mixed-interpolated finite elements.

A third-order mixed finite-element method for the numerical solution of the biharmonic problem

Abstract This paper is devoted to the introduction of a mixed finite element for the solution of the biharmonic problem. We prove optimal rate of convergence for the element. The mixed approach allows the simultaneous approximation of both displacement and tensor of its second derivatives.