The comparative efficiency of three flat triangular shell elements is being assessed for analysing non‐linear behaviour of general shell structures. The bending formulation of the…
Abstract
The comparative efficiency of three flat triangular shell elements is being assessed for analysing non‐linear behaviour of general shell structures. The bending formulation of the three elements is based on a discrete Kirchhoff model (namely the well‐known 3‐node DKT element and a new 6‐node DKTP element). The in‐plane behaviour is defined by constant (CST), linear (LST)and quadratic (QST) strain approximations. The super‐position of bending and membrane elements leads to the 3‐node DCT element (DKT plus CST), the 3‐node DQT element (DKT plus QST) and the 6‐node DLT element (DKTP plus LST). The geometrically non‐linear formulation is based on an approximate updated Lagrangian formulation (AULF) and the solution is obtained by using the Newton‐Raphson method with an automatic arc‐length control method. Illustrative examples include pre‐ and post‐buckling of different shell structures showing, in particular, some bifurcation points, large rotations and displacements and very important membrane‐bending coupling.
H. Stolarski, T. Belytschko, N. Carpenter and J.M. Kennedy
A simple triangular shell element which incorporates the effects of coupling between membrane and flexural behaviour and avoids membrane locking is described. The element uses a…
Abstract
A simple triangular shell element which incorporates the effects of coupling between membrane and flexural behaviour and avoids membrane locking is described. The element uses a discrete Kirchhoff bending formulation and a constant strain membrane element. For the purpose of permitting inextensional modes and thus avoiding membrane locking, a decomposition technique, which can also be viewed as a strain projection method, is used. The method is illustrated first for a beam element and then for a triangular shell element. Results are presented for a variety of linear static problems to illustrate its accuracy and some highly non‐linear problems to indicate its applicability to collapse analysis.
Jean‐Vincent Daurelle, René Occelli and Marc Jaeger
The radiation conduction coupling leads to particular problems due to computation time and high heat fluxes. Because of the hemispheric nature of the radiation, it is difficult to…
Abstract
The radiation conduction coupling leads to particular problems due to computation time and high heat fluxes. Because of the hemispheric nature of the radiation, it is difficult to take into account symmetric or periodic conditions for the reduction of the modelled domain. We developed a finite element model of radiative heat transfers between grey diffuse surfaces with a nonparticipating medium with periodic or symmetric boundary conditions. The approaches used to decrease the computation time allowed the modelling of moving radiative surfaces. We introduced this model into a finite element convection diffusion code in order to simulate heat transfers in an electrical rotating engine. The main originality of this study lies in the use of periodic radiative conditions with moving surfaces and in the use of a method which is not based on the isothermal approximation.
Details
Keywords
Abstract
A model coupling magnetic and electric equations is developed in this paper. The unknowns in this model are the magnetic vector potential and the mesh currents. The nature of the resulting matrix system does not permit the use of the classical Cholesky algorithm. A new decomposition, based on the Cholesky method, is introduced to solve it. To illustrate the efficiency of the method, an example of an iron core coil is modelled. Both calculation time and memory storage are compared with classical methods.
The finite element technique has been used to develop a software package which, with the help of an IBM/PC 386 computer, can determine two dimensional magnetic fields and power…
Abstract
The finite element technique has been used to develop a software package which, with the help of an IBM/PC 386 computer, can determine two dimensional magnetic fields and power loss in transformer cores. The program takes into account both the non‐linear B‐H characteristic and the anisotropy of the core material.
This paper presents a new 9 DOF triangular element for plate bending. It is an analytically integrated improved version of the simplest member in the hierarchy of numerically…
Abstract
This paper presents a new 9 DOF triangular element for plate bending. It is an analytically integrated improved version of the simplest member in the hierarchy of numerically integrated elements. These elements have been based on the so‐called Hybrid—Trefftz model (HT), a recently developed hybrid model associated with enforcing interelement continuity on locally based displacement fields chosen such that they a priori verify the Lagrange plate equation over the element. In the process of development of the element stiffness matrix in a standard HT model, one has to invert the so‐called natural stiffness matrix, a 7 × 7 matrix associated with the expression of the strain energy in terms of the Trefftz's functions of the element. The inversion of this fully populated matrix represents the most expensive part of the calculation of the element. The basic improvement of the standard Hybrid—Trefftz 9 DOF triangle consists in replacing the original Trefftz's functions by new ones which are energy orthogonal and consequently, result in a diagonal natural stiffness matrix. This not only alleviates considerably the computer cost, but also significantly simplifies the algebra making analytical integrations possible. The practical efficiency of the new element which passes the patch test is demonstrated through numerical examples including the difficult simply supported skew plate problem with a strong singularity at its 150° obtuse corner.
C. Benoit, P. Coorevits and J.‐P. Pelle
A method for controlling the quality of finite element analyses for plate structures is proposed herein. It is based on the concept of error in the constitutive relation as well…
Abstract
A method for controlling the quality of finite element analyses for plate structures is proposed herein. It is based on the concept of error in the constitutive relation as well as on associated techniques for constructing admissible displacement‐stress fields with respect to a reference model. In this study, the chosen model is either Reissner‐Mindlin’s or Kirchhoff‐Love’s model. The finite element used is the DKT element; these error estimators allow us to determine that this element converges for Kirchhoff‐Love’s model. Once these error estimators have been identified, techniques of adaptive meshing developed in 2D are applied and several examples are presented.
Details
Keywords
A new triangular shell finite element ‘TNTE.1’ (Ten Node Triangular Element, Model 1) is presented. The formulation is based on Sanders' theory which involves the inclusion of the…
Abstract
A new triangular shell finite element ‘TNTE.1’ (Ten Node Triangular Element, Model 1) is presented. The formulation is based on Sanders' theory which involves the inclusion of the normal rotation Φn in the bending‐strain relations only. The element displacement functions are complete cubic polynomials for inplane displacements u and v. For out‐of‐plane displacement w, three new singular rational shape functions were added at the element corners. Thus a conforming triangular element with twenty seven degrees‐of‐freedom is obtained after eliminating the internal displacements by static condensation. The formulation of this element is new in that an integration technique is developed and applied to the element stiffness matrix and load vector. This technique is based on performing all the necessary integrations externally (i.e. outside the main computer program) and then modifying the formulation of the element matrices to account for this change. Hence, such a method allows the inclusion of higher‐order integration rules without any loss of economy, due to computer time, in the main program. Results using this element showed good agreement with other finite element and closed form solutions.
H.G. Rábade, P. Vellando, F. Padilla and R. Juncosa
A new coupled finite element model has been developed for the joint resolution of both the shallow water equations, that governs the free surface flow, and the groundwater flow…
Abstract
Purpose
A new coupled finite element model has been developed for the joint resolution of both the shallow water equations, that governs the free surface flow, and the groundwater flow equation that governs the motion of water through a porous media. The paper aims to discuss these issues.
Design/methodology/approach
The model is based upon two different modules (surface and ground water) previously developed by the authors, that have been validated separately.
Findings
The newly developed software allows for the assessment of the fluid flow in natural watersheds taking into account both the surface and the underground flow in the way it really takes place in nature.
Originality/value
The main achievement of this work has dealt with the coupling of both models, allowing for a proper moving interface treatment that simulates the actual interaction that takes place between surface and groundwater in natural watersheds.
Details
Keywords
Francis Sabourin and Michel Brunet
The aim of this paper is to present an enriched formulation of a rotation‐free (RF) triangular shell element in order to use it for shells of general shapes while, up to now, it…
Abstract
Purpose
The aim of this paper is to present an enriched formulation of a rotation‐free (RF) triangular shell element in order to use it for shells of general shapes while, up to now, it is limited to shells without branching surfaces and progressive variations in terms of material behavior and thickness.
Design/methodology/approach
The formulation keeps the main characteristic of Morley's element: bending effects can be expressed with three “bending angles” only. But, for a RF element, these angles are defined with the rigid body rotations of the element itself and those of its neighbours. This usual formulation of a RF shell element can be extended provided that curvatures‐displacements relation involves the material characteristics of the element itself and of its neighbours and the same goes for thickness.
Findings
Numerous examples with regular and irregular meshes of structures involving branching surfaces point out convergence and accuracy. Large displacement analyses – including crash simulations – show the effectiveness, too. A deep‐drawing of a “U” shape and the following springback prediction highlight the fact that the curvatures are captured more exactly (when nodes slide on die radius) since they are imposed in terms of translations whereas they are traditionally computed with nodal rotations not managed by contact conditions on the tooling.
Practical implications
The “S3” element detailed here is implemented in RADIOSS® software. The general conclusions are that this triangle often gives almost the same result as “DKT18” but is two times less cheaper and it is found interesting for sheet forming simulations.
Originality/value
Specificity of such an element clearly appears while lifting the initial restrictions quoted before.