Jean-Louis Batoz

Recent developments on the analysis and optimum design of sheet metal forming parts using a simplified inverse approach

Abstract A simplified efficient finite element method called the inverse approach (IA) has been developed to estimate the large elasto-plastic strains in thin metallic panels obtained by deep drawing. This paper deals with the main recent developments introduced by the authors on the IA to improve its efficiency in the analysis and optimum design of blank contours of complicated industrial parts. The IA mainly exploits the knowledge of the 3D shape of the final workpiece. An iterative scheme is used to find the original position of each material point in the initial flat blank after which it is possible to estimate the strains and stresses in the final workpiece. Important assumptions are adopted regarding the constitutive equations (the deformation theory of plasticity) and the action of the tools (the punch, die and blank holders). The IA implies only two degrees of freedom per node even if bending effects are considered. In this paper, we present several recent developments: (1) The bending effects are taken into account using a simple triangular shell element without increasing the number of dof per node. (2) Some analytical formulas are introduced to consider the restraining forces due to the drawbeads. (3) Some improvements of resolution algorithms such as the introduction of a relaxation coefficient, a damping factor and a good initial solution are realized. (4) Shape optimization of blank contours is performed using a numerical procedure based on the coupling of the IA and a sequential quadratic programming method (SQP). In this work, all sensitivities are computed analytically using the adjoint variable method. The numerical results of the IA on two benchmark tests are compared with experimental and other numerical results. The optimization procedure is applied to the blank optimum design of the Renault/Twingo dashpot cup where the objective function is defined to minimize the maximum of the thickness variations.

Optimization of drawbead restraining forces and drawbead design in sheet metal forming process

Abstract This paper presents an optimization procedure of drawbead restraining forces in order to improve the sheet metal formability in deep drawing process. A simplified finite element method called inverse approach (IA) has been developed for sheet forming analysis with the consideration of the drawbead restraining forces. This IA is combined with a mathematical programming algorithm to optimize the restraining forces and then to design the drawbeads. The obtained optimization procedure is very efficient due to the simplified assumptions of the IA and the analytical sensitivity analysis. The Square cup of Numisheet’93 and the Renault Twingo dashpot cup are presented to demonstrate the usefulness of the proposed optimization procedure for industrial applications. Verifications of the obtained results have been carried out using a precise incremental commercial code OPTRISTM based on explicit dynamic approach to show the effectiveness of our approach.

An efficient DKT rotation free shell element for springback simulation in sheet metal forming

This paper presents two simple and efficient DKT triangular shell elements for the springback simulation after the sheet forming process. The first is the DKT12 shell element resulting from the superposition of the CST membrane element and the DKT6 plate element. The second element is called DKTRF element (DKT rotation free element) involving its three neighboring elements and six corner nodes, with only three translations dof per node. The three rotations around the sides are expressed in terms of the 18 nodal translational dof. The present formulation is more general and accurate than existing rotation free elements, particularly in the case of deep shells. A static implicit algorithm using a simple updated Lagrangian formulation is adopted for the springback simulation. Some academic examples and benchmark tests show the accuracy and efficiency of these two shell elements.

A torsion bending element for thin-walled beams with open and closed cross sections

Abstract A finite element is formulated for the torsion problems of thin-walled beams. The element is based on Benscoters beam theory, which is valid for open and also closed cross-sections. The non-polynomial interpolation presented in this paper allows the exact static solution to be obtained with only one element. Numerical results are presented for three thin-walled cantilever beams, one with a channel cross-section and the two others with rectangular cross-sections. The influence of the transverse shear strain is investigated and the different models of torsion are compared. For one example, the results obtained with one-dimensional torsion elements are compared with those obtained using shell elements.

Modeling of connections in the analyses of thin-walled space frames

Abstract This paper deals with the finite element formulation for the analysis of space frames. A numerical method is presented to take into account the deformation of the joint connections in linear, non-linear and stability analyses of three-dimensional thin-walled beam structures.

On the linear analysis of plates and shells using a new-16 degrees of freedom flat shell element

Abstract In this paper, we present a new quadrilateral discrete Kirchhoff flat shell element (called DKQ16) with 16 degrees of freedom (three displacements U, V, W at each corner, a rotation θs at each mid-side) for the linear analysis of plates and shells. This new element is formulated on the basis of the so-called rational element method proposed by Zhong et al. [Zhong WX, Zeng J. J Computat Struct Mech Appl 1996;13:1–8 [in Chinese]]. In this new formulation, the rational quadrilateral plane element (RQ4) is employed for the membrane part and a new discrete Kirchhoff plate element DKQ8 proposed by the present authors [Batoz JL, Hammadi F, Zheng CL, Zhong WX, submitted for publication] is taken for the bending part. The DKQ16 element is one of the most simple quadrilateral flat shell elements. It can be combined with the DKT12 (or Morley) element. Numerical results for some typical problems demonstrate the overall good performance of the new shell element.

Formulation and evaluation of a finite element model for the linear analysis of stiffened composite cylindrical panels

A finite element model for linear static and free vibration analysis of composite cylindrical panels with composite stiffeners is presented. The proposed model is based on a cylindrical shell finite element, which uses a first-roder shear deformation theory. The stiffeners are curved beam elements based on Timoshenko and Saint-Venant assumptions for bending and torsion respectively. The two elements are developed in a cylindrical coordinate system and their stiffness matrices result from a hybrid-mixed formulation where the element assumed stress field is such that exact equilibrium equations are satisfied. The elements are free of membrane and shear locking with correct satisfaction of rigid body motions. Several examples dealing with stiffened isotropic and laminated plates and shells with eccentric as well as concentric stiffeners are analyzed showing the validity of the models.

Analyse non linéaire de coques minces élastoplastiques avec l’élément DKT12

Abstract.A shell element called DKT12 is presented for non-linear analysis which includes large displacements and elasto-plasticity. The triangular facet shell element with 12 degrees of freedom is obtained by the superposition of the membrane element T3 (or CST) and the Discrete Kirchhoff plate bending element DKT6. An Updated Lagrangian Formulation at each Iteration (ULFI) greatly simplifies the non-linear analysis by considering the global deformation as a superposition of a rigid body motion and a movement involving small displacements and rotations. A global criterion is taken into account to consider the elasto-plastic behaviour. The non-linear systems of equations are solved by using various algorithms and strategies based on the Newton method. Several examples are presented to demonstrate the efficiency and precision of the formulation and algorithms.

On a Simplified Model for the Tool and the Sheet Contact Conditions for the SPIF Process Simulation

The numerical simulation of the Single Point Incremental Forming process (SPIF) is time consuming due to the necessity to take into account various non-linearity such as the material behaviour, large strain deformation and the evolution of the tool-flange contact. Classical contact algorithms give good agreement with experimental results, but are time consuming. In this paper, we investigate the development of a procedure to simplify the management of the contact interface between the tool and the sheet. Nodes with imposed displacements are determined by a geometrical approximation of the deformed sheet. In order to have a better approximation of the local stresses in the flange, a pressure is applied on the tool side of the elements in the contact zone. The pressure value is obtained by an analytical model. A classical contact algorithm and the present simplified approach are compared in terms of an incremental forming benchmark. It has been shown that, for the benchmark problem studied here, a CPU time reduction of approximately 65% can be achieved while at the same time have good simulation results.

Un élément quadrilatéral de plaque basé sur une formulation mixte-hybride avec projection en cisaillement

ABSTRACT This work deals with the formulation of a new 4-node quadrilateral element for the linear analysis of plate bending with transverse shear effect. This formulation is based on a variational hybrid-mixed model called MiSP (Mixed Shear Projected). The approximation of the shear forces results from that of bending moments with an explicit satisfaction of two of the three equilibrium equations. The TS deformations are defined in terms of the edge deformations which are projected on the nodal degrees of freedom. The convergence of the element is checked on a series of constant and rigid mode patch-tests, and its precision is evaluated on two series of benchmark examples related to skewed and circular plates.