Arbitrary Lagrangian–Eulerian method in plasticity of pressure-sensitive material: application to powder forming processes

In this paper, an application of Arbitrary Lagrangian–Eulerian (ALE) method is presented in plasticity behavior of pressure-sensitive material, with special reference to large deformation analysis of powder compaction process. In ALE technique, the reference configuration is used for describing the motion, instead of material configuration in Lagrangian, and spatial configuration in Eulerian formulation. The convective term is used to reflect the relative motion between the mesh and the material. Each time-step is divided into the Lagrangian phase and Eulerian phase. The convection term is neglected in the material phase, which is identical to a time-step in a standard Lagrangian analysis. The stresses and plastic internal variables are converted to account the relative mesh-material motion in the convection phase. The ALE formulation is then performed within the framework of a three-invariant cap plasticity model in order to predict the non-uniform density distribution during the large deformation of powder die pressing. The plasticity model is based on a hardening rule with the isotropic and kinematic material functions. The constitutive elasto-plastic matrix and its components are derived by using the definition of yield surface, material functions and non-linear elastic behavior, as function of hardening parameters. Finally, the numerical examples are performed to illustrate the applicability of the computational algorithm in modeling of powder forming process and the results are compared with those obtained from Lagrangian simulation in order to demonstrate the accuracy of proposed model.     

A Multiscale/stabilized Formulation of the Incompressible Navier–Stokes Equations for Moving Boundary Flows and Fluid–structure Interaction

This paper presents a multiscale/stabilized finite element formulation for the incompressible Navier–Stokes equations written in an Arbitrary Lagrangian–Eulerian (ALE) frame to model flow problems that involve moving and deforming meshes. The new formulation is derived based on the variational multiscale method proposed by Hughes (Comput Methods Appl Mech Eng 127:387–401, 1995) and employed in Masud and Khurram in (Comput Methods Appl Mech Eng 193:1997–2018, 2006); Masud and Khurram in (Comput Methods Appl Mech Eng 195:1750–1777, 2006) to study advection dominated transport phenomena. A significant feature of the formulation is that the structure of the stabilization terms and the definition of the stabilization tensor appear naturally via the solution of the sub-grid scale problem. A mesh moving technique is integrated in this formulation to accommodate the motion and deformation of the computational grid, and to map the moving boundaries in a rational way. Some benchmark problems are shown, and simulations of an elastic beam undergoing large amplitude periodic oscillations in a viscous fluid domain are presented.     

Flow‐induced vibrations of non‐linear cables. Part 1: Models and algorithms

In this paper, we develop governing equations for non‐linear cables as well as a formulation for the coupled flow‐structure problem. The structure is discretized with second‐order accuracy while the flow is discretized using spectral/hp elements in the context of the arbitrary Lagrangian–Eulerian formulation (ALE). Several benchmark problems are considered and the computational implementation is detailed. In the second part of this work large‐scale simulation examples are presented. Copyright © 2002 John Wiley & Sons, Ltd.     

On using enriched cover function in the Partition-of-unity method for singular boundary-value problems

 Amongst the various approaches of `meshless' method, the Partition-of-unity concept married with the traditional finite-element method, namely PUFEM, has emerged to be competitive in solving the boundary-value problems. It inherits most of the advantages from both techniques except that the beauty of being `meshless' vanishes. This paper presents an alternative approach to solve singular boundary-value problems. It follows the basic PUFEM procedures. The salient feature is to enhance the quality of the influence functions, either over one single nodal cover or multi-nodal-covers. In the vicinity of the singularity, available asymptotic analytical solution is employed to enrich the influence function. The beauty of present approach is that it facilitates easy replacement of the influence functions. In other words, it favors the `influence-function refinement' procedure in a bid to search for more accurate solutions. It is analogous to the `p-version refinement' in the traditional finite-element procedures. The present approach can yield very accurate solution without adopting refined meshes. As a result, the quantities around the singularity can be evaluated directly once the nodal values are solved. No additional post-processing is needed. Firstly, the formulation of the present PUFEM approach is described. Subsequently, illustrative examples show the application to three classical singular benchmark problems having various orders of singularity. Results obtained through mesh refinements, single-nodal-cover refinements or multi-nodal-cover refinements are compared. Received: 5 November 2001 / Accepted: 21 May 2002     

Numerical simulations of large deformation of thin shell structures using meshfree methods

 In this paper, meshfree simulations of large deformation of thin shell structures is presented. It has been shown that the window function based meshfree interpolants can be used to construct highly smoothed (high order “manifold”) shape functions for three-dimensional (3-D) meshfree discretization/interpolation, which can be used to simulate large deformation of thin shell structures while avoiding ill-conditioning as well as stiffening in numerical computations. The main advantage of such 3-D meshfree continuum approach is its simplicity in both formulation and implementation as compared to shell theory approach, or degenerated continuum approach. Moreover, it is believed that the accuracy of the computation may increase because of using 3-D exact formulation. Possible mechanism to relieve shear/volumetric locking due to the meshfree interpolation is discussed. Several examples have been computed by using a meshfree, explicit, total Lagrangian formulation. Towards to developing a self-contact algorithm, a novel meshfree contact algorithm is proposed in the end.     

An ALE hydrodynamic lubrication finite element method with application to strip rolling

A method that incorporates the hydrodynamic lubrication analysis into the arbitrary Lagrangian Eulerian (ALE) finite element analysis is developed for steady-state strip rolling simulation. By employing the ALE formulation, only part of the workpiece, which is subjected to large plastic deformation within the roll-bite region, is modelled, so that the computational cost is substantially reduced. In the hydrodynamic lubrication formulation, the effect of surface roughness on the lubricant flow is taken into consideration by the use of an average flow model. The friction stress is expressed in terms of forming variables such as surface roughness, lubricant and workpiece properties, film thickness, forming speed and process geometry. Furthermore, the elastic deformation of rolls is also analysed by the boundary element method to avoid the finite element discretization inside the rolls. Two numerical examples, aluminium and steel strip rolling processes, are presented to demonstrate the merits of the proposed method.     

An arbitrary Lagrangian–Eulerian finite element method for finite strain plasticity

This paper presents a new arbitrary Lagrangian–Eulerian (ALE) finite element formulation for finite strain plasticity in non‐linear solid mechanics. We consider the models of finite strain plasticity defined by the multiplicative decomposition of the deformation gradient in an elastic and a plastic part ( F = F ^e F ^p), with the stresses given by a hyperelastic relation. In contrast with more classical ALE approaches based on plastic models of the hypoelastic type, the ALE formulation presented herein considers the direct interpolation of the motion of the material with respect to the reference mesh together with the motion of the spatial mesh with respect to this same reference mesh. This aspect is shown to be crucial for a simple treatment of the advection of the plastic internal variables and dynamic variables. In fact, this advection is carried out exactly through a particle tracking in the reference mesh, a calculation that can be accomplished very efficiently with the use of the connectivity graph of the fixed reference mesh. A staggered scheme defined by three steps (the smoothing, the advection and the Lagrangian steps) leads to an efficient method for the solution of the resulting equations. We present several representative numerical simulations that illustrate the performance of the newly proposed methods. Both quasi‐static and dynamic conditions are considered in these model examples. Copyright © 2003 John Wiley & Sons, Ltd.     

The ALE‐method with triangular elements: direct convection of integration point values

The arbitrary Lagrangian–Eulerian (ALE) finite element method is applied to the simulation of forming processes where material is highly deformed. Here, the split formulation is used: a Lagrangian step is done with an implicit finite element formulation, followed by an explicit (purely convective) Eulerian step. The purpose of this study is to investigate the Eulerian step for quadratic triangular elements. To solve the convection equation for integration point values, a new method inspired by Van Leer is constructed. The new method is based on direct convection of integration point values without intervention of nodal point values. The Molenkamp test and a so‐called block test were executed to check the performance and stability of the convection scheme. From these tests it is concluded that the new convection scheme shows accurate results. The scheme is extended to an ALE‐algorithm. An extrusion process was simulated to test the applicability of the scheme to engineering problems. It is concluded that direct convection of integration point values with the presented algorithm leads to accurate results and that it can be applied to ALE‐simulations. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical analysis of the inclusion-crack interactions using an integral equation

 A general-purpose integral formulation is proposed for the analysis of the interaction between inclusions and cracks embedded in an elastic isotropic homogeneous infinite medium subjected to a remote loading. This formulation is tailored for the inclusions of arbitrary shapes with the presence of cracks. The discretization is limited to the inclusions (with continuous quadratic triangular and quadrilateral elements) and the cracks (using discontinuous quadratic elements). For the calculation of the stress intensity factors at the crack tips, special crack tip elements are used to model the variation of the displacements near the crack tips. Maximum circumferential stress criterion is adopted to determine the crack propagating direction. Numerical results of benchmark examples are compared with other available methods. Received: 8 January 2002 / Accepted: 24 September 2002     

A boundary elements pseudo heat source method formulation for inverse analysis of solidification problems

 An efficient methodology is presented to solve inverse solidification problems. In the procedure, the latent heat effects are implemented by introducing pseudo heat sources near the moving interface. The material properties can be temperature dependent. To account for the nonlinear part of the governing differential equations, a finite-boundary element formulation is employed. To reduce the oscillations in the solution, a sequential regularization scheme is used. A procedure for proper selection of regularization parameters is presented. To smooth the solutions further, a secondary regularization scheme is introduced and employed. Two complete examples are presented to demonstrate the applicability and the accuracy of the methods. Received: 1 March 2002 / Accepted: 10 February 2003     

Interface projection techniques for fluid–structure interaction modeling with moving-mesh methods

The stabilized space–time fluid–structure interaction (SSTFSI) technique developed by the Team for Advanced Flow Simulation and Modeling (T★AFSM) was applied to a number of 3D examples, including arterial fluid mechanics and parachute aerodynamics. Here we focus on the interface projection techniques that were developed as supplementary methods targeting the computational challenges associated with the geometric complexities of the fluid–structure interface. Although these supplementary techniques were developed in conjunction with the SSTFSI method and in the context of air–fabric interactions, they can also be used in conjunction with other moving-mesh methods, such as the Arbitrary Lagrangian–Eulerian (ALE) method, and in the context of other classes of FSI applications. The supplementary techniques currently consist of using split nodal values for pressure at the edges of the fabric and incompatible meshes at the air–fabric interfaces, the FSI Geometric Smoothing Technique (FSI-GST), and the Homogenized Modeling of Geometric Porosity (HMGP). Using split nodal values for pressure at the edges and incompatible meshes at the interfaces stabilizes the structural response at the edges of the membrane used in modeling the fabric. With the FSI-GST, the fluid mechanics mesh is sheltered from the consequences of the geometric complexity of the structure. With the HMGP, we bypass the intractable complexities of the geometric porosity by approximating it with an “equivalent”, locally-varying fabric porosity. As test cases demonstrating how the interface projection techniques work, we compute the air–fabric interactions of windsocks, sails and ringsail parachutes.     

Dynamic and scalar turbulent fluctuation in a diffusion flame of an-axisymmetric methane jet into air

 A study of turbulence/combustion interactions in a relatively large turbulent diffusion flame of an axisymmetric methane jet into air is presented. A first order k–ɛ turbulence closure model is used along with two different models (equal scales and non-equal scales) for the submodel describing the scalar dissipation rate. The flamelet concept is used to model the turbulent combustion along with a joint mixture fraction/strain rate probability density function (PDF) for the prediction of the average parameters of the turbulent diffusion flame. The numerical approach is that of Patankar and Spalding, while the flamelet simulations are obtained from the RUN-1DL code of Rogg and co-workers based on a 17 species detailed reaction mechanism. The chosen configuration is that of the experimentally studied turbulent diffusion flame of Streb [1]. A comparison between these experimental results and the obtained numerical ones is thus presented. Relatively good agreements are obtained which show the usefulness of the two-scale model compared to the classical one-scale model for predicting turbulent diffusion flames. Nonetheless some discrepancies are obtained in the outer and downstream regions of the jet, especially in comparison with the experimental data. These are attributed to short coming of the considered turbulence model and soot radiation which is not accounted for. Received: 2 May 2002 / Accepted: 31 January 2003     

Contact with friction in multi‐material arbitrary Lagrangian‐Eulerian formulations using X‐FEM

A contact method with friction for the multi‐dimensional Lagrangian step in multi‐material arbitrary Lagrangian–Eulerian (ALE) formulations is presented. In our previous research, the extended finite element method (X‐FEM) was used to create independent fields (i.e. velocity, strain rate, force, mass, etc.) for each material in the problem to model contact without friction. The research presented here includes the extension to friction and improvements to the accuracy and robustness of our previous study. The accelerations of the multi‐material nodes are obtained by coupling the material force and mass fields as a function of the prescribed contact; similarly, the velocities of the multi‐material nodes are recalculated using the conservation of momentum when the prescribed contact requires it. The coupling procedures impose the same nodal velocity on the coupled materials in the direction normal to their interface during the time step update. As a result, the overlap of materials is prevented and unwanted separation does not occur. Three different types of contacts are treated: perfectly bonded, frictionless slip, and slip with friction. Example impact problems are solved and the numerical solutions are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A co-rotational 8-node assumed strain shell element for postbuckling analysis of laminated composite plates and shells

 The formulation of a nonlinear composite shell element is presented for the solution of stability problems of composite plates and shells. The formulation of the geometrical stiffness presented here is exactly defined on the midsurface and is efficient for analyzing stability problems of thin and thick laminated plates and shells by incorporating bending moment and transverse shear resultant forces. The composite element is free of both membrane and shear locking behaviour by using the assumed natural strain method such that the element performs very well as thin shells. The transverse shear stiffness is defined by an equilibrium approach instead of using the shear correction factor. The proposed formulation is computationally efficient and the test results showed good agreement. In addition the effect of the viscoelastic material is investigated on the postbuckling behaviour of laminated composite shells. Received: 6 February 2002 / Accecpted: 6 January 2003 ID=" Present address: School of Civil Engineering, Asian Institute of Technology     

A semi-analytical approach to three-dimensional normal contact problems with friction

 We present a semi-analytical approach for three-dimensional elastostatic normal contact problems with friction. The numerical approach to iteration on contact area and stick zone size is supported by an underlying analytical solution relating normal and tangential surface tractions to surface displacements in three coordinate directions. The governing equations are fully coupled. The analytical surface displacement solutions for a basic loading element have been derived elsewhere (Li and Berger 2001), and the total surface displacements are constructed as a superposition of deflections due to overlapping pyramid load segments. This approach requires no interpolation scheme for the field variables, which distinguishes it from other numerical techniques such as the FEM, BEM, and meshless methods. A background grid is defined only on the contact surfaces, and iteration approaches are used to determine a convergent configuration for contact domain and stick zone size. The approach is exercised on several normal contact problems, with and without friction, and the results compare favorably to existing analytical and numerical solutions. Received: 10 July 2002 / Accepted: 3 December 2002 The authors appreciate the support of the UC Department of Mechanical Engineering and the UC Office of the Vice President for Research, who jointly provided funds for this work.     

Solution of large deformation contact problems with friction between Blatz-Ko hyperelastic bodies

The present paper is devoted to the analysis of the contact problems with Coulomb friction and large deformation between two hyperelastic bodies. One approach to separate the material nonlinearity and contact nonlinearity is presented. The total Lagrangian formulation is adopted to describe the geometrically nonlinear behavior. Nondifferentiable contact potentials are regularized by means of the augmented Lagrangian method. Numerical examples are carried out in two cases: rigid-deformable contact and deformable-deformable contact with large slips. The numerical results prove that the proposed approach is robust and efficient concerning numerical stability.     

A complete finite element treatment for the fully coupled implicit ALE formulation

This paper presents a complete derivation and implementation of the Arbitrary Lagrangian Eulerian (ALE) formulation for the simulation of large deformation quasi-static and dynamic problems. While most of the previous work done on ALE for dynamic applications was mainly based on operator split and explicit calculations, this work derives the quasi-static and dynamic ALE equations using a fully coupled implicit approach. Full expression for the ALE virtual work equations and finite element matrices are given. Time integration relations for the dynamic equations are also derived. Several quasi-static and dynamic large deformation applications are solved and presented.     

Elastic buckling analysis of 3-D trusses and frames with thin-walled members

 A finite element method is presented for the determination of the elastic buckling load of three-dimensional trusses and frames with rigid joints. The beam element stiffness matrix is constructed on the basis of the exact solution of the governing equations describing the coupled flexural-torsional buckling behaviour of a three-dimensional beam with an open thin-walled section in the framework of a small deformation theory. Large deformation effects are taken into account approximately through consideration of P−Δ effects. The structural stiffness matrix is obtained by an appropriate superposition of the various element stiffness matrices. The axial force distribution in the members is obtained iteratively for every value of the externally applied loading and the vanishing of the determinant of the structural stiffness matrix is the criterion used to numerically determine the elastic buckling load of the structure. The effect of initial member imperfections is also included in the formulation. Comparisons of accuracy and efficiency of the present exact finite element method against the conventional approximate finite element method are made. Cases where the axial force distribution determination can be done without iterations are also identified. The effect of neglecting the warping stiffness of some mono-symmetric sections is also investigated. Numerical examples involving simple and complex three-dimensional trusses and frames are presented to illustrate the method and demonstrate its merits. Received: 2 May 2000 / Accepted: 15 July 2002