Eulerian elasto‐visco‐plastic formulations for residual stress prediction

A stabilized, Galerkin finite element formulation for modeling the elasto‐visco‐plastic response of quasi‐steady‐state processes, such as welding, laser surfacing, rolling and extrusion, is presented in an Eulerian frame. The mixed formulation consists of four field variables, such as velocity, stress, deformation gradient and internal variable, which is used to describe the evolution of the material's resistance to plastic flow. The streamline upwind Petrov–Galerkin method is used to eliminate spurious oscillations, which may be caused by the convection‐type of stress, deformation gradient and internal variable evolution equations. A progressive solution strategy is introduced to improve the convergence of the Newton–Raphson solution procedure. Two two‐dimensional numerical examples are implemented to verify the accuracy of the Eulerian formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimization of thermo‐elasto‐plastic processes using Eulerian sensitivity analysis

A computational scheme for the analysis and optimization of quasi‐static thermo‐mechanical processes is presented in this paper. In order to obtain desirable mechanical transformations in a workpiece using a thermal treatment process, the optimal control parameters need to be determined. The problem is addressed by posing the process as a decoupled thermo‐mechanical finite element problem and performing an optimization using gradient methods. The forward problem is solved using the Eulerian formulation since it is computationally more efficient compared to an equivalent Lagrangian formulation. The design sensitivities required for the optimization are developed analytically using direct differentiation. This systematic design approach is applied to optimize a laser forming process. The objective is to maximize the angular distortion of a specimen subject to the constraint that the phase transition temperature is not exceeded at any point in the model. Copyright © 2003 John Wiley & Sons, Ltd.     

A three‐invariant hardening plasticity for numerical simulation of powder forming processes via the arbitrary Lagrangian–Eulerian FE model

In this paper, a three‐invariant cap plasticity model with an isotropic hardening rule is presented for numerical simulation of powder compaction processes. A general form is developed for single‐cap plasticity which can be compared with some common double‐surface plasticity models proposed for powders in literature. The constitutive elasto‐plastic matrix and its components are derived based on the definition of yield surface, hardening parameter and non‐linear elastic behaviour, as function of relative density of powder. Different aspects of the new single plasticity are illustrated by generating the classical plasticity models as special cases of the proposed model. The procedure for determination of powder parameters is described by fitting the model to reproduce data from triaxial compression and confining pressure experiments. The three‐invariant cap plasticity is performed within the framework of an arbitrary Lagrangian–Eulerian formulation, in order to predict the non‐uniform relative density distribution during large deformation of powder die pressing. In ALE formulation, the reference configuration is used for describing the motion, instead of material configuration in Lagrangian, and spatial configuration in Eulerian formulation. This formulation introduces some convective terms in the finite element equations and consists of two phases. Each time step is analysed according to Lagrangian phase until required convergence is attained. Then, the Eulerian phase is applied to keep mesh configuration regular. Because of relative displacement between mesh and material, all dependent variables such as stress and strain are converted through the Eulerian phase. Finally, the numerical schemes are examined for efficiency and accuracy in the modelling of a rotational flanged component, an automotive component, a conical shaped‐charge liner and a connecting‐rod. Copyright © 2005 John Wiley & Sons, Ltd.     

A particle‐in‐cell solution to the silo discharging problem

The problem of flow of a granular material during the process of discharging a silo is considered in the present paper. The mechanical behaviour of the material is described by the use of the model of the elastic–plastic solid with the Drucker–Prager yield condition and the non‐associative flow rule. The phenomenon of friction between the stored material and the silo walls is taken into account—the Coulomb model of friction is used in the analysis. The problem is analysed by means of the particle‐in‐cell method—a variant of the finite element method which enables to solve the pertinent equations of motion on an arbitrary computational mesh and trace state variables at points of the body chosen independently of the mesh. The method can be regarded as an arbitrary Lagrangian–Eulerian formulation of the finite element method, and overcomes the main drawback of the updated Lagrangian formulation of FEM related to mesh distortion. The entire process of discharging a silo can be analysed by this approach. The dynamic problem is solved by the use of the explicit time‐integration scheme. Several numerical examples are included. The plane strain and axisymmetric problems are solved for silos with flat bottoms and conical hoppers. Some results are compared with experimental ones. Copyright © 1999 John Wiley & Sons, Ltd.     

New development in extended finite element modeling of large elasto‐plastic deformations

This paper presents new achievements in the extended finite element modeling of large elasto‐plastic deformation in solid problems. The computational technique is presented based on the extended finite element method (X‐FEM) coupled with the Lagrangian formulation in order to model arbitrary interfaces in large deformations. In X‐FEM, the material interfaces are represented independently of element boundaries, and the process is accomplished by partitioning the domain with some triangular sub‐elements whose Gauss points are used for integration of the domain of elements. The large elasto‐plastic deformation formulation is employed within the X‐FEM framework to simulate the non‐linear behavior of materials. The interface between two bodies is modeled by using the X‐FEM technique and applying the Heaviside‐ and level‐set‐based enrichment functions. Finally, several numerical examples are analyzed, including arbitrary material interfaces, to demonstrate the efficiency of the X‐FEM technique in large plasticity deformations. Copyright © 2008 John Wiley & Sons, Ltd.     

Design sensitivity analysis in large deformation elasto‐plastic and elasto‐viscoplastic problems

Implicit time integration algorithm derived by Simo for his large‐deformation elasto‐plastic constitutive model is generalized, for the case of isotropy and associative flow rule, towards viscoplastic material behaviour and consistently differentiated with respect to its input parameters. Combining it with the general formulation of design sensitivity analysis (DSA) for non‐linear finite element transient equilibrium problem, we come at a numerically efficient, closed‐form finite element formulation of DSA for large deformation elasto‐plastic and elasto‐viscoplastic problems, with various types of design variables (material constants, shape parameters). The paper handles several specific issues, like the use of a non‐algorithmic coefficient matrix or sensitivity discontinuities at points of instantaneous structural stiffness change. Computational examples demonstrate abilities of the formulation and quality of results. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Optimization of thermal processes using an Eulerian formulation and application in laser surface hardening

A systematic design approach has been developed for thermal processes combining the finite element method, design sensitivity analysis and optimization. Conductive heat transfer is solved in an Eulerian formulation, where the heat flux is fixed in space and the material flows through a control volume. For constant velocity and heat flux distribution, the Eulerian formulation reduces to a steady-state problem, whereas the Lagrangian formulation remains transient. The reduction to a steady-state problem drastically improves the computational efficiency. Streamline Upwinding Petrov–Galerkin stabilization is employed to suppress the spurious oscillations. Design sensitivities of the temperature field are computed using both the direct differentiation and the adjoint methods. The systematic approach is applied in optimizing the laser surfacing process, where a moving laser beam heats the surface of a plate, and hardening is achieved by rapid cooling due to the heat transfer below the surface. The optimization objective is to maximize the rate of surface hardening. Constraints are introduced on the computed temperature and temperature rate fields to ensure that phase transformations are activated and that melting does not occur. Copyright © 2000 John Wiley & Sons, Ltd.     

Velocity‐based formulations for standard and quasi‐incompressible hypoelastic‐plastic solids

We present three velocity‐based updated Lagrangian formulations for standard and quasi‐incompressible hypoelastic‐plastic solids. Three low‐order finite elements are derived and tested for non‐linear solid mechanics problems. The so‐called V‐element is based on a standard velocity approach, while a mixed velocity–pressure formulation is used for the VP and the VPS elements. The two‐field problem is solved via a two‐step Gauss–Seidel partitioned iterative scheme. First, the momentum equations are solved in terms of velocity increments, as for the V‐element. Then, the constitutive relation for the pressure is solved using the updated velocities obtained at the previous step. For the VPS‐element, the formulation is stabilized using the finite calculus method in order to solve problems involving quasi‐incompressible materials. All the solid elements are validated by solving two‐dimensional and three‐dimensional benchmark problems in statics as in dynamics. Copyright © 2016 John Wiley & Sons, Ltd.     

Treatment of contact separation in Eulerian high‐speed multimaterial dynamic simulations

A frictionless contact separation treatment in a sharp‐interface Eulerian framework is presented to handle the general situation of high‐speed impact and separation of materials. The algorithm has been developed for an established Eulerian‐based Cartesian grid multimaterial flow code in which the interfaces are tracked in a sharp manner using a standard narrow‐band level set approach. Boundary conditions have been applied using a modified ghost fluid method for elasto‐plastic materials. The sharp‐interface treatment maintains the distinct interacting interfaces without smearing the contact zone while also removing the difficulties associated with Lagrangian moving mesh entities in contact‐separation situations. The algorithm has been tested and verified against experimental and numerical results for three different problems in the high strain rate regime, which involve contact, separation and sliding of materials. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

A hybrid variational‐collocation immersed method for fluid‐structure interaction using unstructured T‐splines

We present a hybrid variational‐collocation, immersed, and fully‐implicit formulation for fluid‐structure interaction (FSI) using unstructured T‐splines. In our immersed methodology, we define an Eulerian mesh on the whole computational domain and a Lagrangian mesh on the solid domain, which moves arbitrarily on top of the Eulerian mesh. Mathematically, the problem reduces to solving three equations, namely, the linear momentum balance, mass conservation, and a condition of kinematic compatibility between the Lagrangian displacement and the Eulerian velocity. We use a weighted residual approach for the linear momentum and mass conservation equations, but we discretize directly the strong form of the kinematic relation, deriving a hybrid variational‐collocation method. We use T‐splines for both the spatial discretization and the information transfer between the Eulerian mesh and the Lagrangian mesh. T‐splines offer us two main advantages against non‐uniform rational B‐splines: they can be locally refined and they are unstructured. The generalized‐α method is used for the time discretization. We validate our formulation with a common FSI benchmark problem achieving excellent agreement with the theoretical solution. An example involving a partially immersed solid is also solved. The numerical examples show how the use of T‐junctions and extraordinary nodes results in an accurate, efficient, and flexible method. Copyright © 2015 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.     

An extended finite element formulation for contact in multi‐material arbitrary Lagrangian–Eulerian calculations

Multi‐material Eulerian and arbitrary Lagrangian–Eulerian methods were originally developed for solving hypervelocity impact problems, but they are attractive for solving a broad range of problems having large deformations, the evolution of new free surfaces, and chemical reactions. The contact, separation, and slip between two surfaces have traditionally been addressed by the mixture theory, however the accuracy of this approach is severely limited. To improve the accuracy, an extended finite element formulation is developed and example calculations are presented. As a side benefit, the mixture theory is eliminated from the multi‐material formulation, eliminating the issues associated with the equilibration time between adjacent materials. By design, the new formulation is relatively simple to implement in existing multi‐material codes, parallelizes without difficulty, and has a low memory burden. Copyright © 2006 John Wiley & Sons, Ltd.     

Elasto–acoustic and acoustic–acoustic coupling on non‐matching grids

Flexible discretization techniques for the approximative solution of coupled wave propagation problems are investigated, focussing on aero–acoustic and elasto–acoustic coupling. In particular, the advantages of using non‐matching grids are presented, when one subregion has to be resolved by a substantially finer grid than the other subregion. For the elasto–acoustic coupling, the problem formulation remains essentially the same as for the matching situation, while for the aero–acoustic coupling, the formulation is enhanced with Lagrange multipliers within the framework of mortar finite element methods. Several numerical examples are presented to demonstrate the flexibility and applicability of the approach. Copyright © 2006 John Wiley & Sons, Ltd.     

A full‐discontinuous Galerkin formulation of nonlinear Kirchhoff–Love shells: elasto‐plastic finite deformations,parallel computation,and fracture applications

Because of its ability to take into account discontinuities, the discontinuous Galerkin (DG) method presents some advantages for modeling cracks initiation and propagation. This concept has been recently applied to three‐dimensional simulations and to elastic thin bodies. In this last case, the assumption of small elastic deformations before cracks initiation or propagation reduces drastically the applicability of the framework to a reduced number of materials. To remove this limitation, a full‐DG formulation of nonlinear Kirchhoff–Love shells is presented and is used in combination with an elasto‐plastic finite deformations model. The results obtained by this new formulation are in agreement with other continuum elasto‐plastic shell formulations. Then, this full‐DG formulation of Kirchhoff–Love shells is coupled with the cohesive zone model to perform thin body fracture simulations. As this method considers elasto‐plastic constitutive laws in combination with the cohesive model, accurate results compared with the experiments are found. In particular, the crack path and propagation rate of a blasted cylinder are shown to match experimental results. One of the main advantages of this framework is its ability to run in parallel with a high speed‐up factor, allowing the simulation of ultra fine meshes. Copyright © 2012 John Wiley & Sons, Ltd.     

Implicit solid,flow and flow–solid finite element approaches in blade forging simulation

A comparative theoretical and numerical study of the flow (rigid‐plastic) and solid (elasto‐plastic, in rate form) simulation approaches applied to the isothermal blade forging process for closing and opening of the dies is presented. In the rigid‐plastic approach the solution is followed by an elasto‐plastic solution in unloading (opening of the dies). The deformation of the billet, effective plastic strains and stresses were compared between the two solutions in the closing of the dies and when the dies were opened. Furthermore, the flow–solid approach was used in simulating the process and then the results, including the time–load and time–volume curves, were compared with previous results. The cost was lower and the efficiency was superior in the suggested flow–solid approach. Copyright © 2003 John Wiley & Sons, Ltd.     

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.