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.     

Updated Lagrangian/Arbitrary Lagrangian–Eulerian framework for interaction between a compressible neo‐Hookean structure and an incompressible fluid

We propose a numerical method for a fluid–structure interaction problem. The material of the structure is homogeneous, isotropic, and it can be described by the compressible neo‐Hookean constitutive equation, while the fluid is governed by the Navier–Stokes equations. Our study does not use turbulence model. Updated Lagrangian method is used for the structure and fluid equations are written in Arbitrary Lagrangian–Eulerian coordinates. One global moving mesh is employed for the fluid–structure domain, where the fluid–structure interface is an ‘interior boundary’ of the global mesh. At each time step, we solve a monolithic system of unknown velocity and pressure defined on the global mesh. The continuity of velocity at the interface is automatically satisfied, while the continuity of stress does not appear explicitly in the monolithic fluid–structure system. This method is very fast because at each time step, we solve only one linear system. This linear system was obtained by the linearization of the structure around the previous position in the updated Lagrangian formulation and by the employment of a linear convection term for the fluid. Numerical results are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

On the numerical implementation of variational arbitrary Lagrangian–Eulerian (VALE) formulations

This paper is concerned with the implementation of variational arbitrary Lagrangian–Eulerian formulations, also known as variational r‐adaption methods. These methods seek to minimize the energy function with respect to the finite‐element mesh over the reference configuration of the body. We propose a solution strategy based on a viscous regularization of the configurational forces. This procedure eliminates the ill‐posedness of the problem without changing its solutions, i.e. the minimizers of the regularized problems are also minimizers of the original functional. We also develop strategies for optimizing the triangulation, or mesh connectivity, and for allowing nodes to migrate in and out of the boundary of the domain. Selected numerical examples demonstrate the robustness of the solution procedures and their ability to produce highly anisotropic mesh refinement in regions of high energy density. Copyright © 2006 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.     

A coupled BEM and arbitrary Lagrangian–Eulerian FEM model for the solution of two‐dimensional laminar flows in external flow fields

This paper describes a new computational model developed to solve two‐dimensional incompressible viscous flow problems in external flow fields. The model based on the Navier–Stokes equations in primitive variables is able to solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the pressure projection method. The external flow field is simulated using the boundary element method by solving a pressure Poisson equation that assumes the pressure as zero at the infinite boundary. The momentum equation of the flow motion is solved using the three‐step finite element method. The arbitrary Lagrangian–Eulerian method is incorporated into the model, to solve the moving boundary problems. The present model is applied to simulate various external flow problems like flow across circular cylinder, acceleration and deceleration of the circular cylinder moving in a still fluid and vibration of the circular cylinder induced by the vortex shedding. The simulation results are found to be very reasonable and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

ALE stress update for transient and quasistatic processes

A key issue in Arbitrary Lagrangian–Eulerian (ALE) non-linear solid mechanics is the correct treatment of the convection terms in the constitutive equation. These convection terms, which reflect the relative motion between the finite element mesh and the material, are found for both transient and quasistatic ALE analyses. It is shown in this paper that the same explicit algorithms can be employed to handle the convection terms of the constitutive equation for both types of analyses. The most attractive consequence of this fact is that a quasistatic simulation can be upgraded from Updated Lagrangian (UL) to ALE without significant extra computational cost. These ideas are illustrated by means of two numerical examples. © 1998 John Wiley & Sons, Ltd.     

A Lagrangian–Eulerian method with adaptively local ZOOMing approach to solve three-dimensional advection–diffusion transport equations

We present a Lagrangian–Eulerian method with adaptively local ZOOMing and Peak/valley Capturing approach (LEZOOMPC), consisting of advection–diffusion decoupling, backward particle tracking, forward particle tracking, adaptively local zooming, peak/valley capturing, and slave point utilization, to solve three-dimensional advection–diffusion transport equations. This approach and the associated computer code, 3DLEZOOMPC, were developed to circumvent the difficulties associated with the Exact Peak Capturing and Oscillation-Free (EPCOF) scheme, developed earlier by the authors, when it was extended from a one-dimensional space to a three-dimensional space. The accurate results of applying EPCOF to solving two one-dimensional benchmark problems under a variety of conditions have shown the capability of this scheme to eliminate all types of numerical errors associated with the advection term and to keep the maximum computational error to be within the prescribed error tolerance. However, difficulties arose when the EPCOF scheme was extended to a multi-dimensional space mainly due to the geometry. To avoid these geometric difficulties, we modified the EPCOF scheme and named the modified scheme LEZOOMPC. LEZOOMPC uses regularly local zooming for rough elements and peak/valley capturing within subelements to resolve the problems of tetrangulation and boundary source as well as to preserve the shape of concentration distribution. In addition, LEZOOMPC employs the concept of ‘slave points’ to deal with the compatibility problem in the diffusion zooming of the Eulerian step. As a result, not only is the geometrical problem resolved, but also the spirit of EPCOF is retained. Application of 3DLEZOOMPC to solving an advection-decay and a boundary source benchmark problems indicates its capability in solving advection transport problems accurately to within any prescribed error tolerance by using mesh Courant number ranging from 0 to infinity. Demonstration of using 3DLEZOOMPC to solve an advection–diffusion benchmark problem shows how the numerical solution is improved with the increment of the diffusion zooming factors. 3DLEZOOMPC could solve advection–diffusion transport problems accurately by using mesh Peclet numbers ranging from 0 to infinity and very large time-step size. The size of time-step is related to both the diffusion coefficients and mesh sizes. Hence, it is limited only by the diffusion solver. The application of this approach to a two-dimensional space has been demonstrated earlier in the paper entitled ‘A Lagrangian–Eulerian method with adaptively local zooming and peak/valley capturing approach to solve two-dimensional advection–diffusion transport equations’. © 1998 John Wiley & Sons, Ltd.     

An efficient solution method for finite element ring‐rolling simulation

Finite element ring‐rolling simulation gives rise to poor conditioned non‐linear equations that require repeated solution. The associated computational costs are extreme making analysis impracticable in industry. This paper is concerned with a solution strategy that addresses this problem and involves the combined use of an arbitrary Lagrangian–Eulerian (ALE) formulation and a successive preconditioned conjugate gradient method (SPCGM). This approach, coupled to a finite element flow formulation, is shown to offer considerable computational savings. Through the combined use of the ALE flow formulation and the SPCGM the stability and condition of the non‐linear systems is enhanced. This purely iterative approach takes advantage of the slowly evolving velocity field and the self‐preconditioning offered by the SPCGM. The performance of the solver is compared against well‐known alternatives for varying problem sizes. The approach is shown to be generic but in particular makes ring‐rolling simulation a more practicable proposition. Copyright © 2000 John Wiley & Sons, Ltd.     

Investigation of fluid–structure interactions using a velocity‐linked P2/P1 finite element method and the generalized‐ α method

A velocity‐linked algorithm for solving unsteady fluid–structure interaction (FSI) problems in a fully coupled manner is developed using the arbitrary Lagrangian–Eulerian method. The P2/P1 finite element is used to spatially discretize the incompressible Navier–Stokes equations and structural equations, and the generalized‐ α method is adopted for temporal discretization. Common velocity variables are employed at the fluid–structure interface for the strong coupling of both equations. Because of the velocity‐linked formulation, kinematic compatibility is automatically satisfied and forcing terms do not need to be calculated explicitly. Both the numerical stability and the convergence characteristics of an iterative solver for the coupled algorithm are investigated by solving the FSI problem of flexible tube flows. It is noteworthy that the generalized‐ α method with small damping is free from unstable velocity fields. However, the convergence characteristics of the coupled system deteriorate greatly for certain Poisson's ratios so that direct solvers are essential for these cases. Furthermore, the proposed method is shown to clearly display the advantage of considering FSI in the simulation of flexible tube flows, while enabling much larger time‐steps than those adopted in some previous studies. This is possible through the strong coupling of the fluid and structural equations by employing common primitive variables. Copyright © 2012 John Wiley & Sons, Ltd.     

Comparison of artificial dissipation and limited flux schemes in Arbitrary Lagrangian–Eulerian finite element formulations

A comparison is made between Arbitrary Lagrangian–Eulerian (ALE) finite element formulations for simulation of forming processes based on an artificial dissipation scheme and a limited flux scheme. The first ALE algorithm is based on an averaging procedure used in post-processing of finite element calculations. The second ALE algorithm stems from a finite difference method for compressible fluid dynamics. Both approaches have complementary characteristics with respect to accuracy and implementation. © 1998 John Wiley & Sons, Ltd.     

Numerical simulation of flapping wings using a panel method and a high‐order Navier–Stokes solver

The design of efficient flapping wings for human engineered micro aerial vehicles (MAVs) has long been an elusive goal, in part because of the large size of the design space. One strategy for overcoming this difficulty is to use a multifidelity simulation strategy that appropriately balances computation time and accuracy. We compare two models with different geometric and physical fidelity. The low‐fidelity model is an inviscid doublet lattice method with infinitely thin lifting surfaces. The high‐fidelity model is a high‐order accurate discontinuous Galerkin Navier–Stokes solver, which uses an accurate representation of the flapping wing geometry. To compare the performance of the two methods, we consider a model flapping wing with an elliptical planform and an analytically prescribed spanwise wing twist, at size scales relevant to MAVs. Our results show that in many cases, including those with mild separation, low‐fidelity simulations can accurately predict integrated forces, provide insight into the flow structure, indicate regions of likely separation, and shed light on design–relevant quantities. But for problems with significant levels of separation, higher‐fidelity methods are required to capture the details of the flow field. Inevitably high‐fidelity simulations are needed to establish the limits of validity of the lower fidelity simulations.Copyright © 2012 John Wiley & Sons, Ltd.     

Mesh motion techniques for the ALE formulation in 3D large deformation problems

Efficient mesh motion techniques are a key issue to achieve satisfactory results in the arbitrary Lagrangian–Eulerian (ALE) finite element formulation when simulating large deformation problems such as metal‐forming. In the updated Lagrangian (UL) formulation, mesh and material movement are attached and an excessive mesh distortion usually appears. By uncoupling mesh movement from material movement, the ALE formulation can relocate the mesh to avoid distortion. To facilitate the calculation process, the ALE operator is split into two steps at each analysis time step: UL step (where deformation due to loading is calculated without convective terms) and Eulerian step (where mesh motion is applied). In this work, mesh motion is performed by new nodal relocation methods, developed for eight‐node hexahedral elements, which can move internal and boundary nodes, improving and concentrating the mesh in critical zones. After mesh motion, data is transferred from the UL mesh to the relocated mesh using an expansion of stresses in a Taylor's series. Two numerical applications are presented, comparing results of UL and ALE formulation with results found in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

Adaptive finite element approximation of multiphysics problems: A fluid–structure interaction model problem

In this paper we consider finite element simulation of the mechanical response of an elastic solid immersed into a viscous incompressible fluid flow. For simplicity, we assume that the mechanics of the solid is governed by linear elasticity and the motion of the fluid by the Stokes equation. For this one‐way coupled multiphysics problem we derive an a posteriori error estimate using duality techniques. Based on the estimate we propose an adaptive algorithm that automatically constructs a suitable mesh for the fluid and solid computational domains given a specific goal quantity for the elastic problem. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimal mesh design for the finite element approximation of reaction–diffusion problems

We consider the numerical approximation of singularly perturbed problems, and in particular reaction–diffusion problems, by the h version of the finite element method. We present guidelines on how to design non‐uniform meshes both in one and two dimensions that are asymptotically optimal as the meshwidth tends to zero. We also present the results of numerical computations showing that robust, optimal rates can be achieved even in the pre‐asymptotic range. Copyright © 2001 John Wiley & Sons, Ltd.     

Adaptive finite element analysis of mixed‐mode crack problems with automatic mesh generator

Fully automatic advancing front type mesh generator to take care of crack and fracture problems has been presented. It is coupled with Zienkiewicz and Zhu error estimator, the refinement methodology depends on the concept of strain energy concentration for adaptive analysis of mixed‐mode crack problems. No investigation is reported in this direction so far. It has been found that the above combination proved to be very powerful for adaptive finite element analysis of mixed‐mode crack problems in two‐dimensional isotropic solids. Very accurate stress intensity factors have been obtained for a target error of 10 per cent with a minimum number of steps. Copyright © 2000 John Wiley & Sons, Ltd.     

The partition of unity finite element method for elastic wave propagation in Reissner–Mindlin plates

This paper reports a numerical method for modelling the elastic wave propagation in plates. The method is based on the partition of unity approach, in which the approximate spectral properties of the infinite dimensional system are embedded within the space of a conventional finite element method through a consistent technique of waveform enrichment. The technique is general, such that it can be applied to the Lagrangian family of finite elements with specific waveform enrichment schemes, depending on the dominant modes of wave propagation in the physical system. A four‐noded element for the Reissner–Mindlin plate is derived in this paper, which is free of shear locking. Such a locking‐free property is achieved by removing the transverse displacement degrees of freedom from the element nodal variables and by recovering the same through a line integral and a weak constraint in the frequency domain. As a result, the frequency‐dependent stiffness matrix and the mass matrix are obtained, which capture the higher frequency response with even coarse meshes, accurately. The steps involved in the numerical implementation of such element are discussed in details. Numerical studies on the performance of the proposed element are reported by considering a number of cases, which show very good accuracy and low computational cost. Copyright © 2006 John Wiley & Sons, Ltd.     

A Lagrangian finite element approach for the analysis of fluid–structure interaction problems

A Lagrangian finite element method for the analysis of incompressible Newtonian fluid flows, based on a continuous re‐triangulation of the domain in the spirit of the so‐called Particle Finite Element Method, is here revisited and applied to the analysis of the fluid phase in fluid–structure interaction problems. A new approach for the tracking of the interfaces between fluids and structures is proposed. Special attention is devoted to the mass conservation problem. It is shown that, despite its Lagrangian nature, the proposed combined finite element‐particle method is well suited for large deformation fluid–structure interaction problems with evolving free surfaces and breaking waves. The method is validated against the available analytical and numerical benchmarks. Copyright © 2010 John Wiley & Sons, Ltd.     

Adaptive superposition of finite element meshes in non‐linear transient solid mechanics problems

An s‐adaptive finite element procedure is developed for the transient analysis of 2‐D solid mechanics problems with material non‐linearity due to progressive damage. The resulting adaptive method simultaneously estimates and controls both the spatial error and temporal error within user‐specified tolerances. The spatial error is quantified by the Zienkiewicz–Zhu error estimator and computed via superconvergent patch recovery, while the estimation of temporal error is based on the assumption of a linearly varying third‐order time derivatives of the displacement field in conjunction with direct numerical time integration. The distinguishing characteristic of the s‐adaptive procedure is the use of finite element mesh superposition (s‐refinement) to provide spatial adaptivity. Mesh superposition proves to be particularly advantageous in computationally demanding non‐linear transient problems since it is faster, simpler and more efficient than traditional h‐refinement schemes. Numerical examples are provided to demonstrate the performance characteristics of the s‐adaptive method for quasi‐static and transient problems with material non‐linearity. Copyright © 2007 John Wiley & Sons, Ltd.