The adaptive Lagrangian particle method for macroscopic and micro–macro computations of time-dependent viscoelastic flows

We propose a new numerical technique, referred to as the Adaptive Lagrangian Particle Method (ALPM), for computing time-dependent viscoelastic flows using either a differential constitutive equation (macroscopic approach) or a kinetic theory model (micro–macro approach). In ALPM, the Eulerian finite element solution of the conservation equations is decoupled from the Lagrangian computation of the extra-stress at a number of discrete particles convected by the flow. In the macroscopic approach, the extra-stress carried by the particles is obtained by integrating the constitutive equation along the particle trajectories. In the micro–macro approach, the extra-stress is computed by solving along the particle paths the stochastic differential equation associated with the kinetic theory model. At each time step, ALPM automatically enforces that all elements of the mesh have a number of Lagrangian particles ranging within a user-specified interval. Results are given for the start-up flow between highly eccentric rotating cylinders, using the FENE and FENE-P dumbbell models for dilute polymer solutions.     

On energy conservation for finite element approximation of flow-induced airfoil vibrations

In this paper the numerical approximation of a two-dimensional fluid–structure interaction problem is addressed. The fully coupled formulation of incompressible viscous fluid flow interacting with a flexibly supported airfoil is considered. The flow is described by the incompressible system of Navier–Stokes equations, where large values of the Reynolds number are considered. The Navier–Stokes equations are spatially discretized by the finite element method and stabilized with a modification of the Galerkin Least Squares (GLS) method; cf. [T. Gelhard, G. Lube, M.A. Olshanskii, J.-H. Starcke, Stabilized finite element schemes with LBB-stable elements for incompressible flows, Journal of Computational and Applied Mathematics 177 (2005) 243–267]. The motion of the computational domain is treated with the aid of Arbitrary Lagrangian Eulerian (ALE) method and the stabilizing terms are modified in a consistent way with the ALE formulation.     

Eulerian shape design sensitivity analysis and optimization with a fixed grid

Conventional shape optimization based on the finite element method uses Lagrangian representation in which the finite element mesh moves according to shape change, while modern topology optimization uses Eulerian representation. In this paper, an approach to shape optimization using Eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed. A continuum geometric model is defined on the fixed grid of finite elements. An active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization, in which each element has a unique shape density. The shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements. Using this transformation, it has been shown that the shape design problem can be treated as a parameter design problem, which is a much easier method than the former. A detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation. Very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary. The accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement. Two design optimization problems are presented to show the feasibility of the proposed design approach.     

A mixed Eulerian–Lagrangian method for modelling metal extrusion processes

A mixed Eulerian–Lagrangian approach for the computational modelling of metal extrusion processes in complex three dimensional geometries is presented. The approach involves the representation of the workpiece as a pseudo-fluid, and requires the solution of non-Newtonian fluid flow equations in an Eulerian context, using a free-surface algorithm to track its extreme deformation during its extrusion. The solid mechanics equations associated with the tools are solved in a conventional Lagrangian context. Thermal interactions between the workpiece and tools are modelled and a fluid–structure interaction technique is employed to capture the effect of the fluid traction load imposed by the workpiece on the tools, and especially the subsequent adaption of the Eulerian mesh to account for the impact of die deformation. Two extrusion test cases are investigated and the results obtained show the potential of the model with regard to representing the physics of the process, the advantages of the model over a more loosely coupled approach, and the parallel scalability of the resulting software.     

Numerical solutions of nonlinear problems of continua—I : Survey of formulation methods and solution techniques

A comprehensive survey is provided for the formulation and solution techniques of finite element applications in nonlinear continuum mechanics problems. The survey discusses the Lagrangian, the updated Lagrangian, and the Eulerian formulations. It is shown that many analysts describe relative or updated Lagrangian formulation under the name of Eulerian formulation. Hence, little effort has been devoted to the development of a consistent Eulerian formulation. The applications, limitations and suitability of each formulation to both material and geometrical nonlinear problems are discussed. In the solution methods, exact equilibrium, approximate equilibrium and self-correcting techniques are discussed. An emphasis is given to the applicability of these methods to particular nonlinear problems and to recent developments and modifications of each method to suit a particular nonlinear field.     

Incompressible flow computations over moving boundary using a novel upwind method

In this paper a novel method for simulating unsteady incompressible viscous flow over a moving boundary is described. The numerical model is based on a 2D Navier–Stokes incompressible flow in artificial compressibility formulation with Arbitrary Lagrangian Eulerian approach for moving grid and dual time stepping approach for time accurate discretization. A higher order unstructured finite volume scheme, based on a Harten Lax and van Leer with Contact (HLLC) type Riemann solver for convective fluxes, developed for steady incompressible flow in artificial compressibility formulation by Mandal and Iyer (AIAA paper 2009-3541), is extended to solve unsteady flows over moving boundary. Viscous fluxes are discretized in a central differencing manner based on Coirier’s diamond path. An algorithm based on interpolation with radial basis functions is used for grid movements. The present numerical scheme is validated for an unsteady channel flow with a moving indentation. The present numerical results are found to agree well with experimental results reported in literature.     

A variational principle in terms of stream function for free-surface flows and its application to the finite element method

Steady free-surface flows under the influence of gravity are considered in this paper. The appropriate variational principle is derived, where the stream function within the flow and the free-surface elevation are independently subjected to variation. The formulation of the principle in terms of linear finite elements is presented, and the resulting set of non-linear equations is reduced to the iterative solution of a linear set. The computations are shown to be convergent regardless of whether the Froude number of the stream is greater or less than unity, and independent tests show that the accuracy of the results is well within the usual range expected for internal flows when linear elements are used.     

Response of structures to planar blast loads – A finite element engineering approach

Design and validation of structures against blast loads are important for modern society in order to protect and secure its citizen. Since it is a challenge to validate and optimise protective structures against blast loads using full-scale experimental tests, we have to turn our attention towards advanced numerical tools like the finite element method. Several different finite element techniques can be used to describe the response of structures due to blast loads. Some of these are: (1) a pure Lagrangian formulation, (2) an initial Eulerian simulation (to determine the load) followed by a Lagrangian simulation (for the structural response) and (3) a hybrid technique that combines the advantages of Eulerian and Lagrangian methods to have a full coupling between the blast waves and the deformation of the structure. Ideally, all blast simulations should be carried out using the fully coupled Eulerian–Lagrangian approach, but this may not be practical as the computational time increases considerably when going from a pure Lagrangian to a fully coupled Eulerian–Lagrangian simulation. A major goal in this study is to investigate if a pure Lagrangian formulation can be applied to determine the structural response in a specified blast load problem or if more advanced approaches such as the fully coupled Eulerian–Lagrangian approach is required for reliable results. This is done by conducting numerical simulations of an unprotected 20 ft ISO container exposed to a blast load of 4000 kg TNT at 120 m standoff distance using the three different approaches presented above. To validate and discuss the results, the simulated response of the container is compared to available data from a full-scale blast test under such conditions.     

Preliminary Extension of the Unified Coordinate System Approach to Computation of Viscous Flows

The classical Lagrangian approach will lead to infinite deformation of meshes, especially for viscous flows. In this paper, we will use the unified coordinate system approach to compute viscous flows. This approach, invented by Hui and his coworkers, contains an inner parameter leading to a continuous switch between the classical Lagrangian approach and Eulerian approach. Severe grid deformation can be avoided by controlling this inner parameter.     

Lagrange–Galerkin method for unsteady free surface water waves

We present a new numerical technique to approximate solutions to unsteady free surface flows modelled by the two-dimensional shallow water equations. The method we propose in this paper consists of an Eulerian–Lagrangian splitting of the equations along the characteristic curves. The Lagrangian stage of the splitting is treated by a non-oscillatory modified method of characteristics, while the Eulerian stage is approximated by an implicit time integration scheme using finite element method for spatial discretization. The combined two stages lead to a Lagrange–Galerkin method which is robust, second order accurate, and simple to implement for problems on complex geometry. Numerical results are shown for several test problems with different ranges of difficulty.     

A Parallel Implementation of ALE Moving Mesh Technique for FSI Problems using OpenMP

This paper investigates a high performance implementation of an Arbitrary Lagrangian Eulerian moving mesh technique on shared memory systems using OpenMP environment. Moving mesh techniques are considered an integral part of a wider class of fluid mechanics problems that involve moving and deforming spatial domains, namely, free-surface flows and Fluid Structure Interaction (FSI). The moving mesh technique adopted in this work is based on the notion of nodes relocation, subjected to a certain evolution as well as constraint conditions. A conjugate gradient method augmented with preconditioning is employed for solution of the resulting system of equations. The proposed algorithm, initially, reorders the mesh using an efficient divide and conquer approach and then parallelizes the ALE moving mesh scheme. Numerical simulations are conducted on the multicore AMD Opteron and Intel Xeon processors, and unstructured triangular and tetrahedral meshes are used for the 2D and 3D problems. The quality of generated meshes is checked by comparing the element Jacobians in the reference and current meshes, and by keeping track of the change in the interior angles in triangles and tetrahedrons. Overall, 51 and 72% efficiencies in terms of speedup are achieved for both the parallel mesh reordering and ALE moving mesh algorithms, respectively.     

An Improved Eulerian Approach for the Finite Time Lyapunov Exponent

We propose a new Eulerian numerical approach to compute the Jacobian of flow maps in continuous dynamical systems and subsequently the so-called finite time Lyapunov exponent (FTLE) for Lagrangian coherent structure extraction. The original approach computes the flow map and then numerically determines the Jacobian of the map using finite differences. The new algorithm improves the original Eulerian formulation so that we first obtain partial differential equations for each component of the Jacobian and then solve these equations to obtain the required Jacobian. For periodic dynamical systems, based on the time doubling technique developed for computing the longtime flow map, we also propose a new efficient iterative method to compute the Jacobian of the longtime flow map. Numerical examples will demonstrate that our new proposed approach is more accurate than the original one in computing the Jacobian and thus the FTLE field, especially near the FTLE ridges.     

Volumetric mesh generation from T-spline surface representations

A new approach to obtain a volumetric discretization from a T-spline surface representation is presented. A T-spline boundary zone is created beneath the surface, while the core of the model is discretized with Lagrangian elements. T-spline enriched elements are used as an interface between isogeometric and Lagrangian finite elements. The thickness of the T-spline zone and thereby the isogeometric volume fraction can be chosen arbitrarily large such that pure Lagrangian and pure isogeometric discretizations are included. The presented approach combines the advantages of isogeometric elements (accuracy and smoothness) and classical finite elements (simplicity and efficiency).Different heat transfer problems are solved with the finite element method using the presented discretization approach with different isogeometric volume fractions. For suitable applications, the approach leads to a substantial accuracy gain.     

A practical procedure for mesh motion in arbitrary Lagrangian-Eulerian method

Conventional Lagrangian and Eulerian type formulation methods are applied to simulate some metal forming processes. These formulations possess certain difficulties when applied to finite strain deformation problems, especially when boundary conditions need to be updated during the course of deformation. An Arbitrary Lagrangian-Eulerian (ALE) formulation method is employed to overcome these difficulties. This paper presents an efficient mesh motion scheme for the ALE formulation method. A practical and more efficient scheme for handling supplementary constraint equations, arising from mesh motion algorithms, on the element level is presented. A procedure for handling boundary motion within the scheme and ensuring homogeneous mesh results is described. The presented scheme is employed to simulate punch forging and plane strain metal extrusion processes.     

An analysis of metal forming processes using large deformation elastic-plastic formulations

A concise survey of the literature related to the large deformation elasto-plasticity problems including unilateral contact and friction is presented together with an extension of the friction law for large deformation analysis.Starting from the principle of virtual work, the so-called total Lagrangian and updated Lagrangian formulations are derived based on some fundamental assumptions in linearizing the nonlinear equations. By introducing the Zaremba-Jaumann (co-rotational) increment to the Cauchy stress tensor, the classical Prandtl-Reuss equations are generalized for describing the elastic-plastic material behavior. To allow a proper consideration of the contact conditions in the incremental analysis, a general friction law with an associated isotropic Coulomb sliding rule is obtained by the similarity between dry friction and plasticity. Finite element discretizations and approximations are applied to the resulting formulation of the updated Lagrangian approach.Four example problems are solved to test the formulations developed in this paper. The emphasis is made toward the numerical accuracy of the finite element solutions.     

Singular integral equations in potential flows of open-channel transitions

The free-surface profile of potential flows is calculated in open-channel transitions by using singular integral equation methods. Thus in such free-surface hydraulics applications the analysis of fluid motion is very complicated, as both the subcritical and supercritical flows are presented simultaneously. For the numerical solution of the singular integral equations are used both constant and linear elements.An application is finally given to the determination of the free-surface profile in an open-channel contraction and comparing the numerical results with corresponding results by finite differences.     

Finite element analysis of viscous fluid flow

A finite element model for the analysis of two dimensional viscous flows is formulated using the virtual work method. The model is in part based on a finite element shell model, using the same reduced integration of quadratic interpolations for all variables[1]. Differences from preceding formulations are that integration by parts is applied to the continuity equation, yielding different loading terms which are more easily defined in some problems, and a new approach is used for the convective inertia terms, giving a clearer interpretation of their effects which are distributed to both sides of the nonlinear recurrence relation. In the case of compressible flow, for which comparatively few formulations have been proposed to date, the thermal energy equation is used to form a two stage solution and here this seems the most natural and economical approach.