Updated Lagrangian free surface flow simulations with natural neighbour Galerkin methods

In this paper, a new method to simulate free surface fluid flows within an updated Lagrangian framework is described. It is based on the use of a meshless technique coined as natural element method (NEM) or, more recently, as natural neighbour Galerkin method. The position of the flow front or the geometry of the fluid domain is handled by invoking the geometrical concept of α‐shape of the cloud of points, thus avoiding the explicit definition of the boundary of the domain as it evolves. This method also avoids the traditional need of remeshing typical in finite element simulations of this kind of processes. Three types of fluid behaviour have been considered, namely a purely Newtonian fluid, a non‐Newtonian short fibre‐reinforced thermoplastic, and finally a Norton–Hoff viscoplastic behaviour. Benchmark examples showing the performance of the technique are included in the paper. Copyright © 2004 John Wiley & Sons, Ltd.     

Lagrangian finite element analysis of Newtonian fluid flows

A fully Lagrangian finite element method for the analysis of Newtonian flows is developed. The approach furnishes, in effect, a Lagrangian implementation of the compressible Navier–Stokes equations. As the flow proceeds, the mesh is maintained undistorted through continuous and adaptive remeshing of the fluid mass. The principal advantage of the present approach lies in the treatment of boundary conditions at material surfaces such as free boundaries, fluid/fluid or fluid/solid interfaces. In contrast to Eulerian approaches, boundary conditions are enforced at material surfaces ab initio and therefore require no special attention. Consistent tangents are obtained for Lagrangian implicit analysis of a Newtonian fluid flow which may exhibit compressibility effects. The accuracy of the approach is assessed by comparison of the solution for a sloshing problem with existing numerical results and its versatility demonstrated through a simulation of wave breaking. The finite element mesh is maintained undistorted throughout the computation by recourse to frequent and adaptive remeshing © 1998 John Wiley & Sons, Ltd.     

Two‐fluid model with interface sharpening

Two‐fluid models are applicable for simulations of all types of two‐phase flows ranging from separated flows with large characteristic interfacial length scales to highly dispersed flows with very small characteristic interfacial length scales. The main drawback of the two‐fluid model, when used for simulations of stratified flows, is the numerical diffusion of the interface. Stratified flows can be easily and more accurately solved with interface tracking methods; however, these methods are limited to the flows, that do not develop into dispersed types of flows. The present paper describes a new approach, where the advantage of the two‐fluid model is combined with the conservative level set method for interface tracking. The advection step of the volume fraction transport equation is followed by the interface sharpening, which preserves the thickness of the interface during the simulation. The proposed two‐fluid model with interface sharpening was found to be more accurate than the existing two‐fluid models. The mixed flow with both: stratified and dispersed flow, is simulated with the coupled model in this paper. In the coupled model, the dispersed two‐fluid model and two‐fluid model with interface sharpening are used locally, depending on the parameter which recognizes the flow regime. Copyright © 2010 John Wiley & Sons, Ltd.     

An implicit boundary element solution with consistent linearization for free surface flows and non‐linear fluid–structure interaction of floating bodies

In this work, a new comprehensive method has been developed which enables the solution of large, non‐linear motions of rigid bodies in a fluid with a free surface. The application of the modern Eulerian–Lagrangian approach has been translated into an implicit time‐integration formulation, a development which enables the use of larger time steps (where accuracy requirements allow it). Novel features of this project include: (1) an implicit formulation of the rigid‐body motion in a fluid with a free surface valid for both two or three dimensions and several moving bodies; (2) a complete formulation and solution of the initial conditions; (3) a fully consistent (exact) linearization for free surface flows valid for any boundary elements such that optimal convergence properties are obtained when using a Newton–Raphson solver. The proposed framework has been completed with details on implementation issues referring mainly to the computation of the complete initial conditions and the consistent linearization of the formulation for free surface flows. The second part of the paper demonstrates the mathematical and numerical formulation through numerical results simulating large free surface flows and non‐linear fluid structure interaction. The implicit formulation using a fully consistent linearization based on the boundary element method and the generalized trapezoidal rule has been applied to the solution of free surface flows for the evolution of a triangular wave, the generation of tsunamis and the propagation of a wave up to overturning. Fluid–structure interaction examples include the free and forced motion of a circular cylinder and the sway, heave and roll motion of a U‐shaped body in a tank with a flap wave generator. The presented examples demonstrate the applicability and performance of the implicit scheme with consistent linearization. Copyright © 2001 John Wiley & Sons. Ltd.     

An algorithm for modelling the interaction of a flexible rod with a two‐dimensional high‐speed flow

We present an algorithm for modelling coupled dynamic interactions of a very thin flexible structure immersed in a high‐speed flow. The modelling approach is based on combining an Eulerian finite volume formulation for the fluid flow and a Lagrangian large‐deformation formulation for the dynamic response of the structure. The coupling between the fluid and the solid response is achieved via an approach based on extrapolation and velocity reconstruction inspired in the Ghost Fluid Method. The algorithm presented does not assume the existence of a region exterior to the fluid domain as it was previously proposed and, thus, enables the consideration of very thin open boundaries and structures where the flow may be relevant on both sides of the interface. We demonstrate the accuracy of the method and its ability to describe disparate flow conditions across a fixed thin rigid interface without pollution of the flow field across the solid interface by comparing with analytical solutions of compressible flows. We also demonstrate the versatility and robustness of the method in a complex fluid–structure interaction problem corresponding to the transient supersonic flow past a highly flexible structure. Copyright © 2005 John Wiley & Sons, Ltd.     

A particle method for two‐phase flows with large density difference

A new numerical approach for solving incompressible two‐phase flows is presented in the framework of the recently developed Consistent Particle Method (CPM). In the context of the Lagrangian particle formulation, the CPM computes spatial derivatives based on the generalized finite difference scheme and produces good results for single‐phase flow problems. Nevertheless, for two‐phase flows, the method cannot be directly applied near the fluid interface because of the abrupt discontinuity of fluid density resulting in large change in pressure gradient. This problem is resolved by dealing with the pressure gradient normalized by density, leading to a two‐phase CPM of which the original singlephase CPM is a special case. In addition, a new adaptive particle selection scheme is proposed to overcome the problem of ill‐conditioned coefficient matrix of pressure Poisson equation when particles are sparse and non‐uniformly spaced. Numerical examples of Rayleigh–Taylor instability, gravity current flow, water‐air sloshing and dam break are presented to demonstrate the accuracy of the proposed method in wave profile and pressure solution. Copyright © 2015 John Wiley & Sons, Ltd.     

Fixed mesh front‐tracking methodology for finite element simulations

A direct front‐tracking method using an Eulerian–Lagrangian formulation is developed in two space dimensions. The front‐tracking method is general in that it can track any type of interface once its local velocity is specified or has been determined by calculation. The method uses marker points to describe the interface position and tracks the interface evolution on a fixed finite‐element mesh, including growth, contraction, splitting and merging. Interfacial conditions are applied directly at the interface position. The method is applied to three scenarios that involve different interface conditions and are based on energy and mass diffusion. The three calculations are for the dendritic solidification of a pure substance, the cellular growth of an alloy, and the Ostwald ripening of silica particles in silicon. Numerical results show that very complicated interface morphologies and topological changes can be simulated properly and efficiently. Copyright © 2004 John Wiley & Sons, Ltd.     

A particle‐based moving interface method (PMIM) for modeling the large deformation of boundaries in soft matter systems

The mechanics of the interaction between a fluid and a soft interface undergoing large deformations appear in many places, such as in biological systems or industrial processes. We present an Eulerian approach that describes the mechanics of an interface and its interactions with a surrounding fluid via the so‐called Navier boundary condition. The interface is modeled as a curvilinear surface with arbitrary mechanical properties across which discontinuities in pressure and tangential fluid velocity can be accounted for using a modified version of the extended finite element method. The coupling between the interface and the fluid is enforced through the use of Lagrange multipliers. The tracking and evolution of the interface are then handled in a Lagrangian step with the grid‐based particle method. We show that this method is ideal to describe large membrane deformations and Navier boundary conditions on the interface with velocity/pressure discontinuities. The validity of the model is assessed by evaluating the numerical convergence for a axisymmetrical flow past a spherical capsule with various surface properties. We show the effect of slip length on the shear flow past a two‐dimensional capsule and simulate the compression of an elastic membrane lying on a viscous fluid substrate. Copyright © 2015 John Wiley & Sons, Ltd.     

An updated Lagrangian method with error estimation and adaptive remeshing for very large deformation elasticity problems

Accurate simulations of large deformation hyperelastic materials by the FEM is still a challenging problem. In a total Lagrangian formulation, even when using a very fine initial mesh, the simulation can break down due to severe mesh distortion. Error estimation and adaptive remeshing on the initial geometry are helpful and can provide more accurate solutions but are not sufficient to attain very large deformations. The updated Lagrangian formulation where the geometry is periodically updated is then preferred. However, it requires data transfer from the old mesh to the new one and this is a very delicate issue. In this paper, we present an updated Lagrangian formulation where the error is estimated and adaptive remeshing is performed in order to reach high level of deformations while controlling both the accuracy of the solution and mesh distortion. Special attention is given to data transfer methods and a very accurate cubic Lagrange projection method is introduced. A continuation method is used to automatically pilot the complete algorithm including load increase, error estimation, adaptive remeshing, and data transfer. A number of examples will be presented and analyzed. Copyright © 2014 John Wiley & Sons, Ltd.     

Adaptive finite element remeshing in a large deformation analysis of metal powder forming

In this paper, a general framework for the finite element simulation of powder forming processes is presented. A large displacement formulation, based on a total and updated Lagrangian formulation and an adaptive finite element strategy based on error estimates and automatic remeshing techniques are utilized. To describe the constitutive model of the highly non‐linear behaviour of powder materials, an elliptical cap model based on a hardening rule to define the dependence of the yield surface on the degree of plastic straining is applied. The interfacial behaviour between the die and powder is modelled by using a plasticity theory of friction in the context of an interface element formulation. Finally, the powder behaviour during the compaction of a set of complex shapes are analysed numerically. The simulation of the deformation is shown as well as the distribution of relative density contours at different time stages. The results clearly indicate that the algorithm makes it possible to simulate the powder forming problems efficiently and automatically. Copyright © 1999 John Wiley & Sons, Ltd.     

An L∞–Lp mesh‐adaptive method for computing unsteady bi‐fluid flows

This paper discusses the contribution of mesh adaptation to high‐order convergence of unsteady multi‐fluid flow simulations on complex geometries. The mesh adaptation relies on a metric‐based method controlling the L ^p‐norm of the interpolation error and on a mesh generation algorithm based on an anisotropic Delaunay kernel. The mesh‐adaptive time advancing is achieved, thanks to a transient fixed‐point algorithm to predict the solution evolution coupled with a metric intersection in the time procedure. In the time direction, we enforce the equidistribution of the error, i.e. the error minimization in L ^∞ norm. This adaptive approach is applied to an incompressible Navier–Stokes model combined with a level set formulation discretized on triangular and tetrahedral meshes. Applications to interface flows under gravity are performed to evaluate the performance of this method for this class of discontinuous flows. Copyright © 2010 John Wiley & Sons, Ltd.     

A saturated discrete particle model and characteristic-based SPH method in granular materials

Based on the discrete particle model for solid-phase deformation of granular materials consisting of dry particulate assemblages, a discrete particle–continuum model for modelling the coupled hydro-mechanical behaviour in saturated granular materials is developed. The motion of the interstitial fluid is described by two parallel continuum schemes governed by the averaged incompressible N–S equations and Darcy's law, respectively, where the latter one can be regarded as a degraded case of the former. Owing to the merits in both Lagrangian and mesh-free characters, the characteristic-based smoothed particle hydrodynamics (SPH) method is proposed in this paper for modelling pore fluid flows relative to the deformed solid phase that is modelled as packed assemblages of interacting discrete particles. It is assumed that the formulation is Lagrangian with the co-ordinate system transferring with the movement of the solid particles. The assumed continuous fluid field is discretized into a finite set of Lagrangian (material) points with their number equal to that of solid particles situated in the computational domain. An explicit meshless scheme for granular materials with interstitial water is formulated. Numerical results illustrate the capability and performance of the present model in modelling the fluid–solid interaction and deformation in granular materials saturated with water. Copyright © 2007 John Wiley & Sons, Ltd.     

A monolithic approach for interaction of incompressible viscous fluid and an elastic body based on fluid pressure Poisson equation

This paper describes a new monolithic approach based on the fluid pressure Poisson equation (PPE) to solve an interaction problem of incompressible viscous fluid and an elastic body. The PPE is derived so as to be consistent with the coupled equation system for the fluid‐structure interaction (FSI). Based on this approach, we develop two kinds of efficient monolithic methods. In both methods, the fluid pressure is derived implicitly so as to satisfy the incompressibility constraint, and all other unknown variables are derived fully explicitly or partially explicitly. The coefficient matrix of the PPE for the FSI becomes symmetric and positive definite and its condition is insensitive to inhomogeneity of material properties. The arbitrary Lagrangian–Eulerian (ALE) method is employed for the fluid part in order to take into account the deformable fluid‐structure interface. To demonstrate fundamental performances of the proposed approach, the developed two monolithic methods are applied to evaluate the added mass and the added damping of a circular cylinder as well as to simulate the vibration of a rectangular cylinder induced by vortex shedding. Copyright © 2005 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.     

An enriched finite element method and level sets for axisymmetric two‐phase flow with surface tension

A finite element method for axisymmetric two‐phase flow problems is presented. The method uses an enriched finite element formulation, in which the interface can move arbitrarily through the mesh without remeshing. The enrichment is implemented by the extended finite element method (X‐FEM) which models the discontinuity in the velocity gradient at the interface by a local partition of unity. It provides an accurate representation of the velocity field at interfaces on an Eulerian grid that is not conformal to the weak discontinuity. The interface is represented by a level set which is also used in the construction of the element enrichment. Surface tension effects are considered and the interface curvature is computed from the level set field. The method is demonstrated by several examples. Copyright © 2003 John Wiley & Sons, Ltd.     

A coupling of multi‐zone curved Galerkin BEM with finite elements for independently modelled sub‐domains with non‐matching nodes in elasticity

When the different parts of a structure are modelled independently by BEM or FEM methods, it is sometimes necessary to put the parts together without remeshing of the nodes along the part interfaces. Frequently the nodes do not match along the interface. In this work, the symmetric Galerkin multi‐zone curved boundary element is a fully symmetric formulation and is the method used for the boundary element part. For BEM–FEM coupling it is then necessary to interpolate the tractions in‐between the non‐matching nodes for the FEM part. Finally, the coupling is achieved by transforming the finite element domains to equivalent boundary element domains in a block symmetric formulation. This system is then coupled with a boundary element domain with non‐matching nodes in‐between. Copyright © 2004 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.